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of Mathematical Monographs Volume 43
Stability of Solutions of Differential Equations in Banach Space by Ju. L. Daleckii and M. G. Krein 1'\£:\..';-1',.')',.... ~fl~ , f\L, !ft':
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AMS (MOS) subject classifications (1970). Primary 34D05, 34G05; Secondary 34C30, 47 A50.
Library of Congress Cataloging in Publication Data
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Daletlkif, fUrii L'vovich. Stability of solutions of differential equations in banach space. (Translations of mathematical monographs, v. 43) Translation of Ustoi'chivost' reshenif differenO:ial'nykh uravnenit v banakhovom prostranstve. Bibliography: p. 1. Differential equations. 2. Banach spaces. 3. Linear operators. I. Krern, Mark Grigor'evich, 1907joint author. II. Title. III. Series. QA37l.D255l3 515'.73 74-8403 ISBN 0-8218-1593-0
Copyright © 1974 by the American Mathematical Society
TABLE OF CONTENTS ';;;;,',:;,- ~,---~L_:_-:~, ;
AUTHORS' PREFACE TO ENGLISH EDITION . . . . . . . . . . . . . . : ..'~:':":': . . . . . . . . . : ..
": r.z:T
vi
PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
INTRODUCTION
5
CHAPTER
I.
.......................................................
SOME INFORMATION FROM THE THEORY OF BOUNDED
9 1. General propositions on the geometry of Banach spaces and their linear mappings ................................................ 10 2. Functions of bounded linear operators ............................ 15 3. Solution of some linear operator equations ........................ 21 4. Exponential operator function ................................... 25 5. Generalized Ljapunov theorem .................................. 31 6. A theorem of B. von Sz.-Nagy ................................... 34 7. Hilbert space with an indefinite metric ............................ 37 8. Stable W-unitary operators ...................................... 45 9. Elements of nonlinear analysis ................................... 52 OPERATORS IN BANACH SPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
II. THE LINEAR EQUATION WITH A CONSTANT OPERATOR . . . . . . . . . . . 69 1. Solution of the homogeneous and inhomogeneous equations ......... 69 2. The behavior of the solutions of the homogeneous equation at infinity ....................................................... 71 3. Boundedness of the solutions of the homogeneous equation ......... 75 4. Conditions for the existence of a bounded solution of the inhomogeneous equation ........................................ 79
CHAPTER
III. THE NONSTATIONARY LINEAR EQUATION. BOHL EXPONENTS ..... 94 1. Evolution operator and formulas for the solution of linear equations ..................................................... 94 2. Integral inequalities. Comparison of evolution operators .......... 105 3. Stability and bistability. Comparable equations .................. 111 4. Ljapunov and Bohl exponents of the homogeneous equation on [0, (0). . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 116
CHAPTER
iii
TABLE OF CONTENTS
IV
5. Condition for boundedness on [0, 00) of the solutions of the inhomogeneous equation ...................................... 127 6. Equations with precompactly valued operator functions ............ 132 CHAPTER
IV.
EXPONENTIAL SPLITTING OF THE SOLUTIONS OF THE LINEAR
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Conjugation operators for projections ........................... 149 Kinematically similar equations ................................. 157 Exponentially dichotomic equations ............................. 162 Exponential splitting ......................................... 174 Stability of the Bohl exponents of an exponential splitting ......... 178 Exponential splitting in a Hilbert phase space ..................... 187
EQUATION
1. 2. 3. 4. 5. 6.
V. THE EQUATION WITH A PERIODIC OPERATOR FUNCTION . . . . . . . . . 198 1. Monodromy operator. Reducibility ............................. 198 2. Exponential dichotomy of the solutions of the periodic equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 203 3. Localization theorems on the spectrum of the monodromy operator (~=.p) .............................................. 213 4. Canonical differential equations ................................. 218 5. Second order equations (infinite-dimensional analog of Hill's equation) .................................................... 225 6. Expansion of the logarithm of the monodromy operator in powers of a small parameter .................................... 231
CHAPTER
VI. LINEAR DIFFERENTIAL EQUATIONS IN THE COMPLEX PLANE ...... 251 The equation with a regular singularity. Simplest case ............ , .251 Case of integral differences of eigenvalues ........................ 255 Finite-dimensional case (dim ~
CHAPTER
1. 2. 3. 4.
VII. NONLINEAR EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 1. EXIstence of solutions. Basic stability concepts .................... 277 2. Stability and instability of the solutions of the nonlinear equation with a stationary principal part ................................. 285 3. The nonlinear equation with a nonstationary principal part ......... 293 4. The nonlinear equation with a principal part the spectrum of which does not intersect the imaginary axis ....................... 296 5. Stable integral manifolds ....................................... 305 6. Integral manifolds bounded for t-+ + 00 (or t-+ - 00) ............ 315
CHAPTER
TABLE OF CONTENTS
v
7. Averaging principle ........................................... 319 CHAPTER
VIII. ASYMPTOTIC REPRESENTATION OF THE SOLUTIONS OF A LINEAR
DIFFERENTIAL EQUATION WITH A LARGE PARAMETER . . . . . . . . . . . . . . . . . . 334
1. Approximate decomposition of the equation ...................... 334 2. Estimate of the error .......................................... 341 3. The equation with a rapidly oscillating coefficient ................. 353 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 NOTATION INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . • . . . . . . . 377 AUTHOR INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
AUTHORS' PREFACE TO ENGLISH EDITION
We were delighted to learn that immediately following the appearance of our book in Russian the American Mathematical Society decided to put out an English edition of it. We are very grateful to the Editor of Translations of the American Mathematical Society, Dr. S. H. Gould, for the initiative, energy and interest which he displayed in this regard. We are well aware of the fact that we have not fully succeeded in observing the precept of that wise poet, Samuel Marsak (see the epigraph): the "warm exuberance of the rough draft" is far from being everywhere in evidence, while the slips characteristic of a rough draft have turned out to be more numerous than we would have liked. However, thanks to the efforts of the translator, Dr. S. F. Smith, the number of slips has been substantially reduced. Unfortunately, it was not possible for us to note everywhere where we should have the considerable amount of work performed by Dr. Smith; the statements and proofs of theorems were sometimes changed as a result of it. We take pleasure in expressing our sincere gratitude to Dr. Smith for his responsible, careful and conscientious attitude toward the work of the translation. Odessa, Accadia July 17, 1972 The authors
vi
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"It's good if printed line, austere and strict, retains the lively spirit of the script." S. MarS{lk*
PREFACE
The development of this book into its present form has been largely accidental. One might say that it was created by the method of successive approximations applied over intervals spanning many years. We begin by noting that neither of the authors has considered himself to be a true specialist in the field of stability theory. At any rate this field is now a sort of "hobby" for each of us. First the senior and then the junior a uthor developed the desire to interpret the various methods and situations in stability theory from the point of view of the infinite-dimensional reaches of function spaces, the operators acting in them, their norms and their definite and indefinite metrics. This "hobby" has turned out to be quite fascinating. The attempts directly to generalize the traditional methods which make essential use of the phenomena of linear algebra (e.g. the discreteness of the spectrum, Jordan cells, etc.) run up against insurmountable difficulties because of the lack of a spectral theory for general nonselfadjoint operators. One is therefore compelled to seek other approaches, which on the one hand frequently lead to new problems in functional analysis and on the other hand shed new light on known facts. As so often happens, a view from a different perspective permits one to make new discoveries in an old field. It all began as follows. In the Spring of 1947 M. G. Krein commenced a short lecture course for undergra~uates, graduate students and faculty members of the University of Odessa on the theory of linear operators in Banach spaces and the theory of normed rings. This course was to have been completed by the Fall of 1947. And so it would have been ifM. G. Krein had not attempted to show in his lectures that a number ofthe methods and situations in the Ljapunov theory of stability for solutions of differential equations can be carried over and generalized to the case of differential equations in Banach spaces. This project met with such a lively response from the students that the course was completely given over to its further development and was consequently extended to April of 1948. *The foregoing is a free translation of Marsak into English by o. G. Dorohin, who is a teacher of English in the U.S.S.R.
2
PREFACE
During this same period M. G. Krein gave a paper (M. G. Krein [2]) at a meeting of the Moscow Mathematical Society in which he told of his first attempts to develop a stability theory for systems with an infinite number of degrees of freedom and of the problems encountered in this connection. M. G. Krein later supplemented the lecture course with talks at his seminars. A number of results along the lines laid down by these talks were subsequently obtained in the master's thesis of D. L. Kucer [1, 2], as well as in the work of M. A. Rutman [1]. In 1954 lu. L. Daleckii, who at the time was occupied with investigations in the field of asymptotic methods for operator equations with a small parameter, undertook to give a systematic account of the material that had accumulated from the course of M. G. Krein. In the process he improved and enlarged the discussion of a number of points and in addition wrote a chapter on his own results. This account, on which the "Lectures" (M. G. Krein [9]) were subsequently based, was finished by the Spring of 1955. It was originally planned to submit it for publication in the journal, "Uspehi Matematiceskih Nauk," as soon as some apparently minor tasks were performed: the writing of an introduction, the compilation of a bibliography and a comparison of the results with those then known for equations in a finite-dimensional space. But the need for this tedious addition to the "hobby" brought matters to a standstill for ten years. At the same time individual copies of the future "Lectures" were retyped and circulated among scientists in Odessa, Voronezh, Kiev, Kishinev, Leningrad, Moscow and possibly other cities. The second stage in the writing of the book dates from 1964, when in response to a growing interest in the "Lectures" it was decided to have them mimeographed. The previous ten years, however, had seen a very intensive development of the theory of differential equations. A comparison with the literature was now less advantageous to us and much more difficult (see the 69 pages of the frighteningly large bibliography in the well-known survey of L. Cesari [1] originally published back in 1959). We decided on the basis of this work to give up and publish the "Lectures" after only partially revising them. The most important changes had the following history. In the previously referred to paper given at a 1948 meeting of the Moscow Mathematical Society M. G. Krein called attention to the fact that the remarkable investigations of A. M. Ljapunov on second order linear differential equations with periodic coefficients had not heen extended to equations in Hilbert space or even to the n-dimensional case. After two years M. G. Krein [3-8] succeeded in finding a new approach to the establishment of criteria for the boundedness of the solutions of a canonical linear system of differential equations with a periodic Hamiltonian (in particular, of a certain class of systems of second order equations). This approach received further development in the papers of various Soviet writers (see the survey of M. G. Krein and V. A. lakubovic [1]). Several results in this direction were presented in the mimeographed edition of the "Lectures" for the case of Hilbert spaces directly.
PREFACE
3
An unpolished form of the "Lectures" appeared at the end of 1964 and instantly became a literary rarity (an edition of 400 copies). Beginning at this point the authors became more receptive to the idea of getting out a much larger edition. Nowadays no one is astonished by a functional analytic approach to differential equations nor by the equations themselves being in Banach spaces; these phenomena have penetrated even university texts (see, for example, the book of J. Dieudonne [I])! We have ceased to be monopolists even in the matter of discussing the stability of solutions of differential equations in a Banach space. During this time there appeared the fundamental book of J. Massera and J. Schaffer [1], which summarized their investigations on the stability of solutions of linear differential equations in Banach spaces. There also appeared the short but excellent book of the Australian mathematician W. Coppel [1] and the fundamental book ofP. Hartman [1], in both of which the presentation is conducted by applying the methods of functional analysis in a finite-dimensional space but on a level often permitting an almost direct extension to a Banach space. A number of interesting questions are discussed in the books of E. A. Barbasin [1] and N. N. Krasovskii [1]. We note further the well written intensive university text ofB. P. Demidovic [3]. In recent years members of the Kiev school of N. N. Bogoljubov on nonlinear mechanics (Ju. O. Mitropol'skii, o. B. Likova and others) have become occupied with extending their results to the case of equations in Hilbert space. Closely related investigations have been carried out by the Czechoslovakian mathematician J. Kurzweil [1]. All of these sources have in one way or another had an influence on our selection of material for the enlarged edition and occasionally on the presentation. During this time a substantial contribution to stability theory was discovered in the works of the remarkable Riga mathematician P. G. Bohl (1865-1921).1) who has dwelt so long in obscurity. We found it necessary as a result to substantially revise the brief bibliographical notes appearing in the "Lectures". Thus there arose the present book, which is more than three times as long as the "Lectures" . Unfortunately, our book suffers from various "limitations". To keep its size within bounds we omitted many important topics. In particular, our treatment is limited to boul}ded operators only. The authors clearly recognize that the arena of infinite-dimensional spaces requires the presence of unbounded operators, without which the present stability theory for systems with an infinite number of degrees of freedom would be impossible. Such a theory has not yet been created. Not so long ago there did not even exist
1) For more details see the Notes at the end of Chapter III as well as at the ends of Chapters IV and VII.
4
PREFACE
a basis for it, viz. a theory of linear differential equations with unbounded operator coefficients in Banach spaces. During the past decade and a half the situation has radically changed and it can now be said that such a basis exists (see the book of S. G. Krein [1], where the many investigations in this field are summarized). Individual papers have already appeared in the field of stability theory for unbounded operator coefficients (S. Agmon and L. Nirenberg [1], Z. I. Halilov [1, 2], Ju. I. Domslak [1, 2], V. I. Derguzov [1, 2], V. N. Fomin [1, 5] and others). We would like to hope that our book might serve as a jumping off point for some of the investigations in this more advanced field (in some of the exercises a generalization to the case of unbounded operators can be trivially obtained without particular difficulty). When raw lectures are processed into a book there can arise a danger which in another connection was wonderfully depicted by H. Weyl: "The stringent precision attainable for mathematical thought has led many authors to a mode of writing which must give the reader the impression of being shut up in a brightly illuminated cell where every detail sticks out with the same dazzling clarity, but without relief. I prefer the open landscape under a clear sky with its depth of perspective, where the wealth of sharply defined nearby details gradually fades away towards the horizon."* Few possess a gift of picturing "landscapes under a clear sky ... " similar to the gift of H. Weyl, that indefatigable seeker and discoverer of harmony in mathematics. The authors had to settle for a system of shadows and halfshadows by means of the standard tools known as remarks, exercises and notes. To this we nevertheless add that on the horizon of our landscapes, besides the above mentioned mountainous massif of differential equations with unbounded operators, there can be seen in the distance smooth manifolds with dynamic streams and cascades on them. We consider it our pleasant duty to note that during the last stage of work on the manuscript, for a period of about a year, essential help was rendered us by the editor of the book, M. G. Sonis, to whom we express our very sincere gratitude. Odessa, Arcadia August 6, 1969 The authors
*Preface to the first edition of The classical groups.
INTRODUCTION
In the book, as the title suggests, we are concerned with differential equations for vector functions with values in infinite-dimensional spaces. Therefore, to completely understand its contents, the reader should be familiar with what is known about the geometry of Hilbert and Banach spaces and the theory of linear operators acting in them. In the first chapter we recall the basic definitions and classical theorems of the theory of Banach spaces. It is assumed, however, that the fundamentals of the geometry of Hilbert spaces are known to the reader. We have attempted to avoid any use of the theory of spectral resolutions of selfadjoint and unitary operators, although we do not completely succeed in doing this (see § 8). At the same time we note that a considerable part of the apparatus used in the book, which, beginning with § 3, is presented with more or less complete proofs, has not yet entered even the fundamental courses of functional analysis. This applies especially to questions in the theory of spaces with an indefinite metric, but also, for example, to the methods of solving linear operator equations. And what is more, even the interpretation in finite-dimensional spaces of a number of propositions cited by us has not managed to find a place for itself in books on linear algebra (i.e. the theory of matrices). We wish to emphasize here that a reader unfamiliar with functional analysis can read the book by assuming everywhere that it is dealing with finite-dimensional spaces. We hope that even in this case a reader qualified in the theory of differential equations will find something new in the book. The easiest chapter to understand is the second chapter, in which stationary linear equations are considered. It contains the simplest version of many of the ideas and results presented in the other chapters. A reader becoming acquainted with the subject for the first time should first thoroughly study this chapter. For this purpose he.will not need the results concerning an indefinite metric presented in §§ 7 and 8 of Chapter I (except for the simplest definitions these results are used only in the exercises) nor the contents of § 9 of Chapter I. In the third and fourth chapters we present the general theory of nonstationary linear equations; in particular (Chapter IV), we carry out a detailed investigation of the important notion of an exponential splitting of the solutions of an equation. More restricted classes of linear equations are studied in Chapter V (equations 5
6
INTRODUCTION
with periodic coefficients) and in Chapter VI (equations with a regular singularity in the complex plane). In particular, a large part of Chapter V is given over to the study of conditions for the strong stability of canonical equations. In this connection we make essential use of the methods of the theory of operators in spaces with an indefinite metric. The seventh chapter is devoted to the consideration of various problems connected with the behavior of the solutions of nonlinear equations. Finally, in the last, the eighth, chapter we return to linear equations and consider special asymptotic developments of the solutions with respect to a large parameter of the equation. The reader can obtain a more precise picture of the contents of the book from the Table of Contents and the brief introductions which preface the chapters of the book. In addition to these introductions each chapter is provided with a set of exercises and some bibliographical notes. The exercises are of a diverse character and pursue different ends. Some of them are routine drill problems for the main text. Others together with hints to their solution contain brief developments of important additional questions which were not included in the main text in order to avoid unduly increasing the size of the book. Finally, many exercises contain additional historical and bibliographical commentary and formulations of the results of various authors. Sometimes the notions introduced in the exercises are subsequently used (with an appropriate reference) in the main text of the book. A serious analysis of the material contained in the exercises will enable the reader to go more deeply into various aspects of the theory being studied and sometimes to understand its connection with other branches of functional analysis. Furthermore, it is only in the exercises that we occasionally give a mechanical interpretation of the results being presented and explain the role of individual methods in the theory of oscillations. At the present time, in addition to this book, only the book of J. Massera and J. Schaffer [1] is especially devoted to the study of differential equations in Banach spaces. Its contents overlap those of the third, (more particularly) the fourth and to some extent the fifth chapters of this book. For the reader who wishes to become deeply acquainted with the notion of a dichotomy of the solutions of a linear differential equation the book of Massera and Schaffer will serve as an important supplement to ours (although we do have something new to say in this field). On the other hand, we present a more complete account of the theory of periodic linear equations and a number of other questions. Moreover, the book of Massera and Schaffer considers only linear equations. To better partition the material and to more clearly distinguish the places of greater importance we enumerate not only the lemmas and theorems but also a large number of remarks and corollaries. They are always numbered by two non-
INTRODUCTION
7
negative integers (for example, Lemma 3.5) the first of which indicates the section and the second, the lemma, theorem, etc. in the given section. We also number formulas in the same way, a formula number of the form (0.3), for example, referring to a formula in the exercises. In referring to a formula, theorem or section of another chapter we add a Roman numeral to indicate the chapter (for example, Theorem 1. 3.5).
CHAPTER 1 SOME INFORMATION FROM THE THEORY OF BOUNDED OPERATORS IN BANACH SPACES In this chapter we present various bits of information from the theory of bounded operators in Banach spaces, which will be systematically used in the sequel. Some of this information has long since found a place for itself in the textbooks and much of the rest is well known to the specialist. But some of the facts presented here are quite new. In § 1 we present a number of basic definitions and theorems concerning Banach spaces and the operators acting in them. We then use Cauchy's formula to define a function of an operator ( § 2) and to consider the most important operator function for our purposes, viz. the exponential operator function eAt ( § 4). This function is a one parameter commutative group of bounded operators. Its order of growth at infinity has a simple connection with the spectrum of the operator A. In § 6 we consider bounded commutative groups of operators in Hilbert space that are similar to groups composed of unitary operators. Many questions connected with differential equations reduce to the consideration of "algebraic" linear operator equations. We consider such equations in § 3, where we find conditions for their solvability and integral formulas for their solutions. Other expressions for the solutions of these equations can sometimes be obtained with the use of the exponential operator function ( § 4). In a number of cases the behavior of the solutions of differential equations becomes more visible if the norm of the space is replaced by one of a special form. Such norms are introduced in § 4 with the use of the exponential operator function eAt. When the space is a Hilbert space there correspond to such norms positive quadratic forms that can be calculated by means of linear operator equations if the spectrum of the operator A lies in the interior of the left halfplane ( § 5). We assume that the reader is familiar with the fundamentals of the geometry of Hilbert spaces. An essential role in the investigations in the sequel is played by the indefinite metric. In §§ 7 and 8 we study in a Hilbert space possessing an indefinite metric introduced with the use of a certain Hermitian operator W the important classes of W-dissipative, W-contraction and W-unitary operators. The results in§§ 7 and 8 are of a less elementary nature. But except for the simplest definitions connected with an indefinite metric they are used only in the fifth chapter. The reader can therefore postpone the study of these sections until he reaches Chapter V. In § 9 we present some well-known facts from nonlinear analysis that play an important role in the study of nonlinear equations. At the end of the chapter we give a rather large number of exercises of various types and pursuing different ends. Some of these are routine drill problems for a better understanding of the text, while others present important notions and results that will frequently be used in subsequent chapters in the main text (with an appropriate reference).
9
1.
10
BOUNDED OPERATORS IN BANACH SPACES
§ 1. General propositions on the geometry of Banach spaces and their linear mappings l. Normed and Banach spaces. A set 2 is called a real (complex) normed space if 1) It is a linear vector space over the field ofreal (complex) numbers, and 2) for each vector E 2 there is defined a nonnegative number II, viz. the norm of x, having the following properties: a) I ax I = I a III x I for any x E 2 and any real (complex) number a; b) I x + y I ;;;; I x I + I y I for any x, y E 2 (the triangle inequality); c) Ilxll = 0 if and only if x = o. The function p(x, y) = I x - y I in a normed space defines a metric, so that 2 is a metric space. A sequence { x n } c 2 is called a Cauchy sequence if limn.m~oo I Xn - Xm I = o. A normed space is called a Banach space if every Cauchy sequence in it has a limit point x for which limn~oo I Xn - x I = 0 (in other words, a Banach space 2 ( = 58) is complete in the metric p (x, y) = I x - y II). Every incomplete normed space 2 can be completed, i.e. can be "identified" with a dense linear manifold in a Banach space 58. This is done in a well-known way by considering the set of all Cauchy sequences in 2. If another norm I x 112 is given in a normed space 2 with norm I x 111, every convergent (in the norm I x 111) sequence converges in the norm I x 112 if and only if there exists a positive constant Cl such that I x 112 ;;;; Clil x Ik Two norms I x 111 and I x 112 given in a linear space 2 are said to be topologically equivalent if convergence in one of them implies convergence in the other. For this to be so it is necessary and sufficient that there exist positive constants Cl and C2 such that
x
Ilx
C2 ;;;; I x 112/ I x Ih ;;;; Cl
(x f= 0).
(1.1)
Two Banach spaces are said to be isomorphic if there exists a one-to-one continuous linear map of one space onto the other. This map is called an isomorphism. If it preserves the norm, it is called an isometric isomorphism. Clearly, a Banach space 58 goes over into an isomorphic space if its norm is replaced by a topologically equivalent one. 2. Linear operators. Let 581 and 58 2 be Banach spaces. A map A: 581 ~ 58 2 is called a linear operator if A(ax + (3y) = aAx + {3Ay for any numbers a, {3 and any x, y E 581. A linear operator is continuous if it is continuous at the point x = o. Continuity is equivalent to the property of boundedness of an operator A, i.e. of finiteness of the quantity
I
A
I ~sup { I
Ax liz! I x = sup { I Ax 1121 x
Ih Ix E 581> X f= 0 } E 581. I xiiI = 1 }.
(1.2)
1. GEOMETRY OF BANACH SPACES
11
The set of bounded linear operators A: 181 --+ 182 will be denoted by [181> 182]. This set is a Banach space with norm (1.2) under the natural definitions of the operations of adding operators and multiplying an operator by a number. (A
+ B)x
= Ax
+ Bx;
(aA)x = a(Ax).
If A: 182 --+ 183 and B: 181 --+ 18z, the formula Cx = A(Bx) (x E 181) defines an operator C: 181 --+ 183, which is called the product C = AB of the operators A and B. If the operators A and B in this connection are bounded (A E [182 , 183], B E [181> 182]). Then the operator C is bounded (C E [18 1, 183]) and II C I = I AB I ~
IIA IIIIBII·
An operator B: 182 --+ 181 is called the inverse of an operator A : 181 --+ 18z, and one writes B = A-I, if AB = 12 and BA = II, where Ik is the identity operator in 18k : Ikx = x for any x E 18k (k = 1,2). The set of bounded linear operators mapping a space 18 into itself is briefly denoted by [18]. Thus [18] def [18, 18]. Under the operation of multiplication defined above [18] is not only a Banach space but also a Banach algebra. We formulate without proof the following fundamental proposition. THEOREM 1.1 (BANACH). Suppose an operator A E [181> 182] is a one-to-one mapping of a Banach space 181 onto a Banach space 182 • Then its inverse A-I is bounded and linear: A-I E [18z, 18d. This justifies the above definition of isomorphic Banach spaces. COROLLARY 1.1. Suppose given in a linear space B two norms I x 111 and II x 112 that convert it into Banach spaces 181 and 18z respectively, and suppose I x liz ~ c111 xiiI (x E B), where C1 is a positive constant. Then there exists a positive constant Cz such that II x 111 ~ czll X liz, and consequently the norms II x 111 and II x liz are topologically equivalent. For by Banach's theorem the identity mapping x --+ x of 181 onto 18 2 , being bounded, has a bounded inverse. The method by which Theorem 1.1 is normally proved permits one to prove the following impor~,ant proposition. THEOREM 1.2 (uniform boundedness principle). Suppose from [181> 18z] such that for each x E 181 sup {
II
Ax II I A
Then ~ is bounded, i.e. sup { II Ax III A
E
E 2( }
~}
<
<
~
is a set of operators
00.
00.
From this theorem it follows in particular that if a sequence of operators An
E
I.
12
BOUNDED OPERATORS IN BANACH SPACES
= limn_oo Anx defines a continuous linear operator A E [~b ~]. 3. Direct sums of subspaces and projections. A closed linear manifold of a Banach space ~ is called a subspace of~. One says that a Banach space ~ decomposes into a direct sum of subspaces ~l and ~2:
l~b ~i1 converges at each element x E ~b the equality Ax
(1.3) if each element x
E ~
admits a unique representation in the form (1.4)
where Xl E ~l and X2 E~: Each ofthe subspa~es ~l and ~2 is said to be a direct complement of the other. The decomposition (1.3) defines two linear operators P k : ~ --. ~k (k = 1, 2) by means of the equalities P,tX = Xk, where Xl and X2 are the components of X in the decomposition (1.4). The operators PI and P 2 obviously have the properties
Pi = P k ; They are also continuous: Pk
PI
+
P2
E [~].
I X lit = I Xl I
= I;
P lP 2
=
P 2P l
= o.
(1.5)
To prove this let
I xd ( = I PIX I + I P 2x I ) Xl + X2, we have I X I ~ I Xl I + I X2 I = I X Ih· +
for any X E~. Since X = It is not difficult to see that the completeness of the spaces ~l and ~2 implies the . completeness of ~ in the norm I X (Therefore Corollary 1.1 implies the existence of a constant c> 0 such that I X III ~ cll X II, whence I p."x I ~ cll X II, i.e. P k E [~] (k = 1, 2). An operator P E [~] is called a projection if p2 = P. If an operator PI is a projection then the operator P 2 = I - PI is also a projection, all of the relations (1.5) being satisfied. The operators PI and P 2 are called mutually complementary projections. As noted above, every direct decomposition (decomposition into a direct sum) defines a pair of mutually complementary projections. Conversely, every pair of mutually complementary projections defines a direct decomposition (1.3), where ~l = Pl~ and ~ = P2~' A subspace 21 c ~ is said to be complemented if there exists at least one subspace 22 c ~ such that ~ = 21 2 2, From what has been said it follows that a subspace 21 c ~ is complemented if and only if there exists at least one projection P E [~] that projects ~ onto 2 1 :
!It-
+
P~=
2 1,
It is easily seen that every nnite-dimensional subspace as well as every subspace 2 of finite codimension (i.e. such that dim ~ /2 < (0) is complemented.
1.
13
GEOMETRY OF BANACH SPACES
Clearly, in a Hilbert space each subspace'£? is complemented and, what is more, has a unique orthogonal complement ,£?~. An analogous definition exists for the decomposition of a Banach space Q3 into a direct sum Q3 = Q3l + ... + Q3n of several subspaces, and it can be shown that every such decomposition defines and is defined by a decomposition of the identity into a sum of projections: I = PI + ... + P n , that are pairwise disjoint: PjPk = PkPj = OjkPk
1 .= k
(Ojk =
{o:~ =F
k is the Kronecker Symbol).
(1.6)
We also note that I P I ~ I P 11 2 , i.e. I P I ~ 1, for any projection P by virtue of the equality p2 = P. 4. Direct sums of Banach spaces. Let Q3 h ···, Q3n be Banach spaces with norms I • 111,,'" I • lin. These spaces can be regarded as subspaces of a Banach space Q3 such that
Q3
=
Q3l
+ Q3z + ... + Q3n.
(1.7)
The space Q3 in this connection will be defined to within an isomorphism. In fact, let,£? denote the linear space of ordered collectionsll (Xk
E
Q3k; k = 1,2,.··,n)
with the natural definition of a sum of two collections and multiplication by a scalar. We convert,£? into a Banach space Q3 by defining a norm I • I in,£? so that (1.8) for every collection x all of whose components except the kth are equal to zero. This can always be done by putting I x I = I x 11M, for example, where n
I xilM = k=l 1:: I
Xk
(1.9)
Ilk'
After this we can isometrically imbed the spaces Q3k in ,£? by identifying any element Xk E Q3k with the collection (0,.··, 0, Xk, 0, ... , 0). The decomposition (1.7) is then clearly obtained. We note that of all of the norms I x I satisfying condition (1.8) the norm (1.9) is the maximal norm, i.e. one always has I x I ~ I x 11M for each x E Q3. For if the decomposition (1.7) holds then x = Xl + ... +Xn , and hence
I x II ~ I xIII + I x211 + ... + I Xn I
=
I XliiI + I x2112 + ... + I x" II"
=
I x 11M.
It is not difficult to give a description of all of the norms satisfying condition Il We will find it more convenient at times to write this collection in the form of a column or in the form x = Xl X. (we recall that this sum depends on the order of the summands).
+ ... +
I. BOUNDED OPERATORS IN BANACH SPACES
14
(1.8) by making use of the norms in an n dimensional space (the Minkowski norms). We confine ourselves to mentioning the family of norms (I
~
p ~
00
).2)
5. Direct(orthogonal)sum of Hi/bert spaces. If the 58k (k = I, "', n) are Hilbert spaces ~k' a scalar product can be defined in their direct sum 58 so that 58 is also a Hilbert space ~ and the ~k are mutually orthogonal subspaces of ~. To this end we put n
(x, y) = 1:; (Xk' Yk)k
(X,y
E~).
k=l
Clearly, this definition of a scalar product is the only one possible. One then calls ~ an orthogonal sum of Hi/bert spaces and writes ~ =
'Pl EB ~2 EB ... EB ~n'
From what has been said we can make the following useful REMARK 1.1. Suppose I = PI + ... + P n is a decomposition of the identity into a sum of pairwise disjoint projections acting in a Hilbert space~. Then the scalar product in ~ can be replaced by a topologically equivalent one with respect to which all of the projections P k are orthogonaP) projections. To this end it is necessary to construct the orthogonal sum
sy
= Pl~
EB
P2~
EB .. · EB
Pn~
and identify it in the appropriate manner with ~. This leads to the following definition of the desired scalar product: (x,y), = 1:;~ (PkX,Pky). 6. Complex hull of a real space. It is often useful to consider in conjunction with a real Banach space 58 a complex Banach space >8, which is called the complexijication of 58. The space >8 consists by definition of all possible pairs (Xh X2) of elements of 58, with algebraic behavior defined by the symbolic form (Xb X2) = Xl + iX2 and with norm
III Xl + iX2111 = max" Xl cos 8 + X2 sin 811· o It is easy to verify that under this definition >8 is actually a complex Banach space. The complex hull ~ of a real Hilbert space ~ is more naturally defined by specifying, instead of a norm, a scalar product in such a way that ~ will be a complex Hilbert space. This can be done by setting
(Xl
2) 3)
+
iX2' Jh
+ iY2)
=
(Xb Yl)
+ (X2' Y2) + i [(X2, Yl)
- (Xb Y2)].
II x II ~ limp-;oo II x II (p) = max (II XIII b II xzllz,"" II x. II.J. A projectionP E [.))1 is called an orthogonal projection if x - Px .1. Px or, equivalently, P = P*. 00
2.
15
FUNCTIONS OF BOUNDED LINEAR OPERATORS
§ 2. Functions of bounded linear operators 1. Spectrum and resolvent. Let lB be a complex Banach space. 4) A point A of the complex plane is called a regular point of an operator A if[lB] contains the operator (the value of the resolvent of A at A)
RA
= (A -
E
[lB]
AI)-I.
The set p(A) of all regular points of an operator A is open. Its complement p(A) is called the spectrum of A. The spectrum a(A) is always nonempty, closed and lies in the disk I A I ~ I A ". More precisely, the spectrum a(A) lies in the disk of radius rA
=!~ ~"An"
(the existence of the limit readily follows from the relation I Am+n I ~ "Am" • I An II). In fact, when I A I > r A the series l:;;" A-(Hll Ak converges absolutely in the metric of [lB] inasmuch as the corresponding series of norms is ultimately (for k sufficiently large) majorized by the geometric progression { (rA + C )k / I A for any c > o. Upon multiplying this series by A/ - A, we obtain l. Thus, when I A I > rA, the resolvent always exists and, moreover,
IHI }
RA
= -
l: A-(k+1)Ak.
(2.1)
k=O
It can be shown that the circle IA I = rA always contains a point of the spectrum ~l4nlr is called the spectral radius of A. As can be directly verified, the resolvent equation
a (A). Therefore the limit limn~oo
(A, fl
E
p(A»)
(2.2)
holds for R A• The resolvent RA is an analytic function of Ain a neighborhood of each regular point fl' the absolutely convergent expansion
being valid for I A - fll < 1/ I RI' II· An expansion of the resolvent in a neighborhood of the point at infinity has just been indicated in (2.1). We end this subsection with the following special rule, notable for its simplicity: if I A " < 1 (more generally, rA < 1), then the operator / - A is invertible and (/ - A)-I
= l:
Ak.
k=O
2. Behavior of the spectrum and resolvent of an operator under small perturbations. <)
Analogous notions can be introduced for a real space by going over to its complex hull.
I.
16
BOUNDED OPERATORS IN BANACH SPACES
Let G be an open set covering the spectrum (J(A) of an operator A E [~]. Then in every case the resolvent R),(A) = (A - AI)-1 is continuous on the complement F of G in the closed complex plane. This implies the existence of
I R;,{A) IlIA ¢ G }( < a-neighborhood Ua(A) = { X III A M = max {
We consider a Since for any X E
00 ).
I
X
< a} of A in
[~].
[~]
X - AI = (A - AI) + (X - A) = (A - AI)[f - R), (A)(A - X)]
I
(A ¢ G),
I
X - AI will be invertible whenever R;,{A)(A - X) < 1. Consequently, if a < 11M, an operator X E Ua(A) will have a resolvent R),(X) for all A ¢ G. Thus, under the indicated choice of a, the spectrum (J(X) c G for any X E Ua(A). Let us estimate the difference R),(A) - R),(X) for A ¢ G. From the above representation for X - AI it follows that R),(X)
For X
E
= [I - R),(A) (A - X)]-1 R),(A).
Ua(A) and A ¢ G we have
I R),(A)(A -
X)
I
< Ma«
1),
so that
I R), (X) -
R),(A)
11= II E1 R~+1(A) (A -
AI)k [I
~ 1 ~2~ a'
We have arrived at the following proposition. THEOREM 2.1. Suppose A E [~] and G is an open set covering the spectrum (J(A). Then there exists a a-neighborhood Ua(A) of A such that (J(X) c G for all X E Ua(A). Moreover, for any > 0 there exists a a such that R),(X) - R),(A) < for X E Ua(A) and A ¢ G.
c
I
I
c
3. Analyticfunctions of operators. Let x(t) (a < t < b) be a vector function with values in ~. If this function is continuous, the integral b
Jx(t) dt a
exists in the usual (Cauchy) sense as a limit of integral sums. Similarly, if A(t) (a ~ t ~ b) is a continuous operator function with values in [~h ~2]' we have the existence of A(t)dt, which is an element of [~h ~2]' The following estimates hold in this connection:
J!
f
II x(t) dt I
~ { Ilx(t) Ildt,
f
I A(t) dt II
~ J IIA(t) I dt.
2. FUNCTIONS OF BOUNDED LINEAR OPERA TORS
17
Analogous assertions hold for integrals of vector and operator functions along curves in the complex plane. The classical theory of Cauchy on the vanishing of an integral around a closed contour can be extended to analytic vector and operator functions. Suppose A E [58]. Let KA denote the class of all functions ¢().) of a complex variable that are piecewise analytic on the spectrum a(A). Theis means that a function ¢ E KA if it has the following properties. 1) The domain of definition Dq, of ¢().) consists of a finite number of open connected components whose union contains the spectrum a(A) of A, each component containing at least one point of the spectrum. 2) The function ¢().) is piecewise holomorphic, i.e. holomorphic in each component of its domain of definition Dq,. Clearly, if two functions ¢l().), ¢z().) E KA coincide on an open neighborhood of the spectrum a(A), they will be analytic continuations of each other. Let us agree not to distinguish between such functions. The class KA can be converted into an algebra with unit by introducing in a natural way the operations of addition, multiplication and multiplication by a scalar. According to a well-known principle of F. Riesz there exists an algebraic isomorphism between the algebra KA and a certain commutative algebra of operators under which the function ¢().) == ). corresponds to the operator A (and hence ¢().) = 1/(). - z) (z ¢ a(A) corresponds to the resolvent Rz = (A - zI)-l). This isomorphism is established in the following way. For ¢().) E KA there can always be found a smooth and, in general, composite contour A that surrounds the spectrum a(A). The latter statement means that A decomposes into a finite number of boundaries of certain open sets whose union belongs to Dq, and covers a(A). Each of the Jordan contours making up A can be oriented so that the corresponding open set remains on the left as the contour is traversed in the positive direction. After this, we put for ¢().) E KA
r
r
r
1. ¢(A) = - - 2
f ¢().) R;.d().).
11:l r
(2.3)
A
It follows from Cauchy's theorem that the integral in (2.3) is independent of the ." choice of A. The correspondence ¢().) ~ ¢(A) is clearly linear. Its multiplicativeness follows from the more general formula (2.4), which we prove below.
r
LEMMA 2.1. Let Fl ().) and Fz()') be operator functions that are piecewise analytic on the spectrum a(A) and whose values are operators in [58].5) Then 5) The definition of an operator function that is piecewise analytic on the spectruml1(A) is analogous to the definition in the scalar case.
18
I.
BOUNDED OPERATORS IN BANACH SPACES
PROOF. Let r 1 and tz be two contours of the type of ing r z. From (2.2) it.follows that
rA, the contour r 1 surround-
1 1 - 2. § Fl (A) R;. dA - 2. § Rj1 Fz(fl) dfl 11:1 r, 11:1 r,
+ / 11:z r,§ r,§ FdA)
A~
fl
Fz(fl) dA dfl
From here we get formula (2.4), since ,~Wd J;:-=-fl r, fl
. = 0 for A E r l whlle
1 ~~ § A _ fl dA = FI (fl) 211:i r,
The lemma is proved. By choosing (h(A)I, ¢z(A)I as FlU), Fz(A), we get that ¢(A) ¢(A)
=
for fl
E
r z·
= ¢1(A) ¢z(A) implies
¢1(A)¢z(A).
The correspondence A+-+ A follows from the relation 1. - -2
§ A(A
11:1 r A
- Al)-ldA
=
A,
r
which is obtained from (2.1) by taking as A any Jordan contour surrounding the disk A ~ rA· From the linearity and multiplicativeness of the correspondence ¢(A) +-+ ¢(A) it follows that ¢(A) = 2:~ CkAk implies the formula
I I
By passing to the limit, this formula can be extended to power series that converge in a disk containing the spectrum I1(A). In particular, the operator
2.
19
FUNCTIONS OF BOUNDED LINEAR OPERATORS
eAt =
00
1:::
k=O
Akt k k.
1
-~,- = - - .
§eAtRAdJ... 2m r A
corresponds to the function eAt. It is known that there exists a simple connection between the spectrum a(A) of an operator A and a spectrum a( ¢'(A)), where ¢'(J...) is a function of class KA • This connection is described by the relation (the spectral mapping theorem) (2.5)
a(¢'(A)) = ¢'(a(A)). We note a particular consequence of the latter formula.
LEMMA 2.2. If II eA II = q( II eA II < q), the spectrum a(A) of A lies in (in the interior of) the closed halfplane Re J... ~ In q.
For according to (2.5)
a(eA ) = {
pi p = eJ., J... E a(A) }. Ipi
On the other hand, by hypothesis, a(eA ) lies in (in the interior of) the disk ~ q. 4. The F. Riesz integral. Let us consider an operator A E pal whose spectrum forms a disconnected set. A closed subset of a(A) whose complement in a(A) is also closed is called a spectral set. Suppose m
a(A)
= U ak(A), k=l
where the ak(A) (k = 1, "', m) are disjoint spectral sets. In considering such a situation below we will assume that the contour FA consists of disjoint parts Fk each of which is the boundary of a domain G k containing the corresponding spectral set ak(A). Let A. 'l'k
(J...) = { 1 for J... E Gk, 0 for J... E Gj, j =I- k.
The functions ¢,k(J...) (k = 1, . ", m) belong to the class KA , and we can therefore let (2.6)
Inasmuch as the relations
(Okj is the Kronecker symbol) are clearly satisfied in the domain G = we have the equalities
Ur=l Gk,
20
I.
BOUNDED OPERATORS IN BANACH SPACES
PkPj
= 0 (k #-j);
P~
m
= Pk; 1:: Pk =
(2.7)
l.
k=l
Thus the operators P k form a decomposition of the identity into a sum of pairwise disjoint projections. From formula (2.6) it follows that the Pk commute with A; PkA = APk
(k = 1,2,.··, m).
(2.8)
We recall that a subspace 2 c 18 is said to be invariant under an operator A if A2 c 2. It is easily seen that if 2 is a complemented subspace: 18 = 2 Wl, i.e. ifthere exists a projection P S3 projecting 18 onto 2, then the invariance of 2 under A is equivalent to the relation P S3 APS3 = APS3. And A commutes with P S3 if and only if 2 reduces A, i.e. both 2 and its complement Wl = (/ - PS3)18 are invariant under A.
+
From what has been said it is evident that the validity of the commutativity relations (2.8) is equivalent to the assertion that each of the subspaces 18k = Pk18 is invariant under A. It can be verified that the spectrum Il(A I18k) of the restriction of A to the subspace 18k coincides with the spectral set Ilk(A). In the formula Pk
1
= - -2 .- f R, d)' 7rl
r.
the right side is known as a Riesz integral. The space 18 is the direct sum 181 + ... + 18m , as follows from the last relation in (2.7). Let us agree to call the projection Pk and the subspace 18k = Pk18 the spectral projection and invariant subspace corresponding to each other and the spectral set Ilk(A).
In the sequel we will often consider an operator A whose spectrum does not intersect the imaginary axis. In this case we will always let 1l_(A) and 1l+(A) denote the spectral sets of A lying in the interiors of the left and right halfplanes respectively. This situation will be briefly expressed by the equality Il(A) = 1l+(A) U 1l_(A). In particular, the equality Il(A) = Il-(A) will mean that Il(A) lies in the interior of the left halfplane. The equality Il(A) = 1l+(A) has an analogous meaning. The corresponding spectral projections will be denoted by P _ = P _(A) and P+= P+(A).
If each of the sets Il ±(A) is nonempty, then for reasons to be given in Chapters II and IV we will sometimes say that A is exponentially dichotomic (e-dichotomic). 5. Theorem on the continuous dependence of a spectral projection on the operator. Suppose A E [18] and is a closed contour not intersecting Il(A) and surrounding a spectral set llo(A). We will briefly say that r distinguishes the set llo(A). As follows from subsection 2, the contour r will then not intersect the spectrum Il(X) of any
r
3.
21
SOLUTION OF LINEAR OPERATOR EQUATIONS
operator X E Ua(A) for asufficiently small (the role of the open set G can be played, for example, by the complement of in a sufficiently large open disk i\ < R). Moreover, a a > 0 can be chosen so small that the resolvents R).(X) and R).(A) will differ arbitrarily little from each other on r. But then the projections
I I
r
PA =
-
1 . .f R). (A) di\ -2 nl r
Px
and
= - -2-~.f R). (X) di\ nl r
will differ arbitrarily little from each other. Thus the following proposition holds.
r
THEOREM 2.2. Suppose A E [)H] and is a contour distinguishing a spectral set l1o(A) corresponding to a spectral projection P A. Then for any c > 0 there exists a a-neighborhood Uo(A) such that for all X E Uo(A) the contour will distinguish a spectral set l1o(X) corresponding to a spectral projection P x in the c-neighborhood of PA: II PA - P x II < c.
r
§ 3. Solution of some linear operator equations 1. General formula for the solutions. In certain problems connected with differential equations it is necessary to investigate linear operator equations similar to those of the form AX + XB = Y. We consider the more general equation n
1:
j,k=O
Cjk AjXBk
= Y,
(3.1)
where the Cjk ~:re complex numbers, X is the desired operator and A, Band Yare given operators. In order to consider equation (3.1) in the most general setting we will assume that B E [)Hd, A E [)Hz], Y E [)HI, )Hz] and the desired operator X E [)HI. )Hz]. Let AI and Br denote the linear operators acting on any operator X E [)HI. )H2] that are induced by multiplying X on the left by A and on the right by B respectively:
B
It is easily seen that the operators AI and Br commute for any A [)HI]' With the use of the polynomial
E
[)H2] and
E
n
P(i\, (1)
= 1:
j,k=O
Cjk i\j(1k
we form the operator P A,B ~ P(AI, Br)
n
= 1:
j,k=O
Cjk A{ B;
(E [)HI. ~2])'
(3.2)
I.
22
BOUNDED OPERATORS IN BANACH SPACES
Equation (3.1) now takes the form PA,B X for the existence of an inverse of P A,B. Suppose Art a(A). Then the operator
Y. We will first determine a condition
=
(AI - AI)-l = (A - AI)I- 1
clearly exists. This means in particular that a(AI)
C
a(A).6) In exactly the same
waya(Br) c a(B).
We can now explicitly express the operator P(AI, Br) in terms of the polynomial
peA, fl). In fact, by substituting expressions obtained from (2.3) for the powers of the operators AI and Br in (3.2), we have P(AI, Br)
1
= - - 42 § § P(A, fl) (AI 7r
rAre
- AI)-l (Br - fll)-l dA dfl·
(3.3)
Formula (3.3) suggests the following generalization of the problem under consideration. Let KA,B be the set of all single valued functions ¢(A, fl) that are defined in a neighborhood of the set a(A) x a(B) and analytic in A (fl) in a neighborhood of each point of a(A) (a(B») for fixed fl E a(B) (A E a(A»). For each function ¢(A, fl) E KA,B we define an operator acting in the space [5lh, 5l3 2]:
Formula (3.4) establishes a correspondence, between the functions of class and a certain commutative set of operators acting in [5l31o 5l3zJ, that has the following properties: a) If ¢(A, fl) == 1 then ¢(AI' Br) = l. b) If ¢(A, fl) = a1¢1(A, fl) + a2¢2(A, fl), where a1 and a2 are constants, then KA,B
¢(AI' Br) = a1¢1(AI, Br)
c) If ¢(A, fl)
=
+ a2¢2(AI, Br).
¢1(A' fl)¢2(A, fl) then, as follows from Lemma 2.1, ¢(AI' Br) = ¢l(AI, Br) ¢2(AI, Br).
d) If limn~c>o ¢nCA, fl) = ¢(fl, A) uniformly in a neighborhood of the Cartesian product a(A) x a(B) then lim ¢n(AI, Br) = ¢(AI' Br). n~oo
We note that if X ¢(AI' Br) X
E
[5l3r, 5l3 2] then
-= - 4~2
§ § ¢(A, fl)(A
- AI)-l X (B - fll)-l ddfl.
FA FH
The above remarks imply the following proposition . •llt can in fact be shown that IJ(A/)
= IJ(Ar) = IJ(A).
23
3. SOLUTION OF LINEAR OPERATOR EQUATIONS
THEOREM 3.1. Suppose afunction
4n
f
FA FB
E
[>8 1, >8 2] for each Y E [>81> >8 2], It is represent-
(A - Al)-l Y(B - pI)-l dA dp.
(3,6)
We formulate this result for equation (3.1) in particular. THEOREM 3.2.
If the condition n
P(A, p)
=
«A, p) E (l(A)
1:; Cik Aipk "# 0 i,k=O
is satisfied, equation (3.1) has for each Y which is representable in the form X = - ~
4n
ff
E
x (l(B))
[>8 b >8 2] a unique solution X
(A - Al)-l Y(B - pI)-l dA dp.
E
[>8 b >8 2], (3.7)
peA, p)
FA FB
REMARK 3.1. Suppose A, BE [>8] and a function
8 1 denote the invariant subspace of A that corresponds to the spectral set (ll(A), let PI denote the corresponding spectral projection and let Al denote the restriction of A to >8 1, In exactly the same way we define for the operator B the invariant subspace >8 2 corresponding to the spectral set o2(B), the spectral projection P 2 and the restriction B 2• If the operator Y in (3.1) satisfies the condition Y = PI YP2 , equation (3.1) can be reduced to the form n
.
~
k
~
1:; Cik Ai XB 2 = Y,
i,k=O
by considering X and Y as operators from >8 2 into >81 , Formula (3.7) then permits one to find a solution X E [>8 2, >81], This operator can be extended to the whole space >8 by putting X = P 1 XP2 • The operator X will be the unique solution of equation (3.1) satisfying the condition X = P 1XP 2 • Clearly, all of what has been said also applies to the case of the general equation (3.5). 2. Special cases. a) We consider the equation
AX- XB= Y.
(3.8)
I.
24
BOUNDED OPERATORS IN BANACH SPACES
Here Po., p) = A - p. Thus under the condition A - p "# i.e. under the condition O"(A)
n 0" (B) =
°
(A, p)
E
O"(A) x O"(B» ,
0,
(3.9)
this equation has a unique solution, and it is given by the formula X = __ 1 £ £ (A - AI)-1 Y(B - p/)-1 dA d . 4",2 'J 'J A - r" P FA r B
(3.10)
Suppose now
= O"+(A) U O"-(A),
O"(A)
O"(B)
= 0" +(B) U 0" -(B).
According to Remark 3.1 we can determine the operator X - P (A) XP (B) - __1_ £ £ (A - AI)-1 Y(B - pl)-1 dA d + 4",2 'J 'J A - r" p, FA+ F
(3.11)
A_
which is the unique solution of equation (3.8) satisfying the condition P +(A)XP -(B) = X, provided the condition P +(A) YP -(B)
(3.12)
= Y
is satisfied. An analogous result is obtained under the condition P _(A) YP +(B) = Y. b) We now consider the equation (3.13)
AX- XA = Y.
= A - p and condition (3.9) is obviously not satisfied. We assume, however, that the spectrum O"(A) decomposes into two spectral sets: O"(A) = 0"1(A) U 0"2(A). When A E 0"1(A) and p E 0"2(A), we have A - p "# 0, and hence under the condition Y = P1YP2 the operator In this case Po., p)
X
= ___ 1
£ £
(A - AI)-1 Y(A - p1)-1 dA d
4",2 'J 'J
r r2 l
A _ r"
p
(3.14)
is the unique solution of equation (3.13) satisfying the condition X = P1XP2. c) We also consider the equation b
JeAt Xe Bt dt =
(3.15)
Y.
a
It can be reduced to the form (3.5) if one makes the substitutions
eAt
1
= - - 2. § eAt (A - AI)-1 dA, "'l r A
eBt
1
= - - . § efll (B - pl)-1 dp. 2",l
r.
In fact, upon making these substitutions and changing the order of integration, we get from (3.15)
4.
25
EXPONENTIAL OPERATOR FUNCTION
1 - 42 § § ¢(A, p.) (A - A/)-l X(B - p.I)-l dA dp. = Y, rc r A r.
where
¢(A, p.) =
eU+/l)b _ e(.l+/l)a
b
Je(.l+/l)t dt = -~---a A+p.
This function does not vanish for (A, p.)
I Im(A + p.) I < 2rc/(b -
E
a)
o-(A) x o-(B) if, for example, the condition
(A, p.)
E
o-(A) x o-(B)
(3.16)
is satisfied. Consequently, under condition (3.16) equation (3.15) has a unique solution, which is representable in the form (3.17)
§ 4. Exponential operator function 1. Definition of the exponential operator function. In the theory of differential equations an especially important role is played by the operator function eAt, which can be introduced by means of either one of the two relations 1
eAt = - . JI eAt (A - AI)-l dA , 2rcl rA Aktk eAt = I; ~-, k=O k.
(4.1)
00
(4.2)
From the multiplicativeness of the correspondence ¢(A) the operators eAt form a one parameter group:
{
eAteAT
I
=
eAt t=O
eA(t+T) =
( -
00
< t;
7:
+-+
¢(A) it follows that
< (0),
1.
We note, incidentally, that in general eA +B =I- eA eB • The equality sign holds here whenever AB = BA. Further, by differentiating relation (4.1) with respect to t under the integral sign, we obtain the formula (4.3)
In fact,
={ - _1_. § A(A _ 2rcl r A
)./)-1
dA} { _ _ 1. § eAt(A - A/)-l dA} 2rcl r A
=AeAt.
26
I. BOUNDED OPERATORS IN BANACH SPACES By making use of the series (4.2) it is not difficult to obtain the estimate
I eAt I ~ I;
k=O
"Ak" tk
k!
= eliAlit
(t ;?; 0).
This estimate, however, is too rough; it can be replaced by a more exact one if the location of the spectrum of the operator A is known. We first introduce an important notion. 2. Estimate of the growth of the exponentialfunction. Let pet) (0 ~ t < (0) be (] positive function. As is well known, the (upper) Ljapunov exponent of pet) is the quantity I>
= lim lnp(t) . t
This quantity coincides with the greatest lower bound of real numbers p for which there exists a constant Np such that pet) ~ Npe pt for all t ;?; O. If the limit limhOO In p(t)/t exists, I> is called the strict Ljapunov exponent of pet). The strict Ljapunov exponent (if it exists) also coincides with the least upper bound of real numbers p' such that pet) ;?; eP'1
for sufficiently large t. THEOREM 4.1. For any A exponent 1>, and I>
E
= lim In I 1-->00
[18] the function" exp (At) " has a strict Ljapunov
eAt t
I =
PROOF. We put pet) = " exp (At) In p(t
+ s)
I
max {Re A A E O'(A)}.
II. Clearly,
~
In pet)
p(t
+ s)
(4.4)
~ p(t)p(s), i.e.
+ In pes).
For any c > 0 there exists an he such that In p(h) < -1'- In p(t) h = 1-->00 1m t +c
(h ;?; he).
For any t ;::.. 0 there exists an integer n ;?; 0 such that t Then In pet) ~ n In p(h) + In per) ~ n In ~(h) + nh + r n +r and hence -1' In pet)
1m
1-->00
t
< In p(h) --
h
nh
c (c =
+ r (0
~
r
~
h).
max per) ), O~r~h
(4.6)
.
Comparing (4.5) with (4.6), we conclude that the limit I>
=
(4.5)
=
liml-->oo (p(t)/t) exists.
4. EXPONENTIAL OPERATOR FUNCTION
27
To prove equality (4.4) we note that it is equivalent to the following assertion: in order for a real number p to have a positive number Np corresponding to it such that
I eAt I
~ Np e pt
(t ~ 0),
(4.7)
it is necessary that Re A ~ p for all A E a(A), and sufficient that Re A < P for all
AE a(A). Suppose estimate (4.7) is satisfied. Then according to Lemma 2.2 the spectrum a(At) ( = t a(A» lies in the halfplane Re A ~ In Np + pt (t ~ 0). Hence for any t > 0 the spectrum a(A) lies in the halfplane Re A ~ In Np / t
+ p.
In view of the arbitrariness of t > 0 this implies the first part of the assertion. For a proof of the second part we write eAt in the form eAt
=
-
_1_. feAt RA dA 211:1 r A
'
rA lies completely inside the halfplane Re A < p. Then I eAt I ~ 2~ FAJIeAt III Rd I dA I ~ Np ePt , maxAEr I RA II, I being the length of the contour rA. The
assuming that the contour
where Np = (//211:) theorem is proved. REMARK 4.1. Estimate (4.7) does not hold in general if the spectrum a(A) lies in the closed halfplane Re A ~ p. In order to see this it suffices to consider a finite-dimensional space)8 (0 < dim)8 < 00) and an operator A in it consisting of a single Jordan cell (see Exercises 4 and 16). An application of equality (4.4) to the operator -A gives A
lim
t-.-oo
~~ = t
COROLLARY 4.1. If the estimate a(A) lies on the imaginary axis.
I eAt I
I
min {Re A AEa (A)}. ~ c holds for all t E ( - 00, 00), the spectrum
The converse of tvis assertion is not true (see Remark 4.1). 3. Estimate connected with the theorem on a boundary point of the spectrum. We will need below another estimate of the behavior of eAt that is connected with the location of the spectrum of A. We first note that if Ais an eigenvalue of A and x is a corresponding normalized eigenvector: (A - AI)x = 0, then, as readily follows from (4.2), for example, eAtx = eAlx, and hence I eAtx I = eiRe A. In the case of an arbitrary point of the spectrum the situation is more complicated, although certain estimates can still be given when ). is a boundary point of a(A). Their derivation is based on the following important proposition.
28
I. BOUNDED OPERATORS IN BANACH SPACES THEOREM ON A BOUNDARY POINT OF THE SPECTRUM. For any boundary point E [58]) and any c > 0 there exists a normalized vector
A of a spectrum a(A) (A x (II x I = 1) such that
I (A
- AI) x
I
< c.
PROOF. Suppose A E a(A) and fl E p(A). Then would otherwise get from the obvious identity (A - Al)Rfl - J
=
(4.8)
I Rfl I
~ 1
II fl
- A I. For we
(fl - A)Rfl
that I (A - AI)Rfl - J I < 1, and this would imply (see the special rule mentioned at the end of § 2.1) that (A - AI)Rfl and hence A - AI are invertible, which is impossible. If A is boundary point of a(A), there exists for any c > 0 a fl E p(A) such that I fl - A I < c 12. Then I Rfl I > 21 c and there accordingly exists an element Y ( I y I = 1) such that I RflY I > 21 dor I fl - A I < c 12. Therefore, setting x = RflY I I RflY II, the above identity implies
I (A
- AI)x
I
=
I (A
- Al)Rfl Y III RflY
II I ~ 1/11RflY I + I fl
- A I < c.
A proposition on the estimates referred to above can be formulated as follows. LEMMA 4.l. Let A be a boundary point of a spectrum a(A). For any 0 > 0 and T > 0 there exists a normalized vector x for which
et ReA (1 - 0) ~
I eAt x I
~ et ReA (1
+ 0)
(0 ~ t ~ T).
PROOF. The relations
(eAt - eAt J) x = e.l.t [e(A-,l.I)t - J] x = eAt
t
I e(A-.l.I)r d7: • (A
- AI)x
o
imply the estimate
I eAt x
- eAtx
I
~ et ReA
t
I I e(A-,l.I)r I d7: • I (A o
- AI)X
II·
Irll
Let M = e(A-.l.I)r I d7:. It but remains to choose a normalized vector x that ensures the fuUilment of (4.8) when c = 0 1M.
4. Expressionfor the solutions of the operator equations in terms of the exponential function. We will make use of the properties of the operator function eAt studied above in order to restate some of the results of § 3 in a new form. We consider the equation
AX+ XB= Y, which is obtained from equation (3.8) by substituting - B for B. Suppose that the spectra a(A) and a(B) lie in the interior of the left halfplane. Then the condition
4.
A + fl- #- 0 for (A, fl-)
29
EXPONENTIAL OPERATOR FUNCTION
E
(l(A) x (l(B) is satisfied, and hence equation (4.9) has the
unique solution
__
X -
.c (A :r :r
~1~.c
4
2
1T:
A1)-1 Y(B - fl-1)-1
A
FA F.
(4.10)
dA dp..
+ fl-
We take advantage of the equality
I e(.l+p)t dt,
1 A + fl- = -
00
which is valid for Re A < 0 and Re fl- < O. Substituting this expression in (4.10) and changing the order of integration, we find that
00
= -
JeAtYeBt dt.
(4.11)
o
It can also be verified directly that the integral (4.11),which exists by virtue of estimate (4.7), satisfies equation (4.9). In fact,
- [A ofeAtYeBtdt + 7eAtYeBt dt· BJ = - 7d(eAtYe 0
Bt )
0
= - eAt Ye Bt I';)' = Y. Analogously, if the sets (l(A) and (l(B) lie in the interior of the right halfplane, the solution of equation (4.9) is given by the formula
X =
Je-AtYe- Bt dt.
(4.12)
o
5. Some important renormings of~. Certain norms defined by the exponential operator function playa critical role in stability questions. a) We first assume that the spectrum (l(A) lies in the interior of the left halfplane. Then for any ).I such that Re A <
-).I
for all AE (l(A)
there exists an N :;:., 0 for which
I eAt I <
Ne- vt
This estimate permits one to put for any x
IX
IIA.r· =
(t
~
(4.13)
0).
E ~
I I eAtx Ilr dt }1/r 00
~
1).
(4.14)
From the Minkowski inequality it at once follows that norm. Moreover, we have the following fact.
I x IIA.r, can serve as a
{
(r
30
1.
THEOREM 4.2.
BOUNDED OPERATORS IN BANACH SPACES
I
The norm X IIA,r, is topologically equivalent to theoriginai norm II x II.
PROOF. 1;'he assertion of the theorem is equivalent to the existence of positive constants mr and Mr such that mrll x II ~ II x 11M ~ Mrll x II. The right inequality immediately follows from (4.13) and the relations
On the other hatid, since
II x II = II e-AteAt x II
~
II
II II
e- At
eAtx
II
~ et lIAll
II
eAtx II,
we have
II x 11M =
{of II eAtx
IIr dt}l/r
~ II x II {of e-rtllAIl dt}lIr =
II
x
II
Z!11Al'
Thus the theorem is proved, it being possible to put
= 1/ V r II A II.
mr
The norm
II
X IIA,r is interesting in that for any x
II
eAt x lIA,r = {
E
~ the function of t
JII e&x IIr ds
t
r
monotonically decreases with increasing t and, moreover, has a negative (when
x #: 0) continuous derivative. This fact will be used in Chapters II and VII. When ~ is a Hilbert space ~ the most natural of the II x is defined by the scalar product
11M is II x 1IA,2' This norm
00
(x, y)A,2
=
J(eAtx, eAty) dt.
o
b) Theorem 4.2 is easily carried over to a more general case. We assume that the spectrum of the operator A does not intersect the imaginary axis and is separated by it into two parts, so that O"(A)
=
0" +(A)
UO"-(A).
Let ~+ and ~_ be the invariant subspaces of A corresponding to these parts of the spectrum, and let P + and P _ be the corresponding spectral projections. A norm can be introduced in each of these spaces by using formula (4.14) (with the replacement of A by - A in ~+):
I XIlA.,r = I" eAtx 00
{
II
II-A.,r = I" 00
X
{
IIr dt
}l/r
e-Atx IIr dt
}l/r
5.
31
GENERALIZED LJAPUNOV THEOREM
We now introduce a new norm in 18 by putting
I x 11M =
lip+ x
II-A+,r
+I
p- x
(4.15)
k,r'
Since in each of the subspaces 18± the norms I x II-A+,r, I x Ikr are topologically equivalent to I x II, the norm I x 11M is topologically equivalent to the norm I p + + I p - x II, which in turn is equivalent to the norm I x II. When 18 is a Hilbert space .p it is natural to consider the norm I x IIA defined by the scalar product
xii
(x, Y)A
=
(P + x, P + Y)-A+,2
+ (P -
x, P - Y)A-,2,
which is topologically equivalent to the original scalar product (see § 1.5). The new norms that have been introduced are interesting in that the function I eAtx IIA,r ( I eAtx IIA when 18 = .p) monotonically decreases for x E 18+ but monotonically increases for x E 18-. § 5. Generalized Ljapunov theorem In the next four sections we will be dealing with a Hilbert space: 18 = .p. We assume that the reader is familiar with the fundamentals of the geometry of Hilbert spaces and the theory of operators in them. The scalar product of a pair of elements x, Y E.p will be denoted by (x, y). In order for a linear operator A:.p --+ .p to be bounded (A E [.p]) it is necessary and sufficient that there correspond to it a linear operator A*: .p --+ .p such that (Ax, y) = (x, A*y) for any x, y E .p. If A E [.p] then A* E [.p] and (A*)* = A. The operator A* is called the adjoint of
A. We note the following simple properties of adjoints: 1) (A
+ B)* =
A*
+ B*forA,BE[.p];
2) (aA)* = aA*, where a is a scalar; 3) (AB)* = B*A*; 4)
IAI
=
I A* II;
5) the spectra I7(A) and I7(A*) are distributed symmetrically with respect to the real axis. An operator H E [.p] is asid to be Hermitian if H = H*. A Hermitian operator H is characterized by the fact that its Hermitian form (Hx, x) (x E.p) takes only real values. The spectrum a(H) of a Hermitian operator H is a bounded closed set on the real axis. The least segment that contains a(H) will be denoted by [Am(H), AMCH)]. As is well known, Am(H)
= inf {(Hx,
III x I = I}; AM(H) = sup {(Hx, x) III x I = I H I = max {AM(H), - Am(H)}.
x)
1};
An operator HE [.p] is said to be positive (nonnegative) if its form (Hx, x) is positive (nonnegative) for any x -:f. O. Whenever H is nonnegative one has I H I = AMCH).
I.
32
BOUNDED OPERATORS IN BANACH SPACES
An operator H is said to be uniformly positive, and one writes H» 0, if its form (Hx, x) is uniformly positive on the unit sphere S = {x III x I = I} in .p, i.e. if Am(H) > o. Negative, nonnegative and uniformly negative operators (and the meaning of the relation H «0) are defined analogously. Clearly, in order for a nonnegative operator to be invertible it is necessary and sufficient that it be uniformly positive. It is significant that every uniformly positive operator H permits one to introduce in .p a new scalar product (x, y)H = (Hx, y) with respect to which .p remains a complete Hilbert space. This occurs because the new norm I x IIH = (Hx, x)1/2 is topologically equivalent to the original one by virtue of the estimates Am(H) II x 112 ~ II x 111 ~ II H 1111 x 112 We will say below that two different scalar products defined on one and the same set are topologically equivalent if the norms defined by them are topologically equivalent. It can be asserted that any scalar product (x, y)l that is topologically equivalent to the original one can be obtained by means of a formula of the form
( = (Hx,y»),
(x, y)l = (x, Y)H
where H is a uniformly positive operator. We recall that the real part of an operator A Am == ReA ==
E
[.p] is the Hermitian operator
HA + A*),
while its imaginary part is the Hermitian operator A3 == 1m A == (l/2i)(A - A*), so that A = Re A
+
i 1m A.
THEOREM 5.1 (GENERALIZED LJAPUNOV THEOREM). In order for the spectrum of an operator A to lie in the interior of the left halfplane it is necessary and sufficient that there exist a uniformly positive operator W such that
Re(WA) <{
o.
(5.1)
Moreover, if a(A) lies in the interior of the left halfplane, for any H» 0 there exists an operator W» 0 such that Re(WA)
«
= -
H.
(5.2)
Thus, if Re A 0, the spectrum a(A) lies in the interior of the left halfplane. In linear algebra this proposition is a corollary of Hirsch's theorems. It does not admit a converse even in the case of an n-dimensional space .p with n > 1.
33
5. GENERALIZED LJAPUNOV THEOREM
PROOF OF THE THEOREM. We will show that if the spectrum O'(A) lies in the interior of the left halfplane, equation (5.2) has a solution W»O for any H» O. The "necessary part" of the theorem will thereby be proved. We rewrite equation (5.2) in the more explicit form A*W + WA = - 2H.
Its solution can be represented on the basis of formula (4.11) in the form 00
W = 2
I eA
o
since the spectrum O'(A*), like O'(A), lies in the interior of the lefthalfplane. The uniform positiveness of this operator is implied by the following estimate, which is derived with the use of Theorem 4.2: co
(Wx, x) = 2
00
I (HeAtx,
eAtx) dt ~ 2Am(H)
o
I I eAtx liz dt
0
= 2Am(H)
IX
11~.z ~ 2Am(H)m~
I X liz.
w»
Suppose now that there exists an operator 0 satisfying condition (5.2) for H» O. According to Theorem 4.1 the sufficiency part of the theorem will follow if we obtain the estimate
(v> 0). It suffices to obtain estimate (5.3) in the norm is equivalent to the original norm. Let pet) =
I eAt x
(5.3)
I x I w = (Wx, x)lIZ, since this norm
II~ = (WeAtx, eAtx).
Then p'(t) = (WAeAtx, eAtx) = «WA
~
+ (WeAtx, AeAtx)
+ A*W)eAtx,eAtx) = -
- 2Am(H)
I eAtx 112 ~ IIA~I) I eAtx II~ =
where v = Am(H) 111 W I > O. Integrating the obtained inequality p'(t) pet) _
In p(O) -
2(HeAtx,e Atx)
~
It pi (I:) 0
- 2vp(t), we get
<
p(l:) d'l: = - 2vt,
which implies pet) ~ p(0)e- 2"t and, finally, IleAtx I w ~ REMARK 5.1. An operator A is said to be dissipative if
I x I w r"t.
- 2vp(t),
34
1.
BOUNDED OPERATORS IN BANACH SPACES
(Re A) x, x)
=
Re (Ax, x)
~
O.
«
We will call an operator A uniformly dissipative if Re A 0, i.e. there exists a positive constant such that Re (Ax, x) ~ X (as one can take - AM (Re A)). Since the scalar products (x, y) and (x, y)w are topologically equivalent, condition (5.1) for the uniform dissipativeness of the operator WA can be interpreted as a condition for the uniform dissipativeness of the operator A itself with respect to the scalar product ( " . )w. In fact, if Re (WA) <{ 0,
c
cll liz
c,
cI liz ~ - (cl I W II) Ilx II~· Conversely, if Re(Ax, x)w < - CIII x I ~ (CI > 0), Re(WAx, x) < - CI(Am (W))Z I x liz. Re(Ax, x)w ( = Re(WAx, x)) < -
X
Therefore an operator A satisfying condition (5.1) is naturally said to be uniformly W-dissipative. Using this terminology, the first part of Theorem 5.1 can be rephrased in the following manner. In order for the spectrum of an operator A to lie in the interior of the left halfplane it is necessary and sufficient that it be uniformly W-dissipative for some W» o. § 6. A theorem of B. von Sz.-Nagy 1. Commutative groups of operators. As has already been noted, the operators < t < (0) form a commutative group.
eAt ( - 00
1t can turn out that this group is bounded. In this case it can be asserted that the spectrum (l(A) lies on the imaginary axis (see Corollary 4.1). If)8 is a Hilbert space :p, it is possible to give a complete characterization of the operator A generating a bounded group. This result (Theorem 6.3) will be obtained from a simple extension (Theorem 6.2) of a general stability criterion (Theorem 6.1) of B. von Sz.-Nagy. An operator G E [)8] is said to be stable if it has an inverse and its integral powers are uniformly bounded: Gn ~ (n = 0, ± 1, ± 2, ... ).
I
I
c
THEOREM 6.1 (SZ.-NAGY). In order for an operator G acting in a Hilbert space :p (G E [,p]) to be stable it is necessary and sufficient that it be similar to a unitary operator U, i.e.~G = SUS-I, where U* = U-I and S, S-l E [:Pl.
Since the integral powers of an invertible operator form a commutative group, Theorem 6.1 is a corollary of the following more general proposition. THEOREM 6.2. Suppose (S) = {G} is a commutative group of operators from Then G is bounded:
(G
E
(S)),
[:Pl. (6.1)
6.
35
THEOREM OF B. SZ.-NAGY
if and only if there exists in S) a scalar product (x, y)h topologically equivalent to the original one, that is invariant under the group @:
PROOF.
We consider on
@
(6.2)
(GE@;X,YES»).
(Gx, GY)l = (x, Y)l
the function
rjJ",y(X) = (Xx, Xy)
This function is bounded on
@
I rjJ",y{X) I ~ I X
(X E@).
since 112
I xliii y I ~
(6.3)
c2 11 xliii y II·
From a well-known theorem of A. A. Markov 7) (A. A. Markov [1]; see also M. G. Krein and M. A. Rutman [1] ) it follows that on the space IJJ1 = 1JJ1(@) of all bounded complex valued functions on @ there exists an invariant mean, I.e. a linear functional M(rjJ) (rjJ E 1JJ1) having the following properties: a) M(l)=I; b) M(rjJ(X)) ~ 0 for rjJ(X) ~ 0; c) M(rjJ(GX)) = M(rjJ(X)) for any G E @. From the first two properties it follows that8) IM(rjJ(X))
I~
(6.4)
sup{lrjJ(X)IIXE@}.
As the desired scalar product, one can take (X,Y)l = M(rjJ",y). It can be verified directly that (x, Y)l has the usual properties of a scalar product. From (6.3) and (6.4) it follows that
I (x, Y)l I ~
c2
I xliii Y II; I x
111 ~ c I x II·
On the other hand, we have by virtue of (6.1) that rjJ",,;{X) =
I XX 112
~
I X-I 112 I XX C2
1
112 ~ -c2
I X-1XX 112 =
1 -c2
IX
11
2,
which implies by virtue of a) and b) that
Thus the scalar product ( " . )1 is topologically equivalent to the original one. The 7) We have in mind the following theorem: Suppose Q is an arbitrary set and @ is a commutative family of single valued mappings of Q into itself. Then on the set iln = iln (Q) of all bounded complex valued functions there exists an invariant mean, i.e. a linear functional having properties a) - c). We apply this theorem for Q = @ (here an element G E @ is considered as the mapping X ->GX
(XE @). 8) In fact, if p(X) is real, denoting the right side of (6.4) by s¢, we will have s¢ ± p(X)~O. It follows by virtue of b) that M(s¢ ± p) ~ 0, i.e. I M(p) I -;;;'M(s¢) = s¢. If p(X) is complex valued, putting M(p) = I M(p) le,a, we will have IM(p)1 = r,a M(p) = M(e-;ap) = M(Re(e-;ap». On the other hand, by what has been proved M(Re(r;ap» -;;;, sup I Re(r;ap) I -;;;, sup I p I.
36
I. BOUNDED OPERATORS IN BANACH SPACES
invariance of the scalar product (', ')1 under for every G E @, on the basis of c)
@ is
obtained immediately. In fact,
(Gx, GY)1 = M(o/Gx.GiX») = M(o/xjXG»)
=
M(o/x.iGX»)
= M(o/xjX») = (x, Y)1'
The direct part of the theorem is proved. Suppose now (x, Y)1 is a scalar product that is topologically equivalent to the original one and invariant under the group @. As we know, topological equivalence implies the existence of a uniformly positive operator H such that (x, Y)1 = (Hx, y). Since the operator H is uniformly positive, there exists a uniquely defined uniformly positive square root S = ",/7[ of it (see Exercise 27). From (6.2) it follows that (HGx, Gy)
= (Hx, y) or (S2Gx, Gy) = (S2X, y),
which implies G* S2G = S2
or (S-1G*S) (SGS-1) = /.
Thus the invertible operator U = SGS-1 is a unitary operator. Since G = S-1 US and I U I = 1,
I GI
~
I S-1 IIII S II,
i.e. the group @ is bounded. The theorem is proved. We have simultaneously proved the following assertion, which contains Theorem 6.1. THEOREM 6.2'. In order for a commutative group @ of operators acting in a Hilbert space 5;1 to be bounded it is necessary and sufficient that it be similar to a group of unitary operators. REMARK 6.1. We leave it to the reader to verify that the above arguments imply that the following equality holds for a bounded group @ c [5;1]: sup { I
Gill GE @}
= inf{ I S
IIII S-1 III S E S (@)},
where S(@) is the ~set of all invertible transformations S of @ into the group of unitary operators. 2. Boundedness condition for the exponential operator function in 5;1. We obtain the result referred to in subsection 1 as a simple corollary of Theorem 6.2. THEOREM 6.3. Let A E (5)]. In order for the commutative group @ = {eAt} (- 00 < t < 00) to be bounded it is necessary and sufficient that A be similar to a skew- Hermitian9l operator B. 9)
An operator BE [~l is said to be skew Hermitian if B*= - B, i.e. if the operator iB is Hermitian.
7. HILBERT SPACE WITH INDEFINITE METRIC
37
PROOF. NECESSITY. If the group G is bounded, there exists a uniformly positive operator H such that
(HeAt x, eAty) = (Hx, y). Differentiating this equality with respect to t, we get
°
(HAe At x, eAt y)
+ (HeAt x, A eAt y)
=
0,
which for t = implies HA + A* H = 0. Putting S = -vi H, H = S2, we find that SAS-1 = - S-1 A* S. Since S = S*, the latter equality means that the operator B = SAS-1 is skew Hermitian while A = S-1 BS. The sufficiency follows from the fact that the equality A = S-1BS implies eAt = S-1e Bt S, where eBt is a unitary operator (e Bt)* = e- Bt ). REMARK 6.2. The well-known heorem of Stone, asserting that the general form of a one parameter group V, of unitary operators is given by the formula V, = e iHI , where H is a selfadjoint (possibly unbounded) operator, in combination with Theorem 6.2 permits one to obtain the general form of bounded one parameter semigroups in Wl. Since we are confining our attention to bounded operators, we will not go into this question in the sequel.
§ 7. Hilbert space with an indefinite metric 1. Uniformly W-dissipative operators and the direct decompositions associated l-rith them. A Hermitian operator WE W] is said to be indefinite if the Hermitian form (Wx, x) takes values of different sign for x E .)3. Suppose W is an indefinite operator. We consider the indefinite scalar product (x, y)w = (Wx, y). The index W will often be dropped when this does not lead to confusion. We introduce some notation. An element x E i> is said to be W-positive if (Wx, x) > 0, W-negative if (Wx, x) <0 and W-neutral if (Wx, x) = 0. An element x is said to be W-nonnegative (W-nonpositive) if it is either Wpositive (W-negative) or W-neutral. Two vectors x, y E f' are said to be W-orthogonal if (Wx, y) = 0. A W-neutral vector is orthogonal to itself. Two sets E10 E2 c i> are said to be W-orthogonal if (Wx, y) = 0 for all x E E1 and y E E 2• A subspace 52 c .)3 is said to be W-positive (W-nonnegative) if all of its nonzero elements are W-positive (W-nonnegative). A subspace 52 is said to be uniformly W-positive if there exists a constant a > 0 such that for all x E 52
(x, x)w
~
a (x, x).
(7.1)
I. BOUNDED OPERATORS IN BANACH SPACES
38
W-negative, W-nonpositive and uniformly W-negative spaces are defined analogously. We note that inequality (7.1) and the boundedness of W imply that in a uniformly W-positive subspace the scalar product (x, y)w is an ordinary scalar product that is topologically equivalent to the original one. Suppose W is an arbitrary Hermitian operator. In precisely this case we call an operator A uniformly W-dissipative if Re (WA)>> o. LEMMA 7.1. The spectrum of a uniformly W-dissipative operator A does not intersect the imaginary axis. PROOF. Suppose A = fJ.i (fJ. a real number) is a boundary point of the spectrum (J(A). Then by the theorem on a boundary point of the spectrum there exists a sequence of unit vectors fn (1lfn = 1) such that gn = Ain .:.- Ain -+ 0 for n -+ 00. But this contradicts the uniform positiveness of the operator H = - 2Re (WA) = - (WA + A*W) since
I
o< =
-
= -
An operator T
E
~ (HIn,fn) = - (WAfn.fn) - (Win, Afn) fJ.i (Wfn,fn) - (Wgn,fn) + fJ.i (Wfn,fn) - (Wfn. gn)
Am (H)
[.p] is called a uniform W-contraction if T*WT«
LEMMA 7.2. the operator
If A
I hn 11-+ O.
2Re (Wgn,fn) ~ 211 W
w.
(7.2)
is a uniformly W-dissipative operator, for any positive arf:.(J(A) T = (A
+ al) (A
- al)-l
(7.3)
is a uniform W-contraction whose spectrum does not intersect the unit circle. PROOF. Let ¢(z) = (z + a) /(z - a). Then by the spectral mapping theorem (J(T) = ¢((J(A)). But ¢(z) maps only points of the imaginary axis onto the unit circle, and hence (J(T) does not intersect the unit circle. Further, it follows directly from (7.3) that A = aCT +/) (T - 1)-1. Therefore the equality W A .. + A* W = - H implies
WeT
+ I)(T -
1)-1
+ (T*
-I)
-1
(T*
+ I)
W = - (1/a) H
or
(T* - I) WeT
+ I) + (T* + I) WeT -
I) = - (1/ a)(T* - I)H(T - I) «0.1 0)
Multiplying out the terms in the left side of this inequality, we arrive at (7.2). We now prove a theorem that generalizes Theorem 5.1. 10)
We have made use of the fact that S*HS» 0 for any invertible S if H» O.
7. HILBERT SPACE WITH INDEFINITE METRIC
39
THEOREM 7.1. In order for an operator A E [.p] to be e-dichotomic lll it}s necessary and sufficient that it be uniformly W-dissipative with respect to an indefinite Hermitian operator W: Re (WA)
«0.
(7.4)
Any operator W satisfying condition (7.4) is invertible and such that the subspace .p_ is uniformly W-positive and the subspace .p+ is uniformly W-negative. An operator W can be chosen so that these subspaces are W-orthogonal.
PROOF. SUFFICIENCY. Suppose there exist operators H» 0 and W that WA
+ A*W=
=
W* such
(7.4')
- H.
By virtue of Lemma 7.1 the spectrum of A does not intersect the imaginary axis: a(A) = a +(A) U a - (A). We put.p+ = P +.p,.p_ = p _.p, where P + and P _ are the spectral projections and.p+ and .p_ are the invariant subspaces corresponding to a +(A) and a _(A) respectively. The operator T = (A + aI) (A - aI)-l dealt with in Lemma 7.2 has the same invariant subspaces and projections. Here the set a(TI.p-) lies in the interior, while a(T l.p+) lies in the exterior, of the unit disk. We therefore have the representation P-
I 2m
= -.
f (zI Izl=l
T)-1 dz = -
1
2n
2".
.
Se'9 (e'9I - T)-1 d¢.
(7.5)
0
In order to prove that .p_ is uniformly positive we choose an arbitrary element x E.p_ (P -x = x) and consider the expression
(Wx, x)
(WP -x, x) + (Wx, P -x) - (Wx, x) «WP_ + P*_ W - W)x, x).
= =
It suffices to prove that the operator WP _ WP-
+
+ P *_ W
P*-W - W»
(7.6')
- W is uniformly positive:
o.
(7.6)
Substituting the right side of (7.5) for P _ in the left side of (7.6), we get
+ P!W-
WP-
= _1_
2n
= -
I
2n
r
W
(Weiif> (eiif> I - T)-1
+ e-iif> (e-iif> I
- T*)-l W - W) d¢
0
2"
S(e-iif> 1-
T*)-l H (eiif>I - T)-1 d¢.
0
We recall (see § 2.4) that this means that .p decomposes into a direct sum of invariant (under = .p+ .p_, the spectra IJ± = IJ(A j.p±) of the restrictions of A to.p± lying respectively in the interiors of the right and left halfplanes. 11)
A) subspaces: .p
+
I.
40
BOUNDED OPERATORS IN BANACH SPACES
In view of the fact that all of the points of the unit circle are regular points of T and T* while H»O, the integrand here is uniformly positive.1 2) It follows from (7.6') that (Wx, x) ~ e I x 112 for any x E .p-. The uniform W-negativeness of .p+ is established analogously. We show, finally, that relation (7.4') automatically implies the invertibility of W. In fact,
I
X
112
~
Am tH) (Hx, x)
=
Am tH)
I (Wx, Ax) + (Ax,
Wx)
I~
~~ tH~-
I
Wx
IIII x
II,
and thus an inequality of type I Wx I ~ ell x I is valid. The sufficiency of the condition of the theorem is proved. NECESSITY. Suppose now that the spectrum of A does not intersect the imaginary axis and .p decomposes into a direct sum .p = .p+ .p_ of invariant (under A) subspaces so that the set a+ = a(A l.p+) lies in the interior of the right, while a- = a(A l.p-) lies in the interior of the left, halfplane. Starting from any operator H»O, we form the operator P'S,HP + acting from .p+ into the subspace .p'S, that is invariant under A* and corresponds to its spectral set a'S, = iJ +. According to the results in § 4.4 the operator
+
co
X = -
Se-A*t P'S, HP + e- At dt o
satisfies the equation A* X for fE.p+
+
= -
XA
Pt HP +.
In addition, X
= P'S,XP+ and
00
(XI, f) = -
S(HP + rAt I, P + e- At f) dt < 0. o
Analogously, the operator co
y
=
SeA*tp ~ HP _ eAt dt o
satisfies the equation A* Y + Y A and (YJ, f) > for f E .p_. We put W = X + Y.
°
=
-
P~HP _
and the conditions Y
= P~ YP-
12) We have made use of the following easily proved fact: if F(¢» is a continuous and continuously invertible operator function on [a, b] while H» 0, then f! F*(¢»HF(¢» d¢> O.
»
7. HILBERT SPACE WITH INDEFINITE METRIC
°
Clearly (WI, g) = for fE .fl+, g W-dissipative inasmuch as
.fl-. In addition, the operator A is uniformly
E
A*W+ WA where Hl = P'+ HP +
41
+ P':... HP _ » 0,
=
-
HI>
since
+ (HP _ x, P - x) P + x liz + II P _ x liZ}
(Hl x, x) = (HP + x, P + x)
~ Am (H){ II
~ 1Am (H) {II P+ x II
+
II P- x 11}2 ~ 1Am (H)II
X
liz.
The theorem is proved. In connection with Theorem 7.1 it is useful to have in mind the following general fact. THEOREM 7.2. Suppose W is an invertible Hermitian operator and the space .fl decomposes into a direct sum .fll .flz of W-orthogonal subspaces. If the subspace .fll is W-positive (negative), it is uniformly W-positive (negative).
+
PROOF. Let P 1 denote the projection onto .fll corresponding to the given direct decomposition. Suppose f E .fll and g = W-lj. Then, inasmuch as the form (Wx, y) is positive in .flI> we have Ilf liz = (Wg,f) = (WP1g
+
WPzg,f) = (WP1g,f)
~ ,J (WP1g, P1g)(Wf,f) ~ ,JfWl1. Ilp1W-111·llfll· ,J(Wf,f), which implies (WI, f) ~ m II f liz. 2. u-dichotomic operators and uniform W-contractions. We say that an operator Tis u-dichotomic if its spectrum does not intersect the unit circle and contains both a spectral set (Ji(T) lying in the interior of the unit circle and a spectral set (J.(T) lying in its exterior: (J(T) = (Ji(T) U(J.(T). To this decomposition there corresponds a direct decomposition of the space .fl = .fli .fle into invariant subs paces of the operator T so that the spectra of the restrictions of this operator are
+
-.
(Ji(T)
=
(J(T l.fl,)
and
(J.(T)
=
(J(T l.fle).
In proving Theorem 7.1 we essentially used the properties of an operator T satisfying condition (7.2). It is easily seen that Theorem 7.1 is essentially equivalent to the following proposition on the operators T. THEOREM 7.1'. Suppose TE [.fl] and (J(T) does not cover the unit circle. In order for T to be u-dichotomic it is necessary and sufficient that it be a uniform W-contraction:
42
I.
BOUNDED OPERATORS IN BANACH SPACES
T* WT« W
(7.2)
for some indefinite W ( :;= W* E [.pD. Any operator W (= W*) satisfying condition (7.2) is invertible and such that the invariant subspace .pi of T is W-positive while the invariant subspace .pe of T is Wnegative. An operator W can be chosen so that these subspaces are W-orthogonal. In fact, by hypothesis, the operator T has at least one regular point, on the unit circle. It can be assumed without loss of generality that' = 1 (otherwise we could consider the operator T in place of T). There then exists the operator
,-1 A
= (T + J)(T - 1)-1 (E [.p]),
condition (7.2) for Tbeingequivalent to condition (7.4) for A. 7.1. A uniform W-contraction is either I) u-dichotomic or 2) has a spectrum covering the whole unit circle. COROLLARY
Case 1) is obviously realized. An example will be given below in which case 2) is realized. We first establish a number of propositions. LEMMA 7.3. Let T be a uniform W-contraction. Then all of the points of the unit circle are points of regular typefor T, i.e.for any point, of the unit circle = 1) the operator T is an isomorphism of.p onto a closed subspace.
(1'1
,J
PROOF.
Condition (7.2) means that
(Wx, x) - (WTx, Tx) ~ m II x 112
(x
E
.p),
(7.7)
where m = Am (W - T* WT) > O. We show that this implies the existence of a constant c; > 0 such that (x
E
.p).
(7.8)
In fact, there would otherwise exist a sequence of unit vectors {Xnlll Xn II = I} for which Yn = (T - 'J)xn --+ 0 as n --+ 00. Putting x = Xn, TXn = 'xn + Yn in (7.7), we would then get
mil Xn 112 ~ -- ,(WXn,Yn) - ,(WYn, Xn) - (WYn, Yn) which is impossible. Inequality (7.8) implies that the mapping T closed, B; = (T - 'J).p is also closed.
,J
--+
0,
is an isomorphism. Since
.p is
7. HILBERT SPACE WITH INDEFINITE METRIC
43
REMARK 7.1 t. A comparison of Lemma 7.3 with the theorem on a boundary point of the spectrum (see § 4.3) shows that a point of the unit circle can only belong to a(T) as an interior point. This in turn implies that the unit circle belongs to a(T) if and only if it is contained in an annulus belonging to a( T). THEOREM 7.3. Let T be a uniform W-contraction with Wan invertible operator (W-l E [~)]). In order for T to be u-dichotomic it is necessary and sufficient that T* be a uniform W-l -contraction. PROOF. NECESSITY. If 1 ¢: a(T), the operator A = (T + I)(T - 1)-1 exists. Then A - I = 2(T - 1)-1, i.e. T = (A + I)(A - 1)-1. Substituting the obtained expression for Tin (7.2), we find that WA + A* W «0. Multiplying this inequality from the left and from the right by W-1, we get Re(W-IA*)«O. On the basis of Lemma 7.2 we conclude that (A* + I)(A* - 1)-1 = T* is a uniform W-l -contraction. SUFFICIENCY. If T* is a uniform W-l -contraction, the equality (T* - I)x = 0 implies by Lemma 7.3 that x = O. This in turn implies that ~1 = (T - 1)f) = f), since (T* - I)x = 0 for every vector x .1 ~1' Thus the point A = 1 and hence every other point of the unit circle are regular points of T. In certain cases Theorem 7.3 permits one to verify directly whether the spectrum of an operator T covers the unit circle or does not intersect it. 3. Example of a uniform W-contraction that is not u-dichotomic. Let us construct an example of a uniform W-contraction whose spectrum covers the unit circle. We construct an operator T whose spectrum covers the unit circle and to which at the same time there corresponds even an invertible Hermitian operator W converting it into a uniform W-contraction. Let {ek} ~co be an orthonormal basis in f), f)+ = V ~ ek, f)- = V'(' e_k,13) P + and P _ be the orthogonal projections on .p+ and f)- respectively, and let W = P + P _ ( = W-l). We choose a pair of complex numbers q+ and q-, with 1 q+ 1 < 1 and 1 q- 1 > 1, and define the operator T: n = 1,2, "', n = 0, - 1, - 2, ....
For any x
= L; ~oo
Ck ek E f) we have Tx =
L; q+Ckek-1 k=1 00
(Wx, x)
=
-
(WTx, Tx) =
so that for H 13)
=
L; 1 Ck 12 k=O
+
o L; q- ckek-1,
-1
L; 1 Ck
12 ,
-00
00
L; 1q+ 121 Ck 12 k=1
0
L; 1q- 12 ICk
12,
-00
W - T* WT
The symbol VkEK ek denotes the closed linear span of the system of vectors {ek} kEK.
44
I. BOUNDED OPERATORS IN BANACH SPACES
+ ( I q- 12 -
-1
1) 1:; -00
I
00
Ck
12 ~ 0 1:; -00
I
Ck
12 = O(X, X),
where 0 = min {I - I q+ 12, I q- 12 - I}. Thus H»O, i.e. T is a uniform W-contraction. It is easily seen that T*e n = tj _ e n+1 (n = - 1, - 2, ... ). Therefore (WT*e-1' T*e-1) =
I q- 12 (WeD, eo) = I q- 12 > 0.
On the other hand, (We-h e-1) = - 1 < 0, so that T* is not even an ordinary W-1-contraction (W-1 = W). By Theorem 7.3 (and Corollary 7.1) the spectrum of the constructed operator T covers the whole unit circle. REMARK 7.1. The spectrum of the constructed operator T contains the annulus q+ ~ A ~ q-I (cf. Remark 7.1'). In fact, for any r E q+ q-I) the operator r- 1 T is of the same type as T and hence its spectrum covers the unit circle. This means that the spectrum (J(T) covers the circle I A I = r. We note that the interior points of the considered annulus are points of regular type for Tfor which the co dimension of the space (T - A/)Sj is equal to one. 4. u-dichotomicity conditionsfor uniform W-contractions. The following important assertion is almost trivial.
I I II I
(I I, I
THEOREM 7.4. Suppose W = W* E [Sj] is an invertible operator. Then every uniform (u-dichotomic uniform) W-contraction T E [.p] has a neighborhood in [Sj] consisting of uniform (u-dichotomic uniform) W-contractions. The assertion follows from the fact that the operators W - T*WT and W-1 TW-1 T* are continuous functions of T. THEOREM 7.5. Suppose a uniform W-contraction T is the limit of a sequence {Tn} of u-dichotomic uniform W-contractions. Then T is also a u-dichotomic uniform W-contraction. Proof. Assum~ the contrary. Then the point A = J is a point of the spectrum of T. But according to Lemma 7.3 the operator T - / maps Sj isomorphicallyonto a closed subspace 2 = (T - /)Sj =1= Sj. Therefore by Banach's theorem there exists an operator R in [2, Sj] such that (T - /)R = R(T - /) = /. We choose an n such that T - Tn < 1/ R II· Let us ShOW 14 ) that (Tn - /)Sj =1= Sj.
I
I
I
14) This assertion is a corollary of the following more general proposition of Krein, Krasnosel' ski! and Mil'man on the stability of the deficiency of an operator A EO [lBl> lB 2] injecting lB j onto a closed subspace AlB j • Each operator X in a sufficiently small neighborhood of such an operator A
8.
We have Tn - I
=
45
STABLE W-UNITARY OPERATORS
[I - (T - Tn)R] (T - I), so that (Tn - I) 10
=
(I - Q)B,
where Q = (T - Tn)R. We note that I Q I ~ I T - Tn IIII R I < l. We extend the operator Q onto all of 3) by putting Qx = QP"x, where P" is the orthogonal projection on B. Clearly, I Q I = I Q I < 1. Therefore the operator I - Q is invertible and consequently is a one-to-one mapping of all of 10 onto 10; hence (I - Q)B = (I - Q)B =1= .p. On the other hand, since the operator Tn is a u-dichotomic uniform W-contraction, it follows that 1 1= (J(Tn) and hence (Tn - 1)10 = .p. We have arrived at a contradiction. The theorem is proved. COROLLARY 7.2. Let T(t) (a ~ t ~ b) be a continuous one parameter family of uniform W-contractions. If T(a) is a u-dichotomic operator, so will be T(t) for all t E [a, b].
For suppose there exists a nonempty set of points in [a, b] at which T(t) is not u-dichotomic, and let to denote the infimum of this set. By Theorem 7.5, T(to) is u-dichotomic. But according to Theorem 7.4 there would then exist a neighborhood of to in which T(t) is u-dichotomic, which is impossible.
§ 8. Stable W-unitaryoperators 1. Stability criterion for W-unitary operators. Let W by an invertible Hermitian operator. An operator A will be said to be W-Hermitian if (Ax, Y)w = (x, Ay)w. This condition is equivalent to the equality WA = A*W or WAW-l
=
A*.
(8.1)
An operator U is said to be W-unitary if (Ux, Y)w equivalent to the equality WUW-I
=
=
(x, U-Iy)W, which is
(U-I)*.
(8.2)
We note that the Cayley transform 15) . U
=
(A - iaI) (A
~
+
ial)-l ~.
takes a W-Hermitian operator into a W-unitary one with regular point A = 1. In fact, it is easily seen that U - 1= - 2ia (A
+ il)-I;
(U*)-l - 1= - 2ia (A*
+ il)-I,
also injects ~l onto a closed subspace X~l with codim X~l = codim A~b where codim X~l = dim ~zI X~l. A still more general proposition can be found, for example, in the article [3] of Gohberg and Krein. 15) The real number a*O is chosen sothatthe operator (A + ial)-l will be bounded ( -ia tE o-(A».
I.
46
BOUNDED OPERATORS IN BANACH SPACES
and therefore WUW-I
= 1- 2iWa(A + iI)-I W-I = 1- 2ia(WAW-I + iJ)-I = 1- 2ia(A* + iI)-I = (U*)-I.
In addition, the operator (U - I)-I = (ij2a) (A + iJ) is bounded. One can easily show that the inverse transform A = ai(U + 1) (I - U)-l takes each W-unitary operator with regular point A = 1 into a W-Hermitian operator. We discuss some of the properties of W-Hermitian and W-unitary operators. 1) The spectrum of a W-Hermitian (W-unitary) operator is symmetric with respect to the real axis (unit circle).l6l In fact, the relation WA W-I = A* implies O"(A*) = O"(A). On the other hand, A: E O"(A*) if A E O"(A). The analogous assertion for a W-unitary operator follows in the same way from the similarity of the operators U and (U-I)* (see (8.2»). 2) Let Al and Az be two parts of the spectrum of a W-Hermitian (W-unitary) operator A. We assume that the symmetric image Al of Al with respect to the real axis (unit circle) and Az are separated from each other by nonintersecting contours f'I and r z· Then the invariant subspaces .pI and .pz of A corresponding to the parts Al and Az of its spectrum are W-orthogonal. For a proof it suffices to show that Pi WP z = 0, where I
PI
= - 2. f (AI - A)-I dA
Pz
= -2~ f (AI - A)-I dA
nl r,
and
are the projections on
.pI
nl r,
and
.pz
respectively. Making use of the relation
(AI - A*)-I W = W(Al - A)-I, which is easily deduced from the definition of a W- Hermitian operator, we get
Pi
WP z
= 4~z ="
f f 01 -
A*)-I W(ftl - A)-I dA: dft
FI F2
4~ f f (AI -
A)-I (ftI - A)-I dA: dft
FI F'l.
W f (AI - A)-I (ftl - A)-I dA dft 4n r, r, 1 1 = - -2 f f - - [(AI - A)-I - (ftl - A)-I] dA dft = 0, 4n !" r, A - ft
= - -z f
l')We recall that complex numbers spect to the unit circle if ZI=2 = 1.
ZI
and
Z2
correspond to points that are symmetric with re-
47
8. STABLE W-UNITARY OPERATORS
which follows from Cauchy's theorem (cf. the proof of Lemma 2.1). The analogous assertion for a W-unitary operator can be deduced in the same way (or obtained from what has already been proved by applying the Cayley transform if Whas at least one regular point on the unit circle). The general stability criterion (Theorem 6.1) admits the following sharpening in the case of W-unitary operators. In order to simplify the formulation let us agree to say that a subspace ~ c {> is W"definite if it is either W-positive or W-negative. THEOREM 8.1. In order for a W-unitary operator U to be stable it is necessary and sufficient that {> decompose into a direct sum of two W-orthogonal W-definite subspaces {>1 and {>2 that are invariant under U. PROOF. SUFFICIENCY. By Theorem 7.2., if (8.3)
where {>1 and .fh are W-orthogonal and, for the sake of definiteness, {>1 is Wpositive while {>2 is W-negative, the space {>1 ({>2) is uniformly W-positive (Wnegative). Thus there exists a constant f-l > 0 such that
I(x, x)w I ~ f-l (x, x) The space {>1 «(>z) remains a Hilbert space while U becomes a unitary operator if the scalar product (x, y) in it is replaced by (x, y)w ( - (x, y)w). Thus
which implies II Un I {>k II ~ II W I /f-l (k = 1,2). Inasmuch as the projections P k (k = 1,2) on the {>k are bounded: (k = 1,2), we have
II unll ~ II unP1 11 + II
unP2
11
=
I
Pk
I
~ c
II unl{>lll + I Un 1{>211 ~ 2cll W II /f-l (n = 0, ± 1, ± 2,.··).
NECESSITY. By Theorem 6.1, if U is a stable operator, there exists an invertible operator Stransf6rming Uinto a unitary operator: V = S-l US (V*V = VV* = I). The fact that U is a W-unitary operator implies
W
= U*WU = S*-lV*S*WSVS-1.
Putting G = S*WS, we find that G = V*GV, so that
VG = GV. The operator G is Hermitian, has an inverse and
(8.4)
I. BOUNDED OPERATORS IN BANACH SPACES
48
(Gx, X)
=
(WSx, Sx).
(8.5)
Therefore the real spectrum /Y(G) = /Y+(G) U /Y-(G). Let .p = .p+ EB.p- be the orthogonal decomposition of.p into the corresponding invariant subspaces of G. The invariant (under G) subspaces .p+ and .'0- are G-orthogonal and G-definite (Gx, x) > 0 for x E .'0+ (x =I- 0) and (Gx, x) < 0 for x E.p_ (x =I- 0)). By virtue of (8.4) the subs paces .p± are also invariant under the unitary operator V, so that V.p± = .p±. We now put S.p+ = .p1 and S.p_ = .p2' Then U.pk =.pk (k = 1,2) and according to (8.5) the subspaces .'01 and .p2 will have all of the required properties. 2. Normally W-decomposable operators. Let us agree to say that an operator A is normally W-decomposable if its spectrum /Y(A) can be divided into nonintersecting spectral sets: /Y(A) = /Y1(A) U /Y2(A), so that in the corresponding decomposition .'0
=
.p1
+ .p2
(8.6)
the invariant (under A) subspaces ~.~h and .p2 are uniformly W-definite. Let us agree to use the indexing under which the subspace .p1 will be uniformly W-positive while .p2 will be uniformly W-negative. We note that if A is a W-Hermitian operator (or a W-unitary operator), the subspaces .p1 and .p2 in (8.6) will be W-orthogonal, and by Theorem 7.2 they will be uniformly W-definite whenever they are W-definite. We note some of the properties of normally W-decomposable operators. 1) If a W-unitary operator is normally W-decomposable, it is a stable operator. This assertion follows directly from Theorem 8.1. 2) If an operator A is normally W-decomposable, so will be an operator ¢(A) where the scalar function ¢ E KA is such that ¢(/Y1) ¢(/Yz) = 0. Clearly, the latter condition is always satisfied if the function ¢o.) is one-to-one on the spectrum /Y(A) and, in particular, if ¢ is a linear fractional function. 3) The spectrum of a W-Hermitian normally W-decomposable operator is real. In fact, the Cayley transform of such an operator is a W-unitary operator and hence has a spectrum lying on the unit circle (see Theorem 6.1). 0 (or H» 0), the operator A = WH is normally W-decomposable. 4) If H In fact, assuming, for example, that H 0 (we would otherwise consider the operator -A), we get AW + WA* = 2WHW« O. Applying Theorem 7.1, we obtain the required assertion. We note that the operator A = WH is a W-LHermitian operator.
n
«
«
THEOREM 8.2. The set of all normally W-decomposable operators is open in the Banach space [.p],
In other words, to every normally W-decomposable operator A there corre-
49
8. STABLE W-UNITARY OPERATORS
a
sponds a neighborhood II X - A II < in [.p] consisting of normally W-decomposable operators. Thus the property of being normally W-decomposable is stable. PROOF. Let l and rz be smooth closed contours separating the parts O"l(A) and O"z(A) of the spectrum of A from each other. From Theorem 2.2 it follows that for a sufficiently small > 0 the spectrum of an operator X satisfying the condition II X - A II < also decomposes into parts O"l(X) and O"z(X) separated by the same contours while the spectral projections P{ = Pl(X) and Pf = Pz(X) differ sufficiently little from the corresponding spectral projections PI = Pl(A) and P z = P 2(A). Consequently, for any e > 0 there exists a > 0 such that the inequalities
r
a
a
a
II Pi PI - P{* P{ II < e, II P~ Pz - Pf* Pf II < e, II Pi WPI - P{' WP{ II < e, II n WPz - Pf* WPf II < e are satisfied for I X - A II < (WP{ x, P{x)
=
a.
(WPlx, PI x)
But then
+
([P{* WP{ - Pi WP l ] x, x)
~ m (PIX, PI x) - e
+ m ([Pi PI
II x 112 ~ m (P{x, PIX)
- P{*Pl] x, x) - ell x 112 ~ m IIp{x liz -2ellxll z
and we get (Wx, x) ~ (m - 2e) II x liz when x = PIX. The case when x = Pfx is treated analogously. 3. Strongly stable W-unitary operators. A stable W-unitary operator U is said to be strongly stable if all of the W-unitary operators U' in some a-neighborhood II U ' - U II < of it are stable.
a
THEOREM 8.3. In order for a W-unitary operator U to be strongly stable it is necessary and sufficient that it be normally W-decomposable.· The sufficiency of the condition follows from the preceding theorem and Theorem 8.1. Let us prove its necessity. By Theorem 8.1, if the operator U is strongly stable (and hence stable), the decomposition (8.3) with the known properties will hold. The theorem will be proved if we prove that the strong stability of U implies O"(U
l.pl)
n O"(U I·Pz) =
0·
Let PI and P z denote the projections corresponding to the decomposition (8.3). We introduce in .p the new scalar product (x, y)l
=
(WXl' Yl) - (Wxz, Y2)
where Xk = Pkx, Yk = Pky (k = 1, 2). The new scalar product is definite:
(x, y E
.p),
50
1. BOUNDED OPERATORS IN BANACH SPACES (x =I 0),
and, moreover, as can easily be seen, the norm equivalent to the old one. We put WI = Pi WPI
-
P~
II xiiI
=
(x, x)Vz is topologically
(8.7)
WPz ;
this permits us to write the new scalar product in the form (x, Y
E
oP)·
Since the operator U commutes with the projections Pk (k = 1,2), it is a unitary operator with respect to the new scalar product and its invariant subspaces oPl and oPz are orthogonal to each other. In other words, U is also a Wrunitary operator while the subspaces oPl and oP2 are WI-orthogonal. eO, dE;. (Eo = 0; E;. = E;.-o for 0 < A < 2n) be the spectral resoluLet U = tion of U. In this resolution E;. (0 ~ A ~ 2n) is a Wrorthogonal projection function, i.e. E;. Ep. = Emin(J.,p.) and WlE;. = E1 WI (0 ~ A ~ 2n). We form the operator H = ME;.. Clearly it is WrHermitian, i.e.
J5"
n"
(8.8)
WlH= H*Wl' We show that it is also W-Hermitian, i.e.
WH= H*W.
(8.9)
In fact, since the operators P k (k = 1,2) commute with U, they commute with the projection function E;. and H, and therefore the projections Pt commute with E1 and H*. It therefore follows from (8.8), after mUltiplying it by Pt from the left and by P k from the right, that
Pk WlPkH = H* Pt WI Pk. Nothing that W = Pi WP I + P~ WP 2, we arrive at (8.9). The fact that the operators P k commute with H means that the subspaces oPk are invariant under H. Consequently, to the decomposition (8.3) there corresponds the following representation of H in matrix form:
H = (Hll
o
0 ).
H zz
Then U can be represented in the form U = (exp (iHll )
0 ). exp (iHzz ) ,
a(exp(iHn));
a(U loP2)
o
a(U loPI)
=
= a(exp (iHzz )).
8.
51
STABLE W-UNITARY OPERATORS
The theorem will be proved (by contradiction) if we show that when the spectra (J(H I ~\) and (J( U 1-P2) have a common point i\ there exists in any neighborhood of HaW-Hermitian operator H' whose spectrum contains nonreal points. In fact, this will imply that there exists in any neighborhood of U a W-unitary operator U ' = exp (iH') whose spectrum contains nonunitary points (points not lying on the unit circle) and which are thus not stable. It can be assumed without loss of generality that the common point i\ '# 1, since we could otherwise achieve this by replacing U by a W-unitary operator p U where I p I = 1 and p '# 1. If i\ = exp (iw) (0 < w < 2n) is a common point of the spectra (J(U I-PI) and (J(U 1-P2) then w will be a common point of the spectra (J(Hu) and (J(H22). We first consider the simpler case when w is a common eigenvalue of the operators Hu and H 22 · Let (/JI and CP2 denote corresponding unit eigenvectors: CPk E -Pk, II CPk 111 = 1, Hkk CPk = WCPk (k = 1, 2). For an arbitrary c > 0 we form the operator
_ (Hll H22) _ Gi G, '
H, -
where G, is the operator acting from -Pz into -PI according to the formula and Gi is the WI-adjoint of G, acting from -PI into -Pz according to the formula
Gi x
h(x, CPl)I CP2
= -
= -
ic(Wx, CPI) CP2
(x
E
-PI).
It is easily seen that together with H the operator H, (c > 0) will be W-Hermitian. A simple calculation shows that
H, (CPI
+ CP2)
= (w
+ ic) (CPI + cpz).
Thus the W-Hermitian operator H, (c>O) always has a nonreal eigenvalue. On the other hand, H, will be arbitrarily close in norm to H for sufficiently small c > o. The case when w E (J(Hu) (J(Hzz ) but is not necessarily an eigenvalue for each of the operators Hll and H zz reduces to the preceding one. In fact, using the spectral resolutions of the operators H kk :
n
we form the operators H~%) .
=
(
z". )
w-a
J+ J 0
w+a
a
AdEk ).
+ wi\,
where f\ = J::!:~ dEkJ. (k = 1, 2; 0 < < w). Since, by assumption, w E (J(Hu) (k = 1,2), it follows that Fk '# 0 and w will be an eigenvalue of H j!) with eigenspace Fk-P.
52
I.
BOUNDED OPERATORS IN BANACH SPACES
We have IIHkk -
Hk~)
I
=
/1:1: (A -
w) dEkA II
< 0,
and hence I H - Hij II < 0, where Hij is the W-Hermitian operator defined by the equality Hij = (
~if) H~g)
)
By what has been proved above there exists in any neighborhood of Hij a WHermitian operator with nonreal spectrum; therefore such an operator also exists in any neighborhood of H. The theorem is proved. § 9. Elements of nonlinear analysis
m
1. Contraction principle. In what follows we will assume that is a closed subset of a Banach space ~, although the basic Theorem 9.1 can be formulated for any complete metric space. Let S be a (not necessarily linear) operator mapping m into itself. We call it a contraction if there exists a constant q < 1 such that (9.1)
for any x, y
E
m.
THEOREM 9.1. Suppose the operator Sv is a contraction for some natural number )) ;:::; 1. Then there exists in m one and only one fixed point Xq, of S:
SXq, = Xq,. This point can be obtained from any point Xo Xq,
(9.2) E
mas the following limit:
= lim Snxo. n~oo
PROOF.
Suppose first))
=
1. Then for the sequence
we have according to (9.1)
I Xk+l - Xk II = II SXk - SXk-1 II ~ q II Xk - Xk-l II (k = 1,2, ... ), so that II Xk+l - Xk II ~ qkll Xl - Xo II. Therefore
(9.3)
9.
53
ELEMENTS OF NONLINEAR ANALYSIS
i.e. {x n } is a Cauchy sequence. The fact that 9]( is closed implies the existence of the limit limn~co Xn = Xrp. This limit is a fixed point of S since
I Xrp
- SXrp
I
:<::; I ;;:; I
Xrp - Xm+l Xrp - Xm+l
I + I SXm I + q I Xm -
SXrp I Xrp I ~ 0
form~ 00.
The uniqueness of the fixed point follows at once from (9.1). Thus the theorem is completely proved for the case l.i = 1. Suppose now l.i > 1 and consider the contraction S = S". By what has been proved above S has one and only one fixed point Xrp. On the other hand, since S and S commute, S(SXrp) = S(Sxrp) = SXrp,
i.e. SXrp is also a fixed point of S. Hence by virtue of the uniqueness (9.2) holds. Taking into account that every fixed point of S is a fixed point of S, we conclude that S has no other fixed points. It remains to prove that equality (9.3) holds for any Xo E 9](. To this end we note that every n can be represented in the form n = ml.i + r, where 0 ;;:; r ;;:; l.i - 1. Then Sn Xo = Sr Sm Xo and, since limm~oo Sr Sm Xo = Srxrp = Xrp for fixed r, equality (9.3) is proved. REMARK 9.1. If S itself is a contraction, the difference I Xrp - Snxo I monotonically decreases at the rate of O(qn). We leave it to the reader to formulate what happens when l.i > 1. 2. An application of the contraction principle. Let A E [)B] and let rA = limn~co I An Ill/n be the spectral radius of A (see § 2.1). We consider the equation
+ f, Ax + f, we will have
x = Ax
where f
E )B
is given. If we set Sx
=
(9.4)
n~l
Snx = Anx
+ L:
Akf,
k=O
which implies
I snX2 ..::..
SnXl
I
When r A < 1 and l.i is sufficiently large\ we have I A" I < 1, and hence S" is a contraction. Thus, when rA < 1, equation (9.4) has for any f E )B exactly one solution, which is given by the formula x =
L: A'1 (= lim Snxo)· k=O
n~co
(9.5)
I.
54
BOUNDED OPERATORS IN BANACH SPACES
3. Differentiation offunctions with a vector argument. Let y = f(x) be a function defined in a neighborhood U(xo) of a point Xo of a Banach space j(3o and taking values in a Banach space j(31' The functionf(x) is said to be differentiable (according to Frechet) at Xo if in the neighborhood U(xo) it can be represented in the form f(x)
= f(xo) +
A(x - Xo)
+ 7}1(X, Xo) I x
- Xo II,
(9.6)
where A E [j(3o, j(31] and the function 7}1(X, xo) satisfies the condition lim 7}1(X, xo) = O. X-Xo
The operator A is called the (Frechet) derivative of f(x) at xo, and one writes A
= f'(xo).
As usual, we say that f(x) is differentiable on an open set G if it is differentiable at each point of this set. We will say thatf(x) is continuously differentiable in G if its derivative f'(x) is a continuous function of x with values in [j(3o, j(3d. All of the usual formal rules of differential calculus, which we will not cite here, are retained in the case under consideration. We formulate only the simplest variant of the rule for differentiating a composite function, which will be needed by us in the sequel. If x(t) is a differentiable function of a scalar argument t whose values lie in a domain G of the Banach space j(3o in whichf(x) is differentiable, then (d/dt)f(x(t))
In particular, for x
E
= f'(x(t))
dx/dt.
G and sufficiently small t
+
(d/dt)f(x
ty) = f'(x
+
(y E j(3).
ty)y
Integrating the latter equality from 0 to 1 and changing the notation, we obtain the equality I
f(X2) - f(XI) = Sf' (Xl
+
t(X2 - Xl)) (X2 - Xl) dt,
(9.7)
°
which is valid when the interval Xl + t(xz - Xl) (0 ;::;:; t ;::;:; 1) lies entirely in the set G. REMARK 9.2. It follows from formula (9.7) that if f'(x) is bounded on a convex set G: III' (x) 1J ;: ;:; M (x E G), then Ilf(xz) - f(XI)
I ;: ;:; Mil Xz
- Xl
I
(Xl> Xz E G).
Iff(x) is differentiable in a neighborhood of Xo then the derivativef'(x) itself is a function in this neighborhood with values in the Banach space [j(3o, j(31]. If it too is differentiable at xo, its derivative is denoted by the symbol f"(xo) and called the second derivative of f(x) at Xo. By definition, f"(xo) E [j(3o, [j(3o, j(31]]' In other words, it can be said thatf"(xo) is a bilinear operator taking each pair (hh h z) of elements of j(3o into an element f"(xo)(hh h z) E j(31'
9. ELEMENTS OF NONLINEAR ANALYSIS
55
It is not difficult to obtain the representation f(x) = f(xo)
+ f'(xo)(x
+7jz(x, xo)
Ix
- xo),
- Xo
+ ~
f"(xo)(x - Xo, x - xo)
liz,
(9.8)
where limx~x, 1)z (x, xo) = O. The derivatives of higher order are defined analogously. 4. Inverse function theorem. We now use the contraction principle to obtain an important result of analysis. THEOREM 9.2. Suppose a function y = f(x) is defined and continuously differentiable in a neighborhood U(xo) of a point Xo of a Banach space lBo and takes values in a Banach space lB 1. If the operator T = f'(xo) E [lBo, lB 1] has a bounded inverse T-1 E [lB1, lBo], the function f(x) effects a homeomorphic mapping of a neighborhood V(xo) c U(xo) onto a neighborhood of the point Yo = f(xo). The inverse mapping x = g(y) has in this neighborhood the continuous derivative
g'(y)
= [f'{g (y)]-l.
(9.9)
Iff(x) has continuous derivatives up to order n in a neighborhood ofxo, so will g(y) in a neighborhood of Yo = f(xo). PROOF. We can assume without loss of generality that Xo = Yo = O. The equation f(x) = y is equivalent to the equation
x = x
+
T-1 [y - f(x)] i;! (/) (x).
(9.10)
For fixed y we have
(/)'(x)
= I - T-1f'(x) = T-1 [I' (0) - f' (x)].
Therefore by virtue of the continuity off'(x) there exists a 0 > 0 such that in the closure of the neighborhood Sa,fIl, = {x III x I < o} we have the estimate I (/)'(x) I ~ q < 1, and hence (see Remark 9.2) also the estimate
I (/)(xz)
- (/)(X1)
I
~ q
I Xz
- Xl
I
(Xk E
Sa,fIl,; k = 1,2).
Suppose now
I y I < c=
(1 - q)o /
I
T-1 II·
(9.11)
Then the estimate
I (/) (x) I ~ I (/) (x) - (/) (0) I + II(/) (0) I ~ q I x I + I T-t[[ I y I implies I (/)(x) I ~ 0 provided I x I ~ o. Thus under condition (9.11) the
(9.12)
mapping (/)(x) is a contraction in the closed ball Sa,fIl,. From Theorem 9.1 we deduce that for each y E S"fIl, = {y E lB1 III y I < c} there exists a unique solution x E Sa,fIl, of the equation x = (/)(x) and hence of the equationf(x) = y.
I.
56
BOUNDED OPERATORS IN BANACH SPACES
The inverse function x = g(y) is defined in Se'~l and, since (] can be chosen arbitrarily small, is continuous at y = O. In particular, by taking y sufficiently small, we can obtain an arbitrarily small x = g(y) and hence an operator f'ex) that is close enough to 1'(0) = T to have a bounded inverse. Thus all of the above arguments can still be applied if the pair (xo, Yo) is replaced by the pair (g(y), y), and hence g(y) is also continuous at this point y. In exactly the same way it also suffices to establish the differentiability of g(y) only at y = O. We have y = f(x) = Tx + [f(x) - Tx], where, by virtue of (9.6), f(x) - Tx = r;llxll and r; = 0(1) for x ---+ O. Hence
x = T-l Y - T-l
r; I x I
= T-l y - T-l
r; I g(y)
II·
(9.13)
We now note that (9.12) implies the estimate
I g (y) I
I 1J(g(y) I
=
~ q
I g(y) I + I
T-l
IIII y
II,
from which we get
I g (y) I This shows that for y
---+
ij
~
I T-q I y I / (1
- q).
0
= T-lr;
I g(y) I / I y I
= 0(1).
In this way we obtain the representation (see (9.13) g(y)
= T-l Y
- ij
Iy
II,
showing that g(y) is differentiable at y = 0 and g'(O) = T-l Y = [1'(0)]-1. This proves equality (9.9). The proof of the last assertion of the theorem can be obtained by induction, for which one successively differentiates (9.9). 5. Cone inequality theorem. In this subsection we present a simple assertion, which will be systematically used in more special cases in subsequent chapters. Its proof will be developed for linear operators, although the assertion admits certain generalizations to the nonlinear case. We first intrqduce the following notion. A closed subset Sl: of a Banach space lB is called a cone if it has the following properties: a) Xo E Sl: implies AXo E Sl: for A ~ 0; b) xl. X2 E Sl: implies Xl + X2 E Sl:; c) ± Xo E Sl: implies Xo = O. Suppose a cone Sl: is given in a Banach space lB. We will write
x
(9.14)
57
EXERCISES
if y - X ESt It is not difficult to see that relation (9.14) has the properties of an ordinary inequality,l7) If the cone st is invariant under a linear operator A: Ast c st, then A preserves the inequalities: x y implies Ax Ay.
<
<
THEOREM 9.3. Suppose the cone st is invariant under an operator A spectral radius rA < 1. If a vector x E ~ satisfies the inequality
< Ax +f it also satisfies the estimate x < y
(Ax
x
y
(y
=
+f
< x),
E [~]
with
(9.15)
< x), where y is a solution of the equation
Ay
+ f.
(9.16)
We give a proof of the first variant of the theorem. Since, in addition to A, the operator S: Sx = Ax + J, also preserves inequalities, the inequality x Sx implies x Sx S2X Snx, so that
<
<
< ... < X < Sn X = An X + L: Ak f. k=O
<
n-l
(9.17)
We now note that the condition rA < 1 implies (see (2.1)) the convergence of the series L:;' Ak and the equality (/ - A)-l = L:;' Ak. This means that the element y = L:;' Akf is defined and is the unique solution of equation (9.16). By virtue of the fact that the cone st is closed we can pass to the limit in inequality (9.17). Since the condition rA < 1 implies IIAnl1 --+ 0, we then obtain the required estimate x
< k=O L: Akf= y.
REMARK 9.3. It is not difficult to verify that the assertion of the theorem continues to hold if A is a nonlinear monotone contraction. EXERCISES 1. An operator A E [58] is called an operator of simplest type if its spectrum consists of a finite number of eigenvalues and 58 is the direct sum of its eigenspaces. In other words, an operator A is an operator of simplest type if and only if it is representable in the form
A=I;A.P. j=l
J
J'
(0.1)
where Aj,"', An are mutually distinct complex numbers and the P j (j = 1,"',n) form a complete system (L; P j = l) of pairwise disjoint projections. Show that 17) In particular, we will consider inequalities between operators in a Hilbert space .p defined by the cone of nonnegative operators (see § 5). In this case the relation A
58
I.
BOUNDED OPERATORS IN BANACH SPACES
a) An operator A is an operator of simplest type if and only if there exists a polynomial peA) with simple roots such that peA) = O. Hint. Let p(A) = (A - AJ)"'(A - An). Use the identities
and the relations issuing from them
b) If ~=,p, an operator A is an operator of simplest type if and only if it is similar to a normal operator B (B* B =BB*) and its spectrum consists of a finite number of points. Hint. To prove the necessity of the condition, introduce on the basis of the decomposition (0.1) the new scalar product 2. Let A = L:r AjPj and B = L:r f.1kQk be two operators of simplest type. Condider the operator \1f acting on the elements X of US] according to the formula \1fX = AX - XB. As is well known (cf. § 3), its spectrum consists of the numbers Aj - f.1k (j = I,··, nand k = 1,···,m). Show that the operator \1f is also an operator of simplest type and, if a E 0"(\1f), the spectral projection \l3 a corresponding to a is given by the formula
=
\l3 a X
L:
PjXQk'
).j-p.It=a
Generalize this result to the case of an operator \1f of the form 2(X =
z:; cpq APXBq
(the c pq are scalars).
3. Let A be an operator acting in the finite-dimensional space ~ = Cn, let AJ,··, Ak be the different points of its spectrum, let P j denote the spectral projection corresponding to Aj and let nj denote the dimension of the subspace ~j = Pj~. Show that one has the representation A
k
= z:;1(A.p. + Q.), J
j=
J
(0.2)
J
in which Qj = QjPj = PjQj and Qjf = 0, where mj is an integer not exceeding nj. Hint. Make use of the Jordan normal form of the matrix A and put Qj = (A -Ai)Pj . 4. Retaining the notation of the preceding exercise, verify that the resolvent RA of A is representable in the form RA
Q' = j~J z:;k P ,~O z:; (- 1), _~J _~. (Aj - A)'+J mf-J
...
(0.3)
¢(')~Ai) Qj.
(0.4)
J
Use formula (2.3) to obtain the representation ¢(A)
=
t P mf j
}=1
r=O
r.
Write out, in particular, the representation for eAt. 5. Using the results of Exercise 4, obtain the following representation for a function ¢CA, f.1)
E
KA.B : (0.5)
where
59
EXERCISES k,
A
= .L: (Aj,P;, + Qi'), 11=1
6. Show that in the finite-dimensional case the solution of the equation
L:
SI' 52
(3.1)
c"'" A" XB" = Y
can be written in the form (0.6)
7. When the operators A and B are Hermitian, the operators Qi, and Qi, obviously vanish and the formulas given in Exercises 5 and 6 can be simplified. In particular, the latter of them takes the form (0.7)
Generalize this result to the case when iB is an infinite-dimensional Hilbert space .p and show that the solution of equation (3.1) for Hermitian operators A and B is representable in the form
where E ~ and EZ are the resolutions of the identity corresponding to the operators A and Band L: c"" A"p."*O (for (A,p.) E IJ(A) x IJ(B». 8. Let Ao be an eigenvalue of an operator A E [~l. The linear manifold consisting of all of the vectors x which satisfy the condition (A - AoI)kx= 0 for some integer k ~ 0 is called the root manifold corresponding to Ao (the root space if is closed, in particular, finite dimensional). The dimension of 2,. is called the algebraic multiplicity of the eigenvalue Ao. If for some fixed natural number k
2,.
2,.
(0.8)
one says that Ao is an eigenvalue of finite index, and the least k satisfying this condition is called the index of Ao. If (0.8) is not satisfied for any k, one says that Ao is of infinite index. The index of an eigenvalue does not exceed its algebraic multiplicity. a) Show that an isolated eigenvalue Ao of a linear operator A E [iB] is of finite index r if and only if the resolvent RiA) has a pole of order r at Ao. Hint. Let RiA) = L:':'=An(A - Ao)n. Make use of the relation A- cn +l)=(A -AoI)A-l which follows from the definition of the resolvent, and the fact that A-l = (l/2:rri)§ rRiA)dA is a projection on 2, •. An isolated eigenvalue is said to be normal if it is of finite index. b) Show that an eigenvalue Ao is normal precisely when the algebraic multiplicity of Ao is finite and ~ decomposes into a direct sum iB=2,. ~'"' where ~,. is an invariant subspace of A in which the operator A - AJ is invertible. 9. An operator A E [iB] is called an operator of algebraic type if its spectrum consists of a finite number of eigenvalues Ab ••• , An of finite index. Show that a) An operator A E [iB] is'an operator of algebraic type precisely when the decomposition (0.2) is valid. In particular, every linear operator in a finite-dimensional space is an operator of algebraic type (see Exercise 3). b) An operator A is an operator of algebraic type precisely when there exists a polynomial peA) such that peA) = O.
+
60
I.
BOUNDED OPERATORS IN BANACH SPACES
c) As a generalization of Exercise 2, if A ana B are operators of algebraic type in ~, the operator \!{X = AX - XB in [~] is also an operator of algebraic type. d) The results of Exercises 4 and 5 are retained for operators of algebraic type in an infinitedimensional spa~. 10. a) Suppose the spectrum of an operator Tlies in the interior of the unit circle. Show that the equation W - T* WT = S has for any S E [~] a unique solution, which is representable in the form of the series W = ~ (T*)k STk. k~O
b) Suppose T is a u-dichotomic operator. Show that for any S E [~] satisfying the condition + P':SP.. where Pi and p. are the spectral projections of T on the subspaces ~i and ~ .. the above equation has the solution
S=Pf'SPi
00
W=
L: (T*)k ptSPiTk -
k=O
-I
L: (P,"
k=-oo
+ T*P'n P,SP,(P + TP.)k. j
This solution is unique under the additional condition W = ptWPj
+ P:WP•.
»
The above assertion for S 0 independently implies part of Theorem 7.1'. 11. Suppose the spectrum of an operator A lies in the interior of the left halfplane. Show without using the integral formulas of § 3 that the equation WA + A* W = - H has a unique solution W E [~]. This solution satisfies the condition 0 if o. Hint. Make use of the results of Exercise 10 for T = (A + aJ) (A - aI)-I, where a is sufficiently large.
W»
12. Let G be a triangular operator: G = operator H
(g ~).
H»
If a(A)
n a(C) = 0
= (g g). If on the other hand a(A) n a(C) * 0
then G is similar to the
and AE a(A)
n a(C) is
a normal
eigenvalue of both A and C (of index YA for A and of index r e for C), then Ais a normal eigenvalue of G of index rG ~ rA + Ye. Hint. According to Theorem 3.2, if a(A) n a( C) = 0, the equation AX - XC = B has a solution. A check shows that H Suppose a(A)
= TGT-I,
where T
= (~I Z).
n a(C) *0, GP-IX * 0, GPx = 0 and x = (~~) * o. If X2 = 0 then XI is a root
vector of A andp
~ rA.
On the other hand, if X2 Ar·XI
* 0 andp >
rB
then
+ (Ar.-I B + A,,-2 BC ... + Bcr,-I) X2
*0
is a root vector of A of index p - rB. 13. Suppose A and B are Hermitian operators in a Hilbert space ~ and let AM(A)
=
sup (Ax, x),
/)xll=1
A", (A)
=l!xl1=1 inf (Ax, x).
Obtain the estimate (0.9)
Hint. Differentiate the expression ¢(t) = l!eCA+iB)t X\\2 and obtain an estimate of the logarithmic derivative of ¢(t). 14. Suppose ~ is a Hilbert space and A E [~]. Show that if the spectrum a(Re A), where Re A = (A + A*)/2, lies in the closed halfplane Re A~ a, then the spectrum a(A) satisfies the same condition.
61
EXERCISES
Hint. Make use of inequalities (0.9) and Theorem 5.1. 15. Prove the relation e A+B = lim [eAlneB'n]",
(0.10)
n->=
where A, B EO [~] and the limit is taken in the sense of uniform convergence. Hint. Make use of the relation IlecA+B)t - eAt eBtl1 ~ ct 2 (c = const), which is easily deduced from Taylor's formula. Obtain from (0.10) the upper estimate of (0.9). 16. Show that the following estimate is valid for any operator A acting in en. (t ;:::; 0),
(0.11)
where" = max (Re AI A EO a(A)} (see B. F. Bylov, R. E. Vinograd, D. M. Grobman and V. V. Nemyckii [1], page 131). Hint. The following equality holds for any analytic function f(A) in a domain containing the spectrum a(A) = {Aj} : f(A) = al
+ a2 (A
- AlI)
+ a3(A
- Atf) (A - Azl)
+ ... + an(A
- All) (A - A2l)-··(A - An-II),
where
Hence IaH II complex plane.
~ (Ilk!) max~ I j
(A) I, where 81 is the convex hull of the numbersA b
••• ,
Ak in the
Putf(A) = e A '.
(In the book of I. M. Gel'fand and G. E. Silov [1] the number 1/k! in the estimate for IaHII has been replaced by 1 and therefore the factors 1/ k! in the estimate for I eAt II are missing.) An "exact" estimate for I eAt I in terms of I A I and" is apparently not known. The following Exercises 17-24 contain assertions permitting one to generalize the inequalities of Exercise 13 to the case of operators acting in any Banach space. 17. Prove the existence for any x, y EO ~ of the limit
(>
to which the quantity under the limit sign tends montonically (is non increasing as h 0) decreases). Hint. The assertion follows from the convexity of the function ¢xjh) = Ilx -+- hyll (- co
<
u = lim _~(to_t h) -=-.\"(to>h+O h Prove that the right derivative of the norm Ilx(t)11 exists at to and, in addition, lim Jx(to ± h)II-=I~(to)1L oullx(to)ll. h+O h 19. According to Exercise 17 the following limit exists for any A A(A) = lim III + hAl-=l. h.O h
EO [~]:
r~-'~ I
62
I. BOUNDED OPERATORS IN BANACH SPACES
Clearly, A(aA) A(A
= aA(A),
if a;:::;O; I A(A) I ~ IIAII;
+ B) ~ A(A) + A (B);
[A(A) - A(B)] ~
IIA - Ell.
At the same time the "measure" A is not a norm since it can take negative values. But one always has A(A)
+ A( -
A) ;:::; 0 (= A(O)).
a) Show that, if m= ,p,
and thus - A ( - A) = Am(AiJl). Let mbe a Banach space of sequences of complex numbers: x Em¢=:> x = (~j) I, the ~j=~j (x) (j = 1, 2,.··) being continuous functionals of x E m. Then to each operator A E [m] there corresponds a matrix lIa jk lll such that the equality x' = Ax is equivalent to the equalities (j = 1,2,.··).
Show that b) If m= c (c is the space of all convergent sequences with norm IIxll A(A) = sup (Reajj j
whereas IIAII = c) If then
m= II
SUPj
L:k
= sup
I ~j I),
+k*j L:·lajkl),
I ajkl.
(II is the space of all absolutely summable sequences with norm IIxll A(A)
= sup (Reajj + L: j k*j
= L:j~11 ~
j
I),
lajkl),
whereas IiAII = SUPk L:jlajkl. 20. Extend the result of Exercise 13 by proving that for any A
E [m]
e- AC - A ) ~ lIe A Ii ~ eACAl.
(0.12)
Estimate (0.12) implies the following estimate for a strip containing the spectrum o(A) (see (2.5) and Lemma 2.2): - A( - A) ~ Re A ~ A(A)
for all A E o(A).
(0.13)
Hint. Using the result of Exercise 18, estimate the derivative with respect to t of the norm lIexp (At)xoll for any Xo Em. Estimate (0.9) i; a special case of a more general estimate of Wintner (see Corollary 111.4.3), while estimate (0.12) is a special case of one of the estimates of S. M. Lozinskii referred to in Exercise 111.19. The notion of the functional A(A) as well as the other results indicated in these exercises are due to S. M. Lozinskii [1] (see also the paper of G. Dahlquist [1]; in the papers of these authors mis a finite-dimensional space). S. M. Lozinski! called the measure A(A) the logarithmic norm of A. 21. Estimate (0.13) implies that the spectrum of an operator A lies in the closed convex domain KA of the complex Cplane obtained by taking the intersection of the halfplanes (0
~
e ~ 2n)
63
EXERCISES
A further restriction of this convex domain can be obtained by applying the formulated result to all possible operators of the form Aa = A - aI, where a is a complex number. For ~ = .p the assertion concerning KA yields the well-known Hausdorff theorem KA = {(Ax, x)lx E.p, Ilxll ~ 1).
When applied to operators acting in en with this or that norm the assertion on KA yields one of the well-known theorems of A. Hirsch (see, for example, Marcus and Mine [1], page 140). 22. An entire function x(Q of a complex argument ~ with values in ~ is called an entire function of exponential type 0 (0 ~ 0 ex)) (or, according to S. N. Bernstein, offinite degree 0-) if
<
lim ]rlllx(QII = ,->00
(J.
I~ I
Let x(~) = Xo
~ ~2 ~n + lTXI + 2T X2 + ... + II! Xn + ....
Verify that, as in the scalar case (see B. Ja. Levin [1]), a function will be an entire function of exponential type (J if lim SUPn->oo I Xn II lin = (J. 23. According to a theorem of S. N. Bernstein, if an entire scalar function x(~) (~ = Cl) of exponential type is bounded on the real axis, then Ix'(QI
~
(J
sup
~oo
Ix(t)1
(- ex)
< ~ < ex)).
Verify that the theorem is also valid in the case of an entire function x(~) of exponential type with values in any Banach space ~, i.e. Ilx'(~)11 ~
(J
sup Ilx(t)11 -co
(- ex)
< ~ < ex)).
From estimate (0.13) it incidentally follows that if an entire function of exponential type (J is bounded on the real axis then (J O. Hint. Consider the scalar functionf(x(t )), where fis an element of the conjugate space, and apply the theorem of S. N. Bernstein. 24. Prove that for every nonzero operator A E [~] with spectrum concentrated at the origin (in particular, for a Volterra operator18 )) at least one of the quantities A(A) or A( - A) is positive (while the other is nonnegative). Hint. Construct a proof by contradiction. Consider the entire function exp (~A) and make use of (0.12), (0.13), the theorem of S. N. Bernstein (see the preceding exercise) and the fact that A is a quasi-nilpotent operator: rA = limn->oo IIAnlil/n = O. 25. Let (Ak) (k = 1,.··, n) denote a family of pairwise commutative operators in Hilbert space with spectra lying in the interior of the left halfplane. Show that there exists a uniformly positive operator W such that all of the operators Re (AkW) are uniformly negative (S. G. Krein and S. D. Eidel'man; see S. <1. Krein [1], page 167). Hint. Put W = So ... So exp (I;1skAk) dS I ••• dS n (when n = 1 the assertion is contained in Theorem 5.1). It is assumed in the following exercises that Wis a Hermitian operator in a Hilbert space.p. 26. An operator A E [.))] is said to be W-nonnegative if (Ax, x)w ~ 0 for all x E .p or, in other words, if WA o. a) Show that the spectrum of a W-nonnegative operator A lies on the real axis.
>
>
18) A Volterra operator is a completely continuous operator with spectrum concentrated at the origin.
64
I. BOUNDED OPERATORS IN BANACH SPACES
Hint. Assume the contrary and make use of the fact that for a boundary point ;I. of IY(A) not lying on the real axis there exists a sequence of fn E Sj (lIfnll = 1) such that AI. - ;l.fn -> 0 for n -> 00. b) Suppose A E [Sj] and AW» 0 (in this case the operator A is W-l nonnegative). Prove that Sj decomposes into a W-orthogonal sum Sj = Sj+ Sj_ of subspaces invariant under A such that the restrictions A+ = A ISj + and A_ = A ISj_ are similar to a uniformly positive and a uniformly negative operator respectively. Hint. Make use of Theorem 7.1. 27. Suppose A is a W-Hermitian operator with positive spectrum. Show that there exists a unique operator B with positive spectrum such that B2 = A. This operator is W-Hermitian (V. P. Potapov [1]) and Ju. P. Ginzburg [1]). Hint. To construct the operator B make use of the Poincare-Riesz formula (2.3). 28. An operator Y is called a W-nonenlargement if
+
(WY/' Yf)
~
(W/'f),
«
or, in other words, if y* WY w. Show that if Y is a W-nonenlargement having at least one regular point on the unit circle, its adjoint y* is a W-l-nonenlargement (see, for example, V. P. Potapov [1] and Ju. P. Ginzburg [1]). Hint. Apply the Cayley transform to reduce the problem to one of considering W-dissipative operators. 29. Let U be a W-unitary u-dichotomic operator (the spectrum IY(U) does not intersect the unit circle). Show that Sj decomposes into a direct sum of two W-neutral invariant (under U) subspaces in which the spectrum IY(U) is found respectively inside and outside the unit circle (1. S. Iohvidov and M. G. Krein [1]). Hint. Make use of the arguments employed in the proof of Theorem 7.1. The results noted in the following exercises have been taken from an article of M. G. Krein [11]; they will be used in Chapter V. 30. One calls 0 (0 ~ 0 ~ n:) the angle between nonzero vectors x, y E Sj, and writes 0 = 1:::(x, y) (= 1::: (y, x)) or 0(.0) (0 (y-;:i)), if cosO = Re(x,y)/llxIIIIYII.
An equivalent and perhaps more natural definition of the angle 0 = equality
2 sin (012) = Ile x
-
1::: (x, y) is given by the
eyll,
where ex = xlllxll, ey = ylll yll are the unit vectors in the directions of the vectors x, y. Prove that for any Xj E Sj (Xj *- 0; j = 1,2, 3)
1::: (x" X3)
~
1::: (x" X2) + 1::: (X2, X3),
(0.14)
the equality sign homing if and only if the vector X2 lies "between" the vectors Xl and X3, i.e. X2 = ClXl + C3X3 (c" C3> 0). Hint. Using the unit vectors ej = xj/llxjH and arbitrary real numbers t;; (j = 1,2,3), construct the nonnegative form
Calculate the determinant of this form and explain why it is nonnegative. 31. a) Let e(t) (a ~ t ~ b) be a smooth (i.e. continuously differentiable) arc on the unit sphere S of H. Prove that
65
EXERCISES
£II ~
I dt ~ 1: (e(b), e(a»,
(0.15)
the equality sign holding if and only if e(t) lies between e(a) and e(b) for all t E [a, b]. If the unit sphere of Rn (n-dimensional Euclidean space) is taken as S, inequality (0.15) expresses the fact that the geodesics on S are great circle arcs. Hint. Make use of the result of the preceding exercise. b) Generalize (0.15) to the case when the smoothness condition for e(t) is replaced by the requirement that it be absolutely continuous. Hint. Approximate the spherical curve e(t) by the curves eh (t)
=
*:1:
e(7:)d7:/
I {-:z:
e(7:)d7:
I
(a;;;'t;;;'b),
<
putting eh(t) = eh(a) for t a and ek(t) = eib) for t> b. 32. Suppose A E [.p] has an inverse A-I E [.pl. The quantity dev(A) ~ sup (1:(Ax, x) Ix E is called the angular deviation of A. Thus 0 ;;;, dev (A) ;;;,
.p, X 7r
*'
o}
and
. f { _R~iAx~1 -.L 0 } cos d ev (A) -- In IIAxl1 Ilxll x E fJ., x -r.
>
Clearly, dev (cA) = dev (A) for c o. a) Verify that dev (A) is a continuous function of A E [.p]. Suppose A, B E [.p] are invertible operators. Since 1: (ABx, x) ;;;, 1: (ABx, Bx) we have dev (AB) ;;;, dev (A)
<
+ dev
+ 1: (Bx, x),
(B).
b) Prove that if dev (A) 7r, the spectrum f7(A) of A lies in the sector Iarg AI ;;;, dev (A). Hint. Take a ray C = pe;¢ (0 p co) intersecting f7(A), and take the point CaE 17 (A) on it with maximum modulus. There will exist a sequence (x n) such that AX n - COxn --> 0, Ilxnll = 1. Consider lim 1:(Ax,,, x n). 33. Prove that for a uniformly positive operator H
< <
cos dev (H)
=
2 VAM(H)Am(H) AM(H)
+ Am(H)
,
where Am(H) ;;;, AM(H) are the boundaries of the spectrum f7(H) (see Exercise 13) (L. V. Kantorovic [1]).
34. Suppose A E {.p] is an invertible operator. The quantity am(A)~min
(dev(C-IA)IICI =I)
(0.16)
<
is called the angular amplitude of A. If am (A) 7r then 2am (A) is the angle of the least sector of the C plane contaitling the spectrum f7(A). Let C = Co = eia be the unique point of the circle ICI = I at which the minimum in (0.16) is achieved. Then the indicated sector will be the sector a - am(A) ;;;, arg A ;;;, a + am(A). Clearly, for any scalar c*,O we have am (cA) = am(A). It readily follows from (0.16) that am(AB) ;;;, am(A)
+ am(B).
(0.17)
Prove that a) The inequality am( U) 7r holds for a unitary operator U if and only if the spectrum 17( U) does not cover the whole unit circle. When this condition is satisfied, 2am(U) is the length of the least arc of the unit circle containing the spectrum f7( U).
<
66
I.
BOUNDED OPERATORS IN BANACH SPACES
Hint. Usethefactthat if 17(V) lies on the arc 7 = ly admits the spectral resolution
ei~
(-a
~
¢
~
a, 0
< a < n-) and consequent-
v = -aJ ei ¢ dE(¢), then Re (Vx, x) ~ (x, x) cos a. b) If the spectra of the unitary operators VI and V2 1ie on arcs 71 and 72 of the unit circle the sum of the lengths of which is less than 2n:, the spectrum 17( VI V 2) lies on the product of these arcs 7 = (CIC21 CI EO 71> C2 EO 721 ..
For a further development of this result see Chapter V. Hint. Use assertion a) and inequality (0.17). 35. Let T be a u-dichotomic uniform W-contraction with coefficient q: W - T* wr;p qI.
(0.18)
Show that the annulus of the complex plane
,J 1-
)..jW)
<1A1<,J 1+
)..M(=-W)
(0.19)
does not intersect the spectrum 17( T). We note that the annulus (0.19) contains the annulus
Hint. Consider the boundary points).. of 17(T) and the values of the form «W - T* WT)x, x) on the corresponding sequences (xnl of unit vectors (1lxnll = 1) such that !I(T - )..I)xnll ---> 0 (see the theorem on a boundary point of the spectrum in § 4.3). 36. Let W be an invertible bounded Hermitian operator in -P and let A be a W-Hermitian operator. Show that an equivalent norm can be introduced in ,p in such a way that the operator A becomes Y-Hermitian with y* = Y and y2 = I. Hint. Put (X'Y)I = (I Wlx,y)and y = sign W,where I WI = (W*W)1I2 and sign W = W I WI-I. 37. An indefinite operator W = W* EO [,pl is said to be balanced if under some decomposition of .),) into a direct sum,p = -P+ .p_ of two W-orthogonal W-definite subspaces one has the equality dim -P+ = dim -P-. a) Show that the latter equality will hold under any such decomposition (a special case of the general law of inertia). b) Show that for every balanced operator W EO [,p] there exists an orthogonal decomposition ,p = ·Pl EB ')')2 and an invertible operator S EO [·P] such that
+
S * WS
=
(0 112 ), 121 0
where 112 is an operator of identification of 'P2 with,pl (a unitary mapping of -P2 onto .pI) while 121 = 1121. Hint. Make use of the result of Exercise 36 and then make use of the equality E*
where
(111 0) o -122 '
E =
(0121 112) 0 '
67
EXERCISES
38. Suppose .p = .pI EB.pz and E (c .p) is a closed subspace. Prove that there corresponds to E a unique operator K E [.p], .pz1 such that E = (x
+ Kx Ix
(0.20)
E .pI)
if and only if the restriction to E of the orthogonal projection on .pI is a one-to-one mapping of E onto .pl' If E admits the representation (0.20), the operator K is called the equation operator for E with respect to (.p], .pz) (or angular operator of inclination of E to .pI; see M. G. Krein [10]), and one writes K = tan (E, .pI)' Hint. Make use of Banach's theorem. 39. Let" = P + - P _, where the P ± (E [.p]) are complementary orthogonal projections, and let .p± = P ±.p. A ,,-nonnegative subspace E (c.p) is said to be maximal if it is not a proper part of any other "-nonnegative subspace. Prove (Ju. P. Ginzburg [1, 41 and R. Phillips [2]) that a) In order for a subspace E c .p to be a maximal ,,-nonnegative subspace it is necessary and sufficient that the equation operator KE = tan (E, .p +) exist for it. b) A maximal"-nonnegative subspace will be' (i) ,,-neutral precisely when II KEx I = I x I (x E E), (ii) "-positive precisely when IIKExl1 Ilxll (XE.p, 0) and (iii) uniformly "-positive precisely when IIKEII 1. Hint. The condition ("x, x) ~ 0 for x E E means that liP +xil ~ liP _xII for x EE. This implies the existence of an operator K E [.p~, .p -1 such that KP+x = P -x for x E E and II K II ~ 1; here .p~ = P +E = P +E (the closure of P +E). The maximality of E implies the equality .p~ = P +E = .p+. 40. Suppose" = P + - P _, the.p± are the same as in Exercise 39 and st = st(.p+, .p-)is the unit ball of operators in [.p+, .p-1, i.e. the set of all K E W+, .p-l with IIKII ~ 1. Suppose, further, Tis a u-dichotomic uniform ,,-contraction and, by virtue of the decomposition .p = .p+ EB .p-,
x*'
<
<
Show that the linear fractional mapping ¢T(K)
= (TzI + TzzK) (Tll + T I2 K)-1
<
<
is meaningful on the ball st and maps this ball intoa smaller ball st r = ,st (0 r = rT 1). Hint. Use the fact that if K is the equation operator for a maximal "~nonnegative subspace E, the operator ¢T(K) will be the equation operator for the uniformly "-positive subspace TE. 41. Suppose .p(Z) = .p EB .p and WE [.pcZ)l has the form
W=
(~ ~).
Let nr = nr (.p) denote the right operator halfplane, i.e. the set of all X E Wl such that Re X 0, and let n~ denote the interior of nr (i.e. the set of all X E [.pl such that Re X» 0). Suppose
>-
T= (Tll
TZI
is a u-dichotomic uniform W-contraction with coefficient q (0
W- T*WT>-ql. Prove that the linear fractional tranformation
< q < 1): (0.18)
68
I. BOUNDED OPERATORS IN BANACH SPACES
x ---+ (T21 + T22X)(Tn + T I2 X)-1 (== tPT(X» maps Dr into a bounded closed part of D~. Hint. The transformation Z == (I - X) (I + X)-I maps Dr into the ball Sf == (Z IZ E [~], IIZII~ 1} and D~ onto the open ball Sf 0 == (Z IZ E [~], IjZII 1}. If one makes the substitution X == (I - Z) (I + Z)-I in the mapping X ---+ tPT(X), one obtains the mapping Z ---+ tPT'(Z), where T' == ETE with
<
E==
1 ~
v'2
(I1 - I1) == (E*)-I == E* .
Then (0.18) goes over into W' - T'*W'T'
~
ql, where
After this it remains to make use of the assertion of the preceding Exercise 40. For 'a number of other general theorems on linear fractional transformations of operator balls into halfplanes see the article [1] of M. G. Krein and Ju. L. Smul' jan. NOTES § 3. Formula (3.7) for the solution of equation (3.1) was found by M. G. Krein in 1948 and reported on by him at seminars at Odessa and at the Mathematics Institute of the Academy of Sciences of the Ukrainian Soviet Socialist Republic in Kiev. It was later independently obtained by Ju. L. Daleckii and first published in his paper [1]. Still later an analogous formula for the equation AX + XB = C was obtained by M. Rosenblum [1]. The more general approach presented in § 3 was taken from the paper [4] of Ju. L. Daleckii. Formula (3.17) for equation (3.15) was obtained by Ju. L. Daleckii and first published in the paper [1] of I. Kovtun. § 4. The results of § 4 are due to M. G. Krein and were first presented in the "Lectures." § 5. The algebraic case of Theorem 5.1 was obtained by A. M. Ljapunov [1]. The general case was considered by M. G. Krein in the "Lectures." § 6. Theorem 6.1 was proved by B. von Sz.-Nagy [1]. Theorems 6.2 and 6.3 are natural generalizations of it and were presented in the original lectures of M. G. Krein (see the "Lectures" [9]). § 7. The main results of this section are due to M. G. Krein. These results are related to a number of earlier investigations of M. G. Krein [4, 5, 8], I. S. Iohvidov and M. G. Krein [1], M. L. Brodskii [1], H. Langer [1] and others. Later, results similar to Theorem 7.1 were restated in more precise form in the algebraic case and under other ,more special assumptions by A. Ostrowski and H. Schneider [1], and by O. Taussky [1]. Theorem 7.2 is part of an assertion proved by R. Phillips [1] and Ju. P. Ginzburg [2], but our proof is simpler. It should be noted in connection with Theorem 7.3 that Ju. P. Ginzburg [1] was the first to discover in an infinite-dimensional space the existence of .I-contractions Twhose adjoint T* is not also a .I-contraction, and to investigate their structure. § 8. Theorem 8.1 is due to R. Phillips [1]. In the finite-dimensional case the sufficiency of the condition of Theorem 8.3 was established by M. G. Krein [4], and its necessity, by I. M. Gel'fand and V. B. Lidskii [1]. The generalization of these results to the case of an infinite-dimensional space was obtained by V. I. Derguzov [1, 2] under a less complete formulation (and independently by P. Jonas [1]). The proof presented here is due to M. G. Krein. It is apparently simpler than the others even in the case of a finite-dimensional space.
CHAPTER
II
THE LINEAR EQUATION WITH A CONSTANT OPERATvl'In this chapter, which is propaedeutic in the sense that many of the facts and properties discussed will reappear in the more complicated cases considered in later chapters, we study the simplest equation, viz. the equation x = Ax + J(t) with a constant operator A and a continuous vector function/Ct). In § 1 we deduce formulas permitting one to express the solutions of the homogeneous and inhomogeneous equations in terms of the exponential operator function eAt. We also consider some linear equations in a space of operators. In § 2 we study the effect of the location of the spectrum of A on the behavior of the solutions of the homogeneous equation at infinity. The geometric meaning of the considered facts is discussed in connection with the well-known second method of Ljapunov. In § 3 we use the results of § 6 of Chapter I to obtain bOllndedness conditions for solutions on the real line of the homogeneous equation. Here we also consider a second order equation which reduces to a first order equation with an operator acting in the direct sum of two Banach spaces. The longest section, § 4, is devoted to a study of the inhomogeneous equation. Here we introduce the important notion of a Green operator function, which is used to consider a number of problems concerning existence conditions for solutions bounded on the real line and on a halfline as well as periodic and almost periodic solutions.
§ 1. Solution of the homogeneous and inhomogeneous equations 1. Vector equations. In this chapter we will consider the simplest equation!)
dx/dt=Ax+f(t)
(1.1)
in a Banach space lB with a constant operator A E [lB] and a continuous vector functionf(t). The space lB will often be called the phase space of the equation. All of the following results are valid for a much wider class of functions, which we introduce in the next chapter. In order to attach an elementary character to this chapter we confine ourselves here to the consideration of continuous functions. We first turn to the homogeneous equation dx/dt=Ax.
(1.2)
The solution of the Cauchy problem for this equation with the condition x(to)=xo
(1.3)
l)Here and below, when the interval of variation of t is not indicated, we have in mind an arbitrary finite or infinite interval on which/(t) is defined.
69
70
II.
LINEAR EQUATION WITH CONSTANT OPERATOR
is readily obtained with the use of the operator function eAt. In fact it immediately follows from (1.4.3) that the vector function x(t) = eACt-t,)xo
(1.4)
has a continuous derivative and is a solution of problem (1.2) - (1.3). One can easily verify that this solution is unique in the class of differentiable functions. It suffices to show that if a continuous function x(t) satisfying equation (1.2) vanishes at t = to, it also vanishes in a neighborhood of this point. In fact, such a function must satisfy the equation t
x(t) =
J Ax(s)ds, t,
I
I athe estimate
which implies for t - to ~
II x(t) II ~ a I A II sup I xes) II, Is-t,l~o
leading to a contradiction for
a II A I <
I if
sup II xes) I =I O. Is-t,l
After such a substitution equation (1.1) takes the form eAlt-t')y'(t) = I(t),
from which we get t
yet) = Xo
+ J e-Acs-t')/(s)ds, t,
and, finally, x(t) = eACt-t,)xo
t
+ J eACt-s)f(s)ds.
(1.5)
t,
Expression (1.5) is clearly a differentiable function. The uniqueness of solution (1.5) of the Cauchy problem (1.1)-(1.3) follows from the uniqueness of the solution of the Cauchy problem for the homogeneous equation. 2. Operator equations. The operator function U(t) = eACt - t,) C, where C E [18], satisfies the equation· dU / dt=AU
and the condition U(to) = C.
(1.6)
2.
71
BEHAVIOR OF SOLUTIONS AT INFINITY
We consider the more general operator equation dUjdt
=
AU
+
(A, B
UB
E [~])
(1.7)
in the phase space [~]. Using the arguments in § 3 of Chapter I, we rewrite equation (1.7) in the form dUjdt
=
(AI
+ Br)U.
(1.8)
Since the operators AI and Br commute, a solution of equation (1.8) satisfying the condition U(to) = C has the form U(t) = e(A,+B,) (I-I,) C = eA,(I-I')eB,(I-I,) C = eA(I-t,) CeB(I-I,).
(1.9)
In exactly the same way we can represent the solution of the equation
=
dUjdt
AU
+
UB
+ F(t),
where F(t) is a continuous function with values in U(t)
=
[~],
(1.10)
in the form
I
+ S eA(I-s)F(s)eB(t-s)ds.
eA(I-I,)U(to)eB(t-t o)
(1.11 )
10
§ 2. The behavior of the solutions of the homogeneous equation at infinity 1. Renarmings afthe space. The behavior of the solutions of problem (1.2)-(1.3) at infinity essentially depends on the location of the spectrum of A. Suppose that the spectrum a(A) lies in the interior of the left halfplane. Then it follows from (1.4) on the basis of Theorem 1.4.1 that
II x(t) I
~ Ne-v(l-t o)
II
x(to)
I
(2.1)
for any t ~ to and certain positive constants Nand )). Conversely, if estimate (2.1) holds for every solution x(t) of equation (1.2), the spectrum a(A) lies in the interior of the left halfplane. For it follows from (2.1) that I eA(t-I o) I ~ Ne-v(t-I o), and it remains to apply Theorem 1.4.1. The behavior of the solutions of equation (1.2) under the assumption that a(A) lies in the interior of the left halfplane can be characterized more precisely if one introduces a new equivalent norm in ~ with the use of the formula 00
I
X
IIA
= S I eAsx I ds.
(2.2)
o
It turns out that in this no,m the solutions of equation (1.2) tend monotonically to zero as t -4 00. 2) In fact 00
II 2llt
x(t) IIA =
S I eAsx(t)
o
00
lids =
S II eA(t+s)xo I 0
00
ds =
S I eAsxo I ds I
is assumed without loss of generality in §§ 2 and 3 below that to = O.
II.
72
LINEAR EQUATION WITH CONSTANT OPERATOR
and thus (dldt)
I x(t) IIA = - I eAtxo I < o.
We now pass to case when O"(A) = 0" +(A) U0" _(A), the spectral set 0" +(A) being nonempty. Let P + and P _ be the spectral projections corresponding to this decomposition of the spectrum and let)8 = )8+ + )8- be the corresponding direct decomposition of)8 into invariant subspaces of A. Since )8+ and)8- are also invariant under the operators eAI (0 ~ t < (0), a solution x(t) = eAI Xo of equation (1.2) remains in the subspace containing the initial vector Xo. We introduce in )8 the indefinite norm 00
I X IIA = S {II eAI P -x I - I e- At P +x I }dt.
(2.3)
o
From calculations analogous to those carried out above, we get 00
I x(t) IIA = S {II eAseAI P -Xo I - I e-Ase At P +Xo I }ds o
00
00
S I eAs P -xo I ds - S I e- As P +Xo I ds. I
-I
Therefore (dl dt) I x(t)
IIA = - {II eAI P -Xo I + I eAt P +Xo II} <0
(2.4)
and hence the indefinite norm of any solution of equation (1.2) decreases. We consider two special cases. a) Suppose P +Xo = 0, i.e. Xo E )8_ (= P _ )8). In this subspace I x IIA ;?; 0 and the norms I x I and I x IIA are equivalent. It follows from (2.4) that Ilx(t) IIA tends monotonically to zero. Thus the solutions x(t) = eAlxO of equation (1.2) with initial vector Xo in )8tend to zero. b) Suppose P -Xo = 0, i.e. Xo E )8+ (= P + )8). In this subspace the quantity - I x IIA ;?; 0 is an ordinary norm that is equivalent to I x II. It follows from (2.4) that
- ; I x(t) IIA = I x(t) I
;?; -
-~ I x(t) IIA.
Integrating this inequality, we see that
- I x(t) IIA ;?;
-
I Xo IIA e(l/M) (1-1
0) ,
i.e. the solutions with initial values in )8+ increase unboundedly as t ...... We note that the decomposition
+
00.
2.
73
BEHAVIOR OF SOLUTIONS AT INFINITY
implies that any solution for which P +Xo "# 0 increases unboundedly. In particular, if the whole spectrum lies in the interior of the right halfplane : (j(A) = (j +(A), a nonzero solution of equation (1.2) goes to infinity as t -+ + 00, the norm -II x(t) IIA monotonically increasing. Moreover, this case readily reduces to the original one if we replace A by - A and reverse the time. We recall that an operator A with a spectrum decomposing into two non empty spectral sets lying in the interiors of the right and left halfplanes respectively: (j (A) = (j +(A) U (j _(A), was said to be e-dichotomic. In this case we will also say that the differential equation (1.2) is e-dichotomic. The above arguments clearly imply that the phase space ~ of an e-dichotomic equation decomposes into a direct sum ~ = ~+ + ~_, the solutions initially in ~_ remaining in ~_ and exponentially decreasing and the solutions initially in ~+ remaining in ~+ and exponentially increasing. This accounts for the term edichotomy. We will see later (in Chapter IV) that an analogous situation is encountered in the more general case. 2. Geometric meaning of the renormings. If the phase space ~ is a Hilbert space ~), it is natural to develop all of the arguments with the norms I x IIA replaced by the Hjlbert norms I x IIA.2 (see § 1.4.5). In this case the renorming carried out above has a simple geometric meaning. Suppose first A «0. Then a solution of the equation dx I dt = Ax
(1.2)
satisfies the relation (x'(t), x(t» = (Ax(t), x(t» ;?; - c I x(t)
11
2 ,_
which implies that the angle a between the directions of the radius vector x(t) and the vector x' (t) tangent to an integral curve of equation (1.2) satisfies the estimate cos a=
< _
(x'(t), x(t» x'(t) IIII x(t)
c I x(t) 112 A IIII x(t) 112
I I = I + arc sin (cl I A II) and the vector field of tangents to the
Thus a ~ nl2 integral curves of equation (1.2) is at each point essentially directed toward the interior of the sphere with center at the origin passing through this point. We now consider the more general situation when the spectrum (j(A) lies in the interior of the left halfplane. In this case, by Theorem 1.5.1 there exists a bounded uniformly positive operator W such that Re(WA) «0. The arguments presented earlier remain valid if the estimates are carried out in the new metric (x, y)w = (Wx, y), which is equivalent to the old one. In fact the boundedness and uniform positiveness of W ensure the topological equivalence of the norms I . I and II . I w· On the other hand, if Re (WA) <; - cI (c > 0), we will have Re(x' (t), x(t»w = ([A* W
+
WA] x, x) ;?; - c(x, x) ;?; -
Cl
(x, x)w
74
II.
LINEAR EQUATION WITH CONSTANT OPERATOR
for a solution x(t) of equation (1.2). We recall that the required operator W can be obtained as a solution of the equation A * W + W A = - H, where H is an arbitrary uniformly positive operator. Putting, for example, H = I, we get (see the proof of Theorem 1.5.1) 00
W =
J eA*teAt dt, o
i.e. 00
(x, x)w
= (Wx, x) =
J I eAtx 112 dt.
(2.5)
o
In this case the system of spheres into which the integral curves enter is replaced by the system of ellipsoids (Wx, x) = const with center at the origin. The consideration in a Banach phase space of the norm (2.2), which is analogous to (2.5), generalizes the above mentioned geometric arguments connected with the so-called second method of Ljapunov. Here the role of the ellipsoids is played by the family of centrally symmetric bodies bounded by the surfaces II x IIA = const. A somewhat more complicated but quite clear geometric picture is obtained when the spectrum of A also has a component in the interior of the right halfplane, i.e. when equation (1.2) is e-dichotomic. We first consider the case when the phase space )8 is a Hilbert space Sj. The following simple test is a rephrasing of Theorem 1.7.1. THEOREM 2.1. In order for equation (1. 2) to be e-dichotomic it is necessary and sufficient that the operator A be uniformly W-dissipative with respect to an invertible indefinite Hermitian operator WE [Sj]:
Re WA «0. Under any choice of an operator W satisfying this relation the invariant subspace Sj+ (Sj-) of A corresponding to the part of the spectrum 11+(A) (11-(A) lying in the interior of the right (left) halfplane is uniformly W-negative (W-positive).
The indefinite form (Wx, x) defines two systems of hyperboloids in Sj by the equation (Wx, :x) = c: plus hyperboloids when c > 0 and minus hyperboloids when c < o. It is not difficult to verify that the uniform W-dissipativeness of A implies the relation
d(wx~l; x(t) = 2Re(WAx(t), x(t) < 0
(x(t) "# 0),
which shows that the indefinite form (Wx, x) decreases, for any solution x(t) of equation (1. 2).
3. BOUNDED SOLUTIONS: HOMOGENEOUS CASE
75
If the point Xo is on a minus hyperboloid, the subsequent diminution of (Wx, x) with increasing t means that the trajectory x(t) intersects all of the hyperboloids with larger-in-absolute-value negative values of c and hence goes to infinity. When Xo is on a plus hyperboloid in .f>-, the diminution of (Wx, x), which on .f>- is equivalent to an ordinary norm, leads to an arbitrarily close approach of the trajectory to the center of the hyperboloid. But if Xo is on a plus hyperboloid outside .f>-, the trajectory, after intersecting the hyperboloids (Wx, x) = c with diminishing values of c, enters the cone(Wx, x) = 0 and goes over to a system of minus hyperboloids, receding then to infinity. This follows from the fact that any solution for which P +Xo i= 0 increases unboundedly in norm. We now note that by choosing for the e-dichotomic operator A a Hermitian operator W from the relation 2Re(WA) = WA
+ A*W = -
(P'i,HP+
+ P'!.HP_),
where the P ± are the spectral projections of A corresponding to the invariant subspaces .f>±, we arrive, as follows from the formulas in § 1.4.4, at the form co
(Wx, x) =
J {II HeAtp_x liz - I HrAtP+x IIZ}dt.
o
This form is analogous (when H = J) to the indefinite norm (2.3) in a Banach space 18. It is easily seen that analogous geometric arguments can be developed in a Banach space by considering in place of the hyperboloids the surfaces I x IIA = c. § 3. Boundedness of the solutions of the homogeneous equation 1. First order equation. We wish to determine conditions under which the solutions of the homogeneous equation dx / dt = Ax are bounded on the real line. Since the solutions of this equation are described by the formula x(t) = eAtxo (xo = x(O)), the condition that the solutions be bounded implies the estimate
I eAt Xo I
~ cx,
(- 00 < t < 00),
where the constant Cx , depends only on Xo. Thus the set 01 operators eAt (- 00 < t < 00) is bounded at each element Xo E 18. By the uniform boundedness principle (Theorem 1.1.2) these operators are uniformly bounded:
I eAt I
~
c
(- 00 < t < 00).
(3.1)
The latter estimate, as was shown in Chapter I (Corollary 1.4.1), implies that the spectrum of A lies on the imaginary axis. A more exact result can be obtained if the phase space 18 is a Hilbert space .f>.
76
II.
LINEAR EQUATION WITH CONSTANT OPERATOR
In fact, by virtue of Theorem 1.6.3 condition (3.1) is satisfied if and only if the operator A is similar to a Hermitian operator multiplied by i (a skew-Hermitian operator): A = S-I(iB)S (B* = B). Thus the following result holds. THEOREM 3.1. If each solution of the equation dx / dt = Ax is bounded on the real line, the spectrum a(A) lies on the imaginary axis. If the phase space}B is a Hilbert space, all of the solutions are bounded if and only if the operator A is similar to a skew-Hermitian operator.
2. Second order equation in a Banach space. We consider the second order dif-
ferential equation dZy dt Z
+
_ Ty - 0,
(3.2)
where T E [}B]. Its investigation can be reduced to the investigation of a first order equation in the doubled phase space }B(Z) = }B + }B whose elements are the pairs x = (Xl, Xz) (Xl, Xz E}B) and whose norm is given by the formula X II~ = Xl + Setting y = Xl and dy / dt = Xz, we replace equation (3. 2) by the system
I
dXI - X . dtz,
I liz
I xzllz.
dxz = _ TXl dt '
or by the equivalent equation for a vector X = (XI. xz) in }B(Z) dx/dt
where the operator d
E
= dx,
(3.3)
[}B(Z)] is defined by the operator matrix
It is not difficult to calculate that dZk
The
operato~
= (_ l)k (~k ~k); dZk+1 = (_ l)k( _ ~k+1 ~k).
(3.4)
function edt defining the solutions of equation (3.3) takes the form
co dn co dZk edt - 1: t n - - - 1: t Zk - - n=O n! - k=O (2k)!
dZk+1 tZk+l--k=O (2k + I)!' co
+ 1:
(3.5)
Making use of the correspondence between scalar and operator functions, we put co
cos Tl/Zt
= ~/ -
TktZk l)k (2k)! ;
co
TktZk+1
T-l/Zsin TI/Zt = k'fo(- l)k (2k
In this notation it follows from (3. 4) and (3. 5) that
+
I)!'
3.
77
BOUNDED SOLUTIONS: HOMOGENEOUS CASE
e$t = (
COS P/2t T-l/Z sin P/2t) - T-lIz sin Tl/Z t cos Tl/Z t .
The formula x(t) = e$t Xo, where Xo = (Yo, Yo), now leads us to a representation of the solution of equation (3. 2) satisfying the conditions yeo) = Yo,
(3.6)
y'(O) = Yo,
in the form yet) = (cos P/2t) Yo
+ (T-l/2sin Tl/2t) Yo.
(3.7)
It can be verified by direct substitution that the vector function (3. 7) satisfies equation (3. 2) and the conditions (3. 6). From formula (3. 7) it follows that the boundedness for t E ( - 00, (0) of each solution of equation (3. 2) is equivalent to the boundedness of the operator functions cos Tl/2 t and T-l/Z sin P/z t. We now show that it suffices to require the boundedness of the operator function T-l/2 sin Tl/2t (- 00 < t < (0). We consider the vector function yet) = (T-l/2 sin T1Izt)yo.
Its derivatives are the vector functions y'(t)
= (cos
p/2t)yo
and
y"(t)
= -
(T1I2 sin TlIzt)yo
= -
Ty(t).
By hypothesis, for each fixed Yo E ~ the vector function yet), and hence also = - Ty(t), is bounded. If we prove that y'(t) is also bounded, we can conclude that the set of operators cos p/2 t ( - 00 < t < (0) is bounded at each element Yo E ~, and hence is bounded in norm by virtue of the uniform boundedness principle (Theorem 1.1.2). Thus it remains to show that the boundedness of yet) and y"(t) implies the boundedness of y'(t). We put y"(t)
y" (t) - yet) = J(t).
(3.8)
The function J(t) is bounded on the real line'together with yet) and y"(t). If we consider (3. 8) as a differential equation, we can express the solution of it that is bounded on the real line by means of the easily verified (and well known) formula 3) yet) =
1
2
J e-II-sIJ(s)ds. 00
-00
By differentiating this expression with respect to t, it is easily shown that 3)This formula is a consequence of a general formula involving the Green function of a stationary linear equation (see § 4).
78
II. LINEAR EQUATION WITH CONSTANT OPERATOR sup II y'(t) II ~ SUp Ilf(t) II < I
00.
I
Thus we get that the boundedness of each solution of the second order equation (3.2) on the real line is equivalent to the boundedness of the operator function T-l/2 sin Tl/2 t. We have simultaneously proved the following assertion: in order for all of the solutions of equation (3.2) to be bounded it is necessary and sufficient that the solutions satisfying the condition yeO) = 0 be bounded. 3. Second order equation in Hilbert space. A more precise investigation can be carried out when the space )8 is a Hilbert space .)). THEOREM 3.2. In order for each solution of the equation
-ddt2y2 +
Ty
=
(3.2)
(-oo
0
in a Hilbert space.)) to be bounded on the real line it is necessary and sufficient that the operator T be similar to a uniformly positive operator.
PROOF. For a proof of the sufficiency of the condition we note that if T = S-l TIS, where Tl is uniformly positive, the substitution y = Sx reduces equation (3.2) to the form d 2x /dt 2
+
Tlx =
O.
Thus we can assume from the very beginning without loss of generality that
T» O.
We consider equation (3.3), which is equivalent to (3.2), in the doubled Hilbert space .p(2) = .p EEl .)). A direct calculation shows that d
= i !2PJJ!2-l,
where 0 l' = ( _ TO)'
d
/J7J
= (
OTI/2
;:/iJ
_
0 ) TI/2
is a Hermitian operator in .p(2), and ~
!2 -
(I _
I)
iT1I2 iTl/2
and
!2- l =
1(1I _ i
2
iT-l/2) T-l/2
are bounded operators in .))(2). Thus the operator d is similar to a skew-Hermitian operator and therefore, by virtue of Theorem 3.1, each solution of equation (3.3) is bounded on the real line. This implies the boundedness of each solution together with its derivative of equation (3.2). We now prove the necessity of the condition of the theorem.
4.
BOUNDED SOLUTIONS: INHOMOGENEOUS CASE
I':J
We first note that if a solution of equation (3.2) is bounded, its derivative is also bounded. In fact, it follows from the equation that sup
-00<1<00
I yO(t) II ;£ I T I
sup
-00<1<00
I yet) II,
while it has already been shown above that the boundedness of a function and its second derivative on the real line implies the boundedness of its first derivative. Thus the boundedness of the solutions of equation (3.2) implies the boundedness of the solutions of equation (3.3). By virtue of Theorem 3.1 this means that the operator d is similar to a skew-Hermitian operator. The latter condition can be written in the form Yf d + d* Yf = 0 (see the proof of Theorem 1.6.3), where Yf is a uniformly positive operator in .p(2). Substituting the matrices d = (
I)
0 - T 0
in this equality and multiplying them out, we obtain from a consideration of the element in the second row and first column the equality - H22T + Hn = O. We now note that the uniform positiveness of H in .p(2) implies the uniform positiveness of Hn and H22 in.p. For when x = (Xl> 0) we have (Yfx, X)2 = (Hnxl> Xl)
~
m (Yf) (X, x)z = m(Yf) (Xl, Xl).
The assertion concerning H22 is proved analogously. This permits us to write l2 -l H n= HT = H 22 22 / DHI/2 22'
where D = H;1/2 HnH;1/2 is a uniformly positive operator. The theorem is proved. COROLLARY. In a Hilbert space .p the boundedness of the operator function T-l/2 sin Tl/2 t holds precisely when the operator T is similar to a uniformly positive operator.
§ 4. Conditions for the existence of a bounded solution of the inhomogeneous equation
1. Greenfunctio~s. We consider the inhomogeneous equation
dx/dt = Ax
+ f(t)
(4.1)
with a continuous functionf(t). We assume that the spectrum of the operator A decomposes into two spectral sets: O'(A) = O'l(A) U 0'2(A). Let 181 and 18 2 denote the invariant subspaces of A corresponding to these sets and let PI and P 2 denote the corresponding spectral projections.
80
II.
LINEAR EQUATION WITH CONSTANT OPERATOR
We recall (see 1.2.4) that
Pk
= -
~ § RAdJ... 27Cl
(k = 1,2).
r.
We introduce the Green operator function
eAt PI = - _1_. feAt RAdJ...; G(t) = (
t> 0;
2m r, -eAt p 2 =_1_. f eAtRAdJ...; 27Cl
(4.2) t
r,
< O.
It has the following properties. 1) When t#-O it is continuously differentiable and satisfies the homogeneous equation dG(t)/dt = AG(t). This fact follows directly from (4.2). 2) Its jump at the origin is equal to the identity operator. In fact,
G( + 0) = Ph G( - 0) = - P 2 ; G( + 0) - G( - 0) = PI
+ P2 =
I.
3) The vector function b
x(t) =
J G(t -
s)f(s)ds,
(4.3)
a
wheref(t) is continuous, satisfies the inhomogeneous equation (4.1) when a ~ t For a proof we differentiate the equality
J G(t -
h.
t
b
x(t) =
~
s)f(s)ds
+ J G(t
t
- s)f(s)ds.
a
We get dx dt- - - G( - O)f(t)
+
J dG(~t- s) f(s)ds + G( + O)f(t) + J dG(~; s) f(s)ds t
b
of AG(I
a
+ s)f(s)ds + [G( + 0) -
G( - O)]f(t)
= Ax + f(t).
a
As a rule, in what follows we will consider the case when the spectrum a(A) does not intersect the imaginary axis (in particular, when A is e-dichotomic): a(A) = a+(A) U a-(A). The Green function defined by the formula GA(t) = {
eAtp_ for - eAt P + for
> 0, t < 0,
t
(4.4)
4. BOUNDED SOLUTIONS: INHOMOGENEOUS CASE
81
is called the principal Green function for equation (4.1). We do not assume that both of the sets a ±(A) are nonempty: in formula (4.4) we take P + = I, P _ = 0 when a(A) = a +(A), and P + = 0, P _ = I when a(A) = a-(A). Since the spectrum a(A) does not intersect the imaginary axis, there exist numbers v > 0 and N > 0 for which (see (1.4.7) )
I
GA(t)
II
~ Ne- viti •
(4.5)
2. Solutions bounded on the real line. The principal Green function (4.4) plays an important role in the determination of conditions for the existence of solutions of equation (4.1) that are bounded on the real line. . THEOREM 4.1. In order for there to correspond to any bounded-on-the-real-line continuous vector function f(t) one and only one bounded-on-the-real-line solution of equation (4.1) it is necessary and sufficient that the spectrum a(A) not intersect the imaginary axis. This solution is given by the formula
J
x(t) =
GA(t - s)f(s)ds,
(4.6)
where GA(t) is the principal Green function for equation (4.1). PROOF. Suppose there exists a unique bounded solution for any bounded continuousf(t). We put f(t) = y, where y is a constant vector, and let x(t) be the unique bounded solution of the equation
dxfdt = Ax
+ y.
The vector x(t + -.) is also a solution of this equation for any -., and by virtue of the uniqueness x(t + -.) = x(t), i.e. x(t) = x = const, which implies Ax = - y. From the arbitrariness of y it follows that the continuous linear operator A maps ~ onto itself. By Banach's theorem (1.1.1) such an operator has a continuous inverse A-I, i.e. the point)' = 0 is a regular point of A. Suppose now pi is an arbitrary imaginary number. We consider the equation
dx f dt = Ax The substitution x =
~epit
+ yeP".
gives d~fdt =
(A -
piI)~
+ y.
Repeating the above arguments, we get that the operator A - pi! has a continuous inverse, i.e. pi is a regular point. Thus the necessity of the condition of the theorem is proved. For a proof of the sufficiency we make use of estimate (4.5). It follows from this estimate that the
II. LINEAR EQUATION WITH CONSTANT OPERATOR
82
vector function (4. 6) is bounded:
I x (t) I
~ N
f
I ds ~ v2N
e- V ll-sl Ilf(s)
sup <00
-oo
Ilf(s) II·
The fact that this function satisfies equation (4.1) for t E ( - 00, (0) was established earlier (property 3) of subsection 1). It remains for us to prove the uniqueness of a bounded solution. To this end it suffices to verify that the homogeneous equation dx / dt = Ax does not have any nontrivial solutions that are bounded on the real line. Let us assume that such a solution x(t) = eAI Xo exists. If we let A_ = P _A and A+ = P +A, we can write it in the form x(t)
=
ekl P -Xo
+ eA IP+xo. +
Since the spectrum of the operator A_ in ~L is the set (j_(A) lying in the interior of the left halfplane, the first term, and hence also the second term, is bounded for t
> 0: (t > 0).
But then, taking into account the fact that the spectrum (j +(A) of the operator A+ in 58+ (ifit is not empty) lies in the interior of the right halfplane, we get (t > 0). This inequality for t =
--+ 00
shows that P +Xo =
o.
o. It is shown analogously that P -Xo
The theorem is proved. 3. Solutions bounded on a halfline. Under the assumption that the spectrum of A does not intersect the imaginary axis: (j(A) = (j +(A) U (j _(A), it is possible to completely describe the solutions of equation (4.1) that are bounded on a halftine [to, (0). THEOREM 4.2. Suppose the spectrum (j(A) does not intersect the imaginary axis andf(t) is a continuous function that is bounded on a halfline [to, (0). To each element Xo E 58- there corresponds a unique solution x(t) of equation (4.1) that is bounded on [to, (0) and satisfies the condition P _x(to) = xo. This solution is given by the formula 00
x(t) = eA(t-Io) Xo
+ f
GA(t - s)f(s) ds.
(4.7)
10
PROOF. Equation (4.1) is equivalent to the system of two independent equations { dx+/dt :: A+x+ dx_ / dt - A-x-
+ f+(t),
+ f- (t),
(4.8) (4.9)
where x± = P±x,f± = P±J,A± = P±A. The first of these equations is considered
4.
BOUNDED SOLUTIONS: INHOMOGENEOUS CASE
83
in the subspace ~+ = P +~, and the second, in the subspace ~_ = P _~. Here the spectrum of A+ is the set (J +(A) lying in the interior of the right halfplane while the spectrum (J _(A) of A_ lies in the interior of the left halfplane. Equation (4.9) has a solution that is bounded for t ~ to for any x-(to) = P -Xo. This solution is given by the formula I
x-(t)
eA-(I-I,)P_xo
=
+ S eA-(t-s) f_(s)ds
(4.10)
I,
I
eA(I-I,) p -Xo
=
+ S eA(t-s)
P -f(s) ds.
I,
Equation (4.8) can have no more than one solution that is bounded for t ~ to, inasmuch as the corresponding homogeneous equation cannot have such solutions. It is easily verified that this unique solution is the function 00
x+(t)
00
S eA+(t-s) f+(s)ds
= -
= -
I
S eA(I-s) P+f(s)ds.
(4.11 )
I
Comparing (4. 10) with (4. II), we obtain the required formula (4. 7). The uniqueness of the solution follows from the above arguments. REMARK 4.1. Each bounded solution on [to, 00) of equation (4.1) is uniquely determined by the element xi) = P -x(to). Thus there exists an initial manifold B4 of "the same dimension" as 5lL having the property that those (and only those) trajectories intersecting it at time to remain bounded for t ~ 00. We note that all of these trajectories converge together as t ~ 00. In fact it follows from (4.7) that for any two such trajectories
II xz(t) - Xl (t) II ;;:; II eA(I-I,) P- [xz(to) - Xl (to)] II ;;:; Ne-v(t-I,) II xz(to) - XI(tO) II· lt can be said in particular that all of these trajectories approach the unique trajectory that is bounded on the real line (if the vector function f(t) is bounded on the real line). We note further that if the spectral set (J-(A) is empty: (J(A) = (J +(A),there exists only one trajectory that is bounded on a halftine and for it xi) = O. An analogous situation holds for the trajectories that are bounded for t ~ - 00. They are described by the formula I,
x(t)
=
eA(I-I,) xd
+ S GA(t
- s)f(s) ds
(4.12)
for xri- E ~+. The corresponding initial manifold B;:- has "the same dimension" as ~+. It is obvious that the intersection of the manifolds and B4 consists of the single point S'~oo GA(to - s)f(s)ds.
Bt
84
II.
LINEAR EQUATION WITH CONSTANT OPERATOR
REMARK 4.2. In a finite-dimensional space one can easily deduce from the existence of a bounded solution on a halfline of equation (4.1) for each boundedf(t) that the spectrum a(A) does not have any points on the imaginary ~xis. It is not known whether or not the analogous assertion in the infinite case is true. Under certain additional assumptions on the operator A this assertion follows from the more general results of Chapter IV. 4. Periodic solutions. We consider the equation
dx/dt = Ax
+ f(t)
( - 00
< t < (0),
(4.1)
assuming that the functionf(t) is continuous and periodic: f(t
+
T) = f(t).
(4.13)
We first adduce some formal heuristic arguments permitting one to determine conditions for the existence of periodic solutions of equation (4. 1) and to find these solutions. We consider the expansion of f(t) in a Fourier series 00
f(t) '"
1::
fke2k7rit/T.
(4.14)
k=-oo
We will seek a periodic solution of equation (4.1) also in the form of a sum of a Fourier series: x(t)
=
00
1::
xke2k7rit/T.
(4.15)
k=-oo
Formally4l substituting expansions (4. 15) and (4. 14) in equation (4. 1) and comparing the coefficients of e2k7rit/T, we obtain the system of equations
(A- ~';;i I)
Xk
= - A
(4.16)
We first assume that the following condition is satisfied: 2kJr:i / T E p(A)
(k
= 0, ± 1, ± 2,,,,),
(4.17)
i.e. that the spectrum of A does not contain the points 2k7ii/T, where k = 0, ± 2"", of the imaginary axis. Then (4. 16) implies Xk
2k7ii I = - ( A - --r-
)-1 J",
± 1,
(4.18)
so that (4.19) 4) Formula (4.16) can also be obtained formally, by multiplying (4.1) by exp (2n:ki/T) and integrating from 0 to T.
4.
85
BOUNDED SOLUTIONS: INHOMOGENEOUS CASE
Substituting in the right side the expressions fk
1T
= rSf(s) e- his/ T ds o
for the Fourier coefficients, we obtain the relation X(t)
=
T I:
1 -S
-
To
00
(
2kJr:i
A - -T- I
)-1 e2k"i(t-s)/Tf(s)ds
(4.20)
k=-oo
T
S rT(t - s)f(s) ds. o
We consider in more detail the integrand function rT(t)
= - -1 T
I: 00
(
k=-oo
2kJr:i I A - ~~ T
)-1 e2kJrit/T.
(4.21)
Subtracting the T-periodic function X(t)I
=
_1_._ 27rl
I: ~ e2hit/T I k1=O
( X(t)
k
=
~
-
~
for
0< t<
T)
from both sides of this equality, we obtain the regularized representation
[(A - 2kJr:i 1)-1 + 2k7rl IT. ] e2k"it/T - ~ A-I (4.22) T T ~ A-I + AI: _1_. e2k"a/T( 2k!Ci I _ A)-l. T 2k7rl T
rT(t) = X(t) I - ~ - I: T
=
X(t) I -
k1=O
k1=O
We now note that for k sufficiently large in absolute value
Therefore the series in formula (4.22) converges abolutely and uniformly, and hence its sum is a continuous periodic function. Thus the operator function rT(t) ( - 00 < t < (0) obtains a precise meaning if it is defined by equality (4. 22). This operator function will be called the T-periodic Green function for equation (4.1). It is characterized by the following properties. 1) It is a periodic function: rT(t + T) = rT(t). 2) It is continuous ~in the operator norm for all t with the exception of the points t = Tk (k = 0, ± 1, ± 2,···), where r T(+ 0) - rT(- 0) = I. 3) At points of continuity it is differentiable in norm and satisfies the differential equation r T(t)
=
ArT(t).
(4.23)
The latter equality is readily obtained by differentiating (4.22) (after first removing another singularity in order to improve the convergence).
II. LINEAR EQUATION WITH CONSTANT OPERATOR
lS6
Properties 1) - 3) uniquely determine the function PT(t). Since a continuous solution of equation (4.23) has the form eAtC, where C is a constant operator, we easily find that (0 < t < T).
The operator (/ - eAT)-l exists, since by the spectral mapping theorem (see 1.2.5) the spectrum a(A) would otherwise contain at least one point of the form 2kni/ T. It remains to extend the obtained expression periodically onto all of the real line. Using the periodic Green function FT(t), we at once obtain the following result. THEOREM 4.3. If the spectrum a(A) does not contain the points 2kni/T (k = 0, 1, ± 2,.··) oj the imaginary axis, equation (4.1) Jor any continuous T-periodic Junction J(t) has one and only one T-periodic solution x(t). This solution is given by the Jormula
±
T
x(t) =
J FT(t -
(4.24)
s)J(s) ds.
o
PROOF. The integral (4.24) exists inasmuch as the integrand is continuous. The periodicity of FT(t) implies that x(t) is also periodic. Rewriting (4.24) in the form T
t
x(t)
= J FT(t o
- s)J(s)ds
+ J FT(t
- s)J(s)ds
t
and differentiating with respect to t, we get T
t
x'(t) =
J AFT(t o
s)J(s)ds
+ J AFT(t
- s)J(s)ds
t
+
[FT( + 0) - F T( - O)]J(t) = Ax(t)
+ J(t).
The uniqueness of the solution follows from the fact that under the conditions of the theorem the homogeneous equation cannot have nontrivial solutions that are continuous and T-periodic. For suppose that such a solution x(t) = eAtxo exists. We would then have the relation eTAxo~= xo, which contradicts the existence of the operator (/ - eAT)-l. The theorem is proved. We now drop condition (4.17) and consider the general case. Let KA;T be the set of those indices k for which 2k7r:i/T EO u(A). Since the spectrum u(A) is a bounded set, the set KA;T is finite. The following assertion holds: in order for equation (4.1) to have a T-periodic solution x(t) it is necessary and sufficient that each of the equations
-r
(A - 2kn:i) I
Xk
= -
I" Jk
(4.25)
4.
BOUNDED SOLUTIONS: INHOMOGENEOUS CASE
have at least one solution Xk
87
EO ~.
For, if equation (4.1) has a T-periodic solution x(t), its Fourier coefficients Xk (k = 0, ± 1, ±2,.··) satisfy conditions (4.16). This proves the necessity of the assertion. Suppose now x~ (k EO KA;T) is a solution of equation (4.25). We put x(t) = y(t) +
I:
x~e(2k
kEKA;T
Substituting this expression in (4.1), we obtain for y(t) the equation y' (t) = Ay(t) + rf>(t),
(4.26)
where rf>(t)=f(t)- I:kEKA'TJi,e-2biIT is a function whose Fourier coefficients with index in KA;T are equal to zero: ' (4.27) It remains to note that a solution of equation (4.26) under condition (4.27) can be obtained by making use of formula (4.24), since the terms in (4.21) corresponding to the indices k EO KA-T drop out thanks to (4.27). ' If we introduce the "incomplete" Green function FT(t)= -
--L I:
T kEKA;T
(A _ }k1ri I)-le(2kniIT)t
(4.28)
T
we can write an arbitrary T-periodic solution of equation (4.1) in the form T
x(t)= f fT(t-s)f(s)ds+ o
where the
x~ (k EO
I:
kEKA;T
x2e(2bi!T)t,
(4.29)
KA;T) are solutions of equations (4.16).
5. Almost periodic solutions. In this subsection we consider the differential equation dx/dt
=
Ax
+ J(t)
( - 00
< t < (0),
(4.1)
under the assumption that the function J(t) is continuous and almost periodic. We recall that a continuous functionJ(t) on the real line with values in a Banach space ~ is said to be almost periodic (according to Bohr) if for each c > 0 there exists an Le > 0 such that each interval of the real line of length not less than Le contains a point r = -c(c) (an c-translation number) for which
Ilf(t) - J(t + -c) I < c
( - 00
< t<
00).
According to a well-known theorem of S. Bochner a necessary and sufficient condition for the almost periodicity of a continuous function is the precompactness of its family of translates,h(t) =J(t +-c) (- 00 <-c< 00) in the topology of uniform (on the real line) convergence in norm. It follows in particular from this proposition that the set of all values of an almost periodic function is precompact and hence bounded: sUp-oo
I
<
00.
88
II. LINEAR EQUATION WITH CONSTANT OPERATOR
Suppose that the spectrum li(A) does not intersect the imaginary axis: li(A) = li +(A) U li _(A). Then, as follows from Theorem 4.1, equation (4.1) has the bounded solution 00
S GA(t - s)f(s)ds,
x(t) =
(4.30)
where GA(t) is the principal Green function for equation (4.1). THEOREM 4.4. If the spectrum li(A) does not intersect the imaginary axis, the differential equation (4.1) with an almost periodic function f(t) has one and only one almost periodic solution. This solution is given by formula (4.30). PROOF. Since equation (4.1) has only one bounded solution, we need only prove that the function (4.30) is also almost periodic. To this end it suffices to establish that the set of functions xT(t) = x(t + 'r) (- 00 < 'r < 00) is precompact in the topology of uniform convergence. We consider a sequence xT.(t). By virtue of the almost periodicity off(t) we can select from the sequence of functions f-T.(t) = f(t - 'rk) a Cauchy subsequence Irk! (t). In this connection 00
XTkj (t)
=
x(t +'rk,)
= S GA(t
- s)f(s - 'rk,)ds
00
=
S GA(t
- s)f-Tk,(s)ds.
This relation implies that XTk,(t) is a Cauchy sequence. In fact, 00
II XTk,(t) - XTk,(t) I ~ S I GA(t - s) 1IIIf-Tk,(s) - f-Tk,(S) I ds 00
~ sup Ilf-Tk,(s) - f-Tk,(S) I
S II GA(s) I ds.
s
We have thus proved the precompactness of the set of functions xT(t)
=
x(t +'r), and hence the almost periodicity of x(t).
REMARK 4.~". For the readers familiar with the theory of almost periodic functions we note that the arguments presented in the proof of Theorem 4.4 imply the coincidence of the spectra of x(t) and f(t) as well as a simple connection between the Fourier coefficients of these functions (see Exercise 11). REMARK 4.4. If f(t) is a periodic function (J(t + T) = f(t)), formula (4.30) clearly gives a unique periodic solution x(t) (== x (t + T)). One can easily obtain an expression for the periodic Green function rT(t) in terms of the principal Green function GA(t). In fact, if f(t) is a T-periodic function, (4.30) implies
89
EXERCISES (k+l)T
X(l)
=
J
1:;
GA(t - s)f(s)ds
k=-oo kT T
J GA(t -
1:;
s - kT)f(s)ds
k=-oo 0
I [k=~OO
=
GA(t - s - kT)]f(s) ds.
From a comparison of the latter integral with formula (4.24) we get rT(t)
=
1:; GA(1
k=-oo
+ kT).
(4.31)
The latter series converges uniformly and "rapidly" since its terms decrease exponentially (see (4.5)). Equality (4.31) can also be obtained directly by making use of the well-known Poisson summation formula (see Courant and Hilbert [1]). Despite Remark 4.4, Theorem 4.4 does not contain Theorem 4.3 since its conditions involve the rigid requirement that there be no points of the spectrum of the operator A on the imaginary axis. EXERCISES 1. Let A, B EO [~]. Derive the formula eCA+B)t
=
eAt
+
0::1
~
n= 1
t
SI
S"~l
f f··· f 0 0
eA
0
Hint. Consider the differential equation dcf; / dt = A cf; + f, where f = B cf;, and make use of formula (1. 5). 2. Give an example of an equation dx / dt = Ax all of whose solutions tend to zero as t -> + 00 whereas II eAt II does not tend to zero. Hint. Consider a nonpositive Hermitian operator in Hilbert space.p having A = 0 as a point of its continuous spectrum. 3. Show that the situation described in Exercise 2 is impossible in the finite case. 4. Let {Akl (k = 1, ... , n) be a commutative collection of operators from [~], the spectrum of each of them lying in the interior of the left halfplane. Show that a new norm can be introduced in ~ in such a way that all of the solutions of each of the equations dx / dt = Akx will monotonically tend to zero (S. G. Krein and S. D. EideI'man; see S. G. Krein [1], page 167). Hint. Put
Illxlll =
fo
00
f II exp o
(~Z~ISkAk)
x II dS 1 ••• dSn
(see Exercise 1.25). 5. Suppose W, H EO [.p], H» 0 and W = W* is an invertible Hermitian operator (W-l EO [.p]). Show that a) Independently of whether W is definite or indefinite, all of the solutions of the equation dx/dt = iWHx are bounded on the real line. b) If W is definite, the Ljapunov exponents of all of the solutions of the equation
(0.1)
II.
90
LINEAR EQUATION WITH CONSTANT OPERATOR
(0.2)
dxldt= WHx
lie in the interval (a, b), where a = min (A I A EO O'(WH)) and b = max fA I A EO O'(WH)). Here 0 if W» 0 while b 0 if W O. c) If Wis indefinite, equation (0.2) is e-dichotomic and.f\ decomposes into a W-orthogonal sum of subspaces .)3 = ,)3+ .f\ _ such that every solution of equation (0,2) initially in .f\ + (.f\ _) remains in that subspace and its Ljapunov exponent lies on a certain segment of the positive (negative) halfline. Hint. Make use of the result of Exercise 1.26b). 6. Suppose A, B, C EO [~l and the inverse A-I EO [~l exists. Show that a) For any Yo, Yo' EO ~ the equation
<
a>
«
+
Ad 2Yldt 2 + Bdyldt
+ Cy =
(0.3)
0
has one and only one solution y(t) EO C2(~) (Le. a twice continuously differentiable solution with values in ~) such that y(O) = Yo and y'(O) = Yo. Hint. Reduce equation (0.3) to a stationary equation in the phase space ~(2) = ~ ~ by setting, for example, dy 1dt = z and x = y z. b) For any solution y(t) EO C2(~) the Ljapunov exponents of the functions y(t) and y(t) + dy(t) 1dt coincide. Hint. Make use of the arguments in § 3.2. By the (upper) Ljapunov exponent KL of equation (0.3) is meant the upper bound of the set of Ljapunov exponents of all of the solutions y(t) EO C2(~) of this equation (see § 4 of Chapter III). By the A spectrum of equation (0.3) is meant the set 0' of those complex A for which the operator A2A + A B + C does not have an inverse in [~l. c) For equation (0.3) the upper Ljapunov exponent coincides with the supremum of the set of real parts of the points of the A spectrum: KL = sup (Re A I A EO a). d) Derive a formula for the solution of the equation
+
+
d 2y 1dt 2
+ Ty =
f(t)
(TEO
[~]),
satisfying the condition y(to) = Yo, y' (to) = Yo. 7. Suppose the state of a certain linear autonomous mechanical system S is specified by a vector yEO .f\, its potential energy is given by the form (Cy, y)/2 and its kinetic energy is given by the form (Ay, y)/2 (y = dy / dt). Then its equation of motion is written in the form Aji + Cy = - By (ji = d 2y 1dt 2), where- By gives the effect of the external forces, which are assumed to depend only on the velocity y. For the total mechanical energy E = (Cy, y)/2 + (Ay, y) /2 we will have dEldt = - (Re BY,Y). Thus, if B = - B*, the force - By does not cause a dissipation of energy and is said to be gyroscopic. If B = B* ~ 0, the force - By is called a resistance (or Rayleigh) force, and such a force causes a dissipation of energy. In problems of mechanics one frequently encounters the case when B = Bm + iB3 (B~ = B m, B~ = B 3), Le. the force -By contains both a Rayleigh and a gyroscopic component. Since the kinetit energy of a motion is always positive, the operator A must be positive. We assume that A 0, If the system oscillates about the state y = 0 of stable equilibrium (minimum 0, i.e. C» O. If the motion of the potential energy), the potential energy will be positive for y system began near a state of maximum (a saddle point of the) potential energy, the operator C must be negative (indefinite). These remarks lead toa mechanical interpretation of the results cited in this and the next two exercises (8 and 9). a) Suppose 0 and C EO [.f\l. Show that all of the solutions of the equation Aji + Cy = 0 are bounded on the real line precisely when the operator C is uniformly positive: C» O. Hint. Make use of Theorem 3.2.
»
A»
"*
91
EXERCISES
b) Suppose A » 0, C» 0 and B*
=-
B. Show that all of the solutions of the equation
+ By + Cy =
Aji
0
(0.4)
.p are bounded on the real line. Hint. Reduce the equation to a stationary equation in the phase space .p(2) x = YEElz and
in
z
= .p EEl .p by putting
= A dy/dt + tBy,
(0.5)
and then make use of the assertion of Exercise 5a). 8. a) Suppose A » 0, Re B» 0 and C» o. Show that the upper Ljapunov exponent "L of equation (0.4) is negative and, moreover,
"L :s:: sup -
{Re - (Bf,f)
+ V (Bf,I)2 (AI, f)
4(Af,f)(Cf,f)
liE';
'>',
I-I-O} -j-
•
Hint. Consider a boundary point of the spectrum of the coefficient of the first order equation in .p(2) into which (0.4) is transformed, and make use of the theorem on a boundary point of the spectrum (see § 1.4.3). b) Suppose A = A* is invertible, B = B*, C = C* and for any x E·P (Bx, X)2
~
4 (Ax, x) (Cx, x).
Show that the A spectrum of equation (0.4) is real. 9. Suppose A 0, B = B* and Re C o. Show that a) The differential equation
«
»
(0.6) in [.pl has the two solutions Y± quadratic "algebraic" equation
= exp (tZ±), where the Z± (E [,fl]) are the unique roots of the AZ2 +BZ+ C=O
(0.7)
with the properties Re (AZ+ + B/2)>> 0 and Re (AZ_ + B/2)« o. The spectrum IT(Z+) lies in the interior of the right halfplane: Re 0, while the spectrum IT(Z_) lies in the interior of the left halfplane: Re A o. b) The general solution of equation (0.6) is given by the formula
A>
<
Y(t) = exp (tZ+) C+
+ exp (tZ_) C,
where C+ and C_ are arbitrary operators from [.pl. c) The general solution of equation (0.4) is given by the formula y (t) = exp (tZ+) y+
+ exp (tZ_) y_,
where y+ and y_ are arbitrary 'fectors from ,P. d) If C = C* (<< 0.), then AZ+ + B/2 = - (AZ_ + B/2)*. The operator Z+ is similar to a uniformly positive operator while Z_ is similar to a uniformly negative operator. Hint. Make the substitution (0.5), verify that the coefficient d of the corresponding equation in .p(2) satisfies the condition Re (W d)>> 0 for W = ( ~
+
6) and make use of Theorem 2.1. Under
.p_ corresponding to the decomposition IT(d) = IT +(d) U IT_(d) the decomposition .p(2) = .p+ of the spectrum there will correspond to the W-definite spaces .p± uniformly W-definite operators K± such that
(cf. the proof of Theorem V.2.4).
92
II.
LINEAR EQUATION WITH CONSTANT OPERATOR
The desired roots Z+ are found from the equations AZ+ + BI2 = K+. When C = C* (<< 0)we have W d 0, and the restrictions d I .p + a;:;-d d I .p _ are similar to a uniformly positive and a uniformly negative operator respectively (see Exercise I.26b)). Assertions a) - d) can also be proved by making use of the usual substitution y = z and setting
»
W= (~~). We conclude this group of exercises by noting that more sophisticated investigations of the differential equation (0.4) and the corresponding quadratic equation (0.7) can be found in the papers of M. G. Krein and H. Langer [1, 21, H. Langer [1, 21 and R. Kuhne [1, 2] (see also M. G. Krein [10]). Some of these investigations were generalizations and extensions of the investigations of R. Duffin [1, 2] of equation (0.4) in the finite-dimensional real space .p = Rn. 10. Show that the principal Green function GAt) for an operator A EO [)B] whose spectrum does not intersect the imaginary axis is representable in the form of a Fourier transform: (t
*- 0),
where the integral is understood in the sense of the Cauchy principal value. II. Let f(t) be an almost periodic function. It is known that there exists (uniformly with respect to to) the limit 1J1/(f)
=
1
lim -T T--»oo
to+T
f f(t) dt, to
which is called the mean value of an almost periodic function f(t). Since along with f(t) the function f(t) e- w is also almost periodic for any real A, we can speak of the quantity c,(f) = lJ1/(fe- W ), the nonzero values of which are called the Fourier coefficients of an almost periodic function f(t). The set A(f) of those values of A for which cif) *- 0 is called the spectrum of an almost periodic function f(t). It is known that the spectrum of an almost periodic function is at most a countable set. This permits one to associate with each almost periodic function f(t) the Fourier series f(t) '" L.,c,(f)e iU , Show that under the conditions of Theorem 4.4 the Fourier coefficients of an almost periodic solution x(t) of equation (4.1) are connected with the Fourier coefficients of the functionf(t) by the relation c, (x)
=-
(A - iA1)-1 c, (f).
12. Let A EO [.p]. Prove that in order for the vector function e iAt x to be almost periodic for all x EO [.p] it is necessary and sufficient that the operator A have a complete system of eigenvectors that is orthogonal with respect to a scalar product that is topologically equivalent to the original one. Hint. Make use of Theorem 1.6.1. For a completely continuous operator A EO [:0] the condition that all of the vector functions eiAt x (x EO.p) be almost periodic is equivalent to the simpler condition that they be bounded on the real line. In this case the spectrum of the almost periodic operator function e jAt is concentrated near A= O. The following question can be asked. Suppose A is a completely continuous operator in a Banach space 58 (A EO [)B]) and the operator function e jAt ( - 00 t 00) is bounded. What can be said about the spectral properties of A? This question has been fully and elegantly investigated for a weakly complete (in particular, reflexive) space )8 by Ju. I. Ljubic [1]. 13. (A. I. Perov; personal communication). Suppose A EO [)B]. Prove that in order for the operator function e jAt to be almost periodic (in the topology of the space [)B]) it is necessary and sufficient that the operator A be an operator of simplest type with a real spectrum.
<<
NOTES
93
NOTES In the finite-dimensional case the arguments given in § 2 are due to A. M. Ljapunov [1]. In his paper presented at a meeting of the Moscow Mathematical Society in 1948 M. G. Krein noted that the use by Ljapunov of different quadratic forms can be regarded as a renorming, possibly indefinite, of the space, and indicated how to carry out such a renorming in a Banach space. The simplest variant is considered in this section; the essential importance of similar renormings is brought out in proving theorems on the stability and instability of solutions of nonlinear equations (see Chapter VII). The results of § 3 are closely connected with Theorem I.6.2 and have been taken from the "Lectures" of M. G. Krein. The boundedness conditions for solutions of the second order equation were also indicated in the above mentioned paper of M. G. Krein (see M. G. Krein [2]). The construction of Green functions by means of contour integrals goes back to Cauchy. The principal Green function was used in latent form in the investigations of P. Bohl [1].
CHAPTER
III
THE NONSTATIONARY LINEAR EQUATION. BOHL EXPONENTS In this chapter we consider for the first time homogeneous and inhomogeneous linear equations with a variable operator coefficient A(t). We begin by constructing the evolution operator U(t, 'Z"), which replaces the exponential operator function eACt -<) and reduces to it when A(t) is a constant operator: A(t) == A (§ 1). In § 2 we give some important estimates which are systematically used in this and subsequent chapters. Some elementary notions connected with the behavior of the solutions of a nonstationary equation are considered in § 3. The central notion of this chapter, which is studied in § 4, is that of an (upper) Bohl exponent. The solutions of a nonstationary equation of negative Bohl exponent behave at infinity in the same way as the solutions of a stationary equation with an operator whose spectrum lies in the interior of the left halfplane. In § 5 we discuss the role of the negativeness of the Bohl exponent in the study of bounded solutions of the inhomogeneous equation with a bounded free term. Finally, in § 6 we consider a class of equations with variable coefficients, viz. equations with precompactIy valued operator functions, for which, as in the stationary case, one can estimate and sometimes also calculate the Bohl exponent in terms of the limiting characteristics of the operator function A(t) for t -> 00.
§ I. Evolution operator and formulas for the solution of linear equations I. Bochner integrable functions. In this and several subsequent chapters we will study on a finite or infinite interval $ the linear differential equation dx/dt
= A(t)x + f(t)
(t
E
$)
(1.1)
with variable operator functions A(t) and vector functionsf(t) depending on a real parameter t. We would too greatly restrict the class of equations being studied if we were to consider only strongly continuous functions. In order to avoid complicated theories of integration we will not strive for the greatest generality but rather confine our attention to the sufficiently wide class of equations whose coefficients are strongly measurable and locally Bochner integrable. We recall some definitions and facts connected with these notions (the reader who is unfamiliar with them, however, can as before regard all of the functions encountered as continuous). A vector function x(t) on a finite or infinite interval $ = [a,b] with values in a Banach space 18 is said to be countably valued if it takes on [a, b] no more than a 94
1. EVOLUTION OPERATOR
95
I
countable number of nonzero values Xk (k = I, 2,.··), the sets Ek = {t x(t) = Xk} (k = 1, 2,.··) being Lebesgue measurable. A countably valued function is Bochner integrable on [a, b] if and only if the numerical function IIx(t)lIis Lebesgue integrable on [a, b]. The Bochner integral of a countably valued function is by definition b
00
Jx(t)dt = 1:; XkmEk a
k=l
(mEk is the Lebesgue measure of the set Ek)' A vector function x(t) is said to be strongly measurable on [a, b] if it is the limit on this interval of an almost everywhere convergent sequence of countably valued functions xn(t). If a function x(t) is strongly measurable, the function IIx(t) II is Lebesgue measurable. If Ilx(t) II is also integrable, x(t) is said to be Bochner integrable (strongly integrable) and by definition b
b
Jx(t)dt
=
a
lim Jxn(t)dt. n-.oo a
It can be shown that such an integral does not depend on the choice of the sequence {xn(t)} converging to x(t) and is subject to the estimate
lit x(t)dt I ~ { Ilx(t) Ildt, from which one can deduce various theorems on convergence under the integral sign. We will not enumerate the various properties of the Bochner integral that are completely analogous to properties of the Lebesgue integral (linearity, countable additivity, etc.). We note the relation
A
b
b
a
a
Jx(t)dt = JAx(t)dt
(A
E [~, ~lD.
By considering in place of the space ~ a space [~, ~d of operators, we can transfer all of the above definitions and notions to operator functions. In this connection the product of two integrable functions (operator functions or an operator function and a vector function) of which one is bounded is also an integrable functio~. If x(t) is Bochner integrable, the relation 1 t+Lit lim -;"-t J Ilx(t") - x(t)lldt" = 0 Jt~O i.I
is valid for almost all values of t
t
E
[a, b] and, in particular, the function t
y(t)
=
Jx(t")dt" a
(1.1 a)
96
III.
NONSTATIONARY LINEAR EQUATION
is continuous and almost everywhere differentiable in [a, b]. Further, wherever it is not otherwise stipulated, we will understand by the differentiability of a vector or operator function its representability in the form of an integral of type (1.la) of a strongly integrable function. It is not difficult to verify that all of the basic properties of the derivative (in particular, the rules for differentiating a sum and a product) remain valid in this case. 2. Existence oj solutions. We consider on a finite or infinite interval J the differential equation
+ J(t)
dx/dt = A(t)x
(t
E
J)
(1.1)
in a Banach space ~. In this and the next two chapters, wherever it is not otherwise stipulated, we will assume that the functions J(t) and A(t) with values in ~ and [~] respectively are strongly measurable and Bochner integrable on the finite subintervals of J. By a solution of equation (1.1) we will understand a continuous function x(t) that is differentiable in the sense described above and satisfies (1.1) almost everywhere. Thus a solution of the integral equation x(t)
=
Xo
I
+
S A(-r)x(z,)d-r
+
to
I
SJ(-r)d-r,
(1.2)
to
where Xo = x(to), is by definition a solution of equation (1.1). If, in particular,J(t) is continuous while A(t) is strongly continuous,lJ a solution of equation (1.2) is continuously differentiable at each point t E J and relation (1.1) is satisfied everywhere in J. We consider in place of (1.2) the more general equation
=
x(t)
get)
I
+
S A(-r)x(-r)d-r
(1.3)
I,
with a continuous vector function get) on J, and show that it has a continuous solution on any finite interval [a, b] c J. Let C(~; [a, b]) denote2) the Banach space of continuous functions on [a, b] with values in ~ and norm
Illxlll
=
max Ilx(t)ll·
IE [a, h]
We consider in this space the operator (Sx)(t)
=
g(t)
+
I
S A(-r)x(-r)d-r, I,
l)This means that the vector function A(t)x is continuous for any x EO )S. 2)If it is clear from the context what the interval is, we will use the less cumbersome notation C()S).
1.
97
EVOLUTION OPERATOR
defined by the right side of equation (1.3). This operator takes C(lS) into itself since the function (Sx)(t) is obviously continuous. It is not difficult to verify by induction the formula t
(Snx)(t) = g(t)
+ JA(tl)g(tl)dtl t,
1 I,
+ J JA(t2)A(tl )g(tl )dtl dt2 + ... to to t
[,._1
t2
to
to
to
+ J J ... JA(tn-I)A(tn- 2) ... A(tl)g(tl)dll 1
I.
... dtn- l
(1.4)
I,
+ J J... JA(tn)A(ln-l) ... A(II)x(tl)dll ·· .dIn, to to
to
which implies the relation
til
tz
to to
ta
t
=
J J... -S A(tn)A(tn-l) ... A(II)[X2(t) -
xI(t)]dt l ... dtn
and the estimate
1
;'£
I.
I,
IIIx2 - xliii Jto toJ... toJ IIA(tn)IIIIA(tn-I)II···IIA(II)lldll···dtn.
Since the integrand is invariant under any permutation of the variables tn, we get the equality 1
I.
I,
to to
to
II,··,
J J... J IIA(tn)IIIIA(tn-I)II···IIA(tl)lldtC·dln 1
=flI •
til
Jto Jto ... Jto IIA(tn)IIIIA(tn-I)II···IIA(tI)lldll" .. dtn
(1.5)
~=*U IIA(r)lldrl We finally obtain the estimate
showing that when n is sufficiently large, Sn is a contraction in C(lS). On the basis of Theorem 1.9.1 we conclude that equation (l.3) has exactly one continuous solution in [a, b].
III.
98
NONSTATIONARY LINEAR EQUATION
We note that this solution can be obtained by means of the relation x(t)
= lim Snxo(t) n~oo
for any xo(t) series
E
C(58) and hence, as follows from (1.4), can be represented by the t
x(t) = g(t)
+ JA(t1)g(t1)dt1 t,
tn
t2
n=2 to to
to
t
00
+ 1: S S... SA(tn)A(tn-l)-··A(tl)g(tl)dtl···dtn
(1.6)
00
= g(t)
+ 1: gk(t), k=l
where t
gk(t) =
SA(r)gk-l(r)dr,
go(t) = g(t).
t,
It follows from (1.5) that this series is majorized in norm by the series (1.7)
which implies the estimate (1.8)
We consider in particular the integral equation t
x(t)
=
Xo
+ SA(r)x(r)dr, t,
which is equivalent to the differential equation (1.9)
dxfdt = A(t)x
with the initial condition x(to)
=
(1.10)
Xo·
The solution of the Cauchy problem (1.9)-(1.10) is obtained from (1.6) by setting g(t) ;: Xo. In this connection one obtains the formula t
x(t)
=
Xo
+ SA(t1)xodt1 t, t tn
tz
n=2 to to
to
00
+ 1: S S'" SA(tn)A(tn-l)'"
A(t1)xOdtC ·dtn.
1.
99
EVOLUTION OPERATOR
We introduce the linear operator 1
U(t) = 1+
JA(tl)dt1 I,
t..
t2
n=2 to to
to
co
+ L:
t
J J ... JA(tn)A(tn-l)···A(tl)dtC·dtn'
(1.11)
The series (1.11) is majorized by the series in braces in (1.7) and hence converges uniformly in the operator norm (i.e. in [lB]) on the interval [a, b]. Using the operator U(t), the solution of the Cauchy problem (1.9)-(1.10) can be expressed by the formula (1.12)
x(t) = U(t)xo.
We note that (1.8) implies the estimate
3. Some operator equations. It is not difficult to verify that the operator U(t), which by definition is continuous in [lB] and almost everywhere differentiable, satisfies the equation (1.13)
dU / dt = A(t)U
and the condition U(to) = 1. We consider the more general equation (1.14)
dZ / dt = A(t)Z - ZB(t).
It can be written in the form dZ / dt = W(t)Z,
(1.15)
where W(t): Z ~ AtZ - BrZ = A(t)Z - ZB(t) is a linear operator (transformer) acting in [lB] (see § 1.3.1). All of the results obtained in the preceding subsection can be applied to equation (1.15). It consequently has a unique continuous differentiable solution Z(t) which satisfies the condition Z(to) = Zo and which furthermore can be represented in the form 00
Z(t)
= L: Zk(t),
(1.16)
k=O
where 1
Zn(t) =
1
JW(Z)Zn-l(r)d-r = toJ {A(-r)Zn-l(-r) to
Zn-l(-r)B(-r)}d-r.
We note in this connection that the continuity of Z(t) as a vector function with
100
III.
NONSTATIONARY LINEAR EQUATION
values in [58] means its continuity in the operator norm. These results apply in particular to equation (Ll3), which results from (Ll4) when B = 0, as well as to the equation dZ/dt = - ZB(t),
(Ll7)
resulting from (Ll4) when A(t) = 0. A special case of the latter equation is the equation dV/dt = - VA(t),
(Ll8)
which is called adjoint associate equation of (Ll3). We consider the solution of equation (Ll8) satisfying the condition V(to) = I. It is not difficult to verify that U-l(t) exists and Vet) = U-l(t). For suppose Zl(t) = V(t)U(t) and Z2(t) = U(t)V(t). Then dZ1(t) = dV U dt dt
+
V d~ = - VAU dt
+
VAU = 0,
which implies that Zl(t) == Zl(tO) = I. On the other hand, Z2(t) satisfies the system {
dZ2!!l = dU_ V dt dt Z2(tO) = I,
+
U !IV = AZ2 - Z2 A dt '
the unique solution of which is Z2(t) == I. 4. Formula for the solutions oj the inhomogeneous equation. We return to a consideration of the Cauchy problem for the inhomogeneous equation { dx/dt = A(t)x x(to) = xo·
+ J(t),
(Ll) (LlO)
We will seek its solution, the existence and uniqueness of which have already been proved, in the form x = U(t)y, where U(t) is the operator function (Lll). Making this substitution, we find dy/dt = U-l(t)J(t) and y(to) = xo. Integrating from to to t, we get t
Y = xo
+ JU-l(r)J<'.)dr. t,
Thus t
x(t) = U(t)xo
+ JU(t)U-l(r)J(r)dr.
(Ll9)
t,
5. Evolution (solving) operator oj the equation. Let U(t, r) = U(t)U-l(r). The operator U(t, r) will be called the evolution (solving) operator of the differential equation (Ll) (or equation (1.9».
1.
101
EVOLUTION OPERATOR
Since
dU~~'!l
=
d~~t)
U-l(r) = A(t)U(t)U-l(r)
= A(t)U(t, ..),
this operator satisfies the system dX I dt = A(t)X,
X( ..)
= I
(1.20)
and does not depend on the choice of the value to with respect to which the operator U(t) was constructed. The notation U(t) will be employed in the sequel for the operator U(t) = U(t, 0). This operator will usually be called the Cauchy operator of the equation. An arbitrary invertible solution U(t)C of the operator equation (1.13) is called afundamental operator of equations (1.1) and (1.9). With the use of the evolution operator the solution of the Cauchy problem for the homogeneous equation dx I dt = A(t)x, x( ..) = X T can be written in the form x(t)
U(t, ..)xT ,
=
and, for the inhomogeneous equation, in the form 1
x(t)
=
U(t, to)xo
+ SU(t, ..)f(..)d ...
(1.21)
I,
The following fundamental properties of the evolution operator follow directly from its definition. a) U(t, t) = I, b) U(t, s)U(s, ..) = U(t, ..), c) U(t, ..) = [U( .. , t)]-l. In addition, it follows from the foregoing that d) IIU(t, ..)11 ~ exp[J~IIA(.. ~ t). We note that if the family of operators U(t, ..) (t, .. E J) has properties a) and b), it also has property c). For a proof it suffices to put .. = t in b). We leave it to the reader to verify that if the evolution operator U(t, s) corresponds to the operator function A(t), the evolution operator Uit, s) = ea(l-s) U(t, s) corresponds to the operator function Aa(t) = A(t) + aI (where a is a complex number). When ~ = .p ~e encounter an interesting special case.
)lld..](..
LEMMA 1.1. Let ,p be a Hilbert space in which there has been introduced a (possibly indefinite) scalar product (x, y)w = (Wx, y). If the operator A(t) is W-skew-Hermitian:
(A(t)x, y)w
=
-
(x, A(t)y)w,
the evolution operator U(t, ..) corresponding to it is a W-unitaryoperator.
III.
102 PROOF.
NONSTATIONARY LINEAR EQUATION
Using the differential equation (1.20) for U(t, r), we get d(U(t, r)ify, U(t, r)¢)w dt .
= (AU(t, r)ify, U(t, r)¢)w + (U(t, r)ify, AU(t, r)¢)w == O. Since, in addition, (U(t, r)ify, U(t, r)¢)w
= (ify, ¢)w
for t
= r,
this equality is satisfied for all t, which proves the lemma. It is easily seen that when A(t) == A, i. e. when A(t) does not depend on t, all of the formulas obtained here go over into the corresponding formulas of Chapter II. In this connection U(t, r) = eA(t-T). We note that by virtue of property b) the evolution operator can be written in the form n
U(t, r)
=
U(t, tn)U(tn, tn-I)··· U(tl> r)
= lim IT n---tOO
where tin) = r + k(t - r)/(n operator function A(t)
U(tk~I'
tin)
=
I
+
k=O
U(t k~I' tin),
1). From (1.13) it follows that for a continuous
+ A(tkn). t ~ r +
0(+)= eXP[A(tin). t ~ rJ+ 0(+),
and, since the latter term turns out to be unessential in passing to the limit, U(t, r)
= lim
n~oo
ft exp[A(tkn».~J. n
k=O
The limit on the right bears the name of multiplicative integral and is denoted by the symbol
rexp t
U(t, r)
=
A(O)dO.
T
This construction can be generalized so as to make it suitable for discontinuous operator functions A(t). To this end we note that the differential equation for the operator U(t, r) corresponding to ~n integrable operator function A(t) is equivalent to the equation t
Vet, r) = 1+
JA(O)V(O, r)dO,
which can also be written in the form t
V(t, r) = 1+
J[dF(O)]V(O, r), T
where F(t) = JA(t)dt.
(l.22)
1.
EVOLUTION OPERATOR
103
The latter equation can be generalized by dropping the requirement of differentiability of the operator function F(t) and rewriting it in the more general form t
U(t, r) = 1+
S[Sl(8)dF(8)Sz(8)]U(8, r),
(1.23)
where the Sk(8) (k = 1, 2) are continuous operator functions, F(8) is of bounded variation and the integral is understood in the sense of Stieltjes. The solution of equation (1.23) can be written in the form t
U(t, r) =
Jexp [Sl(8)dF(8)Sz(8)],
(1.24)
if this expression (which is called a multiplicative Stieltjes integral) is understood as the limit of the corresponding product of operators; in particular, the solution of equation (1.22) can be written in the form t
U(t, r) =
Jexp A(8)d8.
A substantial part of the results of this chapter can be generalized along these same lines (see Exercises 1-4). 6. Formulas for the solutions of the operator equations. Here we express the solutions of equation (1.14) and the inhomogeneous equation (1.26) corresponding to it in terms of the evolution operator UA(t, r) of equation (1.9) and the evolution operator UB(t, r) of the equation dx/ dt = B(t)x. We consider the transformer defined by the formula U(t, r)X = UA(t, r)XUB(r, t).
(1.25)
Differentiating with respect to t, we get (d/dt)U(t, r)X = A(t)UA(t, r)XUB(r, t) - UA(t, r)XUB(r, t)B(t) = A(t)U(t, r)X - U(t, r)XB(t).
Since we also have U(r, r)X = X, the transformer (1.25) is the evolution operator for equation (1.15). Therefore the solution of the equation dX / dt = A(t)X - XB(t)
+ F(t)
(1.26)
can be written in the form t
X(t) = U(t, to)X(to)
+ SU(t, r)F(r)dr t,
t
=
UA(t, to)X(to)UB (to, t)
+ SUA(t, r)F(r)UB(r, t)dr. t,
(1.27)
III.
104
NONSTATIONARY LINEAR EQUATION
In particular, for the adjoint equation (1.18) of (1.13) t
X(t) = X(tO)UA(to, t)
+ SFCr)UA(r,
t)d1:.
(1.28)
to
7. Examples. We consider some examples of the equations of type (1.1), (1.9). Suppose the space ~ is finite dimensional (~ = Cn). We choose an orthonormal basis el>"', en in it and let Xk = (x, ek) (k = 1,.··, n) be the coordinates of a vector x E Cn and ajk = (Aek, ej) be the matrix elements of the operator A in this basis. In this case equation (1.1) is equivalent to the system of ordinary differential equations dx. dt
n
-' =
r; ajk(t)xk k=l
(j = 1,2,.··, n).
(1.29)
The evolution operator U(t, 1:) is given by a matrix II Ujk(t, 1:) II satisfying the system
f d~:k
=
S~l ajs(t )Usk'
1Ujk(1:, 1:) = Ojk.
It is easily seen that the columns of this matrix form a fundamental system of solutions of system (1.29). The general solution is a linear combination of these columns: n
xit)
=
r;
k=l
Ujk(t, 1:)Xk(1:).
Finally, the solution of the inhomogeneous system ( 1.30)
is given by the formula n
xit)
=
r;
k=l
Ujk(t, to)Xk(tO)
n
t
k=l
to
+ r; SUjk(t, 1:)h(1:)d1:,
which follows directly from (1.21). A natural generalization of system (1.30) is a countable system of differential equations dx. _L dt
co
= r; ajk(t)xk + Jj(t) k=l
(j = 1,2",),
This system can be considered as a vector equation in various sequence spaces, thereby obtaining various conditions for the existence of solutions. The natural continuous analog of a countable system of differential equations is an integrodifferential equation of, for example, the form
2. INTEGRAL INEQUALITIES dx(t, s) _ -----at- -
Sb
.J.f'(t, s, o")X(t, 0") dO"
105
+ f(t, s).
a
By considering this equation in various function spaces, we can obtain various conditions for the existence of its solutions.
§ 2. Integral inequalities. Comparison of evolution operators 1. Integral inequalities. The following lemma plays a decisive role in obtaining various estimates for the solutions of linear (and, as we will see in Chapter VII, nonlinear) differential equations. LEMMA 2.1. Suppose .J.f'(t, 'r) is a nonnegative kernel on a finite or infinite interval J such that the integral operator (Kx)(t)
=
S.J.f'(t, 'r)x('r)d'r of
leaves the space C(J) of bounded continuous functions on J invariant and has a spectral radius less than one in this space. Then a continuous function p(t) satisfying the inequality p(t) ~ f(t)
+ S.J.f'(t, 'r)p('r)d'r
(t
E
(2.1)
J),
of
wheref(t) is a continuous function on J, satisfies the inequality p(t) where ¢(t) is the solution of the integral equation ¢(t) = f(t)
+ S.J.f'(t, s )¢(s )ds.
~
¢(t) (t
E
J),
(2.2)
of
The assertion is also true when the signs
~
are replaced by
~.
We note that we could formulate an analogous assertion for vector functions. PROOF. The lemma is obtained by applying Theorem I.9.3 to the space C(Rl, J), the cone oR: of nonnegative functions and the operator K. We will give below (see Remarks 2.1 and 2.2 and the corollaries following each of them) a number of concrete applications of Lemma 2.1. The concrete estimates obtained in this connection will be systematically used in this and the following chapters of the hook. REMARK 2.1. Suppose J = [a, b] is an arbitrary finite interval while the kernel .J.f'(t, 'r) is nonnegative on [a, b], continuous for 'r ~ t and vanishes for 'r > t. The integral operator K: t
(Kx)(t) =
S.J.f'(t, 'r)x('r)d'r of
=
S.J.f'(t, 'r)x('r)d'r a
is a Volterra operator and, as is well known, has a spectral radius equal to zero.
III.
106
NONSTATIONARY LINEAR EQUATION
Therefore Lemma 2.1 is applicable and if a function ¢(t) satisfies the inequality t
+ J.%"(t, 'r)¢(r)d'r
¢(t) ~ f(t)
(t ~ a),
(2.1 ')
a
then ¢(t)
~
¢(t), where t
J.%"(t, s)¢(s)ds.
+
¢(t) = f(t)
(2.2')
a
The converse estimates are obtained in exactly the same way. We note that the interval [a, b] can be chosen arbitrarily large and hence the estimates are valid for all t ~ a provided the kernel .%"(t, 'r) and the functionf(t) are defined for a ~ 'r ~ t < 00. In the same way one can consider the inequality
+
¢(t) ~ f(t)
b
J.%"(t, 'r)¢('r)d'r.
(2.1 ")
t
for
t ~
~
b and obtain the estimate ¢(t)
¢(t), where b
+ J.%"(t, s)¢(s)ds.
¢(t) = f(t)
(2.2")
t
COROLLARY
2.1 Suppose t
¢(t) ~
+ Jh('r)¢('r)d'r
C
(2.3)
to
where h(t) is a continuous nonnegative function. Then ~
¢(t)
cefi.h(c)dc.
(2.4)
For if we differentiate the equality t
¢(t) =
C
+ Jh('r)¢('r)d'r to
with respect to t, we obtain the differential equation ¢'(t) condition ¢(to) = c, which implies
COROLLARY
=
h(t)¢(t) with the
2.2. Suppose ¢(t)
~
t
ae-v(Ho)
+ (3 Je-vCt-c)p('r)¢('r)d'r to
where (3p(t) is a continuous nonnegative function. Then
(2.5)
2.
INTEGRAL INEQUALITIES
107
~ ae~"(I~lo)+flf:oP(')d'
(2.6)
cp(t)
For if we set cp(t)
= cplt)e~"I,
we obtain the following inequality for CPl: 1
+ j3 Sp(-,)cpl(-.)d-..
CPl(t) ~ ae"lo
10
Therefore by virtue of (2.4) when c
=
ae"lo
from which the required estimate follows. If we reverse the inequality signs in (2.3) and (2.5), we obtain estimates (2.4) and (2.6) with their inequality signs reversed. REMARK 2.2. Suppose .%(t, -.) is a continuous nonnegative kernel on f and sup S.%(t, -.) d-. = q < 1. 1
Then in C(Rl, f) the spectral radius r(K) ~ II K II 2.1 is applicable. COROLLARY
(2.7)
J
=
q < 1, and hence Lemma
2.3. Suppose for 0 < j3 < v /2 00
cp(t)
+ j3 Se~"IHlcp(-.)d-..
~ ae~"(I~to)
(2.8)
10
Then cp(t) ~ -vPROOF.
Here .%(t, -.)
+
2av . e~ -J"'~2~;'(I~to) '\f'v 2 - 2j3v
= j3e~"II~'1
(2.9)
and
Therefore when j3 < v /2 condition (2.7) is satisfied and Lemma 2.1 is applicable. We consider the equation 00
¢(t) =
ae~"(I~lo)
+ j3 Se~"IHI¢(-.)d-.. 10
Differentiating it twice with respect to t, we obtain the relations 1
¢'(t) = - ave~"(I~lo) - j3v
00
Se~"(H)¢(-.)d-. + j3v Se~"(H)¢(-.)d-., 4
1 00
¢"(t) = av2e~"(Ho)
+ j3v 2 Se~"IHI¢(-.)d-. 10
- 2j3v¢(t),
(2.10)
III.
108
NONSTATIONARY LINEAR EQUATION
which together with (2.10) give the differential equation ¢"(t) - (V Z - 2{M¢(t)
=
o.
Since we are interested in the solution of equation (2.10) that is bounded on [to, (0), we have ¢(t)
= ce~,lv'~z(3v(I~lo) (v z - 2pv ~ 0 for p ~ ~).
The constant c can be found by substituting this expression in (2.10). Carrying z - 2pv) from which out the calculations, we obtain the equality c = 2av!(v + (2.9) follows. Let us consider the more complicated inequality
.vv
00
<jJ(t) ~ ae-v(t-Io)
+ p Se-VII-Tlp(-r)<jJ(-r)d7:
(v > 0, t
~
to).
(2.11)
to
It immediately reduces to the previous inequality if p(t) is a bounded function on [to, (0). We will need an estimate for <jJ(t), however, under the more general assumption that
sup I
1 I+T o S p(7:)d7:
7:0
I
= MTo <
00
(2.12)
for some fixed 7:0. In the present case equation (2.2) has the form 00
¢(t)
= ae-V(Ho) + p Se~vIHlp(7:)¢(7:)d7:.
(2.13)
10
It can be shown in the same way as above that the solution of this equation reduces to finding a bounded solution of the differential equation ¢"(t) (V Z - 2vpp(t»)¢(t) = O. In order to avoid the task of estimating the solutions of this equation, we proceed differently. We put ¢(t) = u(t)e-I1 (H o), where the number fl > 0 will be chosen later. The function u(t) must satisfy the equation 00
u(t) = ae- (V-I1) (I-to)
+ p Se-Vll-TI +11(t-r) p(7:)u(7:)d7:,
(2.14)
to
which we consider in the space C(Rl, [to, (0») of continuous bounded functions on [to, (0). We consider in this space the operator co
(Au)(t) =
PS e-vIHI+I1(t~r)p(7:)u(7:)d7:. 10
It is easily seen that
2. INTEGRAL INEQUALITIES
IIA I
109
00
Je- vl l-Tl+f1.(l-T)p(r)dr.
(3 sup
=
I,
We estimate the integral in this inequality. Let n
=
[t fro]. Then
00
Je-vlt-rl +f1.(I-r) p(r)dr I,
=
L:
to+kTo
J
e-vll-rl +f1.(I-r) p(r)dr
k=l 1,+ (k-l)r,
to'+-kTo
~
L:
max
J
e-vll-rl +f1.(I-r)
p(r)dr
I,+(k-l)r,
k=l [1,+ (k-ljr"I,+kr,]
~ roMr,t~l e-(v-f1.) (n-kjr, + 1 + k~:-2 e-(V+f1.)(k-n-Zlr,} ~ Mr,C(fJ-, )), ro),
where C ) = ro " ( )) , ro r'
ro ro --- < + ---~1 - e-(v-f1.)r, + ---1 - e-(v+f1.)r,
00
for fJ- < )). Thus (2.15)
We require that (3Mr,C(fJ-, )), ro) ~ q
< 1.
(2.16)
Under this requirement the equation u - Au = ae-(v-f1.)(t-I,)
(2.14)
will be solvable in C(Rl, [to, (0), i. e. the function u(t) is bounded: sup u(t)
=
c<
(2.17)
00.
I
This gives the following estimate for ¢(t): ¢(t)
~
cef1.- (t-I,).
(2.18)
We will assume that ro is sufficiently small. Then 1 C(fJ-,)), ro) = - - )) - fJ-
+ -1- + )) + fJ-
2))
O(ro) =
))2 -
and condition (2.16) reduces to the relation (3Mr,[
2)) ))2 -
or
fJ-2
+ O(ro)J <
1
fJ-2
+
O(ro),
III.
110
NONSTATIONARY LINEAR EQUATION
(2.19) Thus an estimate of form (2.18) holds under condition (2.19), and it only remains to find the constant c. This can be done by considering (2.14), which implies 00
c~ a
+ (3c sup Jrvll~TI+I'(I~T)p(7:)d7: 1
~
a
+ (3cMTo C(p,).I, 7:0)
I,
and, finally, c~
The above arguments imply the following result. LEMMA
2.2. Suppose for
¢(t)
> 0 and (3 > 0
).I
~ ae~V(I~to)
+ (3
00
Jr
VIHlp(7:)¢(7:)d7:,
(2.11 )
10
where p(t) is a continuous nonnegative function. For any p < ).I, 7:0 > 0 and q < 1 there exists a 0 > 0 such that if the condition 1 I+To MTo = sup p(7:)d7: < 0 1
J
7:0
1
is satisfied, then ¢(t) ~(a/(I- q) )e~I'(t~lo)
(t~
to).
REMARK 2.3. If 7:0 is sufficiently small, relation (2.19) indicates the values of p for which the desired estimate is valid for a given M To ' If the condition sup p(t) ~ M is satisfied, inequality (2.8) with (3 replaced by (3M will hold and Corollary 2.3 can be applied. 2. Comparison of evolution operators. LEMMA
2.3. Let Uk(t, s) (k
=
1,2; a
~
s, t
~
b) be the evolution operators of the
equations (k = 1,2).
If the estimate (a ~ s, t ~ b)
is satisfied for some N > 0 and real ).11, then for s
I U2(t, s) -
U1(t, s)11
~
(2.20)
t
~ Ne~Vl(t~S)(eNf;IIA'(T)~Al(T)lIdT
- 1),
(2.21)
and consequently
I U2(t, s) I Analogous estimates hold for t
~ Ne~vl(t~s)eNf;IIA.(T)~Al(T)lIdT.
~
s.
(2.22)
3. STABILITY AND BISTABILITY
111
PROOF. It clearly suffices to prove the assertion of the lemma for s The operator U2{t) = U2{t, 0) is a solution of the system
= o.
{ dU2/dt - A 1U2 = (A2 - A 1)U2, U2(0) = I. Using formula (1.21) for the solution of the operator equation
dX/dt - A 1X
= F{t),
X{O)
= I,
where F{t) = [A2{t) - A 1{t)]U2{t), we find that t
+ JU1{t, ..)[A 2(-r)
U2(t) = U1{t,0) Setting ¢J{t)
= II U2{t) II
o
- A 1{..)]U2{..)d...
(2.23)
and using estimate (2.20), we get ~
¢J{t)
t
Ne-",t
+ N Je-",(t-~)p(..)¢J{..)d.. , o
where pet) = IIA 2(t) - A1{t) II. Thus ¢J{t) is subject to estimate (2.6) when a = f3 = N, which implies estimate (2.22). To obtain estimate (2.21) we. note that (2.23) and estimate (2.22) imply
II U2(t)
- U1(t) I ~ ~
t
JII U1{t, ..) II IIA2{..) -
o
A 1(..) I I U2{..) lid..
/ t
N
Je-",(t-~)p{..)Ne-",teNfbP(s)dS d.. o t
= Ne-",t JNp{ ..)eNfbP(s)ds d.. = Ne-",t(eNf6p(S)dS - 1). o Setting A1{t) = 0 and A2{t) = A(t) in Lemma 2.3, we obtain some important estimates. LEMMA 2.4. The following estimate holds for the evolution operator U{t, s) of equation (1.1) when s ~ t:
I U{t, s) -
III ~ ef!IIA(~) IId~
-
1;
(2.24)
in addition, e-f!IIA(~)lId~ ~
IIU±l(t,s)II ~ ef:IIA(~)lId~.
(2.25)
§ 3. Stability and bistability. Comparable equations 1. Right stability. We introduce some notions connected with the behavior of the solutions of the equation
112
III. NONSTATIONARY LINEAR EQUATION dx/dt
= A(t)x
(3.1)
on the halfline [0, 00). Equation (3.1) is said to be stable (more precisely, right stable) if every solution of it is bounded on [0, 00). LEMMA 3.1. A necessary and sufficient condition for the stability of equation (3.1) is the uniform boundedness of its Cauchy operator: supll U(t) I < 00. t;;;O
PROOF. The sufficiency follows at once from the formula x(t) = V(t)xo; the necessity, from the uniform boundedness principle. REMARK 3.1. Rephrasing the condition of the lemma, we say that a necessary and sufficient condition for right stability is the existence of a constant q > 0 such that any solution x(t) of equation (3.1) satisfies the estimate (3.2) The least value qo of this constant is clearly given by the equality qo = sup I U(t) II· 1;;;0
REMARK 3.2. The right stability of equation (3.1) is equivalent to the right stability of the operator equation dX/dt = A(t)X. It is easily seen that the choice of the interval [0, (0) is not essential for the definition of stability. It can be replaced by any fixed interval [to, (0) on which the operator A(t) is defined and satisfies the conditions adopted by us. The role of the operator Vet) in the above formulas will be played by the operator V(t, to) and estimate (3.2) is replaced by the estimate (3.3) where qt, = SUPt;;;t, I U(t, to) II· The constant qt, here generally depends on the choice of to. Equation (3.1) is said to be uniformly (right) stable if there exists a constant N> 0 such that any solution x(t) of this equation satisfies the following estimate for all t ~ s ~ 0: Ilx(t) II ~ Nllx(s)ll·
(3.4)
In exactly the same way as above it can be shown that uniform right stability is equivalent to the condition N = sup IIU(t, t;;;5;;;0
s)11
< 00,
(3.5)
the indicated value of N being the best possible for estimate (3.4). REMARK 3.3. Suppose A(t) = A is a constant operator. In this case Vet, s)
=
3. eA(t-s)
STABILITY AND BISTABILITY
113
and, as is easily seen, the stability condition sup II eAt II <
(3.6)
00
t~O
coincides with the uniform stability condition. As follows from the results of Chapter I, for the fulfilment of (3.6) it is necessary that the spectrum a(A) lie in the closed left halfplane and sufficient that it lie in the interior of this halfplane. 2. Left stability. We say that equation (3.1) is left stable if there exists a constant q' > 0 such that any solution x(t) of this equation satisfies the following estimate for all t ~ 0: Ilx(O) II ~ q' Ilx(t) II
. (t ~ 0).
Left stability is clearly equivalent to the uniform boundedness of the operator U-1(t):
= sup II U-1(t) II <
q'
00,
t~O
the indicated value of q' being the best possible. We note that since the solutions of the adjoint equation dXjdt = - A(t)X are expressed by the formula X(t) = X(0)U-1(t), the left stability of equation (3.1) is equivalent to the right stability of the adjoint equation. In exactly the same way as above we can speak of uniform left stability if we understand by this the validity of the estimate Ilx(s) II ~ Nllx(t) II for all 0 ~ s ~ t and N not depending on sand t. Clearly, uniform left stability is equivalent to the condition (3.7) sup II U(s, t) II < 00. t~s~O
In regard to the left stability of an equation with a constant operator A it is possible to make the same statements concerning the spectrum a(A) as in Remark 3.3, but with the left halfplane replaced by the right halfplane. 3. Bistability. We say that equation (3.1) is bistable on the halfline [0, (0) if it is both right and left stable. Thus equation (3.1) is bistable if and only if sup
{II U±1(t) II} <
00,
O;2;t
or, equivalently, itand only if sup
{IIU(t, s)ll} <
00.
(3.8)
O;2;s,t
A comparison of this condition with (3.5) and (3.7) shows that a bistable equation is uniformly right and left stable. In other words, it can be said that an equation is stable if there exists a constant q > 0 such that any solution x(t) satisfies the estimate Ilx(t) II ~ q Ilx(s)II
(0 ~ s, t < (0).
114
III. NONSTATIONARY LINEAR EQUATION
From the above remarks it follows that in order for a stationary equation to be bistable it is necessary (but not sufficient) that the spectrum a(A) lie on the imaginary axis. It is not difficult to see that the bistability of a stationary equation is equivalent to the boundedness of its solutions on the real line. As we know, when ~ is a Hilbert space J), the spectrum a(A) lies on the imaginary axis precisely when A is similar to a skew-Hermitian operator. 4. Integrally comparable equations. Suppose given on the halfJine [0, (0) the two equations (k
= 1,2).
(3.9)
We say that these equations are integrally comparable if 00
IllAz - Allll =
J IIA 2(t)
o
- Al(t) IIdt <
00.
LEMMA 3.2. If one of the two integrally comparable equations (3.9) is uniformly right or left stable or bistable, so is the other equation. PROOF. The first assertion of the lemma follows at once from estimates (3.5) and (2.22) when ).11 = o. The second assertion is proved analogously. The third assertion is a consequence of the first two. We consider in particular the equation dx/dt
= A(t)x
(3.10)
with an integrable coefficient: 00
IliA III =
J IIA(r)lIdr
(3.11) < co. o This equation is integrally comparable with the bistable equation dx/dt = 0 and hence is bistable itself. Moreover, it can be asserted in this case that the limit x( (0) = limt~oox(t) exists for every solution x(t) of equation (3.10), and for every y E ~ there exists one and only one trajectory x(t) such that x(oo) = y. This fact follows from
THEOREM 3.L Suppose relation (3.11) is satisfied. Then the limit U(co) = exists and is an invertible operator.
limt~ooU(t)
PROOF. By virtue of Lemma 2.4 we have IIU±l(t)1I ~ M = explllAli1 and, further, IIU(t) - U(s)1I
=
II(U(t,s) - I)U(s)11
~
M(ex p !IIA(r)!ldr - 1) <
C
3.
115
STABILITY AND BISTABILITY
for sufficiently large t and s. The latter estimate implies the existence of the limit lim U(t)
=
U( (0).
t~co
The existence of the limit V(oo) = lim U-l(t) t~co
is proved analogously. Since, clearly, U(oo)V(oo) = V(oo)U(oo) = I, the theorem is proved. REMARK. The theorem can also be proved by making the change of variable s = J~ which transforms equation (3.10) into the equation
IIA(-r) lid..,
dx _
A(t(s))
ds - IIA(t(s)) I x
(0
~s~
I
IliA II)
with a bounded operator function on a finite interval. 5. Asymptotically equivalent equations. If equation (3.1) is left stable, then, as follows from the definition, a solution of this equation that tends to zero at infinity is identically equal to zero. For such equations the following definition is meaningful. We say that equations (3.9) are asymptotically equivalent if it is possible to establish a one-to-one attraction correspondence Xl(t) +--+ X2(t) in such a way that lim [X2(t) - Xl(t)]
=
O.
t~oo
THEOREM 3.2. Suppose that equations (3.9) are integrally comparable and one of them (and hence also the other) is bistable. Then these equations are asymptotically equivalent, the attraction correspondence Xl(t) +--+ xzCt) between their solutions being uniquely established and the estimate (3.12) being satisfied by corresponding solutions. In addition, the initial values of a pair of corresponding solutions are connected by a continuous invertible operator from [78]. "
PROOF. The fact that if one of the integrally comparable equations (3.9) is bistable, so is the other, is indicated in Lemma 3.2. We note that the solutions of these equations are bounded. Let Xl(t) be a solution of the first equation. We consider the function 00
X2(t) = Xl(t)
+ JU2(t, s) [Al(s) t
This expression is meaningful since
- A2(s)]xl(s)ds.
(3.13)
III.
116
/If U2(t, S) [Al(S) -
NONSTATIONARY LINEAR EQUATION
A2(S)]xls)ds II
~
IIIXl(S)IIIIIIAl -
A2111St~pll UZ(t, s)ll·
We verify that X2(t) is a solution of the second equation of (3.9). In fact,
x2(t) = xi(t) - [Al(t) - AZ(t)]Xl(t) 00
+ JA Z(t)U2(t, s)[Als) t
= Az(t) {Xl(t) + =
f
- A2(S)]Xl(S)ds
Uz(t, S)[Al(S) - AZ(S)]Xl(S)dS}
A2(t)XZ(t).
From (3.13) it clearly follows that limt~oo[xz(t) - Xl(t)] = O. Suppose now xz(t) is another solution of the second equation of (3.9) such that limt~oo[x2(t) - Xl(t)] = O. Then
= 0,
lim [X2(t) - X2(t)] t~oo
and since, by virtue of the bistability of the equation for any z-, Ilxz(z-) - Xz(z-) II ~
Il x2(t)
- X2(t) II
sup
O~t, T~OO
II U(z-, t)ll,
we get X2(t) == X2(t). Equations (3.9) are equivalent, and therefore equation (3.13) admits the inversion 00
Xl(t) = xz(t)
+ JUl(t, S)[A2(S)
- Als)]X2(S)ds.
t
It remains to note that
X2(0) = { 1+ [ Uz(O, S)[Al(S) - AzCs)] Uls, O)dS} Xl(O).
(3.14)
An analogous expression exists for Xl(O) in terms of X2(0). Estimate (3.12) immediately follows from (3.13). The theorem is proved.
§ 4. Ljapunov and Bohl exponents of the homogeneous equation on [0, 00) 1. Ljapunov exponents. We consider the homogeneous differential equation
dxfdt = A(t)x
(0
~
t < 00).
(4.1)
Ifthe operator function A(t) is bounded: IIA(t)II ~ M (0 ~ t < 00), or, more generally, integrally bounded: HI
sup t
Jt IIA(z-)lldz- ~ M
(0
~
t < 00),
(4.2)
4. LJAPUNOV AND BOHL EXPONENTS
117
then, as follows from (2.25), its solutions are subject to the estimate I x(t)1I ~ eMCt +1) Ilx(O) II, and thus the (upper) Ljapunov exponent I> =
lim In IIx(t) I t
t~co
of each solution x(t) of equation (4.1) satisfies the condition I> ~ M. It is not at all necessary that condition (4.2) be fulfilled in order for all of the solutions of equation (4.1) to have finite Ljapunov exponents (or even to be bounded) (see Exercise 19). Without assuming the fulfilment of condition (4.2), we consider the set:2 of not necessarily finite Ljapunov exponents of all of the possible solutions of equation (4.1). This set is called the (upper) Ljapunov spectrum of equation (4.1). We leave it to the reader to prove that the Ljapunov spectrum Z of the homogeneous equation dx/dt = Ax(A = const) in a finite-dimensional space j8 consists of the real parts of the eigenvalues of A.
By the (upper) Ljapunov exponent of equation (4.1) is meant the quantity sup 1>.
1>1. =
<EI
THEOREM 4.1. The Ljapunov exponent of equation (4.1) coincides with the (upper) Ljapunov exponent of the operator function Vet) = Vet, 0): 1>1. =
lim lnIlV(t)lI. t~co
(4.3)
t
PROOF. Let /l denote the right side of formula (4.3). From the equality x(t) V(t)xo it follows that In IIx(t) I ~ lnIlV(t)1I
+
=
lnllxoll
which implies lim lnllx(t)1I ~ lim In II V(t) II , t~oo
t
-
t~co
t
i. e. I> ~ /l for I> E ~, and hence 1>1. ~ /l. It remains to prove the converse inequality for the case 1>1. < x(t) = V(t)xo of equation (4.1)
I V(t)xo II
00.
For any solution
~ Ne, x,e(KL+e)t,
where c is an arbitrary positive number and Ne,x, is a constant generally depending on c andxo. The latter inequality can be written in the form
II V(t)xoe-(KL+e)tll
~ N"x,.
III.
118
NONSTATIONARY LINEAR EQUATION
The family of operators U(t)e~(KL+e)t (0 ~ t < (0) is bounded at each element Xo E )8. It follows from the uniform boundedness principle that (t
I
~
0).
I
This means that U(t) ~ Nee(KL+e)l, which implies fJ. ~ KL· The theorem is proved. REMARK 4.1. One could analogously introduce the lower Ljapunov exponent K' = lim In lilt of a solution x(t) and then the lower Ljapunov exponent K[ of an equation as the greatest lower bound of lower Ljapunov exponents of its solutions. In this connection K[ = - limHo In U~l(t) It. 2. Bohl exponents. Here we introduce another characteristic of the behavior of the solutions of equation (4.1) which, as will be seen, has more natural properties. Let x(t) = U(t)xo be a solution of this equation. By the (upper) Bohl exponent KB(XO) of this solution is meant the greatest lower bound of all those numbers p for which there exist numbers Np such that
Ilx(t)
I
I
(4.4) for any =
+
'r, t E [0, 00) such that 'r ~
t. If such numbers p do not exist, we put KB(XO)
00.
In exactly the same way the lower Bohl exponent Ks(XO) of a solution x(t) is the least upper bound of those numbers p' for which there exists a number N~ > 0 such that
Ilx(t)
I
~
Np,ep'(H)
Ilx('r) I
(0
~
'r ~
< 00).
t
(4.4')
If K(XO) is a Ljapunov exponent of a solution, clearly, -
00 ~
KB(xo)
~
K(Xo)
~ KB(XO) ~
+
00.
The interval [Ks(Xo), KB(XO)] is called the Bohl interval of the solution in question. The following formulas are easily verified:
KS(XO)
=
lim <~co
t-r-oo
~lxC1lI_-:-_I~llx('r)11 t -
'r
.
Suppose now P is a projection in )8 and )8p = P)8 is the corresponding subspace. We consider the totality of solutions xU) = U(t)xo of equation (4.1) that are initially in )8p: Xo E )8p. By the upper (lower) Bohl exponent KB(P) (KB(P» of this totality of solutions is meant the greatest lower (least upper) bound of the exponents p for which for-
4.
119
LJAPUNOV AND BOHL EXPONENTS
mula (4.4) «4.4')) is valid for all of the solutions x(t) = U(t)xo with Xo E 58 p and a number Np > 0 not depending on Xo. We will call the exponents KB(P) and KB{P) and the interval [KE(P), KB(P)] the upper and lower Bohl exponents and Bohl interval of equation (4.1) corresponding to the projection P (subspace 58 p). In particular, when P = I we will speak simply of the Bohl exponents of equation
(4.1) and use the notation It is clear that an extension of the subspace 58 p can only widen the corresponding Bohl interval and, in particular,
KE
~ KE(P) ~ KB(P) ~ KB·
(4.5)
We cite the following finiteness criterion for Bohl exponents. THEOREM
4.2. In order for the upper (lower) Bold exponent of equation (4.1) to be
finite: KB<
+
00
it is necessary and sufficient that
K =
sup
II U(t, -r) II < sup I U(-r, t) II <
O~t-T~l
(K' =
O~t-T:::;l
00
}
(4.6) 00).
We will prove below the more general Theorem 4.2', which is closely connected with the results presented in Chapter IV. We consider the subspace 58 pCt ) = U(t)58 p • It is the range of the projection pet)
=
U(t)PU-l(t).
An important role will be played in the sequel by the condition mp
= sup I U(t)PU-l(t) II <
(4.7)
00.
O~t
If it is satisfied, we will say that the projection P is uniformly conjugated by the Cauchy operator of equation (4.1). The geometrical significance of this definition will be made clear in Chapter IV. This condition is obviously satisfied if P commutes with U(t) and, in particular, if P = /. THEOREM 4.2'. In order for the estimate KB(P) < 00 (KB{P) > - 00) to be satisfied it is sufficient and,for the projections P that are uniformly conjugated by U(t), also necessary that
Kp
=
(Kp =
sup
II U(t)PU-l(-r) I <
:~;~:: II U(-r)PU-l(t) I
<
00
}
(4.6') 00).
III.
120
NONSTATIONARY LINEAR EQUATION
Suppose estimate (4.4) is satisfied for every solution x(t) l8p. We write it in the form
PROOF. Xo E
II U(t)Pxll
~
= U(I)xo
with
II U(-r)Pxll (I ~ r), where x is an arbitrary element of 18. Setting x = U-l(r)y and using the arbitrariNpep(t-r)
ness of y and estimate (4.5), we obtain the inequality (t ~ r),
(4.8)
which implies that Kp < 00. The necessity of the condition Kp < ro can be shown in the same way if we rewrite (4.4') in the form ~
(r
Suppose now Kp < 00. For 1,2,.··, n) and rn+l = t. Since
t -
I).
r > 0 and n = [t - r] we put rk
=
r
+k
(k =
n+l U(t)PU-l(r) = II U(rk)PU-l(rk-l), k=l
we get
I U(t)PU-l(r) I .
n+l ~ II I U(rk)PU-l(rk-l) I k=l ~
and, consequently, for
Xo E
Kj,+l
~
Kpen InKp
~ Kpe(t-~)In
Kp,
l8p
IIU(t)xoll = IIU(t)PU-l(r)U(r)xoll ~ Kpe(t-~)lnKpIIU(r)xoll,
i. e. K(P) ~ In Kp. The sufficiency of the condition Kp < ro can be shown analogously. REMARK 4.2. We have incidentally obtained the estimates - InKp
~
KP,(P)
~
KB(P)
~
In Kp.
In particular, the Bohl exponents of equation (4.1) (P inf In I U(:, r) II
= -
satisfy the estimates
O~~~~~l In I U(t, r) II
~ KP, ~ KB ~ REMARK
= /)
sup
InIIU(t, r)ll.
O~t-T~l
4.3. Since [U(t)PU-l(S)][U(S)PU-l(r)] = U(t)PU-l(r),
the first of the conditions (4.6') is equivalent to the condition sup
O~t-r~T
II U(t)PU-l(r) I <
00
(4.9)
4. LJAPUNOV AND BOHL EXPONENTS
121
for any T > O. An analogous remark can be made concerning the second condition of (4.6'). As a corollary of Theorem 4.2 we obtain the following result. THEOREM 4.3. If the operator function A(t) is integrally bounded, the Bohl exponents of equation (4.1) are finite. The proof immediately follows from Lemma 2.4 and formulas (4.7) and (4.9). We now find expressions for the Bohl exponents corresponding to a projection P. THEOREM 4.4. If condition (4.7) is satisfied and the Bohl exponents are finite, they are representable by the formulas KB(P)
=
_ KS(P) =
lim InIIU(z-
+ S)PU-l(Z-) II
l
;
r'l:~oo InIIU(z-)P~-l(z- + s)11 ; J S
T, $--+00
in particular,
KB = lim InIIU(z-
,
- KB =
(4.10)
.
l
+ s)ll:
J
+ s, z-)II
r';i: InIIU(/z-
(4.11)
s
T,S-+OO
PROOF. Suppose p > KB(P). Then inequality (4.8) is satisfied. It follows that Inll U(z-
+ s)PU-l(~llL s
~]~Npmp s
+p '
and, further, 1 A
= II'm-- 1nll U(z- + S)PU-l(Z-) II S
T,S--+OO
~
p,
which implies). ~ KB(P) < roo Suppose now p > ).. By the definition of the upper limit there exists a number T = Tp such that for z-, s ~ Tp lnll U(z-
+ S)PU-l(Z-) II
---'----~"-- ~
s
p,
i. e. II U(t)PU-l(Z-) II ~ ep(t-r) for z-, t - z- ~ Tp. By Remark 4.3 the quantity KT = sUPO~t-r~T II U(t)PU-l(Z-) II is finite. Hence for all t ~ z- ~ T II U(t)PU-l(Z-) II ~ Nep(t-r) ,
III. NONSTATIONARY LINEAR EQUATION
122
where N = max(l, K T e 1pl 1). This same estimate is also valid for 0 ~ 'r ~ t ~ Y. Finally, when 0 ~ 'r ~ T < t, we have
II U(t)PU-l('r) II Setting Np
=
~
II U(t)PU-l(T) II II U(T)PU-l(r) II
~
NeP(t-T) NeP(T-T) = N 2ep(t-T).
max(N, N2), we obtain the inequality
II U(t)PU-l('r) II
~ Npep(t-T)
(t ~ 'r ~ 0),
which implies for x('r)
= U('r)Pxo and x(t) = U(t)PU-l('r)X('r) = U(t)Pxo
the estimate showing that KB(P) ~ p. Thus KB ~ }. and the first formula is proved. The second is established analogously. 3. Stationary equation case. In this case we have THEOREM 4.5. The upper (lower) Ljapunov and upper (lower) Bohl exponents of the homogeneous stationary equation dxfdt = Cx (C = const) coincide with each other and are equal to the supremum (infimum) of the real parts of the numbers }. E a(C). PROOF. For the stationary equation U(t)
=
eCI, U(t, 'r)
=
eC(t-T)
and according to Theorem 1.4.1 KL
= lim
I~co
In IleGl II t
= lim
T~CO
t-T_OO
In IleC(I-T) II t - 'r
=
KB
= max{Re}.l}. E a(C)}.
The assertion in parentheses is proved analogously. DEFINITION. By the strict Ljapunov exponent of equation (4.1) is meant the limit lim In II U(t) " t
I~co
if it exists. Analogously, by the strict Bohl exponent of equation (4.1) is meant In II U('r . 11m
+ s, 'r)" s
,
if this limit exists. We have just convinced ourselves that a stationary equation has strict exponents.
4.
123
LJAPUNOV AND BOHL EXPONENTS
4. Perron's example. We will show by an example that the case ble. We consider the scalar Perron equation
"L < "B is possi-
(4.12) dx/dt = (sin In t + cos In t)x. Here ~ is a one-dimensional space and A(t) is the operator of multiplication by sin (In t + n/4). the function The solution of the equation has the form x(t) = etsinlntx(O), which implies U(t) = etsinlnt. We calculate the upper Ljapunov exponent of the equation:
.v2
"r
..
Let us now show that "B
"B = lim
= I~ 1m InIU(t)1 = I~· 1m sIn In t = 1. t t_oo t-HX)
=
.v2. We have
In IIU(t)U-l(r) II = lim t sin In t t - 1: T~OO t -
T~OO t-r-KX)
1: sin 1:
In 1:
t-f-HX>
=
lim (cos In tav
+ sin In tav),
T~OO
t~T-OO
where tav is a point lying between 1: and t (according to the mean value theorem). We now choose a sequence of pairs of points 1:n, tn as follows: In tn = 2nn
+ n/4 + e;
+ n /4. + n /4 + 'fJn, where
In 1:n = 2nn
Then for 1:n < in < tn we have In in = 2nn Under this choice of 1:n, tn for n --+ 00 we have 1:n --+
tn -
1:n
= e2mr +7r/4(e<
-
1)
00
0 < 'fjn < e.
and
--+ 00,
and hence n~OO
= lim [cos (n/4 + 7]n) + sin (n/4 + 7]n)] = n~oo
.v2 + An,
where An --+ 0 as e --+ o. Thus "B ~ .J2. But on the other hand, "B ~ M = .J2. 5. Stability of Bob! exponents. We now discuss the variation of the Bohl exponents of equation (4.1) under certain transformations of this equation (see also in this connection § IV.2). We will consider only upper exponents, since the properties of the lower exponents are established analogously. It will be convenient for us to make use of the following definition. We will say that equation (4.1) has property .?4(v, N),3) where v is real and N 3)This property was denoted by L(v, N) in the "Lectures." The new notation has been introduced in recognition of the fact that this property was first used by P. Bohl (see the Notes).
III.
124
NONSTATIONARY LINEAR EQUATION
is positive, if all of its solutions are subject to the estimate
Ilx(t) II
~ Ne->(H) IIX('t") II
(t
~
~
't").
't")
or, equivalently,
(t
(4.13)
We note that the possession of property .14(0, N) with some N > 0 is equivalent to the uniform right stability of equation (4.1). Clearly, coincides with the least upper bound of those v for which equation (4.1) has property .14(v, N) for N = N>. We note further that if an equation has property .14(v, N) for t, 't" ~ to > 0, it has property .14(v, N') with the same exponent v and some constant N' for t, 't" ~ o. For, the evolution operator U(t, 't") is bounded on a finite interval by virtue of its continuity and it is always possible (by increasing the constant N if necessary) to preserve the validity of (4.13). The following two properties of exponents immediately follow from the definitions. 1) All ofthe exponents of the equation
"B
dy/dt
=
[A(t)
+ aI]y
are obtained from the corresponding exponents of equation (4.1) by means of a shift to the right of magnitude Re a. In particular, we note that the upper Ljapunov and upper Bohl exponents of the equation dx/dt
are the numbers 1 + a and
=
(sin In t
.vT +
+ cos In t + a)x
a respectively. For -
v'T <
a < - 1 we get
"L < 0 and "B > O. Thus all of the solutions of the equation in question decrease
exponentially as t -+ 00, although the upper Bohl exponent of this equation is positive. 2) For any I > 0 the upper Ljapunov and upper Bohl exponents of the equation dx/dt
=
A(t
+ l)x
(0
~
t <(0)
coincide with the corresponding exponents of equation (4.1). We now consider along with equation (4.1) the equation dx/dt = A(t)x LEMMA
+ B(t)x.
(4.14)
4.1. Suppose equation (4.1) has property .14(v, N) and/or some 1 sup 1;;:;0
So
Hs,
S IIB('t")lld't" = 1
M s,
<
0 (4.15)
00.
Then equation (4.14) has property .14(v', N'), where v' NeNMs,s,.
So ~
=
v - NMs, and N'
=
4. LJAPUNOV AND BOHL EXPONENTS PROOF. From estimate (2.22) we deduce that for s
~
125
0
t+s
I U2(t + s, t) I
N
~ Ne-VSe
f
IIB(r) IIdr
t
Lemma 4.1 implies the following important result. THEOREM 4.6. The upper Bohl exponent 11:8 of equation (4.1) has the following stability property: for any e > 0 there exists a number > 0 depending only on e and equation (4.1) such that if
a
_ 1 1+ s lim - J
t,s--oo S
t
IIB(r)lldr < 0,
(4.16)
+ e,
(4.17)
then is ~ II:s
where is is the Bohl exponent of equation (4.14). The lower Bohl exponent has an analogous property.
PROOF. Equation (4.1) has property 86'( - II:s - e /2, N e/ 2). On the other hand, for sufficiently large to and some So > 0 1
sup I~I,
So
I+s,
J 1
IIB(r)lldz- < 20.
Therefore, by virtue of Lemma 4.1, equation (4.14) has property 86'( - II:s- e/2 - Ne/2 ·20, N') with some constant N' > 0, which implies is ~ II:s + cj2 + 20Ne/ 2• It remains to choose < cj4Ne/ 2• REMARK 4.4. Condition (4.16) is satisfied if, beginning with some to, either ~ a or ~ o.
a
IIB(t)II
J:+1 IIB(r) Ildr
COROLLARY 4.1. The upper Bohl exponent of the equation dx/dt = Ax + B(t)x is negative if the spectrum (1(A) lies in the interior of the left halfplane and condition (4.16) is satisfied with a sufficiently small > O.
a
COROLLARY 4.2.
If
liml,s~oo
(l/s) J;+s I B (r) I
dz- = 0, the upper (lower) Bohl exponents of equations (4.1) and (4.14) coincide. < 00 or limT~oo = o. This is true in particular when either Joo
IIB(z-) Ildr
IIB(r) I
Equations whose coefficients differ by a term tending to zero at infinity are said to be asymptotically comparable. Thus the upper (lower) Bohl exponents of integrally or asymptotically comparable equations coincide.
III. NONSTATIONARY LINEAR EQUATION
126
6. Estimates of exponents in Hilbert space. Suppose the space )8 is a Hilbert space .p. We recall that if H is a Hermitian operator then Am(H) and A~H) are respectively the least and greatest numbers of the spectrum (F(H), so that Am(H)(x, x)
~
(Hx, x)
~
AM(H)(x, x)
(x E
.p).
THEOREM 4.7. For any solution x(t) of equation (4.1) in a Hilbert space
(4.18)
.p
the
function ¢M(t) =
Ilx(t)llexp{-J, AM(A!Jl(z»)d'l"}
is non increasing while the function ¢m(t)
=
Ilx(t)llexp
is nondecreasing. In particular, for t
~
{-l
Am[A!Jl('l")]d'l"}
s
Ilx(s)llexp{! Am [A!Jl('l")]d'l" } ~ Ilx(t)11
(4.19)
~ Ilx(s)llexpH AM[A!Jl('l")]d'l"}. where A!Jl
=
(A
+ A *)/2 is the real part of the operator A.
PROOF. Differentiating the function
¢'t:(t) = (x(t),
x(t»)exp{- 2IAM[A!Jl('l")]d'l"},
we obtain the inequality
ft-¢'t:(t)
=
[(x'(t), x(t»)
+ (x(t), x'(t»)]exp { -
- 2(x(t), X(t»)AM[A!Jl(t)]ex p { -
~
21 AM[A!Jl('l")]d'l" }
21 AM[A!Jl('l")]d'l" }
2{(A!Jl(t)x(t), x(t» - AM[A!Jl(t)](x(t), x(t»)}exp {-
21 AM[A!Jl('l")]d'l"},
which by virtue of (4.18) implies the estimate (d/dt) ¢'t:(t) ~ 0, proving the first assertion of the theorem. The second assertion is proved analogously. COROLLARY 4.3. The Bohl exponents of equation (4.1) satisfy the estimates --- 1 t+s _ 1 t+s lim - SAm [A!Jl('l")]d'l" ~ II:B ~ lim - SAM [A!Jl('l")]d'l"; t.5---+ oo
S
t
t,s-+oo S t
5. BOUNDEDNESS OF SOLUTIONS
127
obtained by comparing formulas (4.11) with inequalities (4.19). The Ljapunov spectrum is included in the interval _11 _11 ] [ lim ~t JAm[Am(..)]dr, lim t JAM[Am(..)]dZ' . 0 a t~oo
t-'tOO
These results can be extended to any Banach space lB (see Exercise 19).
§ 5. Condition for boundedness on [0, (0) of the solutions of the inhomogeneous equation
1. Necessity of negativeness of the upper Ljapunov exponent. 4l In this section we discuss the role of negativeness of the upper Bohl exponent of equation (4.1). We consider the equation
~~ =
A(t)x
+ f(t)
(t E [0, (0)).
(5.1)
THEOREM 5.1. If on the halfline [0, (0) the solution x( t) of the Cauchy problem for equation (5.1) with the initial condition x(O) = 0
(5.2)
is bounded whenever the vector function f(t) is bounded and continuous, then there exist positive constants Nand j) such that
(t
~
0)
(5.3)
and every solution of equation (5.1) with a bounded continuous vector functionf(t):
Illflll
=
sup 0;£1",,00
Ilf(t)II
<
00,
(5.4)
is bounded. PROOF. We consider the complete space G(lB) of bounded continuous vector functions f(t) (0 ~ t < (0) with values in lB and norm (5.4). The solution of the Cauchy problem (5.1)-(5.2) is given by the formula I
x(t)
=
JU(t, Z')f(Z')d ..
(0
o
~
t < (0).
(5.5)
The right side of expression (5.5) represents for each t a bounded linear operator Vt acting from G(lB) into lB since
Ilx(t)
I
~
I
JII U(t, Z') lid.. ·111 fill· o
By hypothesis, there corresponds to each
f
E
G(lB) a bounded solution x(t)
'lBy a Ljapunov or Bohl exponent of an inhomogeneous equation (5.1) we will mean the Ljapunov or Bohl exponent respectively of the corresponding homogeneous equation.
III.
128
NONSTATIONARY LINEAR EQUATION
on [0, 00). This means that the operators Vt (0 ~ t < 00) are uniformly bounded at each element of C()8). The uniform boundedness principle therefore implies that
Ilx(t)11
=
IIVdl1 ~ klll/ill
(0 ~ t < 00),
(5.6)
where k = const. We put X(t) = U(t) and consider the function I(t) = U(t)y/X(t), where y is an arbitrary element of )8. It is an element of C()8) since I ~ y < 00. Therefore the corresponding solution of problem (5.1)-(5.2) must by virtue of (5.6) satisfy the condition
I
I
II II
I I
Ilx(t)11 ~ klIYII·
(5.7)
This solution is given by the formula x(t)
[U(t,T)
=
~~~r
dT =U(t)y·if>(t),
where if>(t)= mX(T)]-ldT. From (5.7) and (5.8) we obtain the inequality U(t)yllif>(t) ~ virtue of the arbitrariness of y E )8 and the equality
I
1 if>'(t) = X(t) =
s~p
(5.8)
kllyll, which by
I U(t)yll IIYII
reduces to the relation if>'(t)/if>(t) ~ l/k. Integrating this relation from 1 to t, we get if>(t)
~
if>(l)e(t-l)!k
and hence _1_. X(t)
= rI,'(t) ~ if>(t) > if>(1) e(t-l)!k 'f'
-
k
=
k
.
The latter relation means that (t
~
1),
where)) = 1 /k and Nl = ke1!k /if>(l). . Setting, finally; N = max (Nt. maXO~t~levt I U(t) II), we obtain the required inequality (5.3). The second assertion of the theorem follows upon noting that an arbitrary solution of equation (5.1) differs from the solution distinguished by condition (5.2) by a term U(t)xo: x(t)
=
U(t)xo
+
t
J U(t, T)/(T)dT, o
the boundedness of which follows from estimate (5.3).
5. BOUNDED NESS OF SOLUTIONS
129
The theorem is proved. From Theorem 5.2 below and the fact that there are examples of equations for which A(t) is integrally bounded and KL < < KB (see § 4.5) it follows that the converse of the first assertion of Theorem 5.1 does not hold, i.e. the presence of estimate (5.3) does not guarantee the boundedness of the solution of problem (5.1) -(5.2) for any bounded continuous vector functionf(t) on [0, (0). 2. Role of negativeness of the Bohl exponent in the case of an integrally bounded A(t). A more complete result is obtained under the additional condition of integral boundedness on [0, (0) of the operator function A(t):
°
1+1
J IIA(r) Ildz- ~
M1
(t E [0, 00)).
(5.9)
1
THEOREM 5.2. Suppose condition (5.9) is satisfied. In order for the Cauchy problem (5.1)-(5.2) to have a bounded solution on [0, (0) for every bounded continuousfunction f( t) on [0,(0) it is necessary and sufficient that the Bohl exponent of equation (5.1) be negative. PROOF. The condition is sufficient since the estimate
I U(t, z-) II
~ Ne-v(H)
(v > 0, t
~
z-)
implies
II o U(t, z-)f(z-)dz-II1 ~ Nillfill
J
J0 e-v(/-T)dz- ~ ~ Illflll· v
To prove the necessity of the condition we first show that the boundedness of the solution of the Cauchy problem (5.1)-(5.2) implies the boundedness of the solution of the Cauchy problem { dx(t)/dt = A(t)x x(to) = 0.
+ f(t)
(5.10)
The solution of problem (5.10) is given by the formula 1
x(t) =
JU(t, z-)f(z-)dz-. I,
If the function./(t) were not necessarily continuous, the solution of problem (5.10) could be regarded as the solution of problem (5.1) - (5.2) with free term
°
~ t < to, to.
t ~
In order to avoid working with discontinuous functions we consider the following problem of type (5.1)-(5.2): { dxeCt)/dt = A(t)x,(t) x.(o) = 0,
+ Ie(t),
III.
130
NONSTATIONARY LINEAR EQUATION
where 0
fc(t)
1
{- /(to)(t
=
- to
+ c)
/(t)
t < to - c,
~
for
0
for
to - c
for
t ~
~t<
to,
to·
Its solution is representable in the form t
xe(t)
=
JU(t, 1:)fc(-.)d1:
=
JU(t, 1:)fc(1:)d1: + JU(t, 1:)/(1:)d1:.
o t,
t
to-e
to
By assumption (see (5.6»),
I killfcill kill/III·
IIXe(t) ~ = Suppose now c -+ 0 for fixed t. Then the norm of the first integral tends to zero ~ k and the assertion is proved. and hence xe(t) -+ x(t); consequently, Choosing now in problem (5.10) the function/(t) = U(t, to)yjX(t), where X(t) = U(t, to) and repeating all of the arguments presented in the proof of Theorem 5.1, we get
Ilx(t) I
I
III/III
II,
X(t)
=
I
IIU(t, to) ~ Ne-v(t-t,>,
where N is a constant satisfying the single condition N
~ max { ~(~;, 02!:~1 ev(t-t,)IIU(t, to) II}, 1 ).!
= k' (1) =
t,+1 d1: X(1:)·
f.
It still remains to show that N can be chosen independently of to. To this end we note that
X(t)
=
I U(t, to) I
~ en,IIA(t) Iidt ~ eM,
(to ~
t ~
to
+
1).
But then in the first place (1) ~
t,+l Je- M,d1:
=
e- M ,
t,
and in the second place max O~t-t,~l
I
{ev(t-t,) U(t, to)
II} ~
max
ev(t-t,)+M, = ev+M,.
O~t-t,~l
Thus we can put N = e1/HM'max{1,k}.
5.
131
BOUNDEDNESS OF SOLUTIONS
The theorem is proved. 3. Essentiality of the condition of integral boundedness. Here we give an example showing that condition (5.9) in Theorem 5.2 cannot be dropped. To this end we consider a continuously differentiable positive scalar function u(t) having the following properties: t
Su(s)ds <
a)
o
u(n - an) > nu(n)
b)
u(t),
(n = 1,2,.··; an
+ r(t),
An example of such a function is u(t) = et
r(t) =
o {
nensin2 {2: n (t - n
+ 2an)}
-+
0).
(5.11)
where ~ t
for
n - 1
for
n - 2a n
< n - 2an
~t<
n,
with an = ] J2n+1ne n. If we put a(t)
= u(t)
~
utt)
= -
~~~tl = -
[In u(t)]',
the evolution operator of the scalar differential equation dx Jdt = a(t)x will be U(t, s) = u(s)Ju(t). The Bohl exponent of this equation is obviously nonnegative and in fact infinite since U(n
+ m, n
- an) > ne- m
(m, n = 1,2,.··).
But the solution of the Cauchy problem dx -dt = a(t)x
+ g(t),
x(O) = 0
for a bounded function g(t) will be bounded since t
x(t) = SU(t, s)g(s)ds o implies that Ix(t)1 =
I!
The function
I
I
! u(~~~?) ds ~ ~ax Ig(s) I t
= I
U(t, s)g(s)ds
-
t
Su(s )ds 0
u(t)
~ ~ax Ig(s)l.
S;+1 a(r) dr in this example is of course unbounded.
132
III. NONSTATIONARY LINEAR EQUATION
§ 6. Equations with precompactly valued operator functions I. Criterion for negativeness of the upper Boh! exponent. In the preceding section we demonstrated the importance of the role of negativeness of the (upper5) Bohl exponent of equation (4.1). But our ability to verify that an equation has this property is presently limited to only the simplest cases, when the equation is stationary or differs only slightly from a stationary equation (see Corollary 4.1 of Theorem 4.6). In this section we indicate another class of equations for which the negative ness of the Bohl exponent can be determined from the properties of the coefficient of the equation. We first establish an auxiliary result. THEOREM 6.1. Suppose the Bohl exponent dx -=
(0
A(t)x
dt
irE
of the equation
~
t
< (0)
(6.1)
is finite. In order for it to be negative it is necessary and sufficient that there exist positive numbers T and q < I for which the following condition is satisfied: for every x E 58 and t ~ there exists a number Ox,t!E [0, T] with the property that
°
II U(t
+
Ox,t, t)xll ~ qllxll·
(6.2)
PROOF. NECESSITY. If the Bohl exponent is negative, there exist positive numbers Nand)) for which equation (6.1) has property 9.6'()), N). Then, for any T such that Ne- vT < I, estimate (6.2) is satisfied when Ox. t = T and q = Ne- vT. SUFFICIENCY. Suppose 0 ~ to < t < 00. In view of the "Continuity of the operator U(r, -r') there exists a 0 > 0 such that II U(-r, -r') II < 1/ q for -r and -r' satisfying the conditions to ~ -r,
For these -r, -r' and for any x Ilxll
-r' < 2t, E
l-r' - -rl ~ O.
58 the inequality
= II U(-r, -r')U(-r', -r)xll ~ IIU(-r,-r')II·IIU(-r',-r)xll < (l/q)IIU(-r',-r)xll
implies that IIU(-r, -r')11 > qllxll, and thus Ox.t > 0 provided to ~ -r
2t. Let Xo be an arbitrary element of 58. We put tl t2
= =
to tl
+ Oxo.to; + 0 t, ; Xlo
Xl X2
= =
U(tb to)xo; U (t2, t1)Xl
=
U (t2, to)xo;
5)Later in this section the word upper will be dropped for the sake of brevity.
< -r + Ox.t ~
6. PRECOMPACTLY VALUED OPERATOR FUNCTIONS
l33
(k = 1,2,.··).
After a finite number m (m < (t - to) 10) of steps it turns out that tm < t < tmH' From the equality U(t, to)xo = U(t, tm)xm and the estimate Ilxmll ~ qmllxoll we get (6.3) where k = SUPO:2t-r~T I U(t, z-) I < 00 by virtue of the finiteness of KB (see Theorem 4.2 and Remark 4.3). Since Ox,.t, ~ T (k = 0, 1,. .. , m + 1), we have t ~ to + (m + 1) T and consequently m + 1 ~ (t - to)T. This together with (6.3) implies I U(t, to)xoll ~ (kl q)q(t-t.)!T Ilxoll or
I u(t, to)xoll
~ Ne-v(t-t.) Ilxoll,
where N = klq and)) = T-l In q-l. Q.E.D. REMARK 6.1. The condition of Theorem 6.1 can be verified by verifying it for sufficiently large t (t > t, where t does not depend on x). In fact, whenever the condition is satisfied for t > t, it can be asserted that for some N > 0 and)) > 0 equation (6.1) will have property .%I()), N) in the interval [t, OJ). But then, as is easily seen, under a suitable choice of N > 0 equation (4.1) will have property £!de)), N) on the whole halfline [0, OJ). The usefulness of Theorem 6.1 is illustrated in particular by the following application of it. THEOREM 6.2. Suppose equation (6.1) has a finite Bohl exponent KB andp is any positive number. The Bohl exponent of the equation is negative precisely when there exists a positive constant c for which
I II U(t, z-)xllpdt }l!P ~ c II x I
oo {
(to
~
z- < 00).
(6.4)
PROOF. The necessity of this condition is obvious. Its sufficiency can be proved by showing that it implies condition (6.2) of Theorem 6.1. For suppose the contrary, i.e. suppose that for any q (0 < q < 1) and any T> 0 there exist Xo and Z-o such that .e
I U(t, z-o)xoll > qllxoll
(t
E
[Z-o, Z-o
+
Tn.
Then
=
~+T
J II U(t, z-o)xollpdt ~ J II U(t, Z-o)Xollpdt ~ qPllxollpT.
to
~
Choosing qPT > c P, we arrive at a contradiction with condition (6.3). The theorem is proved. REMARK 6.2. When p ~ 1 condition (6.4) can be replaced by the simpler condition
III.
134
sup o~~
NONSTATIONARY LINEAR EQUATION
S I U(t, -r)xllpdt <
(x
00
E ~).
(6.5)
~
In fact, condition (6.5) can be written in the form 00
sup o~~
S Ilei(t)U(t, -r)xllpdt <
00,
0
where e~(t) is the characteristic function of the interval [-r, 00). This inequality implies the boundedness at each element x of the family of operators V~: x --+ e~(t)U(t, -r)x acting from ~ into the space Lp(~) of p power integrable functions x(t) (0 ~ t < 00) with values in ~. When p ~ 1 the space LP(~) with norm Illxlllp = U;'llx(t)ll pdt}1IP is a Banach space. Therefore, by virtue of the uniform boundedness principle, the set of operators VI is bounded in norm, and this is condition (6.4). REMARK 6.3. The condition that the Bohl exponent be finite encountered in Theorems 6.1 and 6.2 is obviously satisfied if the operator function A(t) is integrally bounded: 1+1
sup S I~O
IIA(-r)lld-r <
00
I
(a fortiori if it is simply bounded). We note further that condition (6.2) is satisfied if the relation lim Ilx(to ~~OO
+ -r)11 = 0
holds uniformly in to and x(to) for any solution of equation (6.1) satisfying the condition Ilx(to) I < 1. Thus from the latter condition and the integral boundedness of A(t) we obtain the negativeness of the Bohl exponent of equation (6.1). Moreover, it is easily seen that for equations with an integrally bounded A(t) the condition in question is equivalent to the negativeness of the Bohl exponent. 2. Auxiliary propositions on compact families of operators.
r
LEMMA 6.1. Suppose is a compact 6 ) set of operators of[~] whose spectra lie in an open set G. The~n the set of resolvents R).(A) = (A - AI)~l, where A ¢ G and A E is bounded in norm. In addition, there exists a bounded closed set F c G containing the spectra of all of the operators of
r,
r.
r
PROOF. Suppose there exist a sequence of operators An E and a sequence of complex numbers An ¢ G such that I R)..(An) I --+ 00. It can be assumed without
'lThis means that every sequence of operators in operator of r.
r
contains a subsequence converging to an
6. PRECOMPACTLY VALUED OPERATOR FUNCTIONS
135
loss of generality that An --+ Ao (E r). But then Theorem 1.2.1 implies I RiAn) I --+ IIR,l(Ao) II· We have obtained a contradiction, which proves the first assertion of the lemma. Suppose now ito is a boundary point of G whose every neighborhood contains a point of the spectra. of the operators of We consider a sequence An --+ ito, where itn E (l(An), An E It can be assumed that An --+ Ao ( E r). But the spectrum (l(Ao) is closed and lies in G. For sufficiently large n the set (l(An) is in an arbitrarily small neighborhood of dAo). This again brings us to a contradiction, which proves the second assertion of the lemma.
r.
r.
r
LEMMA 6.2. Suppose is a compact set of operators of [lB] whose spectra lie in a fixed halfplane Re A < - J.io (J.io > 0). Then there exists a constant No such that for all A E
r
PROOF. As was shown in Lemma 6.1, there exists a domain G bounded by a simple smooth closed contour lying in the halfplane Re A < - J.io (J.io > 0) and containing the spectra of all of the operators of r. By the same Lemma 6.1 the resolvents RiA) are uniformly bounded outside G and, in particular, on the contour
r
r:
(A
Er,
A
En.
From the formula
we obtain the required estimate (A Er),
where I is the length of the contour r. DEFINITION. An operator function A(t) (0 ~ t < (0) will be said to be precompactly valued if its range is precompact in [lB], i.e. if every sequence A(tn) contains a subsequence converging to an operator A of [lB]. Clearly, the co.ntinuity of A(t) implies that it is precompactly valued on each finite interval of variation of t. An operator C will be called an w-limit operator of A(t) if there exists a sequence tn --+ 00 such that A(tn) --+ C. Let = {C} be the set of all w-limit operators of A(t). It is easily seen that is a closed set and moreover, as a subset of the precompact set {A(t)} U r, is precompact. We emphasize that the values themselves of A(t) are generally not contained in If the limit limhcx,A(t) = A( (0) exists, for example, the set of all w-limit operators
r
r
r
r.
III.
136
NONSTATIONARY LINEAR EQUATION
of A(t) consists only of the single operator A( (0). LEMMA 6.3. Suppose the spectra of the w-limit operators of a precompactly valued operator function A(t) lie in one and the same halfplane Re A < - ))0 ())o > 0). Then there exists a number To > 0 such that when t > To
IleA(t)T11 < Noe-v,T, where No and
))0
do not depend on t.
PROOF. By virtue of the preceding lemma it suffices to prove that there exists a To such that the spectra of the operators A(t) lie in the halfplane Re A < - ))0 when t ~ To. Reasoning by contradiction, we suppose that there exists a sequence tn - 00 for which each operator A(tn ) has a point of its spectrum lying outside this open halfplane. It can be assumed without loss of generality that A(tn) - C E Since the spectrum of C lies in the halfplane in question, the spectra of the A(tn ), beginning with some n, also lie in it, which contradicts the supposition. The lemma is proved. 3. Condition of negativeness of the Bohl exponent for an equation with a precompactly valued operator function. We will say that an operator function A(t) satisfies condition Se,L for some c > 0 and L > 0 if there exists a number T > 0 such that the inequality I A(s) - ACt) I ~ cis satisfied when s, t ~ T and Is - t I ~ L. Clearly, if A(t) satisfies condition Se,L, it satisfies condition Se',L' for any c' > c and L' ~ L (L' > 0) as well as condition Sne,nL for any natural number n. A function A(t) is said to be stationary at infinity if it satisfies condition Se,L for any arbitrarily small c > 0 and some positive L (and hence arbitrarily large L). In particular, it is easily seen that a function A(t) is stationary at infinity if one of the following two conditions is satisfied: a) the limit limt~ooA(t) exists; b) A'(t) exists for sufficiently large t and limH)oA'(t) = O. Clearly, a function of the form
r.
A(t)
=
AI(t)
+
Az(t),
(6.6)
where AI(t) satisfies condition a) and Az(t) satisfies condition b), is also stationary at infinity: Condition Se,L is satisfied if a') the condition IIA(t) - Aoll ~ cj2, where Ao is a constant operator, IS satisfied for sufficiently large t; b ') A'(t) exists and the estimate IIA'(t)11 ~ cjL is valid for sufficiently large t. On the other hand, if we set A(t) = AI(t)
+
(6.7)
Az(t),
where 1 t+L
Az(t)
= L S A(z)dr; t
1 t+L
AI(t)
= L S [A(t) t
- A(r)]dr,
6. PRECOMPACTLY VALUED OPERATOR FUNCTIONS
lj
I
we can represent a function satisfying condition S.,L in the form of a sum of two functions such that
IIA (t) I ~ c; IIAz(t) I 1
LJ. -
= II A(t +
A(t) II
~
1
for sufficiently large t. In particular, letting c tend to zero, we get that every stationary at infinity function is representable in the form (6.6), where A 1(t) satisfies condition a) and Az(t) satisfies condition b). THEOREM 6.3. Suppose an operator function A(t) is precompactly valued and the spectra of its w-limit operators lie in one and the same halfplane Re A < - Vo (vo > 0) (then by Lemma 6.3 there exists a To > 0 such that when t > To
IleA(thll
~ Noe- voT ,
(6.8)
where No and Vo do not depend on t). If, in addition, A(t) satisfies condition S.,Lfor sufficiently small c > 0 and sufficiently large L (c < vo/No, L> In No/(vo - Noc») depending only on the set of w-limit operators, the Bohl exponent KB of equation (6.1) is negative,
r
PROOF. By assumption, for sufficiently large 7:
IIA(t) -
I
> to) we have
+ L). the case when a = 7:, b = 7: + Land (7: ~ t ~ 7: + L). (7: ~
A(7:) < c
We apply estimate (2.22) for
(7:
t
~ 7:
Since in the present case U1(t, s) = e(t-s)A(,.) while
Ile(t-S)A(T)
I
~
Noe-vo(t-s) ,
it can be asserted that
I U(t, s) I
~ Noe-v(t-s)
(7: ~ S ~
t ~
7:
+ L),
where v = Vo - Noc > 0 since c < Vo / No. Thus
I U(7: + L, 7:) I
~ Noe- vL (= q).
It is easily seen that q < I for L > In No/(vo - Noc). Therefore condition (6.2) of Theorem 6.1 is satisfied (for Ox,t == L and the indicated q) when t is sufficiently large. Since the precompactly valued operator function A(t) is bounded, Theorem 6.1 is applicable (see Remarks 6.1 and 6.3) and consequently K < O. REMARK 6.4. From the proof of the theorem it follows that the condition of a precompact range of the operator function A(t) can be replaced by the condition that estimate (6.8) exist for sufficiently large t (t > To). The latter condition (i.e. estimate) will hold if the spectrum of A(t) lies in the
138
III. NONSTATIONAY LINEAR EQUATION
halfplane Re A < - Vo for t > To and if, in addition, the resolvents (A(t) - 1./)-1 are uniformly bounded on the straight line Re A = - Vo for t > To. It can be shown directly that when estimate (6.8) is satisfied for t > To the Cauchy problem
f ~~
= A(t)x + f(t) lx(O) = 0
(0
~
t < (0),
has a bounded solution for any continuous vector functionf(t). We recaH that when A(t) is integrally bounded the latter property is equivalent to the negativeness of the Bohl exponent of equation (6.1). We formulate a theorem which in a certain sense can be regarded as the converse of Theorem 6.3. THEOREM 6.4. Suppose an operator function A(t) is precompactly valued and the Bohl exponent IrB of the corresponding equation (6.1) is negative (i.e. the corresponding equation (6.1) has property &?l(v, N) for certain positive N and v). If, in addition, A(t) satisfies condition Se,L for sufficiently small e > 0 and sufficiently large L (e < vlN, L > In NI(v - Ne»), the spectra of its w-limit operators lie in one and the same halfplane Re A ~ - Vo (vo > 0). PROOF. Suppose C is an w-limit operator of A(t), i. e. there exists a sequence tn -+ 00 such that A(tn) -+ C in [)B]. Then for sufficiently large n and arbitrarily small 0 > 0
IIC -
A(tn)
I
< O.
On the other hand, since by assumption A(t) satisfies condition Se,L, for sufficiently large n, IIA(tn) -
A(t)11
~ e
Therefore, for sufficiently large n
IIC- A(t)11 and by Lemma 2.3 (for s
~ e
+0
= tn and t = tn +
IleLCIl
L) we get
~ Ne-v't,
where v' = v - N(e + 0). It follows from Lemma I.2.2 that the spectrum a(C) of C lies in (and hence, by decreasing 0, in the interior of) the closed halfplane ReA ~ In NIL - v'
= - [v - N(e +
0) - In NIL]
= -Vo.
Since by assumption v - N e - L -1 In N > 0, we can choose 0 so small that Vo > O. In view of the fact that the obtained value of Vo does not depend on C the theorem is proved.
EXERCISES
139
As a corollary of Theorems 6.3 and 6.4 we obtain the following proposition. THEOREM 6.5. Suppose A(t) is a precompactly valued operator function that is stationary at infinity. In order for the Bah! exponent II:B of equation (6.1) to be negative it is necessary and sufficient that the spectra of the w-limit operators of A(t) lie in one and the same halfplane Re A < - ))0 ())o > 0). By the right spectral bound of an operator A is meant the supremum of the real parts of the points of its spectrum (l(A). This number coincides with the Ljapunov exponent (and hence the Bohl exponent) of the stationary equation dx/dt = Ax. The following fact is a consequence of the results presented above. THEOREM 6.6. Suppose A(t) is a precompactly valued operator function that is stationary at infinity. Then the upper Boh! exponent II: of equation (6.1) coincides with the supremum of the right spectral bounds of the w-limit operators of A(t). PROOF. Let II: denote the supremum mentioned in the theorem. We pass from equation (6.1) to the equation dx / dt = Aa(t )x,
where Aa(t) = A(t)
+ aI
(a is a real number).
Under such a passage the numbers II:B and II: go over into the numbers II:B + a and II: + a. By virtue of Theorem 6.5 these numbers can be negative only simultaneously. Hence II: = II:B. REMARK 6.5. An analogous property is possessed by the lower Bohl exponent. We note that in the case of a finite-dimensional space Q3 the condition of a precompact range of A(t) is equivalent to the condition of bounded ness of A(t). Clearly, Theorems 6.3 - 6.6 remain valid if the condition of a precompact range of A(t) is replaced by the condition of a precompact range of A(t) beginning with a sufficiently large t, i.e. by the condition of a precompact range of the translate A(t + /) of A(t) with a sufficiently large l. EXERCISES We adopt the following definition in the exercises below (see V. P. Potapov [1] and Ju P. Ginzburg [3]). Suppose given on [a, b] a scalar function J(t) and an operator function F(t) with values in [18]. We consider the product fl, =
e fC ,.) [FCt.)-FCt·-,)J..·efCn) [FCt,)-FCto)]
for an arbitrary subdivision ,d The limit
=
(a
(t j _ l :<::.: r: j :<::.: tj)'
= to < tl < ... < tn = b) .
b
lim fl, = I.1HO
Ia
exp[J(t)dF(t)]
(0.1)
140
III.
NONSTATIONARY LINEAR EQUATION
is called a multiplicative Stieltjes integral if it exists in the norm topology of PS]. 1. Show that the multiplicative integral (0.1) exists if f(t) is a continuous function and F(t) is a continuous function of bounded variation in [)B]. Derive the estimates (v(t) = Var[Q,tlF) b
II
II
II
f
II
! exp[f(t)dF(t)] . ;:::; e b
I/(t) Idv(t)
a
;
!b exp[f(t)dF(t)]-I II ;:::; (b! ff(t)fdv(t))e
!bexp[f(t)dF(t)]-I -
b
jl/(t) Idv(t) a ;
! ff(t)fdv(t) )2 e
(b
II
!f(t)dF(t) ;:::;
jl/(t) Idv(t) a
2. Prove the existence of the more general multiplicative Stieltjes integral b
Ja exp [S1(t )dF(t )S2(t )],
(0.2)
where SI(t) and Sz(t) are continuous operator functions with values in [)B, )Bt] and [)Bj, respectively. Use this integral to obtain a representation of the solution of the integral equation 0.23). 3. Consider the integral equation t
x(t) = f dF('I:)x(r) o
+
)B]
(0.3)
g(t),
where g(t) is a continuous vector function of bounded variation. Obtain formulas analogous to (1.21) for the solution of equation (0.3), using as U(t, r) the multiplicative integral t
U(t, '1:)
= J< exp
[dF(s)].
Extend the results of Exercise 11 below to this case. 4. Let A(t) and B(t) (a;:::; t;:::; b) be continuous operator functions with values in [)B]. We introduce the "alternating" multiplicative integral b
J exp
[A(t)dt] exp [B(t )dt]
(0.4)
a
== lim
n
IT
eACti)Ct.- ti-l)eBCtl.-l)(tk- t i - 1).
/.11-70 k=l
a) Extend Exercise I. 15 by showing that this integral exists and b
J exp
a
b
[A(t)dt] exp [B(t)dt]
= aJ
exp [A(t)
+ B(t)]dt.
(0.5)
Establish the formula t
Jto exp
[A(r)d'l:] exp [B(r)dr]
= U(t, to)
t
f exp [U-1('I:, to)B(r)U('I:, to)dr],
to
where U(t, to) is the evolution operator of the equation dx/dt = A(t)x. Hint. Make the substitution V(t, to) = U(t, to)S(t) in the equation V = A(t) V
+ B(t)V.
141
EXERCISES
b) Extend the results described above to the case of an integral of the form b
Ia
exp [dG(r)] exp [dF(r)],
where G(t) is a continuous operator function and F(t) is an operator function of bounded variation. This integral includes, in particular, the case when G(t)
= f A(t)dt,
F(t)
= f B(t)dt,
where A(t) and B(t) are Bochner integrable operator functions. In this case we agree as before to write the multiplicative integral in the form (0.4). The integral of form (0.4) and formula (0.5) for it can also be extended in certain cases to differential equations with unbounded operator functions. The results obtained in this direction lead to a representation of the solutions of partial differential equations of parabolic and Schrodinger types in the form of functional integrals (see Ju. L. Daleckii [7]). 5. The following results extend a special case of Theorem 3.2. We consider in a Banach space ~ the stationary equation (0.6)
dx/dt = Ax
and the equation dy/dt = Ay
with an integrable operator function R(t)
'Jo
+ R(t)y
(0.7)
E [~]:
IIR(t) [[dt
<
00.
(0.8)
We assume that (0.9)
the spectral set iTo(A) lying on the imaginary axis and iT _(A) lying in the interior of the left halfplane. a) Show that if in addition (0.10)
where Po is the spectral projection of A corresponding to iTo(A), then equations (0.6) and (0.7) are asymptotically equivalent. Hint. An attraction correspondence x(t) H y(t) between the solutions of equations (0.6) and (0.7) is established by the relation x(t o) = y(t o)
+ 'J PoeA(to-<) R(r)y(r)dr, to
where to is a sufficiently large fixed number (the correspondence depends on the choice of to provided iT _(A) *- 0(see Theorem 3.2)). b) The corresponding result in a Hilbert phase space can be formulated more simply: if the stationary equation (0.6) is right stable and condition (0.9) is satisfied, equations (0.6) and (0.7) are asymptotically equivalent. . Condition (0.9) is atuomatically satisfied in the finite-dimensional case, and then the result coincides with Levinson's theorem (E. A. Coddington and N. Levinson [ 1]). It is not clear whether condition (0.9) can be dropped in the infinite-dimensional case. c) Condition (0.10) is satisfied if the operator Ao = PoA is of simplest type (see Exercise 1.1).
III.
142
NONSTATIONARY LINEAR EQUATION
Extend the results described above to the case when Ao is an operator of algebraic type (see Exercise 1.9). In this connection condition (0.8) must be replaced by the condition
fto tnIIR(t)lldt <
(0.11)
00,
where n is the maximum index of the imaginary eigenvalues of Ao (see V. A. Jakubovic [3]). 6. In this and the following exercises we consider in a doubled Hilbert space .p(2) = .p the differential equation on a halfline
= AX + V(r)X
YrdX/dr
(0::;;
r< (0).
EB .p
(0.12)
Here y, =(-9 ~), where 1 is the identity operator in .p, while VCr) is an operator function with values in [.p(2)] which is caned the potential of equation (0.12). It will be convenient to understand by X the column
X( . A) r,
1(r; A») = (x X (r; A) , 2
whose elements are not vector functions with values in .p but operator functions with values in [·P]. When equation (0.12) has a Hermitian potential V = V* it is called a canonical equation (in the case of real .p and real A it is called a Hamiltonian equation (cf. § VA». We note that the operator Yr is an operator of simplest type with its spectrum on the imaginary axis consisting of the two points ±i:
Yr =
iP c+)
-
iP c-)
(p -
»)
c+) --:[ _ 1 (I +l,/r, - . tJ:
(0.13)
while the operator Y = i Y r is an indefinite Hermitian operator. a) Show that the Cauchy operator of equation (0.12) is y-unitary (U YrU* = U* YrU = Yr) for V = V* and 1m A 0; y-nonexpansive (U*(iy r)U::;; iy r) for 1m V ~ 0 and 1m A ~ 0; Y-noncontractive (U*(i Yr) U ~ i Yr) for 1m V::;; 0 and 1m A ::;; 0; unitary (U*U = UU* = I) for yV = V Y and Re A = O. b) Show that a canonical equation can be reduced by means of a unitary transformation X = Q(t)Y (QQ* = Q*Q = I) to a form in which the Hermitian operator V (= V*) is also y-Hermitian (yV = V Y). Hint. Represent V in the form of a sum of y-Hermitian and y-skew-Hermitian terms:
>
VCr)
=
VCr) - ~r VCr),? r
+
VCr)
+ ~r VCr),? r
and make use of the last result of exercise a) (see also § 2 of Chapter IV). 7. Suppose in addition to the conditions of the preceding exercise that the potential VCr) is integrable on [0, (0):
f II VCr) Ildr o
<
00.
Then equation (0.12) and the equation YrdY/dr
= AY
(0.14)
are integrally comparable. a) Show that equations-(0.12) and (0.14) are asymptotically equivalent for 1m A = 0 and that there exists precisely one operator function Av(A) (1m A = 0) such that when X(O; A) = Av(A)Y(O; A)
lim IIX(r; A) - Y(r; A)II = O.
143
EXERCISES
The operator AvO.) is called the asymptotic equivalence operator (A-operator) of equations (0.12) and (0.14). Hint. Make use of Theorem 3.2. b) Prove that an A-operator is a ),-unitary operator when V = V* and a ),-contraction when 1m V ~ 0 (1m l = 0). Hint. Show that Av(l) = lim (U 1 1(r; l)U2(r; l), r->oo
where U 1 and U2 are the Cauchy operators of equations (0.12) and (0.14), and make use of the result of Exercise 6a). c) Show that there exist solutions XCk)(r; l) (k = 1,2) of equation (0.12) satisfying the initial conditions XP)(O; l)
= 0,
X(2)(O; l)
=0
and having the following asymptotic behavior at infinity for 1m l = 0:
_IiI) eiArSp) + (:1) riAr + 0(1); ~) ear S2(l) + ( -/1) ril.r + 0(1), (r 00; I
XCl)(r; l) = ( X(2)(r; l) = (
->
= /p),
where the Sk(A) (k = 1,2) are operator functions with values in [.),'i] (the S-matrices of equation (0.12) in the terminology of quantum mechanics). d) Prove that the S-matrices of equation (0.12) are unitary matrices when V = V* (1m l = 0). H1tzt for Exercises 7c) and 7d). Use the fact that the leading terms of the indicated asymptotic expansions are solutions of the unperturbed equation (0.14) and therefore (X\l)(O; l)
o
0)
X~2)(0;
l)
where IIAjk(A)IIl = Av(A). The Sil) (k = 1,2) are found by equating the nondiagonal terms of the product of matrices on the right side of the equality. Since an A-operator is a ),-unitary operator, the Sk(A) (k = 1,2) are unitary matrices when V = V* (1m l = 0) (see Exercise 6a». It should be noted that when dim :P 00 the S-matrices Sil) (k = 1,2) for the canonical equation (0.12) admit a completely intrinsic characterization and either of them can be used to uniquely recover the normalized potential V (V = V*, )'V = V)') and consequently the matrix Av(A). Moreover, there exist comparatively simple formulas for directly recovering Av(A) by either of the S-matrices. All of these results are in the papers of M. G. Krein and F. E. MelikAdamjan [1,2], from which Exercises 7a)-7d) have been taken. Other aspects of the S-matrix for canonical equations in Hilbert space (dim:P ~ 00) have been studied by V. M. Adamjan [1]. S. Show that Theorem 6.2 remains valid if condition (6.4) is replaced by the condition
<
{f.' II U(t, 7:)xll'd7: }V' ~ c Ilxll or whenp
~
1 by
,
sup
to~t<(X)
f I U(t, 7:)xll'd7:
to
< 00.
9. Prove that the upper Bohl exponent of equation (4.1) is negative if the following conditions are satisfied (E. A. Barbasin [1], Lemma 5.5):
,
sup
to~t<=
f UII(t, 7:)lld7: t'J
< co;
sup
tQ:;;;;'t
II U(t, to) I
<
00.
III.
144
NONSTATIONARY LINEAR EQUATION
Hint. Make use of the result of the preceding exercise. 10. Prove the following theorem (D. L. Kucer [1]): in order for the Cauchy problem (5.1)-(5.2) to have a bounded solution for any vector function f(t) satisfying the condition fO'lIf(t)IIP dt co for some fixed p 1 it is necessary that the following condition be satisfied for certain N> 0, ))>0:
<
>
(l/p+l/q=l,
t>O),
<
and if suptIIA(t)11 co it is necessary and sufficient that the upper Bohl exponent of equation (5.1) be negative. 11. Prove the following theorem (R. Bellman [1] and D. L. Kucer [1]): in order for the Cauchy problem (5.1)-(5.2) to have a bounded solution for any vector function get) satisfying the condition fO' Ilg(t)lldt co it is necessary that the following condition be satisfied:
<
sup I Vet, 0)11 t~O
and ifsuptliA(t)11
< co,
< co it is necessary and sufficient that the following condition be satisfied: sup I Vet, to) I < co. t.to~O
12. Verify that the proofs of the theorems formulated in Exercises 10 and II can be extended to the case obtained by replacing the bounded ness condition suptIIA(t)11 co by the integral bound1 IIA(t)lldt co (E. A. Barbasin [1]). edness condition SUPt 13. Consider in place of equation (5.1) the more general integral equation
<
<
n+
x(t)
t
= Xo + of A('r)X(T)dT -;- get),
(0.15)
where g(t) is a continuous function of strong bounded variation. Prove the following theorem (E. A. Barbasin [1]): in order for equation (0.15) to have a bounded solution for every function get) of bounded variation on the hal/line [0, co) it is necessary and sufficient that the following condition be satisfied: sup I U(t, T)II t.<
<
00.
14. Under the conditions of the preceding exercise prove the following theorem (E. A. Barbasin [1]): in order for equation (0.15) to have a bounded solution for every function g (t) satisfying the cond4tion SUPt>o Var".t+lI get) co it is necessary and sufficient that the upper Bohl exponent of the
<
equation x'(t) = A(t)x(t) be negative. 15. We consider in a Hilbert phase space
.p the differential equation
dx/dt = A(t)x
(O~
t< 00)
(0.16)
the real part of the coefficient of which is integrally bounded: sup
O~t
t+1
f
t
Re A(T)dT
< co.
Show that the following assertions are equivalent. a) The upper Bohl exponent of equation (0.16) is negative. b) The Cauchy problem . { dx/dt = A(t)x x(O) = 0
+ f(t)
(0 ~
t< co),
(0.17)
has on [0, co ) a bounded solution x(t) for each bounded continuous vector function f(t). c) There exists a bounded uniformly positive operator function W(t)>> aI (0 ~ t
145
EXERCISES
that the following condition is satisfied fof any solution x(t) of equation (0.16): (d/dt)(W(t)x(t), xU» ~ - {3l1x(t)1I2
({3>0,0 ~
t< 00).
The equivalence of these assertions under the condition that A(t) be bounded was proved in the finite-dimensional case by I. G. Malkin [3] with the use of Perron's technique of transforming the homogeneous equation into a triangular form [instead of b), Malkin dealt with the stronger assertion that the solutions of equation (0.17) are bounded under any initial condition x(O) = Xo and not just the condition x(O) = 0]. Hint. Go over from equation (0.16) to the equation x = [Re A(t)]x by means of a unitary kinematic similarity transformation (see § IV.2). The equivalence of a) and b) then follows from Theorem 5.2. To determine the operator W(t) consider the differential equation dW/dt
+ WA + A*W =
- H(t)
for a uniformly positive H(t). Show that the bounded solution of this equation is given by the formula 00
W(t) = f U*(r, t)H(t)U('r, t)d-r. I
16. Extend the result of the preceding exercise to the case of a Banach phase space under the assumption that A(t) is integrally bounded. Hint. Introduce in !8 the new norm (depending on t) 00
Ilxll, = ,f I U(r, t)xlld-r (see § 11.2). 17. Suppose that a periodic operator function A(t) = A(t + T) is continuous and that for each t E [0, T] the spectrum Il(A(t» lies in the halfplane Re C~ - v O. a) Show that the equation dx/dt = AA(t)x has a negative Bohl exponent for all sufficiently large positive Aand, moreover, has property .'?a(AVo, N) for some Vo O. This assertion is equivalent to the following: the Bohl exponent of the equation dx/dt = A(ct)x is negative for 0 ~ c co whenever co is sufficiently small. Hint. Make use of Theorem 6.3 (for a direct proof see Lemma VIII.2.1). Suppose the spectrum Il(Ao) of an operator Ao lies in the interior of the left halfplane and B(t) is a continuous periodic operator function. b) Show that there exist constants q > 0 (depending only on Ao) and co > 0 such that the Bohl exponent of the equation
<
>
<
= (Ao + B(ct»x
dx/dt
is negative for IIBII ~ q, 0 ~ c < co. Hint. Make use of the result of Exercise 15a). c) Extend the results given in a) and b) to the case of almost periodic operator functions A(t) and B(t) or to the sti11~more general case of precompactly valued uniformly continuous operator functions. 18. Consider in a Hilbert phase space fl the equation A dd2}
-
t
+B
ddx
t
+ [Co + Clct)] x
(0.15a)
= 0,
where A, Re B and Co are uniformly positive operators and Cj(-r) is a periodic operator function. a) Show that there exist constants q 0 (depending only on the operators A, B and Co) and co> 0 such that the upper Bohl exponent of equation (0.15a) is negative for 0 ~ c co and o
>
<
146
III.
NONSTATIONARY LINEAR EQUATION
This assertion can be interpreted in the theory of parametric resonance as follows: if a damped system is exponentially stable (Re 0), it wilI remain exponentially stable under a sufficiently small and sufficiently slow parametric excitation. Hint. Make use of the results of Exercises 15b) and 11.8. b) In the scalar case it is known that for sufficiently large real B equation (0.15a) has solutions that together with their first derivatives tend to zero for any c O. In the case considered by us the validity of this assertion for small c 0 follows from a); for large c 0 it will follow from the results of Chapter V. The conditions under which it is valid for B = B* and all c 0 are not known (in this case the equation is said to be unlimitedly stable). An interesting investigation of unlimited stability in the scalar case has been conducted by V. A. Jakubovic in [4], where the literature on this question is indicated. 19. Obtain for the solutions of equation (4.1) with a continuous operator function A(t) in an arbitrary Banach space )8 estimates generalizing the Wintner estimates (4.19) (S. M. Lozinskii [1]). Use them to derive estimates for the Bohl exponents of the equation. Hint. Replace the expressions AM(A!Jl('!")) and Am(A!Jl('!")) by A[A('!")] and - A[ - A('!")] respectively (see Exercises 1.18-1.20).
B»
>
>
>
>
NOTES The Ljapunov exponents were introduced back in 1892 under the name of characteristic numbers (differing from them in sign) for the finite-dimensional case of solutions of differential equations in the famous doctoral dissertation of A. M. Ljapunov. The results obtained by him have received further development in many papers (see B. F. Bylov et a!. [1]). The investigations presented in this book are closely connected with the notion of Bohl exponents; in particular, the main results of the present chapter are concentrated about the upper Bohl exponent. This fundamental notion was first introduced by P. Bohl in 1913 in a paper [2] published in a well-known mathematical journal: but for some reason, like many other results of this remarkable and apparently very modest mathematician, it went unnoticed. An attempt by the present authors to establish just what was done by P. Bohl in this memoir has led them to some sensational (at least for them) conclusions.7) This work actually contains a number of results that are presently well known from the works of other authors published up to 15 and more years later (see also the Notes to Chapter VII). The upper Bohl exponent taken with opposite sign was called the index by P. Boh!. And he considered the property &de)), N), calling the numbers Nand ))/N the auxiliary and principal stability coefficients respectively. Bohl arrived at these notions by essentially studying (if one uses modern terminology) the question of stability under constantly acting perturbations (see Exercise VII. 9). He proceeded under the methodological premise that uncontrollable dissipation of energy always plays 7) The contributions of Bohl to the theory of almost periodic functions were made long ago and are well known (he is the creator of the theory of quasiperiodic functions). His results concerning continuous flows on a torus have found a place in textbooks and monographs. A. D. Myskis and I. M. RabinoviC 'Wrote a sensational survey (Uspehi Mat. Nauk 10 (1955), no. 3 (65), 188-192) of his results on continuous vector fields, which contain as a trivial consequence the famous fixed point theorem of Brouwer (these results were published in 1904, i.e. 5 years before the first paper of Brouwer) and were regarded by Bohl as auxiliary results of doubtful originality and of interest mainly for their applications in the theory of differential equations. His work is now known in the Soviet Union thanks to the publication in 1961 of his selected works (P. Bohl [3]) with an introductory article by A. D. Myskis and I. M. Rabinovic, the brochure [1] of these two authors and the jubilee readings devoted to the memory of Bohl that were organized by the Academy of Sciences of the Latvian Soviet Socialist Republic in 1965. In spite of all this, the priority of Bohl in the above mentioned matters connected with the Bohl exponent has yet to be generally recognized.
147
NOTES
an essential role in the problems of geomechanics (Erdische Mechanik) and hence that the behavior of the solutions of correctly posed problems must be stable relative to small perturbations of the equations. It is noteworthy that Bohrs unperturbed equation was generally nonlinear. With this approach he established the stability of the upper Bohl exponent. His arguments essentially made use of an assertion similar to the assertions of Corollaries 2.1 and 2.2 and encountered in the contemporary literature under a wide variety of names. We will not cite the long and unfortunate list of papers especially devoted to the rediscovery of these and similar estimates. The property f!8 (v, N) appeared later in the papers of K. P. Persidskii [1,2,3] in connection with a study of the question of asymptotic stability of the solutions of finite nonlinear systems of equations with a nonstationary principal linear part. Persidskii discovered a criterion similar to the one formulated in Theorem 6.1, introduced the notion of a function stationary at infinity (with weak variation at infinity, in his terminology) and proved that a system of equations has property f!8(v, N) (v> 0) if its coefficients are stationary at infinity and the spectrum of its coefficient matrix lies in the interior of the left halfplane. All of these results are actually contained in the above mentioned paper of P. Boh!. Moreover, this paper also essentially contains (for finite-dimensional systems) the test for negativeness of the Bohl exponent formulated in Theorem 6.3, where the property of being stationary at infinity has been replaced by the condition S,.L for sufficiently small (but not arbitrarily small) c 0 and sufO. This result was later rediscovered by N. ficiently large (but not arbitrarily large) L Ljascenko [1,2] and M. A. Rutman [1]. The latter obtained this result for the infinite-dimensional case, generalizing the investigations of M; G. Krein, who had generalized and sharpened Persidskii's results. The following proposition is an elementary corollary of a basic result (concerning nonlinear equations) in the paper of Boh!. In order for an equation
>
>
dx/dt = A(t)x
(0 ~
t< 00)
with bounded continuous coefficients to have the property that sup , j y(t) - x(t)j
(1)
<
whenever y(t) is a function such that My = sup,jdy/dt - A(t)yj 00 and x(t) is the solution satisfying the condition x(O) = y(O), it is necessary and sufficient that the Bohl exponent of this equation be negative. Under the indicated restrictions on A(t) this result would be completely equivalent to the result of Theorem 5.2 if condition (1) were replaced by the condition
,
supjy(t) - x(t)j
>
00.
(2)
In fact, condition (2) implies condition (1) by virtue of the uniform boundedness principle, which unfortunately was not available to Bohl. Theorem 5.2 and the closely related Theorem 5.1 were established directly for a Banach space by M. G. Krein in [21': where the passage from condition (2) to condition (1) is effected with the use of the uniform boundedness principle. These theorems sharpened and strengthened the corresponding results of I. G. Malkin [3] (see Exercise 15) even for the finite-dimensional case. It should be noted that Bohl's arguments differ from the arguments of Persidskii, Malkin and others in that they do not make use of the cumbersome methods connected with the application of the (now) well-known theorem of Perron [1] (1928) on the reduction to triangular form of a system of differential equations (this theorem was "fortunately" unavailable to Bohl). Unlike Bohl's methods, the methods connected with the application of Perron's theorem (although useful in many problems) do not appear to admit a direct extension to the infinitedimensional case.
148
III.
NONSTATIONARY LINEAR EQUATION
In 1947 M. G. Krein noted (without knowing of Bohl's work) that the methods of functional analysis can be used to significantly simplify all of these results (with some sharpenings even in the finite-dimensional case) and to extend them to the case of equations in a Banach space. 8) A number of results, in particular Theorem 5.2 and Theorem 6.5, which extends the above-mentioned result of Persidskii on equations with operator functions stationary at infinity, were announced by him at a meeting of the Moscow Mathematical Society and published in [2]. They are contained in the "Lectures" along with the other results described in §§ 4-6. In the course of writing up this material we obtained a conversion of Theorem 6.3 (Theorem 6.4) and an exact formula for the upper Bohl exponent in the case of a precompactly valued operator function. Questions similar to these investigations were also considered by V. M. Millionscikov [1].
The method of M. G. Krein was subsequently extended by D. L. Kucer [1,2] (some of his results are given in the Exercises). D. L. Kucer considered, in particular, equations in the wider class of weakly, and not just strongly, measurable functions. This class turns out to be not only more general but also more convenient in the applications, since weak measurability is more readily verified in individual Banach spaces than strong measurability. This is especially true in considering countable systems of differential equations, in application to which the investigations of D. L. Kucer have led in a natural way to significantly more general results than those obtained by K. P. Persidskii in [4]. We note some other results contained in this chapter. The estimates for the solutions of the homogeneous equation in Hilbert space contained in Theorem 4.7 were established by A. Wintner [1] and later repeated by many authors. The example cited in § 4.4 is due to O. Perron [2]. The decomposition (6.7) for the special case of scalar functions stationary at infinity was obtained in a more complicated way by N. I. Gavrilov [1]. The elegant simple argument giving the general decomposition (6.7) directly for vector functions with values in a Banach space is due to M. L. Brodskii. The presentation of the majority of the results of this chapter differs from the presentation of the corresponding results in the "Lectures" by the fact that the requirement that the coefficient of the equation be bounded has been replaced by the requirement that it be integrally bounded. A systematic consideration of differential equations with integrally bounded coefficients was apparently first carried out in the papers and book [1] of J. Massera and J. Schaffer. 8) The important role of the uniform boundedness principle in investigating the solutions of equations in a finite-dimensional phase space was independently discovered by R. Bellman [1]. This paper underwent some criticism by D. L. Kucer [1].
CHAPTER
IV
EXPONENTIAL SPLITTING OF THE SOLUTIONS OF THE LINEAR EQUATION In this chapter we continue our investigation of the solutions of the linear equation. Our main purpose is to conduct a study of the notion of an exponential dichotomy of the solutions and of the more general notion of an exponential splitting of order n. By this we mean, roughly speaking, the existence of a direct decomposition of the space into n subspaces with nonintersecting intervals Df Bohl exponents of the solutions initially in these subspaces. The notion of an exponential dichotomy is introduced in § 3, where we also study its role in questions concerning the existence of bounded solutions of the inhomogeneous equation. In § 4 we ~onsider equations with an exponential splitting of higher order. In § 5 we show that the exponential splittability property is stable with respect to small perturbations of the equation. A deeper investigation of the exponential splittability of an equation can be carried out in a Hilbert phase space; in particular, we are able to establish in this case an exponential splittability test for equations with precompactiy valued operator functions (§ 6). These considerations are essentially based on a new geometric method involving the use of the so called conjugation operators for projections. The fundamentals of these operators are presented in § 1. Conjugation operators naturally arise in the study of certain questions connected with the general theory of perturbations of linear operators. In this book we also make use of them in Chapter VIII. Of importance in the study of the various growth characteristics of the solutions of an equation are the kinematic similarity transformations under which these characteristics are invariant, as well as the related notion of reducibility of an equation. These questions are considered in § 2.
§ I. Conjugation operators for projections I. A conjugation operator function. We consider in a space lB two decompositions of the identity n
n
I; P k = J;
(1.1)
I; Qk = J, k=l
k=l
composed of pairwise disjoint projections: (I
Let
149
~
k,j
~
n).
(1.2)
]50
IV.
EXPONENTIAL SPLITTING OF SOLUTIONS n
S
I; QjPj.
=
(1.3)
j=l
Clearly, this operator has the properties S(P/13) c Q/13;
=
SPj
(j = 1,2,.··, n).
QjS
It can be written in the form S
+
= I
n
I; QiPj - Qj).
j=l
Therefore, if (1.4)
then the operator S has a bounded inverse, the above inclusion relations are converted into equalities: S(P/13)
=
(j
Q/13
= 1,2, ···,n),
and the operators Pj, Qj are found to be similar: (j
= 1,2,.· ·,n).
(1.5)
Suppose now n
I; Pit)
j=l
(a ~ t ~ b)
I
=
(1.6)
is a decomposition composed of continuous pairwise disjoint projection functions on [a, b]: (a ~ t ~ b).
Setting P j
= Pit)
and Qj
= Pit
S(t
+ -r,
(1.7)
+ -r), we form the operator n
t)
= I;
j=l
Pit
+
-r)Pj{t).
We note that the continuity implies the existence of a number 0 > 0 such that when -r < 0
I I
n
~
max.L;
t }=1
I Pit + -r) IIII Pit + -r) -
on the finite interval [a, b]. The operator S(t inverse when -r < o. We consider a subdivision q of [a, b]:
I I
a = to
such that
+
Pj(t)
I
< 1
-r, t) therefore has a bounded
< tl < ... < tk-l < tk < tHl < ... < tm
= b
1.
CONJUGATION OPERATORS FOR PROJECTIONS
I
151
I
max tk - tk~l < 0, k
and construct the operator function Oq{t) = S(t, tk) S(tk,
tk~l)
... S(th to)
(1.8)
for tk ~ t ~ tHl· This operator function has the following properties. 1) Oq(a) = S(a, a) = I. 2) The operators Oq{t) and O;l(t) are bounded and continuous functions of t. The invertibility of Oq{t) follows from the invertibility of each factor in (1.8), while its continuity follows from the continuity of S(t, 'r) and the relation S(t, t) =1. 3) The operator Oq(t) maps the subspace Pj{a)'iB onto Pj{t)'iB (j = 1,. .. , n), the corresponding projections being similar: (l.9)
In fact, Oq{t)Pj{a) = Pj{t)Pj{tk) ... Pj(tl)Pj{a) = Pj{t)Oq{t).
4) If the projections Pj{t) satisfy a Lipschitz condition
II Pit2) - Pj{tl) I
~ c It2 - td
(j
= 1,2,.··,n; th t2 E [a, b]),
so does the operator Oq{l). For it suffices to verify this property in each closed interval [tk~h tk], where it is obviously satisfied. 2. Differential conjugation equation. The constructed operator Oq{t) depends on the choice of the subdivision of the interval [a, b]. This psychologically disturbing defect can be removed by passing to the limit with respect to the filter formed from the subdivisions of [a, b]. The validity of such a limit passage is assured under certain conditions (see Exercise 1). In this way we obtain an operator function O(t), which has certain properties in addition to those of Oq{t). This operator function admits a quite simple description when the projection functions Pj(t) are differentiable. In fact iffollows from (1.8) in this case that n
O~(t)
= I: PJ(t)Pj(tk) ... Pj(a) j=l
=
[£1 P;(t)Pj{tk) JL~l Pitk) ... Pia) ]
= [j~l PJ(t)Pitk) ] Oq(tk).
152
IV. EXPONENTIAL SPLITTING OF SOLUTIONS
If we could pass to the limit in the latter formula under the condition that maxkltk - tk~ll ~ 0, we would obtain the differential equation (1.10) Following these heuristic arguments we study the properties of the solutions of equation (1.10), which is called the conjugation equation for the projections Pit). THEOREM 1.1. Suppose the disjoint projection functions Pit) (j = l, .. ·,n) giving a decomposition of the identity are differentiable. Then the evolution operator D(t, z') (a ~ t, 7: ~ b) of equation (1.1 0) has in addition to the usual properties a) - d) (see § 111.1.5) the intertwining property e)
Pj(t)D(t,7:) = D(t,7:)Pj{7:)
(a
~
t, 7:
~
b;j = 1,2, .. ·,n);
and the property that f) it is a W-unitaryoperator when the space ~ is a Hilbert space ~ and the operators Pj(t) are W-Hermitian with respect to some scalar product (x, y)w = (Wx, y) (possibly indefinite). We note that the intertwining property admits a simple geometric interpretation. To the decomposition (1.6) there corresponds a direct decomposition of the space ~:
(a ~ t ~ b).
(1.11)
By virtue of properties a), c) and e) the operator D(t, 7:) effects a linear isomorphism between the subspaces ~j{7:) and ~it) (j = l, .. ·,n; t, 7: E [a, b]) and thus "tracks" the subspaces ~j{t) under a variation of t. When the decompositions (1.6) and (1.11) are orthogonal (W-orthogonal), this isomorphism, as follows from property f), is isometric (W-unitary). As in (1.9), the operators D(t, 7:) effect a conjugation of the projections: Pj{t) = D(t,
7:)Pj{7:)D~l(t,
7:)
(a
~
t, 7:
~
b;j = 1,2, .. ·,n).
PROOF OF THE THEOREM. We differentiate relation (1.7), multiply it from the right by P j and sum the resulting equality over j. As a result, we get n
P/,(t)
+ L:
j=l
Pk(t)PJ(t)Pj(t) = P/'(t)Pk(t).
We now consider the operator Sk(t, 7:) = Pk(t)D(t, 7:) - D(t, 7:)Pk(7:). It clearly satisfies the condition Sk(7:, 7:) = O. If we can prove that it satisfies the differential equation
(1.12)
1.
CONJUGATIOTION OPERATORS FOR PROJECTIONS
153
(1.13) the uniqueness of the solution of the Cauchy problem for equation (1.13) will imply the identity SkU, r) == 0 equivalent to condition e) of the theorem. It suffices to show that equation (1.13) is satisfied by the first summand of Sk(t, r), since it is obviously satisfied by the second. We differentiate this summand SklU, r) with respect to t and make use of relation (1.12) as well as the equation for the evolution operator O'(t, r)
=
[jtl Pj(t)Pj{t) ] O(t, r).
This gives S/'I(t, r)
=
P/'(t)O(t, r)
=[
+ Pk(t)O'(t, r)
+ ~1 Pk(t)PjU)Pit) ]
P/,(t)
= [jt1Pj(t)Pj{t) ]
Pk(t)O(t, r)
O(t, r)
=
=
P/'(t)Pk(t)O(t, r)
[jt1Pj(t)Pit) ] Skl(t, r).
For a proof of property f) it suffices to show, as follows from Lemma IlL 1.1, that under the considered conditions the operator L;j=1 PjU)Pj{t) will be W-skewHermitian. Since L;j PJ(t) = I, we get (L;j PJ(t)) , = 0, and hence
(J;
PjPjx,y)w +
=
(X, 7 PjPjy)w
(x, ~ PjPjy)w
+
(x, ~ PjPjy)w
= (X,
(7 P7)' Y)w = O.
The theorem is proved. 3. Tracking oj the invariant subspaces oj an operator Juncton. We consider a differentiable operator function F(t) for t E [a, b]. Suppose that, for some to E [a, b], the spectrum of the operator F(to) decomposes into n spectral sets:
(1.14) Let rk(t O) denote a closed curve (or system of curves) separating (Jk(tO) from the rest of the spectrum of F(t o). As before, we see from the continuity of F(t) that its spectrum (J(F(t)) is not intersected by nUo) (k = l,.··,n) for sufficiently small values of It - tol, and we let (Jk(t) denote the part of this spectrum lying inside rk(tO). The spectral projections Pk(t)
1
= - -2-·
f
[F(t) - ).1]-1 d).
11:1 r.(t,)
corresponding to these parts of the spectrum are differentiable with respect to t and
IV. EXPONENTIAL SPLITTING OF SOLUTIONS
154
have all of the properties required by Theorem 1.1. The operator OCt, -r) whose existence is guaranteed by this theorem "tracks" the invariant subspaces 58 k(t) = P k(t)58 of F(t) as t varies in a neighborhood of to. If the spectral sets (Jk(t) (k = l"",n) remain pairwise disjoint for all t E [a, b], the operator OCt, -r) "tracks" the subspaces 58 k (t) on the whole interval [a, b]. 4. Normalization of a conjugation operator. The above constructed operator functions Oit) and Oct, -r) effecting conjugations of the projections of (1.6), although continuous, can be unbounded on an infinite interval even if the projection functions themselves are bounded. If the space 58 is a Hilbert space fl, the situation can be corrected by using the following result. THEOREM 1.2. Suppose Pk(t;) (k = l,.··,n) are pairwise disjoint projections in that compose a decomposition of the identity for each t; of some metric space E and are continuous and bounded functions on E. If there exists a continuous invertible operator function Set;) effecting a conjugation between the projections, there also exists a continuous invertible operator function Set;) that is bounded on E and has the same property:
fl
(t; EE).
If all of the Pk(t;) are orthogonal (viz. Hermitian), the operator function Set;) can be chosen so that it is unitary. PROOF. It can be assumed without loss of generality that the initial projections Pk(t;o) are orthogonal (pt(t;o) = Pit;o»). For one can always introduce an equivalent scalar product in .p with respect to which this property will be possessed (see Remark 1.1.1). We put (1.15)
Suppose f
E
fl. Then
n
=
~
E111
S(t;)Pk(t;o)1 112
nlls!1(t;)112
tt
~
1
n
-IIS-1(t;) 1 2- kL.1 Ilpk(t;o)/IIZ
Pk(t;o)flr = nlfS-\(t;)lli
and hence the operator R2(t;) is uniformly positive.
11/112
1.
155
CONJUGATION OPERATORS FOR PROJECTIONS
Let R(~) be the positive square root of RZ(~). We show that the operator S(~) = S(~)R-l(~) has the required properties. From (1.15) we obtain the equality I
=
R-l(~)S*(~) [ ~ N(~)Pk(~) ] S(~)R-l(~)
and hence the inequality
= 1::
II/liz
(R-l(~)S*(~)N(~)Pk(~)S(~)R-l(~)f,f)
k
=
1::
Ilpk(~)S(~)R-l(~)/llz
k
indicating the boundedness of the operator function S(~):
On the other hand, from (1.15) we obtain the equality n
S*-l(~)RZ(~)S-l(~)
= 1::
k=l
pn~)PM),
which implies
This proves the boundedness of the operator function S-l(~). The continuity of the operator function RZ(~) and hence of R(~) follows directly from formula (1.15). Suppose, finally, Pk(~) = pn~) (~E E). Then (1.15) implies that RZ(~) = S*(~)S(~) and hence that the operator S(~) =;= S(~)R-l(~) = S(~)[S*(~)S(~)]-l!~ is unitary. Since the operator RZ(~) and hence also R(~) clearly commute with the projections Pk(~O), we have S(e)PMo)S-l(e)
=
S(~)R-l(~)Pk(~o)R(~)S-l(~)
=
S(~)Pk(~O)S-l(~)
=
Pk(~).
REMARK 1.1. If the space E is a segment of the real axis and the operators Pk(~) and S(~) are differentiable, the operator S(~) is also differentiable (it suffices to note that a differentiable uniformly positive operator has a differentiable root). 5. Angular distance between disjoint subspaces. In the sequel we will often encounter bounded families of projections in a general Banach space ~. Here we indicate the geometric significance of this boundedness. Let ~1 and ~z be a pair of nonzero subspaces of a Banach space ~ that are disjoint:
156 ~l
n ~2 =
IV. EXPONENTIAL SPLITTING OF SOLUTIONS
{OJ. We introduce the following characterization of their mutual
inclination. Let Sn (~I. ~2) =
inf
II
x.E~.
(k=1.2) Ilx.11 =1
Xl
+
(l.l6)
x211,
where the infimum is taken over all pairs of unit vectors belonging to respectively. One can easily verify that in the case of a Hilbert space Sn(~I.~2) =
~l
and
~2
2 sin({} /2),
where {} is the minimal angle between the subspaces ~l and ~2 (see Exercise 1.30). In the general case we call the quantity Sn(~I. ~2) the angular distance between ~l and ~2 LEMMA 1.1. Suppose the space ~ decomposes into a direct sum ~ = ~l + ~2 of closed subspaces and PI. P z = I - PI are the corresponding projections. The following estimate is valid:
(k
=
1,2).
(1.17)
PROOF. We choose an aribtrary 0 > Sn(~I. ~z). There then exists a pair of unit vectors Xk E ~k (k = 1, 2) for which II Xl + Xz II < o. We put Xl + Xz = x. Then Xk = Pkx and
1 = Ilxkll ~ Ilpkllllxll < Ilpkll 0,
whence l/llpkll < 0 and consequently l/llpk II ~ Sn(~I. ~z)· On the other hand, for aribtrary X E ~
and therefore' .
Sn(~l'~Z) ~ 2x~t
Ilxll _ _2_ Ilplxl - Ilplll'
Analogously, Sn(~-b ~z) ~ 2/llpzll.The lemma is proved. COROLLARY 1.1. The boundedness from above ofa set g> = {P} of projections in a Banach space ~ is equivalent to the boundedness from below of the set
2.
157
KINEMATICALLY SIMILAR EQUATIONS
{Sn (PfB, (/ - p)fB)lp E &} of angular distances between the subspaces PfB and their complements (/ - P)fB. Let P be a projection in a Banach space fB and operator such that U(';o) = I. We consider the projection function (~
U(~) (~ E
B) be an invertible
(1.18)
EB)
and the corresponding subspaces fB(~) = U(~)PfB. If the projection function (1.18) is bounded, we will say that the operator
U(~)
uniformly conjugates the projection P on B.
The significance of this definition is to be found in the fact that, when its condition is satisfied, the set {Sn(P(~)fB, (I - P(~»)fB)l~ E B} of angular distances is bounded from below. § 2. Kinematically similar equations 1. Definition and tests of kinematic similarity of equations. We consider the pair of equations
(t
E
J, k = 1,2).
(2.1)
We say that these equations are kinematically similar on the interval J if it is possible to establish between the totalities of all solutions of these equations a oneto-one correspondence Xz(t) = Q(t)Xl(t)
(t
E
(2.2)
J),
where Q(t) is a bounded linear operator function with a bounded inverse: 1IQ(t) II
~
qb
I Q-l(t) I ~
qz'
(t
E
(2.3)
.1').
Letting Uk(t, -r) denote the evolution operators of equations (2.1), we obtain from (2.2) the relation U 2(t, -r)X2(-r) = Q(t)U1(t, -r)Xl(-r),
which by virtue of the arbitrariness of Xl(-r) and the formula xzC-r) reduces to the equality U2(t, -r)Q(-r) = Q(t)U1(t,-r)
and, in particular, when -r
=
=
Q(-r)xl(r) (2.4)
0, to the equality Uz(t)CU 11 (t) = Q(t),
(2.5)
where C = Q(O). Since the operators 02(t) = U2(t)C and Ol(t) = U1(t) are fundamental operators of equations (2.1), the operator Q(t) can be written in the form ofa ratio offundamental operators of these equations:
158
IV.
EXPONENTIAL SPLITTING OF SOLUTIONS
Dz(t)fJ;I(t)
=
Q(t).
(2.6)
Equality (2.6) implies the differentiability of Q(t) and the equality Q'(t)
Az(t)Q(t) - Q(t)AI(t).
=
(2.7)
Suppose on the other hand that Q(t) is the solution of equation (2.7) satisfying the condition Q(O) = C, where C is an invertible operator. This solution, as follows from (111.1.25), has the form Q(t)
=
Uz(t)Q(0)U1I(t),
and hence the operator Q(t) is invertible for each t E f. Since (2.5) clearly implies (2.2) for xz(O) = CXI(O), all five of the relations (2.2), (2.4)-(2.7) are equivalent. This brings us to the following assertion. 2.1. In order for equations (2.1) to be kinematically similar on an interval it is necessary and sufficient that a bounded and boundedly invertible operator function Q(t) on f satisfy one of the folio wing conditions: a) The evolution operators of the equations are connected by the relation LEMMA
f
(2.4) b) For some invertible operator C connected by the relation
E
[58] the Cauchy operators of the equations are
(2.5)
c) There exist fundamental operators Dk(t) (k = 1,2) of equations (2.I)for which Dz(t)D1I(t)
=. Q(t).
(2.6)
d) The operator Q(t) satisfies the differential equation Q'(t)
= Az(t)Q(t) - Q(t)Al(t).
(2.7)
We note that two equations are kinematically similar on a semi-infinite interval if their coefficients coincide outside of a finite interval. For suppose for the sake of definiteness that f = [0, 00) and AI(t) = Az(t) when t ~ to > O. Then when t, 1: ~ to the evolution operators also coincide: UI(t, 1:) = Uz(t, 1:), and~hence
Consequently, for C
where
~
UI(t)
=
UzCt)
= Uz(t, to)U2(to) = UI(t, to)U2(to).
UI(t, to)UI(tO);
UZI(tO)UI(to) we have
159
2. KINEMATICALLY SIMILAR EQUATIONS when when
0 ~ t ~to, t ~ to,
i.e. condition (2.5) is satisfied. THEOREM 2.1. Equations kinematically similar on a halfline have the same upper Ljapunov and the same Bohl exponents. PROOF. From formulas (2.3) and (2.4) it follows that
(l/qlqz)IIU1(t + 'C, t)11 ~ Iluz(t + 'C, t)11
= QlqzllU1(t + 'C, t)ll,
which implies
InlIUz(t + 'C, t)11 _ In(QlQZ) ~ InlIUz(t + 'C, t)11 'C 'C 'C < In II U1(t + 'C, t) II In(QlQz) + -'C 'C . Passing to the upper limit as -
KB (U) Z -
'C
--+ 00
and t
--+ 00,
we obtain the required relation
-I' InIIUz(t+'C,t)11 -- -I' InIIU1(t+'C,t)11 -1m 1m t,
T--HX)
'r
t,
T-KJ()
KB
(U)
l'
'r
The coincidence of the lower Bohl and upper Ljapunov exponents is proved analogously. 2. Reducible equations. The equation dx /dt = A(t)x is said to be reducible according to Ljapunov if it is kinematically similar to a stationary equation dy /dt = By (B = const). Theorems 111.4.5 and 2.1 directly imply the following result. THEOREM 2.2. The upper Ljapunov and upper Bohl exponents of a reducible equation coincide. REMARK 2.1. It is not difficult to verify that the upper Ljapunov and upper Bohl exponents in the case in question are strict. By definition, the reducibility of an equation is equivalent to the relation
U(t)
= Q(t)eBt ,
(2.8)
where Q(t) and Q-l(t) are bounded continuous operator functions on J (one can easily see that it is always possible in this case to take C = J). We note that in the present case Q(t) satisfies the equation
Q'
= A(t)Q- QB.
(2.9)
3. Reduction to an equation with a Hermitian coefficient. Suppose now that the space ~ is a Hilbert space ~. We represent the operator A(t) by the sum of its real
160
IV.
EXPONENTIAL SPLITTING OF SOLUTIONS
and imaginary parts: A(t) = Aff/(t) x = V(t)y in the equation
+
iA;:lt), and make the change of variables
dx = A(t)x dt '
~
(2.10)
where V(t) is the unitary operator function satisfying the relations
liJ: = iA;:s(t) V,
V(O) = I.
Here y(t) satisfies the equation
iA;}Vy
+
Vy' = Aff/Vy
+
iA;}Vy
or, finally,
y' = V-I(t)Aff/(t)V(t)y.
(2.11)
Since the operator V(t) is unitary for each t, we obtain the following assertion. LEMMA 2.2. In a Hilbert space equation (2.10) is kinematically similar to equation (2.11), the coefficient of which is unitarily equivalent to the real part of the coefficient of the original equation.
4. Theorem on the decomposition of an equation in Hilbert space. We consider a direct decomposition of the space /p:
/P = /PI
+ /Pz + ... + /Pm
(2.12)
and let PI,,··, Pn be the corresponding projections. Applying the Cauchy operator U(t) of equation (2.10) to the decomposition (2.12), we obtain the new decomposition
/P = /PI(t)
+ /Pz(t) + ... + /Pit),
(2.13)
where /Pk(t) = U(t)/Pk (k = 1,.··, n). The projections corresponding to the decomposition (2.13) are similar to the projections P k and are expressed by means of the formulas
(t
E
f).
(2.14)
THEOREM 2.3: Suppose the projections P k are uniformly conjugated by the operator U(t) on an interval f. Then equation (2.10) is kinematically similar to an equatiqn
dy fdt
=
B(t)y,
(2.15)
the coefficient of which commutes with each of the projections P k (k = 1,.··, n) and which therefore decomposes into a system of independent equations in the phase spaces /Pk of the decomposition (2.12). The kinematic similarity transformation can be chosen so that the following estimate is satisfiedfor some constant c depending only on the projections P k:
2.
161
KINEMATICALLY SIMILAR EQUATIONS
IIB(t)II
ell A(t)11
~
(t E J).
(2.16)
PROOF. The operators (2.14) satisfy the conditions of Theorem 1.2 on J. There therefore exists a bounded invertible operator function S(t) = U(t)R-l(t) satisfying the conditions
(k
= 1,2,.··,n).
We will assume that an equivalent renorming of the space to make the operators P k Hermitian (see the proof of Theorem 1.2) has already been carried out. It then follows from (1.15) that n
R2(t)
= 1:; PkU*(t)U(t)Pk.
(2.17)
k=l
Equation (2.10) is kinematically similar to the equation (B(t) = R'(t)R-l(t)),
dy /dt = B(t)y
(2.18)
since R(t) is clearly a fundamental operator of the latter equation and the ratio U(t)R-l(t) = S(t) satisfies the kinematic siimlarity condition (2.6). Formula (2.17) shows that R(t) and hence B(t) commute with each of the Pk. We must still prove the last assertion of the theorem. Differentiating (2.17), we obtain the relation R(t)R'(t) + R'(t)R(t)
n
= 1:; PkU*(t)[A*(t) + A(t)]U(t)Pk. k=l
By virtue of the boundedness of the Hermitian operator A*(t) exist real constants a and f3 such that a(f,f)
~
([A*(t)
+ A(t)]f,f)
~
f3(f,f)
+ A(t)
there
(f E 4)).
Therefore the relation ([RR'
+ R'R]f,f)
n
=
1:; ([A* k=l
+ A]U(t)Pkf, U(t)Pd)
implies the estimate a 1:; (U(t)Pkf, U(t)Pd)
~
([RR'
+
R'R]f,f)
k
• which can be rewritten in the form a(R2f,f)
~
([RR'
+ R'R]f,f)
~
f3 (R2f,f),
or, settingf = R-lg,
allgl1 2~ ([R'R-l + R-IR']g, g) ~ f3llgI12. Thus we see· that the operator
(2.1 9)
162
IV.
EXPONENTIAL SPLITTING OF SOLUTIONS
t (R'R-l + R-lR') = t (E* + E) = Effl
II·
is bounded, it following from (2.19) that !!EffI(t)!! ~ IIA(t) It remains for us to make use of Lemma 2.2 and pass from equation (2.18) to the kinematically similar equation whosecoefficientB(t) = V(t)EffI(t) V-let) is unitarily equivalent to Effl • It is not difficult to see that commutability with the projections Pk is not violated here since the unitary operator Vet) also commutes with them. The operator B(t) which is unitarily equivalent to EffI(t) satisfies the same estimate IIB(t)1I ~ IIA(t)lI· The constant c in (2.16) arises in connection with the renorming of the space. REMARK 2.2: The Cauchy operator of equation (2.15) clearly also commutes with with each of the projections Pk (k = 1,···, n).
§ 3. Exponentially dichotomic equations 1. Basic definitions. In the preceding chapter we were primarily concerned with the equation dxfdt
= A(t)x
(3.1)
when its upper Bohl exponent is negative and hence when all of its solutions decrease exponentially as t ---> 00. We recall that such a situation arises for an equation with a constant coefficient A(t)=A when the spectrum a(A) lies in the interior of the left halfplane. A more complicated situation arises when the spectrum a(A) also contains a component lying in the interior of the right halfplane: (3.2) We recall that the operator A was then said to be exponentially dichotomic. It was shown in § II.2.1 that in this case the space decomposes into a direct sum ~
=
~+
+ Q:L
(3.3)
of subspaces such that the solutions initially in ~+ exponentially increase (decrease) as t---> + 00 (t---> - 00) whereas the solutions initially in ~_, as before, exponentially decrease (increase) as t---> + 00 (t---> - 00). We note that these solutions remain in the fixed subspaces ~+ and ~_ respectively. If P + and P _ are the spectral projections commuting with A that correspond to the decompositions (3.2) and (3.3), there exist positive constants Nand ).i for which
_II
~ Ne-v(t-s)
(t
II eACH) P + I
~ Ne- vCs - t )
(s ~ t).
lIeA(t-S) P
~
s),
(3.4)
We will see at once that an analogous description is possible for an equation with a variable coefficient A(t). And what is more, this description permits us to generalize a number of the results of the preceding chapter.
163
3. EXPONENTIALLY DICHOTOMIC EQUATIONS
DEFINITION 3.1. We will say that a regular ll exponential dichotomy holds on.Jf for the solutions of equation (3.1) (more briefly, the equation is e-dichotomic) if for some to E .Jf the space ~ decomposes into a direct sum (3.5) of closed subspaces such that the following conditions are satisfied: a) The solutions Xl(t) = U(t, to)x~ of equation (3.1) in the subspace ~l(tO) at t = to (x~ E ~1(tO) are subject to the estimate
Il xl(t)II
~ N 1e- v,(t-s)
Il xl(S)II
(t ~
s; t, sE.Jf)
(3.6a)
with some exponent VI > O. b) The solutions xz(t) = U(t, to)x~ of equation (3.1) in the subspace ~z(to) at t = to (x~ E ~z(to)) are subject to the estimate Ilxz(t)
I
~ N ze- V2 (s-t)
IIXz(s)II
(t ~
s; t, sE.Jf)
(3.6b)
with some exponent Vz > o. c) The angular distance between the subspaces ~1(t) = U(t,tO)~I(tO) and ~z(t) = U(t, to)~z(to) cannot become arbitrarily small under a variation of t; more precisely, there exists a constant T > 0 such that (t E .Jf).
(3.7)
Let us make some remarks concerning the above definition. REMARK 3.1. The choice of to is not important. In fact, a change in the value of to in formulas (3.5) - (3.7) will affect only the constants Nk (k = 1, 2). For this reason, in the sequel we will always put to = 0 (assuming that the interval .Jf contains this point) and use the more concise notation ~k(O) = ~k (k = 1, 2). The projections corresponding to the direct decomposition ~ = ~1 ~2 will be denoted by PI and P z. REMARK 3.2. Let Pk(t) (k = 1,2) be the mutually complementary projections on the subspaces ~it) (t E .Jf). Then
+
Pk(t)
=
U(t)Pk U-l(t)
(t E .Jf).
(3.8)
As follows from Lemma 1.1, condition c) in Definition 3.1 is equivalent to the uniform boundedness of the projections Pk(t): (k
= 1, 2; t E .Jf),
(3.9)
i.e. to the uniform conjugatability of the projections P k • REMARK 3.3. Suppose for the sake of definiteness that .Jf contains the halfline 1) For the sake of simplicity we consider a less general situation than the one described in the book of J. Massera and J. Schaffer [1], and for this reason we speak of a regular exponential dichotomy. The word regular will be dropped below.
IV.
164
EXPONENTIAL SPLITTING OF SOLUTIONS
[0, + (0). Then the solutions of equation (3.1) initially in )81 remain bounded on [0, + (0) while the solutions initially in )82 increase unboundedly as t---+ + 00. It turns out, moreover, that all of the solutions x(t) = U(t)xo for which Xo E )81 increase unboundedly as t---+ + 00. For suppose P2XO "# 0. We consider the solution
X2(t)
=
U(t)P2Xo
=
P2(t)U(t)xo
=
P2(t)x(t),
corresponding to an initial value P2Xo E )82' From (3.6b) and (3.9) we obtain the inequality
Ilx(s) I
1
;;;; Ilpz(s)II
Il x 2(S) I
;;;;
1
MN2 e V2 (s-t)
( = JN2 e (s-t) I U(t)P2xoll )
Il x2(t)II
(s;;;; t),
V2
(3.10)
the right side of which unboundedly increases as s---+ 00. Thus the subspace )81 consists of precisely the initial values x(o) of those solutions x(t) of equation (3.1) which remain bounded on [0, + (0). An analogous situation in which )81 is replaced by )82 holds on the left halfline. REMARK 3.4. Suppose J = [0, (0). As was indicated in the preceding remark, the subspace)81 is then uniquely determined by equation (3.1). On the other hand, the subspace )82 can always be replaced by another subspace ~2 that preserves the direct decomposition: )8 = )81 + ~2' It turns out that if condition c) of Definition 3.1 is satisfied, this direct decomposition also determines an exponential dichotomy of the solutions of equation (3.1) in which the constants ].Jk remain the same [it will be shown below that condition c) is a consequence of conditions a) and b) if A(t) is an integrally bounded operator function]. We prove this assertion by verifying condition (3.6b) for the solutions initially in ~2' Suppose Xo E ~2 and x(t) = U(t)xo. Then by virtue of (3.6a) and (3.6b)
Ilx(t)II
~
I U(t)P1x oll + I U(t)P2x oll
~ Nd p 1Xoil + I U(t)P2x oll ~ (Z~ 111;:::\\
+
1) I U(t)P2 x oll·
Combining this inequality with (3.10), we obtain the estimate
Ilx(t)11 ~ M(N1 IIp1X oil + N 2)e. IIp2X oil
V2
(S-t)llx(s)ll,
and ~t only remains to show that the quantity IIp1Xo 11/ IIp2Xo I is bounded for Xo E )82. The latter fact follows directly from Banach's theorem. For the operator P 2 is
165
3. EXPONENTIALLY DICHOTOMIC EQUATIONS
bounded and, as is easily seen, is a one-to-one mapping of)[32 onto ~2. Therefore the inverse mapping is continuous:
and, finally,
We now cite another, in certain cases more convenient, e-dichotomicity condition.
)..11
LEMMA 3.1. In order for equation (3.1) to be e-dichotomic on f with exponents > 0 and)..l2 > 0 it is necessary and sufficient that the conditions
I U(t)P1U-1(S) I I U(t)P2U-1(S) I with certain constants N k (k
~ N1e-v1(t-s)
(t ~ s),
(3.11a)
~ N 2e- Vl (s-t)
(s
~
(3.11b)
t)
= 1, 2) be satisfied on this interval.
PROOF. Inequalities (3.6) immediately follow from (3.11). For, when x(O) = E ~1 and t ~ s, we have
P1X(O)
Ilx(t)11
=
I U(t)P1X(O) I
=
I U(t)P U-1(S)X(S) I 1
~ N1e-v1(t-s)
Ilx(s)ll·
In addition, (3.11) also implies estimate (3.9). Inequality (3.6b) is proved analogously. Conversely, if in (3.6a) we consider the solution X1(S) with initial value x~ = P 1U-1(S)X (x E ~) and take into account (3.9), we obtain the following estimate fort~s:
IIU(t)P1U-1(s)xll
= IIU(t)x~11 ~ N1e-vl(t-S)IIU(s)x~11
I
~ N1e-v,(t-s) U(s)P1U-1(s)xll ~ N1Me-v1(t-s)
Ilxll·
The estimate for t ~ s is obtained in exactly the same way. We can simplify the e-dichotomicity conditions by making the additional assumption that the operator function A(t) is integrally bounded: t+1
J IIA(-c) Ild-c ~
M1
(t
E
f),
(3.12)
t
LEMMA 3.2. If the function A(t) is integrally bounded, condition c) of Definition 3.1 is a consequence of conditions a) and b). PROOF. Estimate (3.12) implies by virtue of Lemma 111.2.4 that I U(t emM1 for integral m.
+ m, t) I
~
166
IV. EXPONENTIAL SPLITTING OF SOLUTIONS
We consider, for some fixed t, a pair of unit vectors Xk(t) E ~h(t) (k = 1,2) and put xi.) = U(., t)Xk(t) (... E f). From (3.6a) and (3.6b) we obtain the estimates !!Xl(t I\xz(t
+ m)1\ + m)1\
~ N 1e- v,ml\xl(t)1\ = N 1e- p ,m, ~ N 2 1e v,mllxz(t)1I
=
N 2 1e v,m,
which show that II XI(t)
+
xz(t) II ~ e- mM, \I U(t
~
+ m, t)Xl(t) + U(t + m, t)X2(t) II e- mM'(l\xz(t + m)1I - IIxl(t + m)l\)
It follows from (1.16) that Sn(5~Mt),
7B z(t»)
~ 1m.
Since the constant 1m > 0 for sufficiently large m, the assertion is proved. 2. Preservation of e-dichotomicity under a kinematic similarity transformation. It is quite easy to establish the following assertion. THEOREM 3.1. If the equations dx fdt
=
Ak(t)X
(k
= 1,2; t Ef)
(3.13)
are kinematically similar and one of them is e-dichotomic on the interval f, this property (with the same exponentS))1 and ))z) is also possessed by the other equation.
PROOF. The Cauchy operators U1(t) and Uz(t) of the kinematically similar equations are connected by a relation of the form (see (2.5») U2(t)
=
Q(t)U1(t)C,
where Q(t) is a bounded operator function with a bounded inverse and C is a bounded invertible operator. Since UZ(t)Pk U 21(S)
=
Q(t)Ul(t)PkU~I(S)Q-l(S),
where Pk = CPkC-l, the assertion of the theorem immediately follows from Lemma 3.1. 3. Green functions. Bounded solutions of the inhomogeneous equation. We now show that an e-dichotomy plays the same role in an investigation of the boundedness of the solutions of the inhomogeneous equation dx fdt = A(t)x
+ f(t)
(t E f)
(3.14)
as condition (3.2) plays for an equation with a constant operator A. To this end we introduce a Green function analogous to the one considered in § IIA
3.
167
EXPONENTIALLY DICHOTOMIC EQUATIONS
Let PI and P z be a pair of mutually complementary projections: PI If U(t) is the Cauchy operator of equation (3.14). we put
G(t -r) ,
U(t)P1 U-l(-r) - U(t)PZU-l(-r) (t, -r E ~).
= {
+ Pz =
t>-r, t < -r
for for
I.
(3.15)
It immediately follows from the definition that the Green function G(t, -r) satisfies the following differential equations on ~ for t i= -r: oG(t, -r)
=
A(t)G(t, -r)
oG~; -r)
=
_
at
(3.16a)
and
G(t, -r)A(-r).
(3.16b)
At t = -r it has a discontinuity such that
G(-r
+ 0, -r) =
- G(-r - 0, -r)
=
+ P Z)U-l(-r) =
U(-r)(P1
+
U(-r)P 1 U-l(-r)
U(-r)PZ U-l(-r)
U(-r)U-l(-r)
= I,
(3.17a)
and analogously
G(t, t
+ 0)
- G(t, t - 0) = - I.
Suppose the functionf(t) is continuous on
g(t)
=
~.
(3.17b)
We consider the integral
S G(t, -r)f(-r)d-r
(3.18)
under the assumption that it exists and permits differentiation under the integral sign. Ifwe set ~ = [a, b] and write t
g(t)
=
b
S G(t, -r)f(-r)d-r + S G(t, -r)f(-r)d-r, t
a
we get
g'(t)
=
G(t, t - O)f(t) - G(t, t
+ O)f(t)
tab a
+ S at G(t, -r)f(-r)d-r + S at G(t, -r)f(-r)d-r a
t b
=
f(t)
+ S A(t)G(t, -r)f(-r)d-r
=
A(t)g(t)
+ f(t).
a
Thus formula (3.18) provides a solution of the inhomogeneous equation (3.14) on ~ as long as the above calculations are permissible. This is always the case when ~ is a finite interval; but special estimates, which can be carried out if equation (3.1) is e-dichotomic, are needed for an infinite or semi-infinite interval.
168
IV. EXPONENTIAL SPLITTING OF SOLUTIONS
Suppose p] and P 2 are the projections corresponding to a direct decomposition = ~] + ~2 for which equation (3.1) is e-dichotomic. The Green function obtained when these projections are substituted in (3.15) is called a principal Green function of equation (3.14). This is the Green function that we will normally make use of in the sequel; for this reason we will usually drop the term principal. Using estimates (3.11), we easily obtain the following estimate for a principal Green function: ~
(3.19) We recall that C(~) denotes the Banach space of bounded continuous functions on J with values in ~ and norm = supJ f(t) II· Estimate (3.19) immediately implies the following result.
I I fill
I
THEOREM 3.2. If equation (3.1) is e-dichotomic on J, the inhomogeneous equation (3.14) has at least one solution x(t) E C(~) for each function f(t) E C(~). This solution is given by the formula x(t) =
S G(t,7:)f(r)dr,
(3.20)
J
where G(t, 7:) is a principal Green function of equation (3.1).
PROOF. Estimate (3.19) implies that the integral (3.20) converges and (3.21) It is easily seen that differentiation under the integral sign is valid in the present case and that therefore, as was shown above, the function (3.20) satisfies equation (3.14) REMARK 3.5. Suppose J = [0, (0). In this case the bounded solution 00
x(t) =
S G(t,7:)f(7:)d7: o
of equation (3.14) on J has the initial value 00
x(O) =
S G(O, 7:)f(7:)d7: = -
o
00
P2
S U-l(7:)f(7:)d7:, 0
belonging to ~2' We obtain the general form of the bounded solutions on [0, (0) of equation (3.14) by adding to the solution already obtained an arbitrary bounded solution of the homogeneous equation (3.1). These are precisely the solutions that are initially in ~1'
3.
EXPONENTIALLY DICHOTOMIC EOUATIONS
169
Thus all of the bounded solutions on [0, 00) of equation (3.14) are represented by the formula x(t) = U(t)y
+ f
(3.22)
G(t, -r)f(r)dr,
o
where y = P1x(0) is an arbitrary element of ~l' REMARK 3.6. The solution (3.20) remains bounded when the boundedness condition for f(t) is replaced by the more general integral boundedness condition t+1
f I f(-r) Ild-r ~ MI'
t
For in this case we have the estimate
Ilx(t) I
~
f I G(t, -r) IIII f(-r) Ild-r J
=
f IIG(t, -r)llllf(-r)lld-r + f IIG(t, -r)llllf(-r)lld-r t?:1: N2 f e-v,(r-t) I f(-r) Ild-r + NI f e-v,(H) Ilf(-r)lld-r
t~t:
~
t~1:
~ N2
f s~o
f~T
e-v,Sllf(t
+ s)llds + Nl f
ev,sllf(t
+ s)llds
s~o
4. e-dichotomicity on a halftine. As was just shown, the e-dichotomicity of equation (3.1) is a sufficient condition for the existence of bounded solutions of the inhomogeneous equation with a bounded free term. In order to explain the extent to which this condition is necessary we must introduce some additional assumptions. We consider the equation on the halfline ~ = [0, 00). The linear manifold ~I consisting of the initial values Xo of the solutions of equation (3.1) that are bounded on [0, 00) is called the (right) !/'-set of this equation. We will assume that ~l is a complemented subspace, i.e. that it is closed and has a direct complement: ~ = ~l + ~2' In the finite-dimensional case this condition is automatically satisfied. In a Hilbert space ,p th~ second part of the condition is superfluous since an orthogonal complement always exists in ,p. We note that this condition is essentially contained in the definition of e-dichotomicity of an equation. It is trivially satisfied for equations with a constant operator whose spectrum does not intersect the imaginary axis. It is easily seen that the closedness and complementedness properties of the !/'-set are not violated under a kinematic similarity transformation. We will subsequently see that they are also preserved under "small" perturbations of e-dicho-
170
IV.
EXPONENTIAL SPLITTING OF SOLUTIONS
tomic equations. All of these transformations permit one to widen the class of equations for which the Sf'-set has the indicated properties. If an equation is also defined on the left halfline, the left Sf'-set (to which all of the remarks made above for the right Sf'-set apply) is introduced analogously. LEMMA 3.3. Suppose that equation (3.14) has for each function f(t) least one solution x that is bounded on [0, (0):
Illxlll
=
sup
o::;::t
Ilx(t)11
<
E
C(58) at
00.
Suppose further that the Sf' -set 58 1 of equation (3.1) is a complemented subspace and that 582 is a complement of it. Then to each function f(t) E C(58) there corresponds a unique solution x(t) that is bounded on [0, (0) and initially in 58 2 : x(O) E 58 2 • This solution is subject to the estimate ~ where K > 0 is a constant not depending on!
Illx I I
Kill fill,
PROOF. Suppose f(t) E C(58). By hypothesis, there exists a solution x(t) E C(58) of equation (3.14). Let Pk (k = 1,2) be the mutually complementary projections on the subspaces 58 k • We denote by Xl(t) the solution of the corresponding homogeneous equation which satisfies the condition Xl(O) = PIX(O). This solution is bounded by definition of the subspace 58 1 • But then the solution X2(t) = x(t) - Xl(t) of the inhomogeneous equation for which X2(0) = x(O) - P1x(0) = P2x(0) E 582 is also bounded. The uniqueness follows from the fact that the difference of two similar solutions would be bounded by a solution initially in 582 of the homogeneous equation, which is possible only for the zero solution. It remains for us to prove the latter assertion of the lemma. To this end we consider the space C1 of all functions x(t) that are solutions of equations of the form
x'(t) - A(t)x(t) = f(t)
under the conditions x(O) E 58 2 andf(t) E C(58). It was essentially shown above that the operator (/Jx(t) = x'(t) - Ax(t) effects a one-to-one mapping of the linear space C1 onto C(58). If in C1 we introduce the norm
<x) =
Illxlll
+
III(/Jxlll,
the operator (/Jx automatically turns out to be continuous. If, in addition, the space C 1 turns out to be complete, the inverse operator (/J-l will also be continuous by Banach's theorem, and the solution x = (/J-lf of equation (3.14) will then satisfy the required estimate
Illx I I
~ <x) ~
11(/J-lI1 I I fill·
3. EXPONENTIALLY DICHOTOMIC EQUATIONS
171
Thus it remains to prove the completeness of C1 . Let {xit)} be a Cauchy sequence in it. Such a sequence is also a Cauchy sequence in C()8) and hence has a limit x(t) in it. In this case, clearly, x(o)
=
lim xn(o)
E )82'
n~co
In exactly the same way it follows that the sequence {fn(t)} = {lPxn(t)} has a limitf(t) in C()8). Therefore, for each t E f, t
x(t) - x(o) = lim n-+oo
f0 x~(r)d'r t
= lim n-too
f0
t
+ A('r)xn('r)]d'r = f
[fn('r)
[f('r)
0
+ A ('r)x('r)]d'r ,
which implies that x(t) satisfies the equation x'(t) - A(t)x(t) = f(t). Thus x(t) E C1 and, as is easily seen, (x - x n ~ for n ~ 00, I.e. C1 is complete. The lemma is proved.
>
°
THEOREM 3.3. Suppose the operator function A(t) is integrally bounded. In order for equation (3.1) to be e-dichotomic on [0, 00) it is necessary and sufficient that its right Y -set be a complemented subspace and that there correspond to each function f(t) E C()8) at least one bounded solution on [0, 00) of the inhomogenous equation. PROOF. The necessity of the second condition follows from Theorem 3.1, while the necessity of the first was noted in defining the Y -set. SUFFICIENCY. Let x(t) be a nonzero bounded solution on f of equation (3.1), i.e. a solution satisfying the condition x(o) E )81' We put t
f X('r)llx('r)11- 1d'rr,
y(t) = x(t)
o
where X(t)
={
I,
'r),
1 - (t - to 0,
O~t~to+'r,
'r
to + ~ t ~ to t ~ to + + 1.
'r
+ 'r +
1,
A simple calculation shows that y(t) is a solution of equation (3.14) for f(t)
=
X(t) x(t)/llx(t)II·
This function is bounded and continuous and such that y(O) = fore, by virtue of Lemma 3.3,
Ily(t)11
I
~ K sup f(t) t
I
~ K
(t
E
f)
°
E )82'
There-
172
IV.
EXPONENTIAL SPILITTING OF SOLUTIONS
and, in particular, when t
= to + 7: t,+~
Ily(to + 7:)11 = Ilx(to
+ 7:)11
J Ilx(s) II-Ids ~ K.
(3.23)
o
We introduce the function t
J Ilx(s) II-Ids.
ifJ(t) =
o
From (3.23) we obtain the inequality
+ 7:)N(to + 7:) ~ ~,
ifJ'{to
which after an integration with respect to 7: over the interval [I, 7:] reduces to the estimate
+ 7:) + I]
ifJ{to Further, for s
E
[to, to
~
ifJ{to
+
l)e(~-l)!K
(3.24)
and hence
ifJ(to
+
t,+l
I)
=
J Ilx(s) II-Ids ~
Ilx(to)II-1e- M,.
t,
Combining this inequality with (3.23) and (3.24), we obtain for 7:
Ilx(lo where Nl
+ 7:)11 ~
K ifJ(to
+ 7:) ~
= Ke1!K+M, Since for 7: Ilx(to
+ 7:)11
~
Ke-(~-l)!K
ifJ{to
+
I)
~
I the estimate
~ Nle-~!Kllx(to)ll,
I
~ eM'llx(to)11 ~ el/K+M'e-~!Kllx(to)ll,
we obtain the estimate
Ilx(t)II ~
Ne-v(t-t,)
Ilx(to)ll,
(3.25)
where).l = I /K and N = max{ I, K}el/K +M " i.e. inequality (3.6a). Suppose now x(t) is a nonzero solution of equation (3.1) with its initial value x(O) E '8 2. Then the function CXl
yet) = x(t)
J Xes) Ilx(s) II-Ids t
is a solution of equation (3.14) for /(1) = - X(I)x(t)/llx(l) II. This solution is bounded (it vanishes for 1 ~ to + 7:) and its initial value y(O) E '82. Therefore
173
3. EXPONENTIALLY DICHOTOMIC EQUATIONS 00
Ily(t)II Letting
7:
Ilx(t)II tSX(s) IIX(S) II-Ids ~ K Illf I I
=
= K.
tend to infinity, we obtain the inequality 00
IIX(S) II-Ids ~ K IIX(t) 11-1,
S t
(3.26)
which, setting CP(t) = S;oo Ilx(s)II- 1ds, is equivalent to the inequality CP'(t) ~ CP(t) / K. An integration of the latter leads to the estimate (3.27) Further, since for
7: ~
t
it follows that
r
Ilx(t)llcp (t) = I x(t)11 t Ilx(s) II-Ids ~ t+k
co
~ ~
S
exp
r
{t+k
t
S IIA(s)llds
-
t
k=1 H(k-l)
J IIA(s) lids tj d7:
exp { -
t
}
1
00
d7: ~ ~ e- kMI = k=1
-M1
e
-1 = c
and, by virtue of (3.26) and (3.27), that
I x(t )11
~
c
c
L(t-to)
CP( t ) ~ cp(to) e K
c
~ K
L(t-to)
eK
I x(to)11 .
We have obtained an estimate of form (3.6b). The theorem is proved. COROLLARY 3.1. In afinite-dimensional phase space equation (3.1) with an integrally bounded A(t) is e-dichotomic on [0, 00) precisely when there corresponds to each function f(t) E C(lB) at least one bounded solution on [0, 00) of the inhomogeneous equation.
5. e-dichotomic equation on the real line : .Jf = (- 00, 00). In this case the subspace consists of the initial values of those solutions that remain bounded as t -+ + 00 (t -+ - 00) Since these subspaces have only the zero element in common, the equation does not have nontrivial solutions that are bounded on the real line. Further, the solutions x(t) for which x(O) ¢ ~1 (x(O) ¢ ~2) unboundedly increase as t -+ + 00 (t -+ - 00). Let us show that a theorem analogous to Theorem 3.3 is valid. We recall that the right and left !7-sets ~1 and ~2 of equation (3.1) are the linear manifolds consisting of the initial values of those solutions of this equation that are bounded on the right and left halflines respectively. ~1 (~2)
174
IV. EXPONENTIAL SPLITTING OF SOLUTIONS
THEOREM 3.3'. Suppose the operator function A(t) (- 00 < t < 00) is integrally bounded. In order for equation (3.1) to be e-dichotomic on the real line it is necessary and sufficient that the right and left.? -sets of this equation be mutually complementary subspaces and that there correspond to each function f(t) E C(~) at least one bounded solution on (- 00, 00) of the inhomogeneous equation (3.14). This solution is the only one bounded on the real line. PROOF. The necessity follows from Theorem 3.2 without the assumption that A(t) is integrally bounded. To prove the sufficiency we note that under the hypothesis of the theorem· there corresponds to every functionf(t) that is bounded on either the right or left halfline a solution x(t) that is bounded on this halfl,ine. Applying Theorem 3.3 to each of these halflines, we see that equation (3.1) is e-dichotomic on each of them for one and the same pair of projections PI and P z• This means that the equation is e-dichotomic on the whole real line. The uniqueness of a bounded solution follows from the absence of nontrivial bounded solutions on the real line of a homogeneous e-dichotomic equation. REMARK 3.7. We note that an equation that is e-dichotomic on both the right and left halflines is not necessarily e-dichotomic on the real line inasmuch as the corresponding pairs of mutually complementary projections need not be the same. COROLLARY 3.2. In afinite-dimensional phase space equation (3.1) with an integrally bounded coefficient A(t) is e-dichotomic on the real line (- 00, (0) precisely when there corresponds to each function f(t) E C(~) at least one (exactly one) bounded solution on (- 00, (0) of the inhomogeneous equation.
§ 4. Exponential splitting 1. Definition and basic properties. We consider the following natural generalization of the notion of an exponential dichotomy. We will say that equation (3.1) admits an exponential splitting of order n on .Jf if there exists a direct decomposition
(4.1) such that the Bohl intervals [KB(Pk), KB(Pk)] of equation (3.1) corresponding to the subspaces ~k are mutually disjoint and each of the corresponding projections p. is uniformly conjugated on .Jf by the operator U(t):
II U(t)PkU-I(t) II ~ M
(k
= 1,2,.··,n; t E .Jf).
(4.2)
The intervals (KB(Pk), KS(Pk+I)) will be called gaps in the Bohl spectrum. 2) By combining subspaces with neighboring Bohl intervals into direct sums in (4.1), we see that the equation also admits an exponential splitting of any smalleI order, in particular, of the second. 2)
We agree to enumerate the projections in the order of increase of the exponents.
4.
175
EXPONENTIAL SPLITTING
It is not difficult to see that an equation is e-dichotomic precisely when it admits an exponential splitting of second order and the point).) = 0 falls into a gap between the Bohl intervals. We note some properties of these notions. 1) The replacement in equation (3.1) of the operator A(t) by an operator of the form A(t) + AI produces an equation all of the Bohl exponents of the solutions of which have been shifted by the amount A.. This maneuver permits one to shift the point).) = 0 into any gap between the Bohl intervals and thereby obtain an edichotomic equation. 2) Kinematically similar equations have the same Bohl intervals. Thus a kinematic similarity transformation does not disturb the exponential splitting property. In particular, this property is preserved on [0, 00) with preservation of the Bohl exponents if the coefficient A(t) is altered on a finite interval. 3) When J is the right (left) haltline, only the subspace ~1 (~n) consisting of the initial values of those solutions "growing" most slowly (rapidly) is uniql,lely determined by the equation. Any other subspace ~k can always be "rotated" (i.e. the corresponding projection Pk can always be conjugated) within the direct sum of subspaces with subscripts not greater (less) than k under the condition that the sum remain direct. 4) From the integral boundedness of the operator A(t) it follows as in Lemma 3.2 that the angular distance between the subspaces ~k(t) = U(t)~k is bounded from below in J. We recall once again that this property of the subspaces ~k(t) is equivalent to the boundedness from above of the projection functions Pit) = U(t)Pk U-l(t). 5) A stationary equation dx fdt = Ax obviously admits an nth order exponential splitting of its solutions if (and only if) the spectrum O"(A) decomposes into n spectral sets lying in mutually disjoint vertical strips ).)k < Re it < ).)k (k = 1,.··,n). We consider separately the case n = 3 when the left (right) Bohl interval lies in the interior of the left (right) haltline and the center interval lies in a neighborhood of the origin. Here, for any - ).)1 > KB(P1), - ).)3 < KB(P3), V3 > KB(P3) and).)2 < KB(P2) ().)1 > ).)3 > 0, ).)2 > V3 > 0), there exist positive numbers Nk (k = 1,3) such that the following estimates are satisfied on J for any x E ~ and t ~ s:
II U(t)P1xll ~ N1e-v,(t-s) II U(s)P1x ll; I U(t)P2x ll ~ N 1ev,(t-s) II U(s)P2x ll; N 31e- v3 (t-s) I U(s)P3x ll ~ II U(t)P3x ll ~ N 3eV3 (t-s) I U(s)P3x ll· .c
z
I
(4.3)
Let us determine some conditions under which in this case the equation
dxfdt = A(t)x has bounded solutions on J. We introduce a Green function by setting
+ /(t)
(4.4)
176
IV. G (t, s)
EXPONENTIAL SPLITTING OF SOLUTIONS
=
s < t, t < s.
{U(t)P1U-l(S) for _ U(t)(P2 + P3)U-l(S) for
(4.5)
Then at least for bounded functionsf(t) with compact support the formula x(t)
=
f G(t, s)f(s)ds
(4.6)
provides certain solutions of equation (4.4). It can be written more explicitly in the form x(t)
f U(t)P1U-l(S)f(s)ds - f U(t)P2 U-l(S)f(s)ds - f =
s<;;,t
s~t
U(t)P3U-l(S)f(s)ds.
(4.7)
s~t
Estimates (4.3) imply that the first two summands of the right side of this expression are bounded for any bounded functionf(t). This is true for the third summand only under the condition thatf(t) decrease sufficiently fast at infinity. In place of the function G(t, s) we consider an "incomplete" Greenfunction G(t, s), which differs from G(t, s) in that the projection P 2 + P 3 in formula (4.5) has been replaced by the projection P 2 • The function x(t)
=
f
G(t, s)f(s)ds
(4.8)
(t EJ)
turns out to be bounded for any bounded functionf(s). One can easily see that it satisfies the equation
= A(t)x(t) + (I - P3(t)f(t), which coincides with (4.4) if P3(t)f(t) = 0 or equivalently P3U-1(t)f(t) = O.
(4.9)
dxfdt
(4.10)
We note that under the satisfaction of this condition the order of growth of the solutions U(t)P3XO is no longer important, i.e. the restrictions ))3 < ))1 and V3 < ))2 are not essential. In particular, the projection P 3 can commute with U(t), i.e. we can have P 3(t) = U(t)P3U-l(t) = P3. In this case condition (4.10) converts into the condition
(4.11)
Pd(t) = 0 of "orthogonality" of f(t) to the constant subspace incomplete Green function is expressed by the formula G(t, s)
= G(t, s)(I - P3).
~3(t)
==
~3
while the
(4.12)
2. Induced exponential splitting of the solutions of an auxiliary equation. We consider the following differential equation in a phase space [~]: dX fdt = A(t)X - XA(t).
(4.13)
4.
177
EXPONENTIAL SPLITTING
We assume that the corresponding equation in lB dx fdt
=
(4.14)
A(t)x
admits an exponential splitting of order n. The Cauchy transform U(t) of equation (4.13) can be expressed in terms of the Cauchy operator U(t) of equation (4.14) by means of the formula (see (Ill. 1.25») U(t)X
=
(X E [lB]).
U(t)XU-l(t)
We consider the projection transforms fYljkX = PjXPk and let fYll
= 1::: fYljk;
fYl2
j
= 1:::
j>k
fYljk;
fYl3
= 1::: fYljj. j
We assume that the projections are enumerated in the order of arrangement of the Bohl intervals from left to right. From the formula [U(t)fYljk]X
=
U(t)Pj U-l(S) [U(s)X]U(S)PkU-l(t)
•
it can easily be seen that the solutions of equation (4.13) of the form (k
=
1,2,3; X
E
[lB])
(4.15)
will exponentially decrease (increase) as t---'> 00 for k = 1 (k = 2), the exponents of decrease and increase being expressed in terms of differences of the Bohl exponents of the vector equation (4.14). Let F(t) be a bounded operator function satisfying the condition of type (4.10) fYl 3U-l(t)F(t)
= 1:::
Pj U-l(t)F(t)U(t)Pj
=0
j
or equivalently (j
= 1,2,.··,n).
(4.16)
By means of a formula of type (4.6) a bounded solution of the equation dxfdt
=
A(t)X - XA(t)
+ F(t)
(4.17)
satisfying the condition fYl 3U-l(t)X(t) = 0
will be obtained independently of the behavior of the solution X3(t). We write out this formula in detail under the assumption that A(t)Pk = PkA(t) (k = 1,···, n), and hence that U(t)Pk = PkU(t). The incomplete Green function in this case has the form
1::: = 1:::
[U(t)fYlIU-l(S)]X =
~(t, s) X =
{
_
[U(t)&l-'2U-l(S)]X
and consequently the formula
jk
PjU(t, s)XU(s, t)Pk (t;;;; s), PjU(t, s)XU(s, t)Pk (t
< s),
178
IV. EXPONENTIAL SPLITTING OF SOLUTIONS
X(t)
J U(t, s)F(s)U(s, t)dsP J U(t, s)F(s)U(s, t)dsP
= 1:: Pj j
(4.18)
k
s;&;t
- 1:: Pj
(t E oF)
k
s~t
j>k
gives a solution of equation (4.17) for a bounded function F(t) satisfying condition (4.16), which now takes the simpler form
PjF(t)Pj = 0
(t
E
oF; j = 1,2,.··,n).
(4.19)
This condition·is also satisfied by the solution: (t
E
oF; j
=
1,2,.··,n).
(4.20)
Formula (4.18) will be employed in the next section to obtain a useful transformation of equations.
§ 5. Stability of the Bobl exponents of an exponential splitting 1. Reduction to an e-dichotomy. Suppose the equation
dx/dt
= A(t)x
(t
E
oF)
(5.1)
with an integrally bounded operator function A(t) admits an exponential splitting of order n ;;:; 2. We will show that this property is stable in the sense that it is preserved in going from equation (5.1) to the perturbed equation
dx/dt
= [A(t) + B(t)]x
(t
E
oF),
(5.2)
when the perturbation B(t) is sufficiently small, i.e. when the condition 1
t+~o
-'&"0 Jt IIB(.) lid'&" < 0
(t
E
oF)
(5.3)
is satisfied for some '&"0 > 0 and a sufficiently small 0 > o. We wish to determine the nature of the response of the Bohl exponents of an equation to a perturbation. Inasmuch as this has already been done for the upper and lower exponents (see Theorem 111.4.6), we need only examine the behavior of the ends of each gap in the Bohl spectrum. By transforming equation (5.1) with the use of the substitution x = yeAt, we can always shift the point].l = 0 to the center of some gap. The equation then becomes e-dichotomic, i.e. its solutions will satisfy estimates (3.6) and (3.11), in which].ll = ].12 (and it can be assumed that Nl = N 2). We see that it suffices to show that the exponents ].11 and ].iz of the equation are stable in this case. 2. Bounded solutions of the perturbed equation. We first consider the case when oF = [0, (0). Suppose an e-dichotomy of the solutions of equation (4.1) exists for a direct
5.
STABILITY OF BOHL EXPONENTS
179
decomposition ~ = ~1 + ~z with corresponding projections PI and P z• We must first establish the existence of a similar decomposition for the perturbed equation. Let x(t) be a bounded solution on [0, (0) of equation (5.2). It can be regarded as a bounded solution of the equation dx/dt
=
A(t)x
+
f(t),
corresponding to the integrally bounded functionf(t) = B(t)x(t). Formula (3.22) permits one to represent this solution in the form 00
x(t) = U(t)y
+
JG(t, 1:)fC'r)dr
(y
E ~1)
o
and thus to obtain the integral equation 00
x(t) = U(t)y
+ JG(t, 1:)B(1:)x(1:)d1:.
(5.4)
o
Here U(t) and G(t, s) are the Cauchy operator and Green function of the unperturbed equation, and y is a vector of the subspace ~1. It is not difficult to verify that one also has the converse assertion: every bounded solution of the integral equation (5.4) also satisfies the differential equation (5.2).
°
LEMMA 5.1. There exists a 0> such that the satisfaction of condition (5.3) implies the existence of a unique bounded solution on [0, (0) of equation (5.4). This solution is representable in the form
x(t) = F(t)y
(5.5)
where F(t) is a bounded operator function on [0, (0). PROOF. We consider in the space C(~) of bounded continuous vector functions on [0, (0) the linear operator Sx(t) = G(t, 1:)B(1:)x(1:)d1:. It maps C(~) into itself, as follows from Remark 3.6. Using formulas (3.11) for Nl = N z = Nand ))1 = ))z = )), we estimate its norm. We have for n = [t /1:0]
J;'
IllSxll1
00
~ N ..
~
N
J0 e~"lt~TIIIB(1:)lld1:IIIXIII
IIIXIII1:00 t~1 e~"(n~k)To + 1 + k l L r"(k~n~Z)To }
~ NIIIXIII1:00 { 1 + Thus under the condition
1_
~~"TO}.
(5.6)
IV.
180
a<
EXPONENTIAL SPLITTING OF SOLUTIONS
~-----,~~~--;
N ..
-0
{I +
( =
2---~}-
1-
J)
=
2N . J)'Co
2
e-vro
2~ [I + 0
(J)'Co)]
for
'Co
->
+
I
J)'Co --'-_-'-'e"----vr-o
(5.7)
0)
we obtain the inequality IIISIII < 1. From this inequality it follows that equation (5.4) has for each y E )81 exactly one solution x(t) on [0, (0) belonging to C()8). Clearly, this solution linearly depends on y, and by virtue of (4.6) (5.8)
The lemma is proved. 3. e-dichotomicity of the perturbed equation. Using Lemma 5.1, we construct for equation (5.2) the direct decomposition. (5.9)
which leads to an e-dichotomy of its solutions. The subspace ~1 must consist of the initial values xo = x(O) of those solutions of equation (5.2) which remain bounded on [0, (0). From formulas (5.4) and (5.5) it follows that these vectors are given by the formula xo = x(O) = y
where y
E )81
+
=
JG(O, 'C)B('C)r('C)yd'C =
o
(I - PZRP1)y,
and R =
=
JP zU-1('C)B('C)r('C)P1d'C
o
is a bounded operator inasmuch as (3.11), (5.3) and (5.8) imply that
-~ 1N'-Cooe
I RII
.. VTo
Nllpd.
I - N 'COO
(
2) + -----1 - e- vro
I
Thus the bounded operator 1 - P ZRP1 maps the subspace operator has the bounded inverse (I - PZRP1)-1 = 1
)81
+ PZRPh
and therefore the subspace ~1 is closed. The operator
f\
= (I =
PzRP1)P1(1 - PZRPl)-l
(I - PZRPl)Pl(I
+ PZRPl )
= P l - PZRPl
(5.10)
onto ~1. This
5.
STABILITY OF BOHL EXPONENTS
181
is a projection whose range coincides with ~l. The complementary projection has the form
P2 = I - PI = Pz + PZRP1 = Pz(I + RP1), showing that ~2 = 78 2 • To obtain the desired result we must estimate the solutions of equation (5.2) that are initially in the subspaces ~l and ~2. We first consider a solution Xl(t) satisfying the condition Xl(O) E ~h i.e. an arbitrary bounded solutions of equation (5.2). These solutions are given by formula (5.4). Since 00
Xl(S)
= U(s)y + S G(s, 7:)B(r)xlr)dr, o
it follows that 00
S G(s, 7:)B(7:)Xl(7:)d7:
y = U-l(S)Xl(S) - U-l(S)
o
00
= P1U-l(S)Xl(S) - P1U-l(S) S G(s, 7:)B(7:)Xl(7:)d7: o
s
= P1U-l(S)Xl(S) - SP 1U-l(7:)B(7:)Xl(7:)d7:. o
Substituting this expression into (5.4), we obtain for t > s the equation s
00
+ S G(t, 7:)B(7:)Xl(7:)d7:
S U(t)P1U-l(7:)B(7:)xl(r)d7:
-
o
0 00
= U(t)P1U-l(S)Xl(S) + S G(t, 7:)B(7:)Xl(7:)d7:. s
By virtue of estimate (3.6) it implies the integral inequality
Ilx(t)11
~
Ne-v(t-s)
IIX(S)II +
co
N
S e- VIH1 1IB(7:)llllx(7:)lld7:.
(5.11)
But this implies according to Lemma 111.2.2 that
Ilx(t) I
~
N1e-If(t-s)
Ilx(s) I
(t ;;;; s),
where Nl -
-
N().!LpZ) --------~).!z-pL2N).!0 + 0(7:00)'
-------
and p satisfies the single condition
(5.l2)
182
IV. EXPONENTIAL SPLITTING OF SOLUTIONS (0 <) f-l < ,.; v2-2N))0
+
(5.13)
O(t"oo).
We note that, by decreasing 0, we can make f-l as close as we like to)). Suppose now xz(t) is a solution satisfying the condition xz(O) We make use of the following formulas under the restriction t < s:
E ~2
=
)82'
t
x(t) = U(t)xz(O)
+ JU(t, t")B(r)x(t")dt", o s
xes) = U(s)xz(O)
+ JU(s, t")B(t")x(t")dt". o
Employing the relation xz(O) = Pzxz(O) = Pzxz(O) in a manner analogous to the way in which we obtained the above estimate of Xl(t), we deduce the equation
x(t)
= U(t)PZU-l(S)X(S) +
t
JU(t)P1 U-l(r)B(t")x(t")dt" o
s
- JU(t)P2 U-l(t")B(r)x(t")d-r t
and then the integral inequality
Ilx(t)
I
~ Ne-v(s-t)
Ilx(s) I
t
+ N Je-v(t-r) IIB(-r) 1IIIx(t") Ild-r o
s
+
N
Je-v(r-t) IIB(t")llllx(-r)lld-r t
s
= Ne-v(s-t) Ilx(s)II + N Je-v't-r'IIB(-r)llllx(-r)lld-r. o
From this inequality it follows in the same way as above that
Ilx(t)
I
~ N 2e- p (s-t) Ilx(s)
I
(t
~
s)
(5.14)
for f-l satisfying condition (5.13) and some N 2 • We thus arrive at the following result. THEOREM 5.1. Suppose equation (5.1) with an integrally bounded coefficient A(t) admits an exponential splitting of order n. For any e E (0, eo), where eo is one half of the minimal width of the gaps in the Bohl spectrum, there exists a number 0 > 0 depending only on equation (5.1) and the number e and such that under condition (5.3) equation (5.2) also admits on exponential splitting of order n. The Bohl exponents /i,B(Pk ) and /i,B(Pk ) of equation (5.2) then satisfy the estimates
IcB(Pk) ~ /i,B(Pk)
-
e,
/i,B(Pk) ~ /i,B(Pk)
+ e.
I
REMARK 5.1. If the operator function B(t) satisfies the condition IIB(t) < 0 (t E Ji), we can put = 0 in estimates (5.7) and (5.13) and thereby obtain the inequalities
t"o
5. STABILITY OF BOHL EXPONENTS
183
;; < vj2N and f1. < .j v2 - 2Nv;;. REMARK 5.2. The integral boundedness condition (5.3) need only be satisfied for t greater than some to > 0, i.e. it can be replaced by a condition of type (111.4.16). REMARK 5.3. An analogous result holds when equation (5.1) admits an exponential splitting on the real line. As before, it suffices to consider the case of an e-dichotomy. We can consider equation (5.2) on each halfline (right and left) separately. By virtue of Theorem 5.1 it can be asserted that for sufficiently small ;; > equation (5.2) is e-dichotomic on each of the halflines, the dichotomy on the right (left) halfline being connected with the subspaces ~I and lB2 (lB l and ~2)' where ~I = (/ - P 2RPI )lB l and ~2 = (/ - P I R'P2)lB 2 • Also the operator
°
S
= (/ -
P 2RPI )PI
+ (I -
P I R'P2)P2
=/-
(P2RPI
+ PI R'P2)
is bounded for a sufficiently small (possibly smaller) ;; > 0, invertible (since the operators Rand R' turn out to be small) and such that ~I = SlB l and ~2 = SlB2. Therefore lB = ~I + ~2' It follows from Remark 3.4 that the pair ~b ~2 is connected with an e-dichotomy of the solutions of(5.I) on each of the halflines and hence on the whole real line. 4. Stably decomposing equations. In this subsection we consider equation (5.1) under the assumption that it admits an exponential splitting on an infinite or semi-infinite interval and that the operator A(t) commutes with the projections PI,,", Pn inducing this splitting: A(t)Pk = PkA(t) (k = 1,.··, n). Such an equation decomposes into a system of independent equations in the phase spaces lBk = PklB:
(k
=
1,2,.··,n),
and the Bohl intervals [Vk, Vk] of these equations are pairwise disjoint. We now consider the perturbed equation (5.2) and show (Theorem 5.2) that if the integral boundedness condition (5.3) is satisfied with a sufficiently small ;; > 0, equation (5.2) is kinematically similar to an equation of the form dx jdt
=
[A(t)
+
C(t)]x,
(5.15)
where the operator C(t) is sufficiently small and commutes with the projections Pk •
From this result it will follow independently of Theorem 5.1 that the exponential splitting property is stable for an equation of the described special (quasidiagonal) form. We note that this permits us to eliminate the requirement of integral boundedness of A(t). It will be shown in the next section that any equation of form (5.1) admitting an
184
IV.
EXPONENTIAL SPLITTING OF SOLUTIONS
exponential splitting in a Hilbert space is kinematicaHy similar to a system of independent equations of the described type. We first note that if equations (5.2) and (5.15) are kinematically similar the, transforming operator Q(t) = U(t)V-l(t), where U(t) and Vet) are fundamental operators of these equations, must satisfy the equation Q'(t)
=
[A(t)
+
B(t)]Q(t) - Q(t)[A(t)
+
C(t)]
(5.16)
= A(t)Q(t) - Q(t)A(t) + B(t)Q(t) - Q(t)C(t).
We need only select the operator C(t) so that equation (5.16) has a solution Q(t) that is bounded together with the operator function Q-l(t). It is necessary in this connection that C(t) be of quasidiagonal form: (5.17) We will seek a solution in the form Q(t) = 1 + Set), where the operator Set) is small enough to guarantee the invertibility of the operator Q(t). For Set) we have from (5.16) the equation S'(t)
=
A(t)S(t) - S(t)A(t) + B(t)[1 + Set)] - [I + S(t)]C(t).
(5.18)
A solution Set) of equation (5.18) is a solution of the equation S'(t)
(5.19)
A(t)S(t) - S(t)A(t) +F(t)
=
where F(t)
= B(t)[1 + Set)] - [I + S(t)]C(t).
(5.20)
Equation (5.19) has been considered in the preceding section. The formula Set)
=
L:
Pj
J U(t, s)F(s)U(s, t)dsPk
s~t
j
- L: j>k
Pj
(5.21)
JU(t, s)F(s)U(t, s)dsPk s;;;;t
represents the bounded solutions of equation (5.19) satisfying the condition (m = 1,2,.··,n),
(5.22)
provided F(t) satisfies the same condition: PmF(t)Pm = 0 (m = 1,···,n). The latter condition, as follows from (5.17), (5.20) and (5.22), reduces to the relation '" PmB(t)[1
+
S(t)]Pm = Pm[I
+
S(t)](f PkC(t)Pk)pm
Pm[1
+
S(t)]PmC(t)Pm.
=
It is satisfied if one puts C(t) = B(t)[1 + Set)] =
L: PkB(t)[1 +
S(t)]Pk.
k
Taking into account relation (5.22), we obtain ~
..
5.
F(t)
= =
B(t)[I + Set)] - [I + S(t)](f PkB(t)[I + S(t)]Pk )
I: PjB(t)[I + S(t)]Pk - Set) I: PkB(t)[I + S(t)]Pk
k#
=
185
STABILITY OF BOHL EXPONENTS
(5.23)
k
I: Pj[I - S(t)]B(t)[I + S(t)]Pk. k*j
The problem consequently reduces to the determination of a solution S(t) (III Sill
< 1) of the integral equation obtained from (5.21) by substituting for F the transform F[S] defined by the right side of expression (5.23). Denoting the transform defined by the right side of (5.21) by L(F), we write this equation in the form
S
=
(5.24)
L(F[SD.
LEMMA 5.2. For any 0 < q < 1 there exists a number 0 > 0 such that under condition (5.3) equation (5.24) has a unique solution subject to the estimate
(t
IIS(t)II ;;i; q
E
J).
(5.25)
This solution satisfies condition (5.22). PROOF. We will prove the lemma for n = 2. Since the reduction of an equation to a quasidiagonal form can be carried out in successive steps in each of which the space is decomposed into a direct sum of two summands, the general case will then follow. In this simplest case equations (5.21) can be written in the form
Set)
=
PI
J U(t, s)[I -
S(t)]B(t)[I
+
S(t)]U(s, t)dsPz
S(t)]B(t)[I
+
S(t)]U(s, t)dSPl~Ll[S].
(5.21')
s~t
- Pz
J U(t, s)[I -
s?;;t
We recall that, inasmuch as equation (5.1) admits an exponential splitting, we have the estimates II PI U(t, s)11 = II U(t)P 1 U-l(S) II ;;i; N 1ev,(t-s) IlpzU(t, s) II = II U(t)PZU-l(S) II ;;i; N 2e- v,Ct-s)
(t ~ s), } (t ;;i; s),
(5.26)
in which ).)1 < ).)z· We consider th~ metric space Kq of operator functions Set) satisfying condition (5.25) with metric IIIS2 - sllll = supIIS2(t) - SI(t)II· 5
From (5.26) and (5.3) we obtain the estimate IIIL1[S] III ;;i; N 1N 2(1
+
IIISIII)2
J e-
27:00
;;i; 1 _ e
<,(v,
v,)
(.
NIN2 1
+ q)2 ,
IV.
186
EXPONENTIAL SPLITTING OF SOLUTIONS
showing that the operator L1 maps the space Kq into itselffor q
(] < (1
q
I_e-r,(v,-v,l
+ q)Z
. N 1N z '
=
2'&'0
(1
+ q)Z
Analogously, using the fact that 11[1 - Sz(t)]B(t)[1
+ Sz(t)]
- [I - Sl(t)]B(t)[1
+
~ 11[1 - Sz(t)]B(t)[Sz(t) - Sl(t)]JJ
~ (1 for S10 S2
E
+ q)IIISz
-
+ Sl(t)]JI
II[Sl(t) - Sz(t)]B(t)[1
+ Sl(t)]JJ
Sdl . IIB(t)11
K q , we obtain under condition (5.27) the estimate
IIIL1[Sz] -
L 1[Sl]
II
~ N 1N 2(l
+ q)III S 2
-
Sdl Je-(v,-v,l't-s'IIB(s)llds f
+
1 N2(l :::;; q) - 2'&'0(]N 1 - e-r,(v,-v,l
IllS - Sill:::;; -q-llls 1+q 2
1
-
2
-
Sill· 1
Thus under condition (5.27) the operator Ll is a contraction in Kq and hence has in this space exactly one fixed point, which is a solution of equation (5.21) having the required properties. Lemma 5.2 and the arguments preceding it imply the following result. THEOREM
5.2. Suppose equation (5.1) admits an exponential splitting on an interval
f with the projections P 1"",Pn commuting with the operator function A(t).
Thenfor afixed '&'0 there exists a number (] > 0 depending only on equation (5.1) and such that under the condition 1
t+r,
-'&'0 Jt
IIB(s)llds < (]
(t
E
f)
(5.28)
a kinematic similarity transformation x(t) = Q(t)y(t) can be found which reduces the perturbed equation (5.2) to the quasidiagonal form (t E f),
(5.29)
with a coefficient differing sufficiently little from the original coefficient A(t). The operator Q(t) is a solution of the nonlinear equation Q'(t) = A(t)Q(t) - Q(t)A(t)
+
B(t)Q(t) - Q(t)
1:: PkB(t)Q(t)Pk, k
I
satisfying the condition III - Q(t) ~ q < 1 and can be obtained from the integral equation (5.21), (5.23) (for S = Q - I) by means of the method of successive approximations.
6. HILBERT PHASE SPACE
187
REMARK 5.4. Under the conditions of the theorem there exists a direct decomposition)8 = )81 )8n such that both of the equations (5.1) and (5.29) decompose into systems of independent equations in the subspaces )8k:
+ ... +
dXk/ dt = PkA(t)Xk
and
dXk/ dt = PkA(t)Xk
+ PkB(t)Q(t)PkXk'
Since the perturbation is sufficiently small for each of them, the shift in the Bohl interval is also small. This result extends to the kinematically similar (to (5.29)) equation (5.2). We note that it does not depend on A(t) being an integrally bounded operator function but requires that A(t)Pk = PkA(t) (k = 1,,,,, n). REMARK 5.5. When ~ is a semi-infinite interval, the condition of integral smallness of the operator function B(t) need only be satisfied for sufficiently large t, i.e. 1. lim t~oo
--:r 0
t+~o
St
IIB(-r)lldzo <
o.
§ 6. Exponential splitting in a Hilbert phase space 1. Reduction to a system of independent equations. When the phase space )8 is a Hilbert space 4), we can strengthen some of the results obtained in recent sections by applying the methods developed in § 1 in connection with the conjugation of projections. We begin by noting the following result, which by virtue of Theorem 2.3 is a direct consequence of the definition of an exponential splitting of the solutions of an equation. THEOREM 6.1.
If the equation dx
dt
=
A(t)x
(t
E~)
in 4) admits an exponential splitting of order n on system of independent equations
~,
(k = 1,2, .. ·,n;
(6.1)
it is kinematically similar to a
t E~)
(6.2)
in the fixed subspaces 4)k = Pk4) (k = 1, .. ·,n) inducing the exponential splitting of the solutions of equation (6.1). REMARK 6.1. Theorem 6.1 in combination with Theorem 5.2 leads to the following result: For afixed ZOo there exists a number 0 > 0 depending only on equation (6.1) and such that if t+TO
S IIB(s) lids < 0 t
the equation
(t
E ~),
(6.3)
IV. EXPONENTIAL SPLITTING OF SOLUTIONS
188
dx fdt
= A(t)x + B(t)x
(t
E
J)
(6.4)
is kinematically similar to a system of independent equations dXkfdt
= Ak(t)xk + Bk(t)Xk
in the same subspaces {>k and in which only on the operators P k (k = 1,. ··,n).
(k
I Bk(S) I
= 1,2,.··,n)
(6.5)
;2; ckIIB(s)ll, where the Ck depend
The latter estimate follows from the formula Bk(t) = PkB(t)Q(t)Pk, where IIQ(t) - I ;2; q < 1 and the relation Ilpkll = 1 holds when Pk = pt. To each of the equations (6.5) we can apply Theorem III.4.6 on the stability of the upper and lower Bohl exponents. This leads us to the following result, which generalizes Theorem 5.1 in the case of Hilbert space in the sense that the integral boundedness of the operator function A(t) is not required.
I
THEOREM 6.2. Suppose equation (6.1) admits an exponential splitting of order n on an infinite or semi-infinite interval J. For any $ E (0, $0), where $0 is one half of the minimal width of the gaps in the Bohl spectrum corresponding to this splitting, there exists a number 0 > 0 depending only on equation (6.1) and the number $ such that under condition (6.3) equation (6.4) admits an exponential splitting of the same order. The Bohl exponents of equations (6.4) then satisfy the estimates ICE(Pk) ;:;; ICE (Pk) -
$;
ICB(Pk);2; ICB(Pk)
+ $.
REMARK 6.2. When J is a semi-infinite interval, condition (6.3) need only be satisfied for sufficiently large t. 2. Exponential splitting of the solutions of equations with a precompactly valued operator function. The importance of the exponential splitting property for equation (6.1) has been demonstrated in the preceding sections. Until now we could assert only that this property is possessed by stationary equations dx fdt = Ax, with a spectrum a(A) decomposing into components lying in mutually disjoint vertical strips, and the equations differing sufficiently little from them. We now obtain for equations in Hilbert space on the halftine J = [0, 00) another test for the presence of this property, which generalizes the test for negativeness of the upper Bohl exponent presented in § IlI.6 We recall that, by definition, an operator function A(t) satisfies condition Sa.L (0 > 0, L > 0) if there exists a number T > 0 such that the inequality A(t) ;2; 0 is satisfied for s, t ;:;; T and t ;2; L. A function A(t) satisfying condition Sa.L can be represented in the form
IIA(s) -
I
Is - I
(6.6) where
6. HILBERT PHASE SPACE
1
Al(t)
= L
1
t+L
J
189
[A(t) - A(r)]dr;
Az(t)
t+L
J
=T
A (r)dr,
(6.7)
t
t
the following conditions being satisfied for sufficiently large t (t
IIA (!)II l
IIAze!) I
~ 0,
~
~
T):
oiL.
(6.8)
THEOREM 6.3. Suppose A(t) (0 ~ t < (0) is a continuous precompactly valued operator function and suppose the spectra of all of its w-limit operators lie in a set U consisting of n mutually disjoint vertical strips Uk =
Pl).lk ~
ReA. ~ ).Ik}
(k = 1,2,.·· ,n ; ).II ~ ).II ~ ).12 ~ ).Iz ~ .. , ~ ).I~ ~ ).In),
each of which contains at least one point of the spectra of these operators. For any $ < min ().Ik+l - ).Ik)/2 there exist numbers > 0 and L > 0 depending only on $ and the w-limit operators of A(t) and such that if A(t) satisfies conditions SiJ,L, equation (6.1) admits an exponential splitting of order n with Bohl intervals satisfying the condition
a
[,,'(Pk)' ,,(Pk)]
C
[).Ik -
$,
).Ik
+ c].
PROOF. For some h > 0 the spectra of all of the w-limit operators lie in the horizontal strip 1m A ~ h (by virtue of the boundedness of this set of operators). We consider the rectangles
I
I
U~' h =
P l).Ik -
$
12 < Re A < ).Ik
+ cj2;
I1m AI < h}.
It follows from Theorem 1.2.1. that, beginning with some t > to, the spectra (J(A(t») lie in the interior of the union of these rectangles while their resolvents are uniformly bounded on the contours r k of these rectangles, the corresponding constants depending only on the quantity $ and the w-limit operators of A(t). Since Az(t) = A(t) - Al(t), it follows from the first estimate of (6.8) for a sufficiently small > 0 that these same properties will also be possessed by the operator function Az(t). By varying A(t) on a finite interval (and hence changing the equation to a kinematically similar one), we can establish that all of these properties as well as estimates (6.8) will he fulfilled on the whole interval [0, (0). In view of Theorem 6.2 it suffices to prove the theorem for the equation
a
dx Idt = Az(t)x.
(6.9)
We introduce the projections (6.10) They are continuous and differentiable together with the operator AzCt); in fact,
190
IV.
EXPONENTIAL SPLITTING OF SOLUTIONS
(6.11)
where
= 21k
Cl
'IT:
sup ).E1,
I [Az(t) -
AI]-I\\
tE [0,00)
(lk
is the length of the contour
rk).
With respect to the projections (6.10) it is possible to construct a conjugation operator O(t) solving the Cauchy problem O'(t) =
n ~
Pk(t)Pk(t)O(t),
0(0)
= I,
(6.12)
k=1
and having the property (6.13)
Since the operator functions Pk(t) (k = 1,,,,, n) are bounded, O(t) uniformly conjugates the projections Pk(O). From Theorem 2.3 it follows that equation (6.12) is kinematically similar to an equation dy fdt = B(t)y the coefficient B(t) and Cauchy operator W(t) of which commute with the projections Pk(O). Let Q(t) be the operator effecting the similarity transformation. Then O(t) = Q(t)W(t)
(6.14)
and from (6.13) it follows that Pk(t) = Q(t)W(t)Pk(O)W-l(t)Q-l(t) = Q(t)Pk(O)Q-l(t).
(6.15)
We note that by the definition of kinematic similarity there exists a constant q > 0 such that
IIQ(t)11 ;£ q, IIQ-l(t)ll;£ q. We now carry out the transformation x into the form ~
tJ:
=
(6.16)
Q(t)y in equation (6.9). It goes over
= [Q-l(t)Az(t)Q(t) - Q-l(t)Q'(t)]y.
(6.17)
The operator Az(t) = Q-l(t)A z(t)Q(t), as follows from (6.15), commutes with the projections Pk(O) , and the spectrum of its restriction to a subspace !Ok = PlO)!O coincides with the part of the spectrum li(Az(t») lying in the interior of
ne,h k
•
It is not difficult to get, using (6.14), that
191
EXERCISES Q'(t)Q~l(t)
=
O'(t)O~l(t)
- Q(t) W'(t) W~l(t)Q~l(t)
n
= I: P£(t)Pk(t) k=l
-
Q(t)B(t)Q~l(t).
From Theorem 2.3 it follows that for some C2 > 0 IIB(t) II
~ c211 ~1 P£(t)Plt) II·
V sing estimates (6.10) and (6.15), we now get that there exists a constant for which
II Q'(t)Q~l(t) I
C3
> 0
~ C30.
We can now estimate the derivative of the operator function A2(t): A2(t) = Q~l(t)A2(t)Q(t) - Q~1(t)Q'(t)Q~1(t)A2(t)Q(t) -
=
Q~1(t)A2(t)Q'(t)
Q~l(t)A2(t)Q(t)
-
-
Q~1(t)Q'(t)Q~l(t)A2(t)Q(t)
Q~l(t)Az(t)Q'(t)Q~l(t)Q(t),
and hence where C4 is a positive constant. With the above estimates we can complete the proof. The equation dy /dt = A2(t)y is equivalent to the system of independent equations (k = 1,2,. ··,n)
in the subspaces {lk = Pk(O){l. We can apply Theorem II1.6.6 to each of these equations and conclude that for a sufficiently small 0 > 0 their Bohl intervals lie within the intervals ().Jk - e /2, ).Jk + e /2). It remains to apply Theorem 5.2 to equation (6.17), which by virtue of (6.15) differs sufficiently little from (6.16). It is not difficult to verify that all of the constants contained in the estimates depend only on e and on estimates of A(t) and its resolvent for sufficiently large t, i.e. in the final analysis on the number e and the w-limit operators of A(t). The theorem is prov€d. EXERCISES 1. Prove the existence of the conjugation operator Q(t, r) described in Theorem 1.1 by assuming (in place of differentiability) that the Pk(t) are continuous functions of bounded variation (Ju. L. Daleckii [3]). Hint. Replace the differential equation (1.10) by an integral equation and make use of the multiplicative Stieitjes integral. 2. Show that the bounded variation condition in Exercise 1 is essential even in a two-dimensional space (L. A. Ivanov [1]).
192
IV.
EXPONENTIAL SPLITTING OF SOLUTIONS
Hint. Consider the operator pet) of orthogonal projection on the straight line forming an angle
oct) with the axis of abscissas. Show that the product P(tn)P(tn-I)···P(tI) has a limit if and only if OCt) is a function of bounded variation. 3. Let P(t) (a ~ t ~ b) be a continuous projection function in a Hilbert space .p. Construct a unitary operator Q q(t) having the property .p(t) = Q It)-p(a), where .p(t) = pet})). Hint. Make use of Theorem 1.2 and the following result (I. C. Gohberg and M. G. Krein [3]): the norm of the difference of two projections on a pair of subs paces achieves a minimum when these projections are Hermitian. 4. Let A(z) be a matrix function of order n that analytically depends on z in some simply connected domain G of the complex plane. Suppose that for each z E G the spectrum O"(A(z» decomposes into spectral sets: O"(z)
== O"(A(z»
= O"I(Z) U O"zez) U ... U O"p(z),
each of which continuously depends on z (in the sense that a contour r k separating 0" iz) from the rest of O"(z) for some z E G will continue to do so under a small variation of z). Show that there exists a matrix U(z) analytically depending on z in G that reduces the matrix A(z) to the quasi diagonal form
Al (z) !
••••• , •• , ••• "', •• ,•• 03 •••••••••••• ",.,, •• ·_
~ A2 (z) ~
"""""""""""~
U-I(z)A(z)U(z) =
where Ak(z) is a matrix function of order nk (k = 1,.··, p) not depending on z (nk= dim ~iz), where ~k(Z) is the invariant subspace of A(z) corresponding to the spectral set O"iz». Hint. Consider the differential conjugation equation (1.10) in a domain of the complex variable z. The matrix U(z) can be expressed in terms of its solution, which is analytic in z. The result noted in this exercise is essentially contained in a paper of Ju. L. Daleckii and S. G. Krein [2]. It was later published by Y. Sibuya (see W. Wasow II]). 5. Show that if the equation x = A(t)x with an integrally bounded coefficient admits an exponential splitting and SO' II B(t) Iidt 00, the equation
<
dxfdt = [A(t)
+ B(t)]x
(0 ~ t
< (0)
admits an exponential splitting with the same Bohl intervals. 6. Suppose the equation x = A(t)x with an almost periodic coefficient A(t) is e-dichotomic on the real line andf(t) is an almost periodic vector function. a) Show that the only bounded solution of the equation dxfdt = A(t)x + f(t) is almost periodic. b) The module M(!) of an almost periodic function is the totality of finite integral linear combinations of the points of its spectrum. The following result is due to J. Favard [1]. Let f and g be almost periodic functions. The relation MCf) C M(g) is satisfied precisely when for 0 such that every a-translation number of g is an e-translation number of any e 0 there exists a f(see § 11.4.5). Show that the module of a solution x(t) of the equation x = A(t)x + f(t) is contained in the ~), which module of the almost periodicfunction (A(t)J(t» (taking values in the direct sum [~] coincides with the minimal module containing the union of the spectra of the almost periodic functions f(t) and A(t). Hint. Consider the family of equations for the translates xr(t) = x(t + -r).
>
a>
+
193
EXERCISES
7. Let A be an operator of simplest type (see Exercise I.1) and let R(t) be an operator function satisfying the condition
f II R(t) II dt o
< 00.
(0.1)
a) Show that the equations dx/dt
= Ax
(0.2)
and dx/dt = Ax
+ R(t)x
(0.3)
are kinematically similar (this means that equation (0.3) is reducible to the form (0.2 ). Hint. By representing the operator Q(t) effecting the kinematic similarity transformation in the form Q(t) = I + Set), obtain for Set) the equation S'(t) = [A
+ R(t)]S(t) -
S(t)A
+ R(t).
(0.4)
<
Prove the existence of a solution of this equation satisfying the condition I S(t) Ii 1 by using the results of § 4.2 and Exercise 5. b) Obtain an analogous result for the case when A is an operator of algebraic type (see Exercise 1.9) and condition (0.1) is replaced by the stronger condition
f
o
t 2"-2 IIR(t)1Idf
< 00,
where n is the maximal index of the eigenvalues of A. c) Extend the results of a) and b) to be case of an equation dx/dt = A(t)x + R(t)x with a varIable operator of simplest type A(t) having eigenspaces not depending on t and eigenvalues ilk(t) with the property that Re [illt) - il/t)] does not change in sign for j, k = 1,···,n. Finite-dimensional systems of this form with an integrable R(t) are customarily called L-diagonal systems (see 1. M. Rapoport [1]). Hint. See the hint to Exercise a). To prove the existence of a bounded solution of equation (0.4) construct a special Green function by dividing the totality of spectral projections 9' jk X = PjXPk of the transform TX = AX - XB into two groups according to the summability of the functions Re[j,lt) - ilj(t)]. 8. Consider in a Hilbert phase space the equation dx dt-
= A ( st)x,
(0.5)
where A(T) is a uniformly continuous precompactly valued operator function. Extend Exercise 111.17 by showing that if the spectra of the ill-limit operators of this operator function lie in a system of vertical strips, equation (0.5) admits an exponential splitting for sufficiently small c O. 9. Theorem 6.3 on exponential splittability conditions for equations with precompactly valued operator functions was proved in a Hilbert phase space only. This would not appear to be completely natural inasmuch as the analogous theorem of Chapter III was proved for an arbitrary Banach space. Investigate the possibility that Theorem 6.3 can be extended to the case of a Banach space. 10. Prove that the equation
>
dx/dt
=
A(t)x
(0.6)
in a Hilbert space i"i is e-dichotomic whenever 1) Re A(t) is an integrally bounded operator function and 2) there exists an indefinite operator WE [,P] such that Re(WA(t» «Ouniformlyint E (- 00,00).
194
IV.
EXPONENTIAL SPLITTING OF SOLUTIONS
The latter condition implies the existence of a positive a such that Re (WA(t»';;; - al
(- 00
< t < 00).
For the stationary case A(t)=const this assertion admits a converse assertion and both assertions together constitute Theorem 11.2.1. In Chapter V this assertion is proved (Theorem V.2.3) for the case of a periodic operator function A(t) = A(t + T). In this case the proof is significantly simpler. When .p is a finite-dimensional real space and A(t) is a bounded real operator function (W is a symmetric real matrix), the assertion is a corollary of a more general theorem of A. D. Maizel' [1] (see Exercise 16). Hint. Make use of the results of the exercise presented below and appropriately extend the arguments used in establishing the assertion of Exercise 14 (it is a special case of the assertion considered here and comparatively detailed hints are given for its proof). 11. Show that a) If condition 2) of Exercise 10 is satisfied, the evolution operator U(t, 'r) of equation (0.6) is a 'r t « 00). u-dichotomic uniform W-contraction for all (- 00 b) If conditions 1) and 2) are satisfied, there exists for any 1 0 a q 0 such that
<) <
>
>
for t - 'r ;::::; I.
W - U*(t, 'r)WU(t, 'r);::::; ql
(0.7)
Hint. Differentiate the left side with respect to 1 (see identity (V.2.4» and make use of the result of Exercise 1.41 as well as the fact that
!
,+I
dt
II U l(t, 'r)11 ;::::;
! exp -! IIReA(s)lIds
r+l
(
t
)
dt
(the latter expression is bounded from below by a constant q(l) >0 for 1>0 by virtue of condition 1». Let .9'(/) (- 00 1 00) be a locally integrable operator function with values in [.p]. Exercises 12-14 below will be formulated for the following equation in .p:
<<
d 2y/dt Z + .9'(/)y = 0,
(0.8)
which by the simple substitution z = dyJdl reduces to the system { dy/dl = z, dz/dl = - .9'(/)y,
(0.9)
i.e. to equation (0.6) for the vector function x = y EEl z in the phase space .p(Z) = .p EEl ~) with d(/) = (_
.~t) ~).
In conjunction with equation (0.8) it is convenient to consider the following equation in d 2Y/d1 2 + .9'(t) Y =
o.
[~')]:
(0.10)
12. Show that if the values of a solution yet) of equation (0.10) are invertible in.p for t in an interval [a, b], the operator function K(/) = Y'(t)y-1(t) (a;£ 1;£ b) satisfies the Riccati operator equation dKJdt
+ K2 + .9'(t) =
O.
(0.11)
We note that a priori a solution K(t) of equation (0.11) can "go to infinity" or "come from infinity" in a finite period of time. One can easily check this by considering the scalar equation (0.11) (when .p is one dimensional). But if a solution K(/) of this equation exists in the whole interval [a, b], it defines to within an
195
EXERCISES
invertible right constant factor in [.p] a solution Y(t) of equation (0.10) on [a, b] with invertible values in [.p]. 13. Show that the following assertions are true. a) Suppose the following condition is satisfied for some a>o: 1) Re.9'(t)« - al (- 00 1< (0). Then for any(- 00<) 'r<1 «00) the evolution operator u(t,'r) of system (0.9) is a u-dichotomic uniform W-contraction, where
<
W=
(~ ~).
b) Suppose in addition that the following integral bounded ness condition for .9'(t) is satisfied: 2) (M =) sup fl+l 1I.9'(s) lids 00. Then for any I> 0 there exists a q such that U(t, 'r) will satisfy condition (0.7).
<
Hint. Obtain this assertion as a corollary of the assertions of Exercise 11. In the following assertions we assume that conditions 1) and 2) are satisfied. c) For any Ko E [.p] withReKo land 'r E (- 00,(0) there exists in the interval ['r, (0) a solution K(t) of the Riccati equation (0.11) satisfying the condition K('r)=Ko; in this connection
»
Re K(t)
~
rl
('r ~ t< (0),
(0.12)
where r depends only on IIKoll and a. Hint. Make use of assertions a) and b) and the assertion of Exercise 1.41. d) Let Kj(t) ('r ~ t oo;j = 1, 2) be two solutions of equation (0.11) satisfying condition (0.12). Then
<
('r~t
Hint. Use the fact that K2 and Kl are solutions of one and the same Riccati equation, calculate the derivative d(K2 - K1)/dt and then show that
where t
Xit) =
J, exp [- Kj(t)]dt
(j = 1,2).
After this it remains to apply Wintner's estimate (III.4.19). 14. Equation (0.8) is said to be e-dichotomic if system (0.9) is e-dichotomic. Show that equation (0.8) is e-dichotomic whenever 2) i?I'(t) is an integrally bounded operator function and 3) Re .9'(/) «0 uniformly in 1 E ( - 00, (0). It is easily seen that if conditions 2) and 3) are satisfied, conditions 1) and 2) of Exercise 10 will be satisfied for system (0.9) and that therefore the above assertion is a corollary of the assertion of Exercise 10. But we insert a hint below leading to an independent solution of Exercise 14 in order to highlight some interesting details. Hint. Choosing au arbitrary 'r E ( - 00, (0), consider the solutionK.(t) ('r ~ t< (0) of the Riccati equation (0.11) satisfying the condition K,('r) = I. The limit K+(t) = lim,->_oo K,(t) will also be a solution of this equation (see Exercise 13) satisfying the condition Re K+(t) 0 uniformly in 1 E (-00, (0). Let 2+ denote the subspace in .p(2) consisting of all x = y EB z such that z = K+(O). Then 2+ will be the left S"-set of system (0.9). The right S"-set can be constructed analogously by reversing the time I. The assertion of this exercise was proved for dim .p 00 and a bounded real operator function .9'(/) in an article of D. V. Anosov and Ja. G. Sinai [1]. In this case it is a direct corollary of a
»
<
196
IV.
EXPONENTIAL SPLITTING OF SOLUTIONS
theorem of A. D. MaizeI' (see Exercise 16). The arguments of Anosov and Sinai carry over directly to the case of a bounded Hermitian operator function .9'(t) = .9'*(t). The assertions of Exercises 13c) and 13d) are generalizations of corresponding propositions in the article of these authors. Exercises 10-14 were taken from an article of M. G. Krein [12]. 15. Suppose the equation (0.13)
dx/dt = A(t)x
in a Hilbert phase space .p is e-dichotomic on the real line. a) Show that there exists a bounded indefinite operator function W(t) such that the following estimate holds for the solutions of equation (0.13): (0.14) An analogous result holds for a halfline. b) Extend the assertion to the case of a Banach space. (J. Massera and J. Schaffer [1]). Hint. See the hint to Exercise 111.15. For an equation on the real line we have W(t)
=
f
U*-'(t)P~ U*('r)H(-c)U(-c)P _ U-l(t)d-c
I
where P± are the projections on the right and left sP-sets of equation (0.13) and H(t) is an arbitrary uniformly positive operator function. 16. An assertion analogous to the assertion 'of Exercise 111.15 is valid for equation (0.13) in a finite-dimensional real space and with an integrally bounded operator Re A(t). Namely, The following assertions are equivalent: a) Equation (0.13) is e-dichotomic on the real line (on [0, b) The equation dx/dt = A(t)x + f(t) has at least one bounded solution on the real line (on [0, for each boundedf(t). c) There exists a bounded indefinite operator W(t) satisfying condition (0.14). The above result was established for bounded A(t) by A. D. Maizel' [1], who generalized a result of I. G. Malkin (see Exercise 111.15) by employing the technique (used also by Malkin) of Perron's transformation. A generalization of this result has been obtained by J. Massera and J. Schaffer [1]. The equivalence of conditions a) and b) in a Banach space under additional assumptions on the sP-sets constitutes the assertion of Theorems 3.3 and 3.3'. These additional assumptions are automatically satisfied in the finite-dimensional case. The fact that a) implies c) was indicated in Exercise 15. As was shown by Schaffer, the converse assertion c)~a) is not true in the general case of a Hilbert space.
(0».
(0»
NOTES e-dichotomic systems of equations with variable coefficients were essentially considered by O. Perron [2], who studied nonlinear perturbations of such equations. Still earlier, an analogous problem for a nonlinear equation with a stationary linear part was considered by P. Bohl. Perron's work was a generalization to the two-dimensional discrete case of a work of J. Hadamard [1]. The condition of e-dichotomicity did not appear explicitly in it. Instead there was given the condition of the existence of bounded solutions of the inhomogeneous linear equation (3.14) for bounded functions f(t).
The equivalence of this condition to the condition of e-dichotomicity was first established by A. D. MaizeI' [1] (see the comments to Exercise 16).
NOTES
197
The e-dichotomicity of solutions of equations in Banach space has been studied by J. Massera and J. Schaffer [1] under more general assumptions than those adopted in the present book (to simplify the presentation we have adopted the unnecessary requirement that the Y -set of the equation be complemented). Our proof of the theorem on the stability of the e-dichotomicity property differs from the one presented by Massera and Schaffer. It should be mentioned that these brief notes do not begin to reflect the very extensive investigations of e-dichotomies performed by Massera and Schaffer for equations of first order, and by P. Hartman [1] for equations of higher order. A lucid account of a number of results in this direction in the finite-dimensional case is contained in the works of W. Coppel [1 - 3]. Coppel was the first to apply the methods connected with conjugation operators for subspaces. To him is due Theorem 1.2 on the normalization of a conjugation operator, Theorem 2.3 on the decomposition of an equation, a proof of Theorem 5.2 based on an idea of N. Ja. Ljascenko [1] and a special case of Theorem 6.3 (when A(t) is a differentiable function with a small derivative). The proofs of all of these results of Coppel made explicit use of the finite dimensionality of the phase space, so that some modification was required in order to transfer them to Hilbert space. The simplest conjugation operators considered in § 1.1 were encountered long ago in papers on perturbation theory. The differential conjugation equation was first introduced by Ju. L. Daleckii and S. G. Krein [2]. These results were later repeated by T. Kato [1]. Ju. L. Daleckii [3] was the first to consider the more general problem when the projections are not assumed to be differentiable. Similarity transformations and the reducible systems connected with them were first considered by A. M. Ljapunov.') A number of results on such systems and the asymptotic behavior of their solutions have been obtained by N. P. Erugin [1], A. Wintner [2], H. Weyl [1], V. A. lakubovic [2,3,5] and others. Unfortunately, we cannot dwell here on the interesting investigations of D. V. Anosov [1] and D. V. Anosov and Ja. G. Sinai [1], in which e-dichotomic equations were used for the study of flows and cascades on a smooth manifold. The monograph of Anosov [11 contains a brief but eloquent historical commentary (which, unfortunately, fails to mention P. Bohl). Some additional historical comments can be found in the exercises for this chapter. 3)
The term kinematic similarity was introduced by L. Markus [1].
CHAPTER
V
THE EQUATION WITH A PERIODIC OPERATOR FUNCTION
In this chapter we consider an important class of linear equations which we will call periodic, i.e. equations with a periodic coefficient A(t). The behavior of the solutions of such an equation is determined by the spectral properties of its monodromy operator. In § 1 we present the more or less traditional material: we introduce the monodromy operator, consider its elementary properties and indicate a condition for the validity of the well-known Floquet representation of a Cauchy operator. In § 2 we study e-dichotomicity conditions for the periodic equation. In § 3 we establish various theorems on the localization of the spectrum of a monodromy operator. The presentation in this section makes use of the notions of deviation and amplitude of an operator introduced in Exercises 1.32-1.34. We next ( § 4) consider the so-called canonical equations, whose coefficients are skew-Hermitian in an indefinite metric. The methods connected with the theory of operators in a space with an indefinite metric that were developed in §§ 7 and 8 of Chapter I underlie the results presented here (these methods are also partially used in § 2, where they permit one to establish a simple edichotomicity test). The basic contents of § 4 are connected with the consideration of stable boundedness (strong stability) conditions for the solutions of canonical equations. We establish exact estimates for the central stability band of a canonical equation with a real parameter. In § 5 we present analogous results for equations of second order. Finally, in § 6 we give a useful method of calculating the monodromy operator for an equation whose coefficient depends analytically on a small parameter, by expanding the logarithm of this operator in powers of this parameter. The creator of the general theory of A-stability bands for the scalar Hill equation, A. M. Ljapunov, attached great importance in his works not only to the establishment of general theorems but also to the deduction from them of individual exact stability tests. We have endeavored to carryon this tradition in the main text as well as in the exercises, which contain important additions to this chapter.
§ 1. Monodromy operator. Reducibility 1. The upper Ljapunov and upper Bohl exponents of the periodic equatwn. In this chapter we consider the important equation dxfdt
= A(t)x,
(1.1)
in which the operator A(t) is a T-periodic operator function, i.e. for some T> 0
A(t + T) = A(t)
(0
~ 1
< 00).
The Cauchy operator U(I) of equation (1.1) solves the Cauchy problem 198
(1.2)
199
1. MONODROMY OPERATOR
I
d~?)
= A(t)U(t),
(1.3)
U(O) = 1.
It is easily seen that the same problem is also solved by the operator
U1(t) = U(t
+
T)U-1(T).
By virtue of the uniqueness of the solution of problem (1.3) we have U1(t) == U(t), which implies U(t + T) = U(t)U(T). The operator U(T) is called the monodromy operator of equation (1.1). We recall that the spectral radius of U(T) is the radius r of the minimal circle containing its spectrum. This radius is given by the formula r = limk~oo {I II Uk(T) II. (see Chapter I, § 2). We now prove the following assertion. THEOREM 1.1. The periodic equation (1.1) has strict upper Ljapunov and strict upper Bohl exponents which coincide with the logarithm of the spectral radius of its monodromy operator divided by the period: IrL
=
IrB
= T-1 In r.
PROOF. By virtue of the continuous dependence on t of the operators U and U-1 there exists a q such that II U(1:) II ~ q, II U-1(1:) II ~ q for 0 ~ 'C ~ T. Suppose t = nT + 'C and t' = mT + 'C' ('C, 'C' E [0, T]). Then U(t) = U('C)Un(T) and U(t') = U('C')Um(T), which implies U(t, t') = U(t)U-1(t') = U('C)Un-m(T)U-1('C'), and hence (l/q2) II Un-m(T) II ~ II U(t, t') II ~ q211 Un-m(T) II. Taking the logarithm of this system of inequalities and dividing each term by
t - t' (> 0), we get In q2
-~ +
In II Un-m(T) II (n - m)T + ('C -'C')
_ < InIIU(t,t')11 <
=
t - t'
Passing to the limit for t - t' formly with respect to t': lim I-t'~oo
~ 00 ,
InIJu(t, t')11 = lim
t - t'
n-m~oo
=
In II un-m(T) II
= (n - m)T +
('C -
Inq2 'C') +~.
we find that the following limit exists uniInll un-m(T) II (n - m)T
lim In {III Uk(t) II Inr k~oo T = ---r"
v. EQUATION WITH PERIODIC OPERATOR FUNCTION
200
In particular, when t' = 0 (t' (Bohl) exponent of the equation.
(0) the left side is the strict upper Ljapunov
-4
COROLLARY 1.1. In order for the periodic equation to have a negative upper Bohl exponent it is necessary and sufficient that the spectrum of its monodromy operator lie in the interior of the unit disk. With the use of the monodromy operator one can easily formulate a condition under which the solutions of the differential equation (1.1) are bounded on the real line. LEMMA 1.1. In order for every solution of the differential equation (1.1) to be bounded on the real line it is necessary and sufficient that its monodromy operator U(T) be stable. PROOF. If a solution x(t)
= U(t)xo of equation (1.1) is bounded, then
sup I un(T)xoll n
= sup liU(nT)xo I = sup Ilx(nT) I n
~
n
sup
~oo
Ilx(t)II
<
00.
Thus the sequence {un(T) }:=~oo is bounded at every element of the space Q3 and hence, by the uniform boundedness principle, is bounded in [Q3]. Conversely, from the formula U(t + nT) = U(t)Un(T) it follows that for I un(T) I ~ c sup
~oo
I U(t)xo I
~ sup I U(t) II· cIlxo II· O;St;ST
REMARK 1.1. It is easily seen that the condition I Un(T) I ~ c (n = 1,2,.··) is equivalent to the boundedness of every solution of equation (1.1) on the halfline t ~
o.
2. Reducibility condition for the periodic equation. Floquet representation. We assume that the monodromy operator U(T) of equation (1.1) has a logarithm. This means that an operator S = In U(T) exists for which U(T) = eS • We introduce'the operator = T~l In U(T). Then U(T) = e Tr . We now put
r
Q(t) =
U(t)e~tr.
(1.4)
The operator function Q(t) is T-periodic:
Q(t + T) = U(t + =
T)e~(t+T)r
U(t)U(T)e~Tre~tr
=
U(t)e~tr
= Q(t).
20]
]. MONODROMY OPERATOR
On the segment 0 ~ t ~ T this operator function is continuous, differentiable and has a continuous inverse Q-I(t}. From (1.4) we obtain a Floquet representation of the Cauchy operator U(t} = Q(t}etr
(1.5)
in the form of a product of a differentiable periodic operator function Q(t} having a bounded inverse Q-I(t} and an exponential operator function etr with a constant operator From the foregoing we obtain a theorem on the Floquet representation.
r.
THEOREM 1.2. In order for the Cauchy operator of the periodic equation (1.1) to admit a Floquet representation (1.5) it is necessary and sufficient that the monodromy operator of this equation have a logarithm. This will be the case, in particular, if the spectrum of the monodromy operator does not surround the originY PROOF. The sufficiency of the existence of the logarithm has been proved above. The necessity follows from the relations Q(T}
= Q(O} = U(O} = I and U(T} = Q(T}e Tr = e Tr •
To prove the last assertion we choose a closed Jordan contour r surrounding the spectrum a(U(T}) but not surrounding the origin, and a branch of In A having a single value on this contour. A logarithm of the monodromy operator can then be constructed with the use of the formula In U(T} = - -2]. § In A[U(T} - AI]-IdA.
(1.6)
'lrlr
The theorem is proved. Some other tests for the existence of a Floquet representation are indicated below in § 3 (with the use of the theorem on the localization of the spectrum of the monodromy operator) and in § 4 (for canonical equations). REMARK ] .2. Recalling a necessary and sufficient condition for the reducibility of an equation (see § 2 of Chapter IV), we obtain the following result of Ljapunov, which is a rephrasing of Theorem 1.2: in order for the periodic equation (I.]) to be reducible by mea1is of the T-periodic operator Q(t} to the form dyfdt = ry, it is necessary and sufficient that there exist a logarithm of the monodromy operator.
Let us consider an important special case. Suppose 58 is a Hilbert space .p and suppose the following condition is satisfied for some invertible Hermitian operator 1) The spectrum 11( U(t) of course cannot surround the origin if the space or if A(t) is an operator function with completely continuous values.
~
is finite dimensional
202
V. EQUATION WITH PERIODIC OPERATOR FUNCTION
WEW]:
WA(t)
+
A*(t)W = 0,
(1.7)
i.e. the operator A(t) is W-skew-Hermitian. By virtue of Lemma III.1.1 we at once obtain the first part of the following theorem. THEOREM 1.3 ..If condition (1.7) is satisfied, the Cauchy operator U(t) of equation (1.1) (in particular, its monodromy operator U(T» is W-unitary. The spectrum of
this operator is symmetric relative to the unit circle. lf, in addition, the spectrum a(U(T» does not surround the origin, then in the representation (1.5) the operator is W-skew-Hermitian while the operator Q(t) is W-unitary.
r
r
PROOF. It remains to prove that is W-skew-Hermitian; the latter assertion then follows from formula (1.4) since in this case the operator e- tr is also W-unitary. We make use of formula (1.6). From the symmetry of the spectrum of the Wunitary operator U(T) relative to the unit circle it follows that the contour r can be chosen so that it is symmetric relative to the unit circle. Inasmuch as the fact that U(T) is a W-unitary operator implies the equality U(T)W-l = W-l[U-l(T)]* (see § 1.8.1), we have
- wr =
2 I'T § In A' W[U(T) - AI]- ldA nl r .
§ In A[U(T)W-l
=
_1._
=
_1._ § In 1.[( U*(T»-1 - AI]-ldA' W. 2nzT r
2nzT r
- AW-l]-ldA
Further,
[(U*(T»-1 - 1.1]-1
=
[I - AU*(T)]-IU*(T)
1
= [I - AU*(T)l-{ (I - (I - AU*(T») =
1
[(I - AU*(T»-I- I]
--}I
=
J
-12( U*(T) -1 Itl
and, as a result of the analyticity of the function A-I In Ain the interior of r,
wr=
1 2niT
In A [ §;:z U*(T)
J-l
1 -;:1
dA' W.
(1.8)
r
Let r I be the contour described by the point f1 = 1 /1. as A runs along r. This contour, like r, is symmetric relative to the unit circle and obviously surrounds the spectrum of the operator U*(T) while separating it from the origin.
2. EXPONENTIAL DICHOTOMY OF SOLUTIONS
203
Carrying out the change of variable 1/A = f1 in the integral (1.8), we obtain the required assertion: wr = -
2 l'T 'Jl:l
§ In f1[U*(T)
- f1I]-ldf1' W = - r*w,
r'
since (1.6) and the symmetry of r imply the equality r*
= _1._ § In A[U*(T) - AI]-ldA 2mT r
=
2 I'T 'Jl:l
§ In f1[U*(T) r
- f1I]-ldf1.
§ 2. Exponential dichotomy of the solutions of the periodic equation 1. e-dichotomicity criterion. With the use of the monodromy operator we can give a simply formulated condition for the e-dichotomicity of the periodic equation on the real line.
THEOREM 2.1. In order Jor the periodic equation (1.1) to be e-dichotomic on the real line it is necessary and sufficient that the monodromy operator U(t) be u-dichotomic (i.e. that the spectrum O'(U(T)) not intersect the unit circle and contain components lying in both the interior and exterior oj the unit circle). PROOF. NECESSITY. If equation (1.1) is e-dichotomic, there corresponds to each bounded continuous vector functionJ(t) on the real line exactly one bounded solution of the equation
~~ =
+ J(t).
A(t)x
(2.1)
From the uniqueness of the solution it follows that there corresponds to a Tperiodic function J(t) a T-periodic solution x(t), since the same equation (2.1) is also satisfied by the function x(t + T) in place of x(t). For an arbitrary y E )B we choose the vector function Jy{t) = - (6/T3)t(T - t) . U-I(T, t)y for 0 ~ t ~ T and periodically extend it on the real line. The unique periodic solution x(Y)(t) of equation (2.1) corresponding to it can be written in the form x(Y)(t)
= U(t)X6Y) +
t
JU(t, 'C)h(r:)d'C o
6
t
= U(t)X6Y) - 'J'3 JU(t, T)r:(T - r:)dr:' y, o
from which for t = T we obtain X6Y) = U(T)X6Y) - y or
204
v. EQUATION WITH PERIODIC OPERATOR FUNCTION [U(T) - I]xaY) = y.
(2.2)
Thus equation (2.2) has exactly one solution for any y E~. This means that [U(T) - 1]-1 E [~] and hence that the point A = I is not contained in the spectrum O"(U(t). We now consider the equation
~~ = A(t)x + f(t)e2JriOJt/T. Setting x(t) = z(t)e21riOJt/T, we reduce it to the form dz/dt = [A(t) - 2niw/T]z
+ f(t).
This equation has the same properties as (2.1), and hence the spectrum of its monodromy operator U(T)e- 21riOJ does not contain the point A = 1. But this means that the spectrum O"(U(T) does not contain the point A = e21riOJ . Since w is arbitrary, the necessity of the condition of the theorem is proved. SUFFICIENCY. Suppose the spectrum of the monodromy operator admits a decomposition O"(U(T) = 0"1 U 0"2 in which the set 0"1 (0"2) lies in the interior (exterior) of the unit disk. We denote by PI, P 2, ~b ~2 the spectral projections and invariant subspaces corresponding to this decomposition. The spectrum of the restriction U1 = U(T)i ~1 coincides with the set 0"1 lying in the interior of the unit disk, and hence the operator U1 has a spectral radius r1 < I. In exactly the same way the spectral radius r2 of the operator U:;\ where U2 = U(T)i~2' turns out to be less than one. There therefore exist numbers N > 0 (0 < q < 1) for which (n = 1,2,.· .).
(2.3)
To prove the required assertion it suffices to show that the upper Bohl exponent "B(P1) is negative while the lower Bohl exponent "f,,(P2) is positive. We make use of formulas (111.4.10). Let t1 = nIT + Z"1 and t2 = n2T + Z"2 (Z"h 'Z"2 E [0, T]). Then
U(t 2)Pk U-1(t 1)
=
U('Z"2)[Un'(T)Pk U-n,(T)]U('Z"1)
=
U('Z"2)[Un,-n,Pk ]U('Z"1),
and conseque1!tly II U(t2)Pk U-1(t 1) II ;£ M21IUn,-n'lllB., where M = sUPo~<~TIIU('Z")II. Therefore "B(P1 )
= lim InIIU(t2)P1U-1(t1)11 t,-t,~co
::; -
t2 - t1
lim __ In(M2Nqn,-n,) (n2 - n1)T + ('Z"2 - 'Z"1)
t,-t,~oo
=
InTCL <
o.
2. EXPONENTIAL DICHOTOMY OF SOLUTIONS
205
The second assertion can be proved analogously. The theorem is proved. It is not difficult to verify that the annulus ICB(P1) < < ICB(P2 ) is the maximal open annulus containing the unit circle and not intersecting the spectrum 11( U( T». 2. e-dichotomicity tests for )8 = ,p. Suppose the space )8 is a Hilbert space ,p. In this case it is possible to indicate a simple e-dichotomicity test for equation (1.1) which is connected with the behavior of the coefficient A(t) when,p is rigged with an indefinite metric. Let WE [,p] be an invertible indefinite Hermitian operator. We first derive an auxiliary relation. Let
I;q
H(t)
= - ReWA(t) (= - -} [WA(t) + A*(t)W]).
Integrating the obvious relation
~ [U*(t)WU(t)] dt
=
dU* WU(t) dt
+
U*(t)W dU dt
= U*(t)A*(t) WU(t) + U*(t) WA(t)U(t) = -
2U*(t)H(t)U(t),
from 0 to T, we obtain the formula T
W - U*(T)WU(T) = 2
JU*(t)H(t)U(t)dt. o
(2.4)
From it one easily deduces the following result. THEOREM 2.2. Suppose for some indefinite W = W* (E [,pD T
JRe
o
(WA(t»dt« O.
Then there exists an c > 0 such that for any). (- c < ). < c, ). i= 0) the equation dx dt
=
)'A(t)x
(A(t
+
T)
=
A(t»
(2.5)
will be e-dichotomic and, in addition, the phase space ,p will decompose into a direct sum ,p
=
~)+().)
+ ,p-().)
of uniformly W-definite subspaces that are invariant under the monodromy operator U(T; ).), the spectra of the restrictions U(t; ).)I,p± lying respectively in the interior and exterior of the unit disk. PROOF. If we set H(t)
= - Re(WA(t», we will get according to (2.4) that
W - U*(T; )')WU(T;)')
=
2)'.Jl'().),
(2.6)
206
V. EQUATION WITH PERIODIC OPERATOR FUNCTION
where T
£'(A)
=
JU*(t; A)H(t)U(t; A)dt.
(2.7)
o
Since (see (111.1.11» t
U(t; A) = I
+ A JA(s)ds + ... =
1+ O(A)
o
uniformly in t E [0, T], the Hermitian operator £'(A) will differ arbitrarily little from the operator JlH(t)dt« 0 for real A that is sufficiently small in absolute value. There therefore exists an e > 0 such that the operator £'(A) will be 0 for - e < A < e. But then by virtue of (2.6) the operator U(t; A) will be a uniform W-contraction (W-expansion) for 0 < A < e (- e < A < 0). Since, moreover, the spectrum of the operator U(T; A) (IAI < e) can be made to lie in any neighborhood of the point 1 and a fortiori not cover the unit circle by taking e sufficiently small, the assertion of the theorem is a corollary of Theorem 1.7.1'. REMARK 2.1. We have simultaneously obtained the following sharpening of the second assertion of the theorem: for 0 < A < e (- e < A < 0) the subspace ~+, for which the spectrum rJ(U(T; A)I~+) lies in the interior of the unit disk, is uniformly W-positive (W-negative) while, conversely, the subspace ~_ is uniformly W-negative ( W-positive).
«
THEOREM 2.3. Suppose the following inequality is satisfied for some indefinite W= W*: Re(WA(t») «0
uniformly in t
E ( - 00,
(2.8)
(0).
Then the periodic equation (Ll) is e-dichotomic and, in addition, the phase space decomposes into a direct sum ~ = .p+ ~_ of subspaces invariant under U(T), of which ~+ is uniformly W-positive while ~_ is uniformly W-negative, the spectrum rJ(U(T)I~+) (rJ(U(T)I.p-») lying in the interior (exterior) of the unit disk.
+
~
PROOF. Condition (2.8) implies the existence of a positive a such that H(t) = Re( W A(t») ~ - aI. It follows according to (2.7) that for £'(A) we will have T
(£'(A)X, x) =
J(H(t)U(t; A)X, U(t; A)x)dt
o
~
T
- a
~ IIU(t; A)xl1 2dt ~ -
! IIU-1(t;dt ~)II
T
a
Ilx112,
so that £'(A) «0 (- 00 < A < (0). Therefore, by virtue of (2.6) the operator U(T; A) is a uniform W-contraction for A > 0; in addition, according to the preceding theorem the operator U(T; A)
2.
EXPONENTIAL DICHOTOMY OF SOLUTIONS
207
is a u-dichotomic W-contraction for sufficiently small A > O. But then it is a u-dichotomic W-contraction for all A > 0 (see Corollary l,7.2). Thus the assertions of Theorem 2.2 and Remark 2.1 hold for the operator U(T;A) for all A > 0 and, in particular, for A = 1. The theorem is proved. 3. The periodic equation of second order. We consider in .p the equation
+ .9(t)y =
0
with a periodic operator function .9(t) (= .9(t Setting
+
d 2y /dt 2
z =
:'i:, x = (~) (=
(2.9) T»).
EB z),
Y
we reduce equation (2.9) to the following equation of first order in the phase space .p(2)
=
.p EB .p:
dx /dt
(2.10)
d(t)x,
=
where sl(t) = (_
~(t) ~)
(I = /.5)'
(2.10')
The monodromy operator U(T) of equation (2.10) will also be called the monodromy operator of equation (2.9). It is easily seen that it has the form U(T)
=
(T»)
IfJ 1fJ'(T) ,
C/J( T) ( C/J'(T)
(2.11)
where C/J(t) and lfJ(t) are the solutions of the equation in [.p] d 2 Y/dt 2
+ .9(t)Y =
0
distinguished by the initial conditions C/J(O) 1fJ(0)
= =
I, C/J'(O) 0, 1fJ'(0)
=
0;
= I.
We introduce an indefinite operator which belongs to [.))(2)] and whose spectrum consists of the points ± 1 : W= (
0
-I
- oI) .
(2.12)
We have
-
Wd
+ d* W
=
(.9 +0 .9* _ 210) .
This relation together with Theorem 2.3 permits us to prove the following proposition.
V. EQUATION WITH PERIODIC OPERATOR FUNCTION
208
THEOREM 2.4. Suppose Re&P(t)
«0
uniformly in t
E ( - 00,
(0).
(2.13)
Then the phase space .p(2) decomposes into a direct sum of two subspaces ~± that are invariant under the monodromy operator U(T) of equation (2.9) and such that the spectrum of U(T)I~+ (U(T)I~_) lies in the interior (exterior) of the unit disk. Each of the subspaces ~± can be given by the equation z = ± K ±y,2l where the operators K± (E [.pD are invertible uniformly dissipative operators.
The operator K+ (K_) will be called the plus (minus) equation operator of the periodic equation (2.9). PROOF. The fulfilment of condition (2.13) implies the fulfilment of the condition of Theorem 2.3 for the corresponding equation (2.10). Therefore the first assertion of Theorem 2.4 is a corollary of Theorem 2.3. Moreover, according to Theorem 2.3 it can be asserted that the subspace ~+ (~_) in the decomposition .p(2) = ~+ + ~_ is uniformly W-positive (W-negative). Thus in regard to ~+, for example, we can assert the existence of a constant m (> 0) such that (Wx, x) ~ m(x, x) for x E ~-'-. Since W has the form (2.12), this property means that - (y, z) - (z, y) ~ m(IYI2
+
(x = y EB
Iz12)
Z E ~+)
or, equivalently Ily - Zll2 =
Ily
+ Zll2
~ 2m (lIyll2
m(lIy - Zll2
+
lIy
+
+
Z1l2)
I z 1l2) (x = y EB ZE~+).
(2.14)
Let S± denote the operators mapping ~+ (c .p(2)) into .p according to the rule S±x = y Z (x E ~+). If S+x = y - Z = 0 for some x E ~+, it will also be true by virtue of (2.14) that y + Z = 0, i.e. x = y = Z = O. Thus the operator S+ is a one-to-one mapping of ~+ onto ilR = S+~+, i.e. to any u E ilR there corresponds a unique pair y, Z E.p such that u = y - Z, y EB Z E ~+. We put +u = y + Z for any u E ilR. Then inequality (2.14) can be written In
+
.Y'
the form lIull2 -
II.Y'+u1l 2 ~ m (lIull2 + II.Y' +uIl 2),
from which if follows that m < 1 and (u
Thus the linear operator Since it follows from y -
E
ilR).
.Y' + acting from ilR into .p is a uniform contraction. Z = u and y + Z = .Y' +u that
2) We say that a subspace E c .pI EB .p2 has the equation X2 = KXb where K E [.ph .p2], if E = [x EB Kx Ix E .pI}' The operator K is called the equation operator for E (more precisely, the equa-
tion operator for E with respect to (.ph .p2»'
2.
209
EXPONENTIAL DICHOTOMY OF SOLUTIONS
21
y =
(u
+ ff +u),
Z
=
21 (ff +u
- u),
we have (2.15)
Let us show that iJR = .p. Suppose iJR is a proper part of .p. Then ff + has an extension ffE[.p] with norm Ilffll = Ilff+II(~[(I-m)/(l +m)]1/2). Such an extension can be obtained, for example, by defining ff on the closure m as the closure of ff + by continuity and setting ff equal to zero on .p iJR. After this, by setting
e
2
=
{(u
+
ffu) EB (ffu - u)lu
E
N,
we obtain a subspace 2 which contains 2+ as a proper part (the operators / ± ff are continuous one-to-one mappings of ,'0 onto itself) and, in addition, is uniformly W-positive (also inequality (2.14) will be satisfied with the same m for all x E 2). But if 2 is an extension of 2+, it follows from 2+ + 2- = .p(2) that the intersection of 2 with 2- contains nonzero elements. Since 2- is a W-negative subspace, we have arrived at a contradiction. Thus iJR = .p, and it therefore follows from (2.15) that 2+
=
{y EB K+yly
E
''o},
(2.15')
«
where K+ = (ff + - /) (/ + ff +)-1. Since Ilff + II < 1, we get Re K+ 0 (see § 1.7). Thus the assertion of the theorem concerning the subspace 2+ is proved. The assertion concerning the subspace 2- can be proved analogously. The theorem is proved. 4. Equation operators as fixed points. We consider a vector x = y the monodromy operator (2.11) into the vector U(T)x
=
Since this vector also lies in
(I/!(T)y
EB K+y E
.I.? + . It
is mapped by
+ IJf(T)K+y) EB (I/!'(T)y + 1Jf'(T)K+y).
.I.? +,
(2.16) In view of its arbitrariness the vector y in this equation can be dropped. We put ¢(Z)
= (I/!'(T) + IF'(T)Z)(I/!(T) + IJf(T)Z)-1
for those Z for which the right side is meaningful. It can be shown (see Exercise 1.41) that under condition (2.13) of Theorem 2.4 the linear fractional function ¢(Z) is meaningful for any Z in the interior 09(f,) = {ZIZE [{ij, ReZ«Oj
of the left operator halfplane and maps it into itself. Relation (2.16) shows that K+ (E 09) is a fixed point of the mapping ¢: ¢(K+) = K+. It can be shown that K+ is the only fixed point of ¢ in 09. Moreover (see Exercises IV.13 and 1.41), for any
v.
210
EQUATION WITH PERIODIC OPERATOR FUNCTION
ZoEm
= n->oo lim
K+
Z.,
where Zn
= ¢(Zn-l)
(n
= 1,2,···).
(2.17)
We consider the linear fractional transformation ¢-I(Z)
=-
(W'(T) - ZW(T»-l((j)'(T) - Z(j)(T»,
= ¢-I(¢(Z» = Z. It maps the interior n~(i) = (Z IZ E [i)], Re Z » 0)
which is the inverse of ¢: ¢(¢-I(Z»
of the right operator halfplane into itself and its only fixed point in n~ is - K_, for which there exists a formula analogous to (2.17).
5. Splitting of the solutions by means of the initial conditions. Let r ± denote the spectral radii of the restrictions of the operators U(T) and U~1(T) to 53+ and 53~ respectively. Then the quantities "B(P +) = In r + « 0) and "B(P~) = - In r ~ (> 0), where P ± are the spectral projections of U(T) corresponding to the subspaces 53±, will be Bohl exponents of equation (2.10) in the phase space .p(2). Thus for any c > 0 there exists an N, > 0 such that Ilx(t) I ~ N,eCKBCP+) +e) (t~s) Ilx(s) I
whenever x(t) is a solution of equation (2.10) with x(O) E 53+ (x(O) E 53~) and - 00 < s < t < 00. The second assertion of Theorem 2.4 permits us to reformulate these relations in terms of the solutions of the original equation (2.9) as follows: suppose y(t) is a solution of equation (2.9) and - 00 < s < t < 00; then if y'(O) = K+y(O), (II y(t) 112
+ I y'(t) 112)112
~ N,eC
while if y'(O)
= -
+ I y'(s) 112)1/2,
(2.18)
+ Ily'(t)112)1/2
(2.19)
K~y(O),
(1Iy(s)112 + IIY'(s)112)1/2 ~ N,e~cK'BCP-)~e)Ct~S)(lly(t)112
Let 53+(t) = U(t)53+. Since U(t + T) = U(t)U(T) while U(T)53+ = 53+, we have 53+(t + T) = 53+(t). By means of arguments similar to the ones that led us to relation (2.15') it is not difficult to see that the subspace 53+(t) has the equation z = K+(t)y, where K+(t) is a continuous operator function with values in (Re K+(t) 0) obtained by substituting K+ for K+(O) in the formula
n
«
K+(t)
=
«(/)'(t)
+ 1/!'(t)K+(O»«(/)(t) + l/!(t)K+(0»~1.
Clearly, K+(t + T) = K+(t). Since y'(O) = K+(O)y(O) in (2.18), it readily follows that the assertion for any c > 0 and a corresponding Ne> 0 estimate (2.18) holds for all solutions y(t) of equation (2.9) with y'(O) = K+y(O) is equivalent to the assertion for any c > 0 and a corresponding M, > 0 the following estimate holds for these solutions:
2. EXPONENTIAL DICHOTOMY OF SOLUTIONS
I yet) I ;:;; Mee(KB(P+) +e) (I-s) II y(s) II
( - 00
211
< s < t < (0).
It is now not difficult to conclude that the least upper bound of Bohl exponents of all solutions yet) of equation (2.9) with y'(O) = K+y(O) coincides with In r +. An analogous assertion can be formulated in regard to the solutions y(t) of equation (2.9) with y'(O) = - K_y(O). 6. The equation of second order with a Hermitian coefficient &P(t). When &P(t) = &P*(t) (- 00 < t < (0), Theorem 2.4 can be sharpened. For suppose &P(t) = &P*(t) (- 00 < t < (0). Then for d(t) of the form (2.10')
we have
where (=
,,*).
It accordingly follows from Theorem 1.3 that U(T) is a ,,-unitary operator:
U* "U
= ,,; U = "-lU*-',,,
and hence that its spectrum (J(U(T» is situated symmetrically with respect to the unit circle. This implies the latter assertion of the following proposition. THEOREM 2.5. Suppose in equation (2.9) &P(t)
= &P*(t)
«0
uniformly in t
E ( - 00,
(0).
(2.20)
Then the equation operators K± of equation (2.9) are uniformly negative Hermitian operators, i.e. K± = K'± O. The spectra of the restrictions of the monodromy operator U(T) to 2+ and 2- are specularly situated with respect to the unit circle.
«
PROOF. Since the fulfilment of condition (2.20) implies the fulfilment of the original condition of Theorem 2.4, the existence of uniformly dissipative operators K± such that z = ± K±y are the equations for the invariant subspaces 2± (c 0\5(2) of U(T) is guaranteed by Theorem 2.4. Since the spectium (J+ = (J(U(T)12+) ((J- = (J(U(T)12-») lies entirely in the interior (exterior) of the unit disk and U(T) is a ,,-unitary operator, it follows from this fact that 2+ and 2_ are ,,-neutral subspaces (see Exercise 1.29). Let Xl = Yl EEl K+Y1 and X2 = Y2 EEl K+Y2 be any two vectors of 2+. The relation ("Xl, X2) = 0 implies (K+Yb Y2) - (Yb K+Y2)
= O.
In view of the arbitrariness of Yb Y2 E~) it follows that K+ ness of K_ can be proved analogously.
= K't-. The Hermitian-
v. EQUATION WITHPERIODIC OPERATOR FUNCTION
212
We give another proof of the Hermitianness of the K± which does not make use of the result of Exercise I.29. Let P + and P _ denote the complementary spectral projections of U(T) corresponding to the decomposition (1(U(T») = (1+ U (1- of its spectrum, so that P ±.p(Z) = £±. Clearly, P"± will be the spectral projections of U*(T) corresponding to the decomposition (1(U*(T») = 0-+ U 0-- in which the spectral sets o-± are the specular images with respect to the real axis of the sets (1±. The corresponding invariant subspaces £1: of U*(T) can be found by means of the formulas £1: = P f.p(Z). It is easily seen that these formulas have the following geometric meaning: £-'i'
= .p(Z)
e £_;
£."'.
= .p(Z)
e £+.
(2.21)
Since, on the other hand, U(T) = /(U*(T»)-l/-l, we have /£."'. = £+ and /£-'i' = £-. In fact, the subspaces /£± are invariant under the operator /(U*(T»)-l/, the restrictions of which to these subspaces have the spectra (o-±)-l = (1+. According to (2.21) a vector Y EB Z (E .p(Z» belongs to £."'. if and only if it is orthogonal to all of the vectors Yl EB K+Yl (Yl E .p), i.e.
From this result we obtain the equation Y = - K-'i'z (y E .p) for the subspace £."'..3) But then, as is easily seen, £+ (= /£."'.) has the equation z = K-'i'y, whereas we already know that it has the equation z = K+y. Therefore K-'i' = K+. Analogously, from the relations £- = /£.+- and £-'i' = .p £_ we can get that K."'. =K_. The theorem is proved. 7. The equation of second order with a parameter. Since for d(t) = (-~(t) ~) and W = (~I we have
e
-J)
JRe(Wd(t»)dt o
= ([ RePJ'(t)dt
oj,
l ° -I
the following proposition can be deduced from Theorem 2.2 by means of the same arguments that were used to obtain Theorems 2.4 and 2.5 (from Theorem 2.3). THEOREM
2.6.
If T
SRePJ'(t)dt» 0,
(2.22)
o
there exists an c >
3)
°such that the assertions of Theorem
I.e. Bot: = (- K:;"z EB z Iz E .p) .
2.4 and a/so, if P(t) is a
3.
LOCALIZATION OF SPECTRUM
213
Hermitian operator function, the assertions of Theorem 2.5 will hold for the periodic equation d 2y/dt 2 when -
E
+ A&>(t) Y
=
(&>(t
0
+
T) = &>(t))
(2.23)
< A < O.
Thus the monodromy operator U(T; A) of the equation will be unstable when A is sufficiently small in absolute value and negative. It will be shown below ( § 6; see also Exercise 8) for a Hermitian operator function satisfying condition (2.22) that the monodromy operator U(T; A) will be strongly stable when A is sufficiently small and positive.
§ 3. Localization theorems on the spectrum of the monodromy operator (>8 = .))) 1. Annular localization. According to Wintner's estimate (lII.4.l9), if >8 = .)), the Cauchy operator of equation (1.1) satisfies the inequalities
II U(T) I II U-l(T) I
G
~ exp A~Am(t))dt ), ~ exp ( -
fAm(Am(t))dt).
This immediately implies the following assertion. 3.1. The spectrum of the monodromy operator U(T) lies in the annulus KA = {(IrA ~ 1'1 ~ R A}, where THEOREM
T
rA
= exp JAm(Am(t))dt, o
T
RA
= exp
JAM(Am(t))dt. 0
Despite the apparent roughness of this localization theorem it has some interesting applications (see Exercise 5 and the commentary on it). 2. Angular localization. If the operator function A(t) is sufficiently "small in the large" on the segment (0, T) then U(t) must be sufficiently close to the identity operator and hence the spectrum of the monodromy operator must lie in the intersection of the ab~ve annulus KA and some sector larg ~ a (a < n). We will cite below two theorems which permit one to distinguish a sector (or a rotated sector) of this type. To deduce these theorems we must make use of the properties of the following functionals, which were introduced in the exercises of Chapter I:
pi
dey U def sup {~(Ux, x)lx ¥- 0, x am U def min {dev(,U)Ii'1
= 1}.
E .))},
v.
214
EQUATION WITH PERIODIC OPERATOR FUNCTION
We recall that the spectrum of U lies in the angle larg pi ~ dev U. Let Ar(Am) = I min (0, - Am(Am), AM(Am» I· THEOREM
3.2. Thefollowing estimate holds: dev (U(T») ~
T
S .v IIA(t) liz -
o
Pr(Am(t»)]2dt (= .9""A).
(3.1)
Thus if.9""A < n, the spectrum of the monodromy operator of equation (1.1) lies in the sector
I arg p I ~ .9""A·
(3.2)
In particular, it lies in the sector T
I arg p I ~ oS IIA(t)lldt.
(3.3)
In proving this and the following theorem (and the intermediate lemmas) we will assume that the operator function A(t) = A(t + T) is continuous. Once the theorems have been proved under this assumption, they can be obtained in the general case by applying them to the approximating equations
and passing to the limit for h ! 0 in the obtained estimates. PROOF. Any solution x(t) = U(t)xo of equation (l.l) clearly satisfies the inequality Ildx/dtll ~
IIA(t)llllx(t)ll·
In addition,4) (3.4)
since by virtue of Wintner's estimates
+ h) I !Am(Am(s))ds ~ In Ilx(tIlx(t)11
t+h
t+h
~! AM(Am(s»)ds .
.,
If we pute(t) = Ilx(t)II-1x(t) or x(t) = Ilx(t)lle(t), we will have
~~
=
:r Ilxll·e + Ilxll ~~.
Since (e, e) = 1, it follows that 4) Whenever x(t) is a continuously differentiable function, IIx(t}11 has a left and right derivative (see Exercise I. 18). It can be assumed for the sake of definiteness that the right derivative of In IIx(t)1I has been taken in (3.4).
3.
215
LOCALIZATION OF SPECTRUM
(de/dt, e)
+ (e, de/dt) =
0,
and consequently 1\
~~ r= (~y+ Ilx 112 ( 4ft Y,
w-
I ~ I = JII%r· ~
v' IIA 112
(~ 1nllx(t)IIY
- [Ar(Affi(t»)p.
Thus (see Exercise 1.31)
1::: (e(T), e(O»)
~ !II ~~
II dt
~ f7 A·
Since 1::: (e(T), e(O») = 1::: (U(T)xo, xo) for an arbitrary nonzero vector Xo E .p, the above estimate implies (3.1) and hence the assertions of the theorem. 3. Angular localization with regard for amplitude. As we know (see § 111.2), equation (1.1) is kinematically similar to the equation
dy/dt = Q-l(t)Affi(t)Q(t)y,
(3.5)
where Q(t) is the unitary operator function solving the Cauchy problem
Q(O) = I.
dQ/dt = iA:iJCt)Q,
Under the substitution y = Qx equation (1.1) goes over into equation (3.5), for which Wet) = Q-l(t)U(t) is a fundamental operator function. The equality U(T) = Q(T) WeT) implies that am U(T)
~
+ am
am Q(T)
WeT)
~
am Q(T)
+ dev
WeT).
We can obtain simple estimates for am Q(T) and dev WeT). The spectral length of a Hermitian operator H, i.e. the length of the shortest segment containing its spectrum (J(H), will be denoted by I(H):
I(H) LEMMA
3.1. Suppose H(t) (0
~
def
t
AM(H) - Am(H). ~
T) is a Hermitian operator function and T
WeT) =
J eH(t)dt.
(3.6)
o
Then dev Wet)
~ ~
JI(H(t»)dt.
(3.7)
°
It can be assumed without loss of generality that H(t) is a continuous operator function. For sufficiently large natural n the product PROOF.
v.
216
EQUATION WITH PERIODIC OPERATOR FUNCTION
n
fi:
(tk = kT/n, Lltk = T/n; k = 1,2,.··,n)
exp (H(tk)Lltk)
1
will be arbitrarily close to WeT). The deviation of this product (see Exercise 1.32) will not be larger than n
I; dey exp (H(t k) Lltk).
(3.8)
k=l
By virtue of the result of L. V. Kantorovic (see Exercise 1.33), for arbitrary G
=
G*
cos (dev (exp G») and consequently sin (dev (exp G»)
=
1 cosh (I(G)/2) ,
tanh (I(G)/2), so that
=
dey (exp G) ~ tan (dev (exp G») = sinh (I(G)/2) = I(G)/2
+
O(/3(G»).
Therefore the sum (3.8) does not exceed
But this implies (3.7). LEMMA
3.2. Suppose H(t) (0
~
t
~
T) is a Hermitian operator function and T
Q(T) =
JeiH(t)dt.
(3.9)
o
Then
dev(~I?Q(T») ~ ~
fI(H(t»)dt,
(3.10)
o
where (0
~
t ~ T).
PROOF. Let Q, ( = O(H») denote the right side of (3.10), which is a trivial estimate unless 0 < 1C. As in the preceding proof, it can be assumed that H( t) is a continuous operator function, so that for sufficiently large n
Qn =
iJ eiH(t.)Jt. == UnUn-l"··U n
1
n;
(tkk = T n' Lltk = T k = 1,2,.··,n)
will differ arbitrarily little from Q(T). The spectrum of Uk = exp (iH(tk) Lltk) lies on the arc
(3.11)
3. LOCALIZATION OF SPECTRUM
217
TR = {e i.l[Am{H(tk»)L1tk ~ A ~ AM{H(tk»)L1tk}' The sum of the lengths of these arcs is equal to I:~=l I{H(tk»)L1tk (= 2wn), which for sufficiently large n will be arbitrarily close to {) and hence less than n when {) < n. Therefore (see Exercise 1.34) the spectrum of the product (3.11) will lie on the product of the arcs Tk (k = 1"",n), i.e. on the arc oflength 2wn with center
'n
=
exp i { ~
iE
+
[AM{H(tk»)
Am{H(tk»)]L1tk }.
Passing to the limit for n -+ 00, we conclude that the spectrum of Q(t) will lie on the arc of length 2{)(H) with center But this is the fact that inequality (3.10) expresses.
'H'
THEOREM 3.3. Suppose the following condition is satisfiedfor equation (1.1): (w
clef)
+J
[/(Am)
o
+
I(A,,)]dt < n.
(3.12)
Then the spectrum of the monodromy operator of equation (1.1) lies in the sector of the complex p-plane
Iarg p where a
E [-
a
I ~ w,
n, n] is determined from the congruence 1
a == -2
T
J[AM(A,,) + Am(A,,)]dt (mod 2n). o
PROOF. We return to the relation U(T) = Q(T) WeT), where Q(t) has the form (3.9) with H(t) = A,,(t) while Wet) has the form (3.6) with H(t) = Q-l(t)Am(t)Q(t). In view of the fact that Q is unitary we have according to Lemma 3.1 that dey WeT) ~
1
2
T I T
JI(Q-IAmQ) dt = 2 JI(Am)dt. o
0
According to Lemma 3.2
dev{e-ir
~
~ ~
dev{e-iaQ(T»)
fI(A,,)dt, o
+ dey
dev{e-iaU(T»)
~
WeT), it follows that
w.
The latter estimate contains the assertion of the theorem. REMARK 3.1. If either of the estimates T
Jo -JIIA(t)[[2 -
[Ar{Am(t))Fdt < n
(3.13)
218
v.
EQUATION WITH PERIODIC OPERATOR FUNCTION
or
; Jo [l(Am(t») + I(Ail(t»)]dt <
7r
(3.14)
is satisfied, the spectrum of the monodromy operator of equation (1.1) will lie in a certain sector, so that the Floquet representation will hold for it, and this equation will consequently be reducible. This fact will be the case if, in particular, T
Jo IIA(t) Iidt < 7r.
(3.15)
§ 4. Canonical differential equations 1. Stable canonical equations. In this section we consider in a Hilbert space a differential equation of the form dx/dt
= iA(t)x,
.p
(4.1)
where A(t) is a periodic operator function that is also W-Hermitian for some indefinite invertible operator W. By suitably renorming the space it is always possible to replace the operator W with an opt;:rator ; such that (see Exercise 1.36) ; = ;* = ;-1. The fact that A is a ;-Hermitian operator means that .!It = ; A is a Hermitian operator. Inasmuch as then A = ;.!It, we conclude that it is possible without loss of generality to consider in place of equation (4.1) the equation dx/dt
= i;.!It(t)x,
(4.2)
where .!It(t) is a periodic Hermitian operator function. An equation of this form is said to be canonical, and the operator .!It is called the Hermitian. The similar (to (4.2» equation dx/dt
= Jr.Yf(t)x
(4.2')
is frequently considered in a real space .p. Here the role of the operator i; is played by a skew-Hermitian unitary operator J r (= - Jt = - J-;;I). The Hermitian operator function .!It(t) in this case is called the Hamiltonian, while the canonical equation (4.2') is called Hamilton's equation. It can easily be shown that to the operator J r there always corresponds an orthogonal decomposition of .p :5)
into subspaces of equal
.p = .pI ED .p2 dimension (dim .pI = dim .p2) and with
5) To this end it is necessary to carry out in the complex hull Exercise III. 6.
respect to which
.i\ of .p the same arguments as in
4.
219
CANONICAL DIFFERENTIAL EQUATIONS
the operator J r has the matrix representation Jr = (
0 lIZ), - IZI 0
where IZI is a unitary operator mapping .pz onto .pI and lIZ = r;/. Thus Hamilton's equation can always be considered in a doubled Hilbert space, assuming that the operator J r has the indicated matrix representation. From Theorem 1.3 it follows that the evolution operator U(t, ..) of equation (4.2), and hence its monodromy operator U(T) = U(T, 0), is .I-unitary, i.e. U*(T).I U (t) = .1,
(4.3)
or, equivalently, U-l(T)
= .IU*(T).I.
(4.4)
In particular, the spectrum of U(T) is symmetric with respect to the unit circle. From formula (4.4) we obtain the equality
II U-n(T) I = I\.IA.I-11\ =
= I\.I Un*(T).I1\ =
II un(T) 1\
(since II.1A.l1\ IIAII for any bounded operator A by virtue of the fact that .1 is a unitary operator), which shows that the stability of a .I-unitary operator is equivalent to the uniform boundedness of its positive powers: sup l\un(t)1\ < 00. n~O
From Lemma 1.1 and Remark l.1 it follows that for canonical equations the condition of boundedness of the solutions on the real line is equivalent to the condition of boundedness of them on [0,00). In accordance with the general definition of Chapter III we will say that equation (4.2) is stable if each of its solutions is bounded on the real line (on [0, 00)). From Lemma l.1 we conclude that the stability of equation (4.2) is equivalent to the stability of its monodromy operator. With the use of Theorem 1.8.1 we at once obtain the following result. THEOREM 4.1. In order for equation (4.2) to be stable it is necessary and sufficient .Pz of subspaces that the space .p decompose into a .I-orthogonal sum .p = .pI that are invariant under the monodromy operator, the subspace .pI (.pz) being uniformly .I-positive (.I-negative).
+
COROLLARY 4.1. A stable canonical equation is reducible; more precisely, there exists a periodic differential operator function Q(t) having a bounded inverse Q-l(t) and such that the substitution x = Q(t)y transforms the given equation into a canonical equation dyJdt = i .IGy with a constant Hermitian G.
By virtue of Theorem 1.2 it suffices to show that the monodromy operator has a logarithm.
220
V. EQUATION
WITH
PERIODIC OPERATOR FUNCTION
Let UI and Uz denote the restrictions of U(T) to -PI and -Pz respectively. With respect to the scalar product (X,y)l = (Ix, y), x, y E -PI (the scalar product (x, y)z = - (Ix, y), x, y E -Pz) the space -PI (-Pz) is a Hilbert space while the operator UI (Uz) is an ordinary unitary operator. Therefore we have the representations Uk = exp(iCk) (k = 1, 2), where Ck is a selfadjoint operator acting in -Pk. The operators Ck can be chosen so that they are positive and their spectra lie on the segment [0, 2n]. We consider in -P the direct sum C of CI and Cz: Cx = CIXI + Czxz if x = Xl + Xz (Xl E ,Ph Xz E -Pz)· The operator C is a -I-Hermitian operator. Setting G = T~I-lC, we get U(T)
= exp (iT -IG).
(4.5)
Later on (see § 6) we indicate a method for calculating the operators Q(t) and G. REMARK 4.1. The spectrum of the monodromy operator of a stable equation lies on the unit circle. In fact, this property is possessed by every stable operator inasmuch as by Theorem 1.6.1. it is similar to a unitary operator. 2. Strongly stable canonical equations. We recall that a -I-unitary operator U is said to be strongly stable if all of the -I-unitary operators in a sufficiently small neighborhood of it are stable. We say that equation (4.2) is strongly stable if there exists a 0 > 0 such that every equation
whose Hermitian satisfies the inequality T
J11Jf'(t) o
Jf'l(t)lldt < 0
(4.6)
is stable. LEMMA 4.1. In order for equation (4.2) to be strongly stable it is necessary and sufficient that its monodromy operator U(T) be a strongly stable operator. PROOF. To prove the sufficiency we note that to the Hermitian Jf'(t) there correspond two constants CI > 0 and Cz > 0 such that
(see Lemma III.2.3). This shows that the mapping Jf'(t) ~ U(T) that takes each Hermitian into its monodromy operator is continuous. Thus under a small variation of the Hermitian we obtain a monodromy operator UI(T) lying in a small
4.
221
CANONICAL DIFFERENTIAL EQUATIONS
neighborhood of V(T). From the strong stability of the latter we obtain the stability of Vl(T). To prove the necessity we must show that any f-unitatyoperator V1(T) lying in a small neighborhood of V(T) is the monodromy operator of an equation with a Hermitian J~''t(t) differing little from £,(t). In other words, we must show that the mapping described above is open.We omit a proof of this fact (see Exercise 4). Lemma 4.1 and Theorem 1.8.3 imply the following fact. THEOREM 4.2. In order for a canonical equation to be strongly stable it is necessary and sufficient that its monodromy operator V(T) be normally f-decomposable, i.e. that under a decomposition6 ) :0 = :01 :02 of:o into uniformly f-definite invariant (under V(T)) subspaces the spectra 17(V(T)I£'1) and 17(V(T)I£'2) do not intersect.
+
3. Central stability band of the canonical equation with a parameter. We now consider the canonical equation with a real parameter A: dxjdt = iAf£'(t)x.
(4.7)
The stability (strong stability) points of this equation are those values of A for which it is stable (strongly stable), i.e. for which its monodromy operator V(T; A) is stable (strongly stable). The set of strong stability points of equation (4.7) is obviously open and therefore, if it is not empty, decomposes into a system of open intervals-the stability intervals of equation (4.7). We note that the following representation is valid: T
V(T; A) = 1+ iAf
J£'(s)ds + O(A2)
(A
-+
0).
(4.8)
o
For if we integrate each side of the equality dVjdt = iA.J'£'(t) Vet; A), we get t
U(t; A) = 1+ iA
Jf£'(s)V(s; A)ds.
o
By iterating this equality and setting t = T, we obtain (4.8) (cf. (111.1.11)). THEOREM
4.3.
If the operator £'av =
1 T
T
J£'(s)ds o
is uniformly positive, there exists a maximal interval (A-1' AJ) (30) (the central stability band of the equation) all of whose nonzero points A are strong stability points of equation (4.7). 6)
See Theorem 4.1.
222
V.
EQUATION WITH PERIODIC OPERATOR FUNCTION
PROOF. It suffices to show that A is a strong stability point if it is different from zero and sufficiently small in absolute value. From (4.8) we obtain the equality
U(T; A) = 1+ iA[T ,1Yfav
+
O(A)].
From the properties of normally W-decomposable operators cited in § 1.8 it follows that under the conditions of the theorem the operator T,I Yfav is normally ,I-decomposable. Since the property of normal ,I-decomposability is stable (Theorem 1.8.2), it is not affected by adding a small operator O(A) to T ,IYfav . It is also obviously preserved under multiplication by a scalar iA i= 0 and the subsequent addition of tlie identity operator. Thus the ,I-unitary operator U(T; A) is normally ,I-decomposable. By Theorem I.8.3 it is strongly stable. The theorem is proved. The purpose of the following arguments is an estimation of the central stability band of a canonical equation whose Hermitian satisfies the conditions Yf(t)
>0
(0 ~ t ~ T);
Yfav
I T
=
~T
SYf(s)ds» O. o
Let us agree to call such an operator function Yf(t), a Hermitian of positive type, and the canonical equation corresponding to it, a canonical equation of positive type. LEMMA 4.2. Let [a, b] be a closed interval containing zero at each point A of which there exists the operator
[U(T; A)
+ 1]-1
(E[.p]).
(4.9)
If the Hermitian Yt'(t) in equation (4.7) is of positive type, every nonzero point of [a, b] is a strong stability point of this equation. PROOF.
We consider the operator V(A) = i[1 - U(T; Ami
+
U(T; A)]-l.
We shall prove that V(A) can be represented in the form V(A)
=
,IN(A)
(A
E
[a, b]),
(4.10)
where N(A) is a uniformly positive operator. It will then follow from the properties of normally decomposable operators (see § I.8.2) that V(A) is normallY,l-decom. posable. Since U(T; A) can be expressed in terms of V(A) in the form of a linear fractional transformation, it too is normally ,I-decomposable. In addition, this operator is ,I-unitary. By virtue of Theorem 1.8.3 this operator will be strongly stable. To obtain formula (4.10) we consider the derivative
4. V(A)
223
CANONICAL DIFFERENTIAL EQUATIONS
= ~~ =
i
~
{2[/ + U(T; A)]-1 - I} (4.11)
= - 2i[/ + U(T; A)]-1 dU~; A) [/ + U(T; A)]-I. Let us calculate U(T; A) = dUCT; A)/dA. Differentiating the equation dUet; A)/dt
= iA,IYt'(t)U(t; A)
and the equality U(O; A) = / with respect to A, we get
dU~/}l =
iA,IYt'(t)U(t; A)
+
i,lYt'(t)U(t; A),
U(O; A)
=
o.
We have thus obtained an operator equation of the form of equation (111.1.1), = i,lYt'(t)U(t; A). Its solution is given by formula (111.1.19):
wheref(t)
t
=
U(t; A)
JU(t; A)U-l(S; A),IYt'(S)U(s; A)ds o
t
= iU(t; A),I JU*(s; A)Yt'(S)U(s; A)ds. o
Introducing the notation T
R(A)
=
J U*(s; A)Yt'(S)U(s; A)ds,
(4.12)
o
we get dUCT; A)/dA
= WeT; A),IR(A).
We substitute this expression in (4.11) and make use of the relation
(/ +
U)-IU,I
= (U-l + /),1 = (,IU*,I + I)-I,1 = ,I(U* + I)-I.
Then V(A)
= - 2i[/ + U(T; A)]-I·W(T; A),IR(A)[/ + U(T; A)]-l = 2,1[U*(T; A) + 1]-lR(A)[T; A) + 1]-1.
Since V(O) = O· it follows that A
V(A)
=
2,1
J [U*(T; p) + 1]-IR(p)[U(T; p) + 1]-ldp,
o
and we obtain relation (4.10) with the operator A
N(A)
= 2 J[U*(T; p) + 1]-IR(p)[U(T; p) + 1]-ldp.
(4.13) o Finally, the operator R(p) in the integrand is positive and, for sufficiently small
224
v.
EQUATION WITH PERIODIC OPERATOR FUNCTION
f1 > 0, uniformly positive. For it follows from (4.12) that R{f1) > 0; and if f1 is sufficiently small, the operator U(1:; f1) is close to I, so that R{f1) is close to Sl£{s) ds, and therefore R{f1)>> O. But this means that N(}..)>> 0 (0 < A < b). We note that the operator N(A) receives a positive increase from an increase in the value of it An analogous examination can be made of the case a < A < o. The lemma is proved. We now establish a connection between the existence of the operator (4.9) and the solvability of a special boundary problem. We consider the boundary problem dxfdt = iAf£(t)x,
x{O)
+ x(T)
=
f
(4.14)
A number A is called a regular point ofproblem (4.14) iffor any f E 4' there exists a unique solution x(t) (O ~ t ~ T) depending continuously on f in the sense of the norm SUPO~t~T Ilx{t) II· The complement of the set of regular points of problem (4.14) is called its spectrum. LEMMA 4.3. In order for a point A to be a regular point of problem (4.14) it is necessary and sufficient that the operator [U(T; A) + 1]-1 (E [4'D exist. PROOF. We write the solution of equation (4.14) in the form x(t) = U(t; A)Xo. Substituting this expression in the boundary condition, we obtain the relation [U(T; A) + I]xo = J, which proves the lemma. REMARK 4.2. Since the operator U(T; A) depends continuously on A, the set of points A for which the operator [U(T; A) + 1]-1 is bounded is open. Consequently, the spectrum of problem (4.14) is closed. Let A-I and Al denote respectively the maximal negative and minimal positive points of the spectrum of problem (4.14).7) The operator [U(T; A) + 1]-1 IS bounded in the interval (A-I> AI). Using Lemma 4.2, we obtain the final result. THEOREM 4.4. Suppose £(t) is a Hermitian of positive type. Every nonzero point of the interval (Ll> AI) of regularity ofproblem (4.14) is a strong stability point of the canonical equation (4.7), i.e.
A-I
~
A-I < 0 < Al
~
AI·
4. Characteristic multipliers of the finite-dimensional canonical equation. We consider the canonical equation under the assumption that the space 4' is finite dimensional. The eigenvalues p of the monodromy operator U(T; A) are called the characteristic multipliers of equation (4.7). When the equation is stable, the characteristic mUltipliers p lie on the unit circle. 7) We will not dwell on a proof of the existence of the spectrum of the problem on each of the halflines. In the finite-dimensional case this fact follows from formula (0.21) of Exercise 12; if, in addition, the equation is a Hamiltonian equation, it can be shown that A±1 = A±I.
5.
225
SECOND ORDER EQUATIONS
Of interest is their behavior under a variation of ALet us discuss the behavior of the characteristic multipliers when A is in the interval (A~h AI)' We consider the operator V(A)
= i[I - V(T; AmI + V(T;
A)]~I.
In the proof of Lemma 4.2 it was shown that V(A) = ,IN(A), where N(A) is a positive operator having a positive increase under an increase in the value of A. The equation for the eigenvalues of yeA) can be written in the form N(A)x = fJ-,1x. As is well known, the eigenvalues fJ- can be found with the use of the so-called minimax relations 1 _. (,Ix, x) mm max (N(A)X, x) ,
-/i- -
where the maximum is taken over all vectors satisfying additional linear relations and the minimum, over all such relations. Since the denominator increases under an increase in the value of A, we obtain the following result: the eigenvalues of V(A) corresponding to ,I-positive (,I-negative) eigenvectors increase (decrease) under an increase in the value of A. It is easily seen that the characteristic multipliers p are connected with the eigenvalues fJ- of yeA) by the formula I-p fJ- = i - - l+p
or
p
_i-II r - i+fJ-'
This leads to the following rule: under an increase in the value of A (A E (A~h AI») the characteristic multipliers of thefirst kind (corresponding to ,I-positive eigenvectors of the monodromy operator) undergo a counterclockwise movement while the characteristic multipliers of the second kind (corresponding to ,I-negative eigenvectors of the monodromy operator) undergo a clockwise movement. 8 )
§ 5. Second order equations (infinite-dimensional analog of Hill's equation) 1. Central stability band. We consider in a Hilbert space ,p the differential equation of second ord~r with a real parameter fJd 2y/dt 2
+
fJ-pjJ(t)y = 0,
(5.1)
where pjJ(t) is a periodic Hermitian valued operator function: pjJ(t + T) = pjJ(t) = pjJ*(t}. As in the case of a first order equation, we say that this equation is stable for a given fJ- if all of its solutions are bounded on the real line, and strongly stable if it 8) Additional rules for the movement of characteristic multipliers in more complicated situations can be found in the papers of M. G. Krein [6] and M. G. Krein and G. Ja. Ljubarskii [1].
226
v. EQUATION WITH PERIODIC OPERATOR FUNCTION
remains stable under a small variation of the operator function g>(t).9) In accordance with this we will refer to the values of fl. as stability or strong stability points. If fl. #- 0, we can use the substitution). = .v~, Yl = ).y, Y2 = dyjdt, x = G~) to reduce equation (5.1) to a first order equation in the space .p(2) = .p EEl.p:
~~- = ). (+g>(~) ~)x, which can also be written in the Hamilton canonical form if g>(t) is real:
dx dt
=
'1(0il
~) x
III
i).fYfx.
=
(5.2)
°
Here the upper sign is taken when fl. = ).2 > while the lower is taken when - ).2 < 0. It is obvious that the solutions of equation (5.1) are bounded if the solutions of equation (5.2) are bounded, and hence, if ). is a strong stability point for equation (5.2), fl. = ±).2 will be a strong stability point for equation (5.1). With the use of this fact we can make a number of deductions concerning second order equations. Theorem 4.3 immediately implies the following result. fl. =
5.1. Ifg>(t) is an operator function with a uniformly positive mean value > such that all of the points of the open interval (0, fl.l) are strong stability points of equation (5.1 ).lOJ THEOREM
g>av» 0, there exists a number fl.l
° > °
We will further assume that fl. and g>(t) is an operator function of positive type.llJ In this case we will also call equation (5.1) an equation of positive type. Its Hamiltonian
will also be of positive type. In order to characterize the number fl.1 we must in accordance with Lemmas 4.2 and 4.3 consider the boundary problem
-~~
=
i)./Yf(t)x,
x(o)
+ x(T)
=
f,
where
f =
(1)
(h,J2 E .p),
- if)
°'
In the metric d(!?l'j, !?l'z) = Sifll!?l'j(t) - !?l'2(t)lldt. 10) A generalization and sharpening of this theorem as well as a mechanical interpretation of it are given in Exercise 8. 11) I.e. !?l'(t) > 0 and !?l'av» O. 9)
5.
and find its regular points A. Recalling that x = (Yl) yz and if£' = ( form
~l yICO)
227
SECOND ORDER EQUATIONS
!!1t~ =
= AYz;
+ Yl(T)
=
,2
h;
0/ ), we reduce this problem to the
;;r
AY'(t)yr.
-
+ yz(T)
Yz(O)
=
fz,
or, finally, by setting y = YI/A and y' = Yz, to the form dZy dt Z
+
_ f-LfjJ(t)y - 0,
(5.3)
+ y(T) = gr. y'(O) + y'(T) where in the present case gl = hlA, gz = fz, A = ..;-;;. y(O)
= gz,
In accordance with the definition adopted earlier it is natural to regard a given value of /1 as a regular point of problem (5.3) if there exists a unique solution y(t) for any gl, gz E.p that together with its derivative y'(t) depends continuously on gl and g2' On the basis of the above arguments we immediately deduce the following result from Theorem 4.4. THEOREM 5.2. Suppose fjJ(t) is an operator function of positive type. If /11 is the minimal point of the spectrum of problem (5.3), every point of the open interval (0, /11) is a strong stability point of the equation
dZxldt Z + f-LfjJ(t)x
= 0.
2. Generalization of Ljapunov's test. In this subsection we derive an effective estimate for the central stability band of equation (5.1). To this end we consider in more detail the boundary problem (5.3). By means of the substitution y(t) = z(t)
+ ¢(t),
where ¢(t) = (h12A - (TJ4)fz) + t tJz, we reduce it to the problem of finding a function z(t) satisfying the equation
_c!Z~_ + rI/fjJ(t)z dt 2
,ufjJ(t)"'(t)
= -
'f'
(5.4)
and the homogeneous boundary conditions z(O)
+ z(T) =
0,
z'(O)
+
z'(T)
= 0.
(5.5)
We now take advantage of the fact that the solution of the differential equation - (d Zzldt 2)
=
h(t)
with conditions (5.5) can be written in the form
(5.6)
v. EQUATION WITH PERIODIC OPERATOR
228
FUNCTION
T
Jg(t -
Z(t) =
s)h(s)ds,
o
where g(t) = T/4 - It 1/2 is the Green function of problem (5.6), (5.5) (see, for example, R. Courant and D. Hilbert [1]). It is easily verified that the Green function g(t) has the series representation 00
L:
g(t) =
(5.7)
Ckei(2k+1)t,
k=-oo
where
Ck
= (T/n 2)(2k
+
1)2. We note that 00
L:
Ck
= g(O) = T/4.
k=-oo
Setting h(t) = f.lf!lJ(t)(z + ¢), we conclude that the boundary problem (5.4), (5.5) is equivalent to the integral equation T
z(t) - p
T
Jg(t -
s)f!lJ(s)z(s)ds
o We consider the Hilbert space in.p and scalar square
=
f-t
Jg(t -
i>
(5.8)
of functions z(t) (0 ~ t ~ T) with values
T
«Z))2
s)f!lJ(s)¢(s)ds.
0
T
J(f!lJ(s)z(s), z(s»)ds = J 11f!lJ1I2(s)z(s)11 2ds.
=
o 0 In addition, we consider the Banach space lB of continuous functions z(t) with norm Illzlll = maXO~t~T Ilz(t)ll· For z E lB we have ((Z))2 ~
T
T
J 11f!lJ(s)llllz(s)11 2ds ~ III zl 12 J 11f!lJ(s)llds. o 0
We consider the operator T
Kz
= Jg(t - s)f!lJ(s )z(s )ds o
and verify that it acts from In fact, for any cp E .p I(Kz, cp)12
=I
i> into lB.
r
g(t - s)(f!lJ(s)z(s), cp) ds l
~ i~ T2
{I
2
11f!lJ1/2(S)Z(s)II·IIf!lJ1/2(S)cpllds
T
T
r
~
-16 0J11f!lJ1/2(s)z(s)11 2ds. 0J11f!lJ1/2(s)cpI12ds
~
16 Ilcp 112 J11f!lJ(s) lids. «Z))2,
T2
T
o
(5.9)
5.
229
SECOND ORDER EQUATIONS
which implies IllKzl1i =
TJT!
O~~ET IIKzl1 ;;;; "4
11&J(s)llds
((z».
(5.10)
It is now obvious how to prove the continuity of (Kz)(t). Let C denote the norm of the operator K acting from ,f> into lB: C = sUPzEvIIIKzIII/((z», and let CI = 11K II denote the norm of this operator in the space,f>. Equation (5.8) can be written in the form
z - fJXz = fJXifJ·
(5.11)
(0 <)p.< l/CI
(5.12)
If the condition
is satisfied, there exists a unique solution of equation (5.11). This solution is represented by the series z = .Er' p.sKsifJ, which converges in the space B by virtue of the majorization Illzlll ;;;;
cS~Ip.sq-l((ifJ));;;;
1
!:~CI
((ifJ))·
(5.13)
Recalling the expression for ifJ(t), we find that ((ifJ))2 =
I{:A
(&J(s)fi,fi) -
; ; v:
T (&Javfi, fi) I
+
r
(&J(s)/z,/z)
4T2 (&Jav/z,/z)
+ ~
(Y'(s)/z,/z)}dS
(5.14)
(A = v. /p.).
Estimates (5.13) and (5.14) show that the solution of problem (5.4), (5.5), and hence also the solution of problem (5.3), depends continuously onfi and/z in the norm III y III = sup II y(t) II· Differentiating relation (5.8) with respect to t, we obtain for the derivative z'(t) the expression T
z'(t) = - p. -
Ssign(t
T
- s)&J(s)z(s)ds - P.
0
S sign(t -
s)&J(s)ifJ(s)ds
0
and the estimate T
Illzz. - z{ III ;;;; p.
S0 11&J(s)ll ds lllz2 -
zrlll,
from which it follows that the derivative y'(t) also depends continuously onfi and /z. But this means that the points p. E (0, l/CI ) are regular points of problem (5.3). On the basis of Theorem 5.2 we can formulate the following result. .
230
v. EQUATION WITH PERIODIC OPERATOR FUNCTION
LEMMA 5.1. The points J1. of the interval (0, l/IIKII) are strong stability points of equation (5.1). From the above arguments (see formulas (5.9) and (5.10) it follows that
IIKII ~
c [ ! 11&(s)llds T
J1/2
T
~4
! 11&(s)llds.
T
On the basis of Lemma 5.1 we conclude that the interval 0< J1. <
1
-~T----
~ S 11&(s)llds o
consists of strong stability points. But this estimate is too rough, and we now show that it can be replaced by the more exact estimate
THEOREM 5.3. Suppose &(t) is an operator function of positive type. Then every point f.J, E (0, 4/T211&avll) is a strong stability point of equation (5.1).
T211&avll/4.
PROOF. By virtue of Lemma 5.1 it suffices to show that IIKII ~ Using the expansion (5.7), we write the operator K in the form 00
K=
L: cjKj,
(5.15)
j=-oo
where
Kjy = ei (2j+1)t
T
Se- i (2j+1)S&(s)y(s)ds. o
The operators Kj have the following properties. a) The range ffi(Kj) of an operator Kj consists of elements of the formfe i (2j+1)t, where f is a constant element of .p. b) On its range ffi(Kj) an operator Kj has the form Kjz = T&avz. For if Z E ffi(Kj) then Z = fei(2j+1)t and
Kjz = ei (2j+1)t
T
Se- i (2j+1)S&(s)fe i (2j+1)sds o T
=ei (2j+1)t S&(s)dsf = o c) The Kj are Hermitian operators in i).
T
S&(s)dsz = T&avz. 0
6.
231
LOGARITHM OF MONODROMY OPERATOR
For
=
f
f
(£37l(t)e i (2jt-1)t e- i (2H l) S£37l(S)Z(S)dS, z(t))dt
= ({ e- i (2HI)S£37l(S )Z(S )ds, {e- i (2 H l) t£37l(t)Z(t)dt)
=
II
I
e-'(2HI)S£37l(S)Z(S)dS
r
In order to calculate the norm of an operator Kj we write an arbitrary element x E in the form x = Xl + X2, where Xl E lR(Kj) and X2 1. lR(Kj). Since then KjX2 = 0 and Xl = fiei(2HI)t, wherefi E.p, we have
i>
= Xl> = T<£37laV Xl> Xl> = T <£37lav ei (2j+I)tfi, ei (2HI)tfi> = T <£37lav fi,JI> T
= T J(£37l(s)£37l av fi,fi)ds
T211£37lavfi112
=
o
and T
<x, x> ~ <Xl> Xl>
=
= J(£37l(s)fi,fi)ds = T(£37l av fi,fi)· o
Therefore IIKjl1
=
~
sup
"
<x, x>
sup
"I
<Xl> Xl>
= Tsup II
II£37lav fil12
(£37l avfi, fi)
=
TII£37lav ll.
From formula (5.15) we obtain the estimate
which proves the theorem.
§ 6. Expansion of the logarithm of the monodromy operator in powers of a small parameter
1. The equation with a periodic operator function analytically depending on a parameter. We consider in a Banach phase space ~ the differential equation
dxfdt
= A(t; il)x
(6.1)
with a periodic operator function
A(t
+
T; il)
= A(t; il)
( - 00
< t<
00;
lill < R)
which in some disk lill < R is a holomorphic vector function with values in the
v.
232
EQUATION WITH PERIODIC OPERATOR FUNCTION
space LT[lB] of all periodic locally integrable operator functions A(t) with norm
=
A(t
+
T)
T
IIIAIIIL = Jo IIA(t)lIdt. Under the indicated assumptions the operator function A(t; A) will have an expansion A(t; A) =
1:::
Ak(t)Ak
k=O
with coefficients Ak E LT[lB] (k = 0,1,.··) which converges absolutely in the metric of LT[lB] and uniformly in A in each disk IAI ~ R - e (e > 0). From the expansion (111.1.11) and the corresponding estimates it readily follows that the Cauchy operator U(t; A) (0 ~ t ~ T) of equation (6.1) will be a holomorphic vector function in the disk IAI < R with values in the space CT[lB] of continuous functions F(t) (0 ~ t ~ T) with values in [lB] and norm IIIFllle
= O<:::t<:::T max IIF(t) II·
Therefore U(t; A)
= 1::: Uk(t)Ak,
(6.2)
k=O
where Uk E CT[lB] (k = 0,1,.··), the series in the right side of (6.2) converging absolutely and uniformly in t E; [0, T] and A in a disk IAI ~ R - e (e > 0). We consider the monodromy operator Uo(T) = U(T; 0) of equation (6.1) for A = o. It will be assumed below that its spectrum does not surround the origin and hence that there exists a simple curve 1 that goes from the origin to infinity without intersecting the spectrum a(Uo(T) of Uo(T). We will further understand that the function In p is a well-defined branch of the logarithm that is single valued in the plane cut by the curve I. Let r be a simple closed contour lying in this cut plane and surrounding the spectrum a(Uo(T). The spectrum of U(T; A) will also lie inside this contour for sufficiently small IAI. For such A it .~s therefore possible to construct the operator function
=
+
In U(T; A) = - 2;iT §1np( U(T; A) - p/)-ld p, (6.3) r which will obviously be a holomorphic function of A in the small neighborhood of the origin in question. Let Rl denote the radius of the maximal disk with center at the origin in which r(A) will be a holomorphic function. We will then have r(A)
r(A)
= Fo + Ar1 + A2r 2 + ... ,
(6.4)
6.
233
LOGARITHM OF MONODROMY OPERATOR
where r k E[j8] (k = 0, 1,.··), and the expansion converges in [j8] absolutely and uniformly in each disk IAI ~ RI - E (E > 0). In general, RI < R. If the space lB is a Hilbert space.p, the localization theorems of § 3 can be used to estimate the radius R j from below (see Remark 3.1). It is not difficult to see that the estimate Rj
~ Ro = sup {p
I sup f .J IIA(t;).)112 Ill::;:p 0
Pr(AiJl(t; ).»j2dt
<
is valid if the set indicated by the braces is not empty. In fact, for the points). of a disk of radius r Ro the spectrum u(U(T; containing the negative halfline. A rougher estimate is
<
Rj
~ sup {p I 1),I';;;;p sup
n:}
).»
lies outside an angle
J IIA(t;).)lIdt
We note that if j8 is a Hilbert space.» and for some Hermitian operator WE [.»] the operator A(t; A) is W-skew-Hermitian for real A it will follow that rCA) is also a W-skew-Hermitian operator (see Theorem 1.3) for real A(IAI < RI)' But then the operators r k = (k!)-lr(k)(O) (k = 1,2,.··) will also have this property. We consider the operator function Q(t; A)
= U(t; il)e-tr(A).
(6.5)
This operator function is periodic with period T (see § 1). From (6.2) and (6.4) it follows that it is analytic for IAI < RI and the series (6.6)
converges uniformly for t We note that
E ( - 00,
(0) and
lill < RI ( - 00
< t<
E (E > 0). 00).
(6.7)
2. Simplest case. We first consider in more detail an equation of the form dx/dt
Here Ao(t) =
9, AI(t)
=
(6.8)
AA(t)x.
= A(t) and Ak(t) = 0 (k = 2, 3,.· .).
In the case of a Hilbert phase space estimate (3.3) implies the following estimate for the radius of convergence of the series (6.4): T
R j ~ n:/ f IIA(t)lldt. o
The monodromy operator corresponding to operator: Uo(T) = I, and hence
il =
ro = Inl = O.
0 is converted into the identity
(6.9)
234
V. EQUATION WITH PERIODIC OPERATOR FUNCTION
Differentiating (6.5), we obtain the equation dQ(t; }.,)/dt
=
}"A(t)Q(t; }.,) - Q(t; }.,)r(}.,).
(6.10)
We substitute expansions (6.4) and (6.6) in it and equate the coefficients of like terms in the left and right sides. Taking into account (6.9), we obtain first of all the relation dQo(t) /dt = 0, which implies the identity Qo(t) == I since Qo(O) = 1. We write the remaining relations in the form
= A(t) - r .
dQ1(t) dt
(6.11 )
1,
(k > 1).
(6.12)
The integrals of these relations should satisfy the conditions (k
= 1,2,,,,),
(6.13)
which follow from the periodicity of Q(t) and the identity Q(O; }.,) == 1. Relations (6.11)-(6.13) permit us to successively determine all of the desired coefficients r k and Qk. From (6.11) and (6.13) it follows that t
Q1(t)
=
J[A(t) -
(6.14)
rddt,
°
and hence, since Q1(T) = 0,
1 r1= T
T
JA(t)dt. ° In the same way, if all of the coefficients up to order k -
(6.15)
1 have been found,
we have (6.16)
and ..
rk=
~
f[
A(t)Qk-1(t) -
:~: Qs(t)rk-
S]
dt.
(6.17)
3. Case when A(t) is holomorphic in the upper halfplane. We consider an interesting special case. Suppose the Fourier series of the periodic operator function A(t) does not contain terms with negative indices: 00
A(t) '"
1:: Amem(zJr;/T)t. m=O
From (6.15) it follows that
r
1
= Ao, while from (6.14) and (6.18) we get
(6.18)
6.
Ql(t) =
too
S 1: o
235
LOGARITHM OF MONODROMY OPERATOR
A mem (271:i/T)t dt =
m=l
00
TA
1: -.,- --""- (e m (271:i/T)t m=l 21CI
m
- 1).
We could carry out the calculations indicated by formulas (6.16) and (6.17), but there is another approach of interest to us. An interesting modification of the method under consideration consists in replacing the conditions Qk(O) = 0 by the conditions (k
= 1,2,.· .).
(6.19)
Relations (6.16) and (6.17) then take the form Qit) =
t[
A(t)Qk-l(t) -
:~: QS(t)f'k-S -
f'k] dt + Qk(O) ,
(6. 16a)
where Qk(O) is determined from conditions (6.19), and f'k = Tl S[A(t)Qk-1(t) - k£1 QS(t)f'k-S] dt. o s=1
In particular, we again have
r
(6. 17a)
A o, but now T A Ql(t) = 1: - . ---'!!. em (271:i/T)t. m=12m m 1
_
=
00
This function differs from Q1(t) in that its Fourier series contains only terms with positive indices. From (6.18) and (6.16a) it follows that this property will also be possessed by all subsequent functions Qk(t). But then the integrand in (6.17a) has a Fourier series without a free term, and therefore f'k = 0 (k = 2, 3,.··). Thus under this method of calculating f'o.) = 'Ar1 = 'AAo·
But, in contrast to the case when conditions (6.13) are adopted, we can no longer assert that the series (6.20)
converges in any neighborhood whatsoever of the point 'A = O. If this series nevertheless converges in some neighborhood of the origin, it gives a solution of the equation dQ(t; 'A)/dt
=
'AA(t)Q(t; 'A) - 'AQ(t; 'A)Ao
(cf. equation (6.10». Hence Q(t; 'A) = U(t; 'A)Q(O; 'A)e-AA,t.
The latter formula leads to the representation
236
V.
EQUATION WITH PERIODIC OPERATOR FUNCTION
U(t; A) = Q(t; A)eAAotQ-l(O; A).
(6.21)
These arguments can be interpreted as follows. We put e21!it/T = z in (6.18). Then equation (6.8) can be written for A = 27T:i/T in the form z -dx dz
=
(00 L: Akzk)X,
(6.22)
k=O
and formula (6.21), when it makes sense, leads to the representation of a fundamental operator of this equation in the form U(z) = V(Z)ZAoC, where V(z) is a holomorphic operator function in a neighborhood of z = O. Similar results are rigorously obtained in the next chapter, where equation (6.22) will be studied in detail under certain additional assumptions in the class of analytic operator functions. 4. Determination of the coefficients of the expansions in the general case. We will assume that the Cauchy operator U(t; 0) of the equation resulting from (6.1) for A = 0 is known, which implies that the operators Fa = T- 1 1n U(T; 0) and Qo(t) are known. As long as there exists a ray emanating from the origin that does not intersect the spectrum a(U(T; 0») it can be assumed that the spectrum of ro has the property
lImO -
(A, ,u E a(ro»)·
,u)1 < 27T:/T
(6.23)
We will indicate below a method of successively calculating the other coefficients of the expansions (6.4) and (6.6). It will be convenient for us to consider in place of Q(t; A) the function Set; A)
=
QOl(t)Q(t; A).
From (6.6) we have the expansion Set; A)
00
=
1+ L: AkSk(t),
(6.24)
k=l
where Sk(t)
=
Qol(t)Qit).
(6.25)
Differentiating (6.5), we obtain the equation dQ(t; A)/dt
=
A(t; A)Q(t; A) - Q(t; A)r(A);
(6.26)
= Ao(t)Qo(t) - Qo(t)ro.
(6.27)
and in particular, for A = 0, dQo(t)/dt Since Q(t; A) = Qo(t)S(t; A), it follows from (6.26) that Set; A) satisfies the equation
(6.28)
237
6. LOGARITHM OF MONODROMY OPERATOR
dS(t; "A)/dt
=
[Qol(t)A(t; "A)QO(t) - QOl(t)QO(t)]S(t;"A) - Set; "A)r("A),
which after an application of (6.27) takes the form dS(t; A)/dt = roS(t; "A) - Set; "A)ro
+[~l "AkQol(t)Ait)Qo(t)] S(t;"A)
- S(t;"A) k~lAkrk'
(6.29)
We substitute the expansion (6.24) in (6.29) and equate the coefficients of like powers of A. This gives us the system of equations (k
=
1,2,.··),
(6.30)
where k-l
(6.31)
k-l
+ 1:: Qol(t)Ait)Qo(t)Sk-it ) - 1:: j=l
Sit)rk- j •
j=l
Inasmuch as the operator function Fk(t) depends only on those Sj(t) with indices j ~ k - 1, the equations (6.30) are recursion equations and can be solved successively. Let us describe how this is done. From the condition S(O; "A) = Q01(0)Q(0, A) == I it follows that equation (6.30) should be considered under the condition 1
~
(k = 1,2,.··).
(6.32)
The solution of equation (6.30) under condition (6.32) is given by formula (11.1.11) and has the form Sk(t) =
t
Ser,(t-T)[Fk(-r)
- rk]e-r,(H)d-.
(k
o
=
1,2,.·.).
(6.33)
We note further that (6.7) and (6.25) imply that the operator functions Sk(t) (k = 1,2,.··) are periodic, and hence by virtue of (6.32) (k
=
1,2,.··).
(6.34)
Substituting expression (6.33) for Sk(t) in (6.34), we obtain the relation T
T
o
0
Ser,(T-T) rke-r,(T-T)d-. = Ser,(T-T) Fi-.)e- r ,(T-T)d7:,
(6.35)
which can be regarded as an equation for the operator r k • Suppose now that all of the operators SiC-.) and j (j = 1,.··, k - 1) have been determined. Then owing to condition (6.23) we can find k from (6.35) with the use of formula (1.3 .17) :
r
r
238
V. EQUATION WITH PERIODIC OPERATOR FUNCTION
I II A-fi _ 4n-2 J J e(A p.)T _ I (Fo - AI) 1
Fk = -
ro To
x (
!
e-FoTFi'r)eFoTd'r )(Fo - fJ,1)-ldA dfi
- ~I~ S f 4n-2 X
0
f
To To
(A - fi)e-U-p.)T (Fa - AI)-1 eU-p.)T - I
(6.36)
Fk('r)(Fa - fil)-ldA dfi d'r,
where fo is a contour surrounding a(Fo) sufficiently closely. After this, formula (6.33) permits us to find Sk(t). Thus we can actually successively calculate the coefficients of the expansions (6.4) and (6.24) with sufficiently large indices and thereby find to within any degree of accuracy approximate expressions for the operators QU; A) and In U(T; A). 5. Use of Fourier series. There exists another method of solving system (6.30) which is based on an application of Fourier series. Suppose that to the operator functions Fk(t) and Sk(t) there correspond the Fourier series I T
co
Fk(t)
1:
~
where Cp = T
Cpe2P7rit/T,
p=-co
SFk(t)e-2Prrit/T dt, o
and co
SkU)
~
1:
Dpe2prrit/T,
(6.37)
p=-co
sr
where Dp = (liT) Sk(t)e-2P7rit/T dt. Multiplying equation (6.30) by e-2P7rit/T and integrating with respect to t from 0 to T, we find that the operator Dp (p t= 0) satisfies the equation (2n-pilT)Dp - FoDp
+ DpFo
= Cp
(p
t= 0).
From condition (6.23) it follows that 2n-pilT - A + fi t= 0 for A, fi therefore equation (6.38) has the unique solution D = - ~I~ p 4n- 2
ff To To
(Fo - AI)-ICp(Fo - fiI)-1 dA d 2n-pilT _ A + r/I fi·
(6.38) E
a(Fa), and
(6.39)
The series (6:37) converges for each t since Sit) is an absolutely continuous function (see (6.30». If ~ =.p and the function (0 ~ t ~ T) belongs to L 2(0, T) (and hence Fk(t)x EL 2(0, T» for each x E.p, the series (6.37) converges absolutely and uniformly in t for each x E.p. For when N> 0 is sufficiently large the inequality
IIA(t)xll
I Cpx I (2n-piIT holds for any A, fi
E
A + fi)-1 ~ N
a(Fo) and integral p.
IICpXlllpl-l
(6.40)
239
EXERCISES
As a result, the series I; Cpx(2rcpi/T - A + fl)-leZ1rPitIT,
(6.41)
p
which is majorized by the series N
(7 IICpXlllpl-l) ~ N (7 P-Zf/Z(7 IICpxllzf/z < const ( f I Fk(t)x IIZdt t z <
00.
converges uniformly in t, A, fl for any x E ~. It remains to note that the series (6.37) is obtained from (6.41) by multiplying by bounded operator functions and integrating. We must still determine the coefficient Do. It is obtained from the relation Sk(O) = I;p Dp = 0, which reduces to the equality Do = - I;#oDp. Finally, integrating (6.30) from to T under the condition Sl (dSit)/dt) dt = 0, we find the operator Fk :
°
Fk =
r-1 JoT [FOSk(t) -
Sk(t)Fo + Fk(t)]dt = FoDo - DoFo
+
Co·
We note that the series (6.41) can be summed. It can be verified that we arrive in this way at formula (6.33). EXERCISES 1. Consider the T-periodic equation dx/dt = A(t)x
(0.1)
in a real Hilbert space .fi. a) Show that the Floquet representation holds for equation (0.1) if the spectrum of its monodromy operator does not contain the point A = - I and does not surround the origin. Hint. Make use of the symmetry of the spectrum of the real operator U(T) relative to the real axis and construct a symmetric contour separating this spectrum from the origin. Use this contour to calculate In U(T). b) Show (see J. Massera and J. Schaffer [1]) that the Floquet representation holds for (0.1) if T
f II A(r) Ild-r < n-. o
Hint. Make use of the result of the preceding exercise. We note that the corresponding fact for a complex space isjndicated in Remark 3.1. c) Suppose the real Hilbert space.fi is finite dimensional. If - 1 E u(U(T», there exists a real Floquet representation of order 2, i.e. a representation U(t) = Q(t) exp (rt) with real Q(t) and r, where Q(t) is a periodic function with period 2T: Q(t + 2T) = Q(t). Hint. Consider the operator U 2(t) and show that it has a real logarithm (see E. A. Coddington and N. Levinson [1]). 2. Show that the Cauchy operator U(t) of a T-periodic equation (0.1) in a phase space .fi (complex or real) has a Floquet representation if at least one of the following two conditions is satisfied. a) For some operator C E [.fi] of simplest type with spectrum contained in the sequence
2jn-i .} {r I J-0,±I,±2,.··,
240
v.
EQUATION WITH PERIODIC OPERATOR FUNCTION
If C* = - C, the latter condition is equivalent to the condition T
f IIA(t) - C Iidt o
<
7r.
Hint. Perform the substitution U = Vexp (Ct) in the equation (j = A U and make use of Remark 3.1 and the result of Exercise Ib). REMARK. It is helpful to think through the meaning of the assertion of the exercise for the case when to an orthogonal decomposition.p = .pI EB.p2EB···EB.pn there correspond the decompositions A(t) = A I(t)EBA 2(t)EB···EBA n(t) and C = CIEB···EtK., where C k = 27riT- I.hh, the .h (k = I,.··, n) being integers. b) There exists an invertible operator C and a periodic scalar function ¢/(t) with zero mean value for which T
f II C-IA(t)C - ¢(t)IIIdt < 7r. o
Hint. Perform the substitution U(t) = Ce fb ¢«)dr V(t)C-I (J. Massera and J. Schaffer [1], page 353). 3. Consider in a real or complex Hilbert space .p with an orthonormal basis en (n = 0, ± I, ± 2,.··) the orthogonal or unitary operator V defined by the relation Ve n = en+1 (II = 0, ± 1, ± 2,···). Let R be the operator defined by the relation
n
~
0, c >0.
The operator (0.2)
differs only by a scalar factor from the operator studied in § 1.7.3. It is known (J. Schaffer [1] and P. Halmos and G. Lumer [1]) that an operator having the property noted there (its spectrum contains an annulus consisting of points of regular type with finite deficiency) does not have a square root and hence a logarithm. Construct a periodic equation for which (0.2) is the monodromy operator and hence for which a Floquet representation does not hold (J. Massera and J. Schaffer [1], page 354). Hint. Consider the periodic equation with coefficient A(t) = ACt + 1) defined by the formula A(t) = -
W - 6t(1 - t)etwRe-tW
(0
~
t
~
1),
where Wis a skew-Hermitian operator satisfying the condition e W = V. 4. Let U(T) be the monodromy operator of a strongly stable equation x = i,l.Yt(t)x and let U I be a ,I-unitary operator satisfying the condition IIUI - U(T)II o. Show that for sufficiently small 0 0 there exists a Hermitian .Ytl(t) such VI = VI(T) is the monodromy operator of the equation
<
>
dx/dt = i,l.Ytl(t)x.
In this connection the condition
f IIJf'Ct) - .YtJCt)lIdt < c
T
o
>
can be satisfied for any e 0 by choosing 0 sufficiently small. Hint. Make use of the Floquet representation and the considerations of Corollary 4.1. 5. Show that the spectrum of the monodromy operator of the equation d 2yJdt 2
+ fl'(t)y =
0
(0.3)
241
EXERCISES
in
.p with periodic coefficient £JI'(t)
(= £JI'(t
+ T»
lies in the annulus l/r
T
r = exp ( T { 1I£J1'(t)lldt
)112
~
P ~ r, where (0.4)
•
>
The estimate is exact, for example, when £JI'(t) = - pI (p 0).12) Hint. By setting dy/dt = lz (l> 0) and x = yEJjz, transform equation (0.3) into equation (0.1) in the phase space .pEJj~\ where s/(t)
=
Oil) (l-I£JI'(t) 0 .
The norm I V(T)II of the monodromy operator of equation (0.1) can be taken as r. It is estimated by means of Wintner's inequality (III.4.19), after which the estimate is minimized with respect to l. Suppose £JI'(t) 0 (- (X) t (X). It is not known in this case whether the assertion remains in force if (0.4) is replaced by the equality
<<
«
(i.e. if the operations of integration and taking of the norm are interchanged in (0.4». A conjecture of Ljapunov's. Consider the scalar HiII equation y"
+ Ap(t)y = 0
(p(t
+ T) = pet»~.
(0.5)
Let ¢(t; A) and
= 1, ¢'(O; A) = 0;
= 0,
= 1.
The characteristic multipliers ptC}.) and pz(A) of equation (0.5) wiII be the eigenvalues of the monodromy matrix ( ¢(T; A) ¢'(T; A»)
Since its determinant is equal to 1, they wiII be the roots of the quadratic equation pz _ 2A(A)p
+ 1 = 0,
(0'.6)
where 2A(A) =
dS) 112,
which implies, sincc,2A(A)
= PI(A) + l/PI(A)
and hence 2IA(A)1 T
IA(A) I ~ cosh ( IAI T { Ip(s) Ids
) 112
~
IptC}.) I
+ l/lpl(A)I,
.
Let
It is easily seen that 12)
In this and only this case forthe scalar equation (see L. Ja Mirocnik [1]).
that (0.7)
v.
242
EQUATION WITH PERIODIC OPERATOR FUNCTION
T T ao = 1, al = ztP(s)ds.
(0.8)
A. M. Ljapunov conjectured l3 ) back in 1902 ([1], page 419) that (TfIlp(s)lds)n l a n Is . (2n)!
(n=O,I, .. ·).
(0.9)
It follows from (0.8) that this estimate is valid for n = 0, 1. Ljapunov showed by means of an ingenious calculation that it is also valid for n = 2, 3. The assumed estimates (0.9) clearly imply estimate (0.7), but not conversely.14) A very simple direct proof of estimate (0.7) has recently been proposed by L. Ja. Mirocnik [1]. 6. Show that all of the points A of the interval
(0.10) are strong stability points for the periodic canonical equation of positive type (Jlf'(t
dx/dt = i A,IJlf'(t)x
+ T) =
Jlf'(t».
(0.11)
Moreover, if condition (0.10) is satisfied, the spectrum of the strongly stable monodromy operator U(T; A) lies on the arc
r = {e;810 < 0 ~ A{ IIJlf'(s)llds} of the unit circle, and to each of the two parts of this spectrum lying above and below the real axis there corresponds a uniformly ,I-definite invariant subspace of U(T; A). Hint. Make use of Remark 3.1, Theorem 4.4 and the arguments in its proof. The assertion was obtained for the case of a Hamiltonian equation in the two-dimensional phase space R2 by V. A. Jakubovic [7], and in a 2m-dimensional space by M. G. Neigauz and V. B. Lidskii [1], who used the rule of M. G. Krein for the "traffic flow" of characteristic multipliers ofthe first and second kind. 7. Show that the equation d 2y/dt 2 + A&'(t)y = 0
(&'(t) = &'(t
+ T»
(0.12)
can be transformed into the canonical form dx/dt
= ,Ir(Jlf'o + AJf\(t»x,
(0.12')
where (£
q
r =
(0 I) _ I 0'
JIf' 0
= (&'av 0) 0 0'
JIf' (t) 1
= (-
Q2(t) - Q(t») Q{t) I'
in which Q(t)
t
= f [&'(t) -
Hint. Put z = Q{t)yEJj{l/A) dy/dt (A> 0)
&'av] dt,
and x
Qav = O.
= (~).
We note that if 13) Ljapunov considered the case when p (t) ~ O. In this case it follows from formulas obtained by him for the an that every an> O. From these same formulas it directly follows that estimate (0.9) holds for complexp(t) as long as it holds for nonnegative pet). 14) Note added/or the translation. The conjecture of A. M. Ljapunov was successfully proved by M. G. Krein [13] at the end of 1971.
243
EXERCISES
&'(t) '"
f
e2<;ktITCk
k=-oo
(Co = &'av),
i.e. Ck =
f
~ r T 0
2<WIT&'(t)dt
(E .p; k
= 0, ±
1,.··),
then
T»
In the following exercises it is assumed everywhere that &'(t) (= &'(t + + are connected by the above relations. 8. Generalize Theorem 5.1 by showing that if in the case &'(1) = &'*(t) = &'(t one of the conditions
(= Q(t
T»
and Q(I)
+ T) at least (0.13)
or (0.14)
< <
is satisfied, the points A will be strong stability points for equation (0.12) whenever 0 A Ab where Al is the least positive number of the spectrum of the skewperiodic boundary problem (see (5.3». Mechanical commentary. Let us represent &'(t) in the form &'(t) = &'av
+ R(t).
Then Rav = O. Setting A = l/w 2 and performing the substitution t (0.12) to the form d 2y/dt 2 + [&'av
--->
wt, we reduce equation
+ R(wt)] y = o.
If &'av» 0, the unperturbed equation ji + &'avY = 0 describes the motion of a stable conservative system. The assertion of the exercise means that, independently of the form and size of the amplitude of a parametric excitation with zero mean value, the system remains stable for an excitation of sufficiently large frequency. Moreover, if the system is even semistable: &' av ;9 0, the above assertion remains valid under the usually satisfied condition (0.14). Hint. Pass from equation (0.12) to equation (0.12'). If at least one of the conditions (0.13) or (0.14) is satisfied, the operator function £0 + A£I is of positive type for any A O. Verify that this circumstance permits one to extend the part of Theorem 4.4 concerning the right half 0 A Al of the interval Qf regularity to the canonical equation (0.12'). The assertion of the theorem was published for the scalar Hill equation by A. M. Ljapunov in "Soobscenijah Har'kovskogo Matematiceskogo Obscestva" back in 1896 (see [1]), and for the n-dimensional case by M. G. Krein [7]. We note that if &'(t) = &'*(1) then Ck = C'i'.k (k = 1,2,.··) and
>
<
fo Q2(t)dt = L
f;
471:"2 k~1
ckct
<
+ etc k
k2
Condition (0.14) will be satisfied if at least one of the coefficients Ck is invertible in [.p]. When dim .p 00 condition (0.14) means that for any x E .p the vector function &'(t)x is different from zero on a set of positive measure.
<
v. EQUATION WITH PERIODIC OPERATOR FUNCTION
244
We recall that according to Theorem 2.4, if the condition !?/>av» 0 is satisfied, equation (OJ will be e-dichotomic and a fortiori unstable for negative A that are sufficiently small in absoh value; and, as we now note for the second time (see Theorem 5.1), this equation will be stronl stable for sufficiently small positive A. 9. Suppose !?/>av = 0 and Q2(t)dt» O. Show that equation (0.12) will be strongly stable wht ever
rr;
(0.:
Hint. Transform equation (0.12) by means of the substitution Iz
= Q(t)y + dyjdt
into a canonical equation of positive type, make use of the test of Exercise 6 and "optimize" with respect to I. The test (0.15) for dim.p 00 was indicated in an article of M. G. Krein [7]. It is exact sinc1 is exact in the scalar case (the latter was proved by T. M. Karaseva [1]). The assertion of the exercise can be generalized to the case !?/>av> 0, but in this case we do I obtain exact tests in the indicated way. For the scalar Hill equation exact tests for the strong s bility of the equation in terms of the two parameters a = !?/>avand b = T2(Q2)av were found T. M. Karaseva [1]. Surprisingly, the periods of hyperelliptic integrals playa role in the investi tions of the boundaries of the "stability domains" in the (a, b)-plane. 10. A paradoxical result in the theory of parametric excitation. Suppose a periodic opera function !?/>(t) satisfies the conditions of the preceding exercise and R(t) = R(t + T) is a boum Hermitian operator function. Consider the equation
<
d 2yjdt 2
+ [R(wt) + sw2!?/>(wt)]y = O.
(0.
>
Show that there exists an So 0 such that for s EO (0, so) equation (0.16) is stable if w is su ciently great: w Q(s), where Q(s) is a number depending on s. Hint. By the substitution '/: = wt reduce the equation to the form
>
2y dd,/:2
1 R('/:)] Y = o. + [ &('/:) + (;)2
(0.
>
The equation ji + s!?/>('/:)y = 0 is strongly stable for sufficiently small s 0 by virtue of the re! of the preceding exercise, while equation (0.17) for large w is obtained from it by means of a sn perturbation. The assertion recorded above shows in particular that, for a non positive R = const, an unsta system described by the equation
can be converted into a stable one by means of a high frequency parametric excitation sw2!?/>( of zero mean value (and consequently of variable sign) (cf. the deduction of Exercise 8 as wei Exercises 111.17, HI.18 and IV.8). In particular, this assertion applies to the linearized equation of motion of a mathemat pendulum of length I with point of suspension vertically vibrating according to the law! sl sin(wt + a). For this equation has the form
if>" + [± gjl- sw2 sin(wt
+ a)] if> = 0,
where if> is the angular displacement of the pendulum from its initial vertical position while th( or - sign is taken before gjl, depending on whether the pendulum is under or over its poinl
245
EXERCISES
suspension. Thus the "linear theory" permits one to assert that an co> 0 can always be found such that for c co the pendulum will perform stable oscillations independently of whether it was originally under or over its point of suspension, provided the vibration frequency OJ of its point of suspension is sufficiently great (OJ> Q(c)). This paradoxical deduction of the linear theory is well known. P. L. Kapica [1, 2] confirmed it experimentally and proposed a nonlinear theory for the phenomenon. An outline of a general nonlinear theory of motion in a high frequency field is given by L. D. Landau and E. M. Lirsic in [I] (but their arguments have remained unintelligible to the authors of this book). 11. Consider the equation
<
(0.18) where [Po» 0 and [PI is a 2n periodic Hermitian operator function. Show (M. G. Krein [6]) that this equation is strongly stable for sufficiently small c OJ
*- (A + /1)/N
(N = 1,2, .. ·),
> 0 if (0.19)
when A2 and /12 independently range over the spectrum 1l([PO)' In particular, the equation is strongly stable for sufficiently large OJ (OJ> 2v' AM([PO)). In the finite-dimensional case, when the spectrum 1l([Po) consists of the squares of numbers Ph"', Pm (>0), a violation of the strong stability of equation (0.18) can occur only for the following isolated values of the frequency: (j, k
= 1,2, .. ·,m; N = 1,2, .. ·).
Hint. Reduce the equation ji + [P oY = 0 to an equation of first order in the doubled space ,p(2). If it is considered as a periodic equation with period T = 2n/OJ, the spectrum of its monodromy operator will consist of the numbers e±iAT (A2 EO 1l([PO), A> 0) and consequently of the two sets (e iAT I A2 EO 1l([Po)} and (e- aT IA2 EO 1l([PO)}' which are the symmetric images of each other with respect to the real axis. To these sets there correspond j'r-definite invariant subspaces of the monodromy operator in question. One should then make use of the results of § 1.8, according to which this operator turns out to be strongly stable if the above two sets do not intersect. For a further development of this theme see the paper of V. A. lakubovic [6] and the survey of M. G. Krein and V. A. lakubovic [I]. In the following problems the space ,p is assumed to be finite dimensional (,p = Cn or Rn). The assertions of these exercises can be extended to the infinite-dimensional case only under special conditions requiring that the values of the operator coefficients be completely continuous operators of this or that class. 12. Consider the skewperiodic boundary problem {dx/dt x(O)
= OJ' yt'(t )x,
+ x(T) = f
(0.20)
in the phase space Cn with a Hermitian :Yt'(t) of positive type. It is elementary to"prove that all of the eigenvalues of this boundary problerri are real (the problem is selfadjoint). Show that the following two equalities hold for the complete sequence (Ai) of eigenvalues of the problem, taken with regard for their multiplicities (M. G. Krein [6, 7]):
7 1] =
~2
lim 1:: r-O IAjl
tr (j':Yt' av)2;
(0.21)
~
(0.22)
Aj
= O.
Hint. To obtain the first equality consider the equivalent (to the boundary problem) integral
246
V.
EQUATION WITH PERIODIC OPERATOR FUNCTION
equation ). T
= -:r {sign(t -
x(t)
s),1.Yt'(s)x(s)ds
and calculate the trace of the second iterated kernel, which turns out to be continuous. The proof of the second equality is complicated; two different proofs of it are given in the articles [6, 7] of M. G. Krein, and a third is given in the book [2] of I. C. Gohberg and M. G. Krein. All three proofs make use of subtle facts of the theory of entire functions of exponential type. In the cited book of Gohberg and Krein there is a generalization of equalities (0.21) and (0.22) to the case of an infinite-dimensional phase space.p when .Yt'(t) and .Yt'av are nuclear operators. We note that equality (0.21) implies that the boundary problem (0.20) always has at least one positive and at least.one negative eigenvalue. 13. Show (M. G. Krein [7]) that a T-periodic canonical equation of positive type x = i).,1.Yt'(t)x is strongly stable whenever (0.23) If the Hermitian Yf(t) is even: .Yt'(t) = Yf( - t), the number 4 in this test should be replaced by
the number 8. Hint. Make use of formula (0.21) and Theorem 4.4 on the central stability band. To obtain the second assertion one should make use of the fact that when the Hermitian is even: Yf(t) = .Yt'( - t), the spectrum of the boundary problem (0.20) is symmetrically arranged with respect to the origin and hence (0.21) implies an upper bound for 2/.l.l. The use of the first or second test is facilitated by bearing in mind that if i,l = "r and .Yt'av = IIAjklll, then
The test (0.23) is not exact. This will follow from the assertion of Exercise 15. On the other hand, the second test for even Hermitians turns out to be exact (see Exercise 17). 14. If .Yt'av 0, the operator T ,1.Yt'av will have exactly n+ positive eigenvalues: wt ;::0; ••• ;::0; w;t 0), and n_ negative ones: - Wi ~ ... ~ - w;;:, where n+ and n_ are respectively the multiplicities of the eigenvalues + 1 and - 1 of the operator ,I = ,1* = ,I -I. The following theorem holds (I. C. Gohberg and M. G. Krein [2]). I. Suppose that for a T-periodic canonical equation of positive type x = i,lYf(t)x the following condition is satisfied for some "0 (0 "0 1): REMARK.
(>
>
< <
I:. wi
j~IJ+"o-1
+ ~ . Wj
j~IJ-"o
<2n".
(0.24)
Then the equation is strongly stable. Furthermore, all of its characteristic multipliers of the first kind lie on the halfopen arc of the unit circle r + = (exp (iO) 10< 0 ~ 2n"'(1) while those of the second kind lie on'the halfopen arc r _ = (exp(iO) I - 2n"(1 - '(2) ~ 0 0), where (0 "I "2 « 1) are the unique pair of roots in the interval (0,1) of the equation
<
n+
L:.
j~1
]
w+
+ "' -
n_
1
(j)7
+ j~1 L: -.-'- = 2n". ] - "
<) <
(0.25)
The proof of this theorem is based on the rules of motion of characteristic multipliers and a number of propositions in the theory of nonselfadjoint operators, a formulation of which would take us too far astray. It is elementary to verify that under condition (0.24) equation (0.25) has a total of two simple roots in the interval (0, 1).
247
EXERCISES
The following proposition is obtained as a simple corollary of this theorem. II. A periodic Hamilton equation x =Jr:Yf'(t)x of positive type in a 2m-dimensional phase space is strongly stable whenever (j)t
m
j'f
l
1[;
2j'-- 1
< 2'
In this case its characteristic multipliers of the first kind lie on the arc r + = (exp(i8) 10< 8 ~ 2n-"IJ while those of the second kind lie on the mirror image of this arc with respect to the real axis, where is the unique root in the interval (0, 1/2) of the equation
"I
~
j~1
wJ {J . +"1 -
1
+ _._1_} ] -"
= 2n-.
Hint. Use the fact that when / = iJr and dim .p = 2m one has n+ = n_ = m and (j = 1, 2, .. ·,m). Verify that if "0 is a root of equation (0.25), the second root is 1 - "0'
wJ =
Wj
15. Show that the first assertion of Exercise 13 is a corollary of Theorem I and, in addition, that the test (0.23) can be replaced by the stronger test
(the right side of this inequality tends to n-2/4 as n± -> co). Hint. Estimate the left side of (0.24) for "0 = 1/2 by means of Cauchy's inequality and make use of the fact that T2 tr (/Yf av)2
= 1: (Wj)2 + 1: (Wj)2.
16. Suppose that in aT-periodic m-dimensional Hill equation Y + .9(t)y = 0 the coefficient .9(t) = .9*(t) is an odd function: .9(- t) = - .9(t), and Q = f.9 dt, Qav = O. Then (M. G. Krein [7]) this equation will be strongly stable whenever (0.26) For the scalar case (m = 1) this proposition was obtained (in a completely different way, of course) by A. M. Ljapunov [1] back in 1899 (see the reference in Exercise 8). Hint. Transform the Hill equation into a canonical equation with an even Hermitian .p(t) and make use of the second assertion of Exercise 13. 17. Show that the test (0.26) for strong stability of the Hill equation is exact (M. G. Krein [7]). Hint. Consider the scalar Hill equation Y + APa(t)y = 0 with a periodic generalized function Pa(t) defined in the interval (- T12, T12) by the equality Pa(t)
= oCt + a) -
oct - a)
(- Tl2
< t < Tl2),
where a EO (0, TI2). Verify that the skewperiodic boundary problem 2 { d 2Yl dt + APa(t)y = 0, yeO) + yeT) = y'(O) + y'(T)
has a total of two eigenvalues A+(a)
(> 0) and L(a) =
=0
- A+(a), and
where q = Po, qav = O. The equation Y + APa(t)y = 0 with real A will be strongly stable (e-dichotomic) for 0 A+(a) (I AI A+(a)).
<
>
< IAI
248
v. EQUATION WITH PERIODIC OPERATOR FUNCTION
REMARK. If the equation ji + AP.(t)y = 0 is transformed into a Hamiltonian equation with an even Hamiltonian (see Exercise 7), the strong stability test indicated in Exercise 13, when applied to the latter equation, becomes a necessary and sufficient condition for A 0 to be a strong stability point. 18. Suppose that in aT-periodic m-dimensional Hill equation ji + f!/J(t)y = 0 the coefficient f!/J(t) = f!/J*(t) has a mean value f!/J.v = 0 while Q(t) is defined as in the preceding exercise. Let q~ (j = 1,.··, m; ql ~ ..• ~ qm) denote the successive eigenvalues of the operator J'2(Q2).v. Show that the Hill equation is strongly stable whenever
*
I: -/b-<~, 2} - 1 2
j=1
and a fortiori whenever (cf. (0.26)) 2
J'2(tr Q ).v
< 4""" (mj'f 1C2
l
1 (2j _ 1)2
)-1 .
Hint. Transform the Hill equation into a canonical one and make use of Theorem I (see Exercise 14). 19. Show that when dim .f'> 00 Theorem 2.4 can be sharpened by replacing the condition Re f!/J(t) «0 in the theorem by the condition Re f!/J(t) < 0, Re f!/J. v «0. Hint. Using the notation in the proof of the theorem, show that the strict inequality
<
(Wxo, xo) - (WU(T)xo, U(T)xo)
*
>0
is satisfied for Xo E.f'>, Xo o. It will then foIlow that W - U*(T)WU(T»> 0 (the relations :Yt' 0 and:Yt'» 0 are equivalent in a finite-dimensional .f'». 20. Let aT(.f'» denote the Banach space of T-periodic Hermitians :Yt'(t) (= .1t'*(t) = :Yt'(t + T)) with values in [.f'>] and norm
>
T
1l1:Yt'11i
=f
1I:Yt'(t)lIdt.
° (E [.f'>]) denote a signature operator: f =
Let f f* = f -I, and let U(f) denote the group of all f-unitary operators U E [.f'>]. The operator f induces a mapping !T: aT(.f'» ....... U(f) which takes a Hermitian :Yt'(t) into the monodromy operator of the corresponding equation x = jf:Yt'(t)x. Let ST(.f'>; f) denote the set of all:Yt' E aT(.f'» whose images under!T are strongly stable f-unitary operators U E U($). According to Lemma 4.1, ST(.f'>; f) is an open set in aT(.f'» and therefore divides into domains (connected components) called stability domains. When .f'> is finite dimensional, these domains form a countable set. For real .f'> = R}m and f = Jr , a complete description of the stability domains for m = 1 has been given by V. A. Jakubovic [7], and for any natural m, by I. M. Gel' fand and V. B. Lidskii in their important paper [1]. The results of tHe latter were supplemented and then generalized by V. A. Jakubovic [8, 9] to the general case of a complex finite-dimensional .f'> and any signature operator f (in connection with these papers we note the paper of V. B. Lidskii and P. A. Frolov [1], in which related questions are solved for the T-periodic equation
dx = (:Yt'(t) _ dt
.l dQ )x 2 dt
(Q*(t) = - Q(t))
in a finite-dimensional .f'> and which contains some corrections to the paper [9] of V. A. Jakubovic). See also the papers of W. A. Coppel and A. Howe [1] and S. Diliberto [1.] Among the stability domains in ST(.f'>; f) there is always (independently of the dimension dim.f'> ~ 00) the domain Qt(.f'>; f) which contains all of the Hermitians :Yt'(t) of positive type for
NOTES
249
>
which Aj(J'f) 1 (see Theorem 4.4 on the central stability band). The domain Qt(s.J; ,I) has the obvious property that, if it contains a Hermitian of positive type J'f(t), it also contains the halfopen "segment" qJ'f (0 q :::;; 1). In addition, if J'f belongs to Qi, then so does every Hermitian Yf j of positive type such that J'fj(t) :::;; J'f(t) (this assertion is offered as an exercise). All of the tests for strong stability of a canonical equation of positive type indicated in the preceding exercises are tests of whether or not Aj(J'f) 1, i.e. tests of whether or not a Hermitian Yf belongs to Qi(f»; ,I). These tests are called
<
>
Z-tests.
There exist techniques permitting one to obtain from the Z-tests tests of whether or not a Hermitian J'f belongs to the other domains of ST(s.J; ,I). In regard to these techniques and other methods of obtaining tests different from the Z-tests for the strong stability of a canonical equation see the paper of M. G. Krein [7] and the survey paper of M. G. Krein and v. A. Jakubovic [1] (see also the survey article of V. M. Stadinskii [1]). NOTES Equations with periodic coefficients were first investigated in a finite-dimensional space by G. Floquet [1], who essentially introduced the monodromy matrix and obtained formula (1.5). A. M. Ljapunov and H. Poincare discovered that in the case of a Hamiltonian equation the monodromy matrix is a Jr-unitary matrix, and it is from this fact that we obtain the symmetry properties of the spectrum of the monodromy matrix. The e-dichotomicity criterion (Theorem 2.1) is taken from the book of J. Massera and J. Schaffer [1]. The subsequent results of § 2, which make use of the properties of operators in Hilbert space with an indefinite metric, are due to M. G. Krein. For the elementary aspects of Theorem 2.2 see his paper [5]. (See also the commentary to Exercise IV.14.) The localization theorems of § 3 are due to M. G. Krein [11]. Theorem 3.2 was obtained by a sharpening and development of the arguments of J. Massera and J. Schaffer [1] showing that under condition (3.15) the spectrum IJ(U(T» does not intersect the negative half of the real axis. Massera and Schaffer managed to find an analogous test for Banach spaces (see their book [1]). The theory of stability bands for the scalar Hill equation with a parameter was constructed by A. M. Ljapunov for both the cases of a coefficient of constant sign and one of variable sign. He was also the first to obtain existence theorems and various estimates for the central stability band. M. G. Krein [3, 6] constructed the theory of characteristic multipliers of the first and second kind, which permitted him to generalize the results of A. M. Ljapunov and obtain a number of new results for finite-dimensional canonical equations. At the present time there is apparently no stability test for the scalar Hill equation which does not have analogs (often several of them) in the multidimensional, and frequently also in the infinite-dimensional, case. Moreover, the methods developed for systems of equations have in many cases provided tests that are also new for the scalar case (see M. G. Krein and V. A. Jakubovic [1]). The important paper of I. M. Gel'fand and V. B. Lidskii [1] is devoted to the classification and calculation of the homotopy invariants of stable Hamiltonian systems (see also the Notes to Chapter I and Exercise 20 ab·ove). The papers [1, 8, 9] of V. A. Jakubovic are devoted to the further development of the theory of periodic canonical equations in this direction. The first investigations of canonical systems in infinite-dimensional Hilbert space were undertaken by V. I. Derguzov [1, 2], who investigated specific classes of canonical equations with an unbounded selfadjoint Hamiltonian J'f(t). It is to canonical systems of this type that concrete problems in the study of parametric resonance of elastic systems with an infinite number of degrees of freedom reduce. The papers [1-5] of V. N. Fomin are devoted to the further investigation of canonical systems of this type.
250
V.
EQUATION WITH PERIODIC OPERATOR FUNCTION
The papers of V. I. Derguzov moved M. G. Krein to investigate the possibility of adapting the methods at his disposal to the infinite-dimensional case. The results obtained were partially published in [9, 10]. These results of M. G. Krein are presented with revisions and additions in § § 4 and 5 and the corresponding exercises. The method in § 6.2 of calculating the logarithm of the monodromy operator for the elementary equation x = AA(t)x was pointed out by N. P. Erugin [1]. A similar method for equations with almost periodic coefficients of the form x = [Ao + AA(t)]x was applied still earlier by I. Z. Stokalo [1] to obtain asymptotic expansions of the solutions. A simpler presentation of these results has been given by N. P. Erugin [2]. The general case considered here for equations with a periodic coefficient A(t; A) was considered by M. G. Krein in a seminar lecture in 1948, but these results were never published. It was in connection with this work that formulas (1.3.7) were first obtained. In 1959 there appeared an article by V. A. Jakubovic [2] that is also devoted to the application of the small parameter method to canonical systems. His results were generalized to the infinitedimensional case by I. I. Kovtun in [1], where, in particular, he proposed a method for calculating coefficients that is based on the use of formula (1.3.17). For the further development of asymptotic methods in application to canonical equations and the relevant literature see the survey of M. G. Krein and V. A. Jakubovic [1].
CHAPTER
VI
LINEAR DIFFERENTIAL EQUATIONS IN THE COMPLEX PLANE In this chapter we consider a linear differential equation the operator coefficient A(z) of which is an analytic function whose only singularity is a pole of first order at the origin. In § 1 we consider the simplest case, when the spectrum of the principal part Ao of A(z) does not contain pairs of points with integral differences. In § 2 we weaken this condition by assuming only that integral differences correspond to isolated points of the spectrum. In § 3 we show how to calculate in the finitedimensional case the coefficients of a series representing a solution by using the Jordan form of the matrix Ao. In § 4 we consider a method of solution which under certain conditions permits one to also obtain a solution of the equation in the case when there are integral differences of points of the continuous spectrum.
§ 1. The equation with a regular singularity. Simplest case 1. The equation with a regular coefficient. In this chapter we consider some questions concerning the Cauchy problem
dx/dz
=
A(z)x, x(zo)
=
Xo
(1.1)
in a complex domain of the z-plane. The case when the operator function A(z) is regular in a simply connected domain G containing the point Zo can be investigated by means of considerations similar to those at the beginning of Chapter III. The solution of problem (1.1) can be represented in the form x(z) = U(z, zo)xo,
(1.2)
where the operator U(z, zo) satisfies the condition U(zo, zo)
=
I
(1.3)
and the equation dUjdz
=
A(z)U
(1.4)
in the phase space [)8]. This operator has the series expansion (1.5)
251
252
VI.
LINEAR EQUATIONS IN THE COMPLEX PLANE
where Wo(z)
= I,
Wk(z)
=
J• A(z) Wk-I(Z)dz,
'0
the integrals being independent of the path of integration in the domain G. The series (1.5) converges uniformly in every closed part F c G whose points can be reached from Zo along paths of bounded length. The solution (1.2) of problem (1.1) is unique. The general solution of equation (1.4) is V(z) = U(z, zo)C, where C (= V(zo» is an arbitrary bounded operator. 2. Simplest equation with a singularity. Of greater interest are equations with a nonregular coefficient A(z). We confine ourselves to a study of the behavior of a solution of equation (1.1) in a neighborhood of a simple pole Zo of the operator function A(z). It can be assumed without loss of generality that Zo = 0 and hence that A(z) has the form A(z)
= (lIz) Ao + Al + Azz + ... ,
(1.6)
Izl
the series (1.6) converging in a domain 0 < < p. It can easily be shown by the usual methods that if a power series with operator coefficients converges in the interior of a disk, it converges there absolutely, i.e. I: IIAklllzlk < 00. Under the above assumptions it is customary in the analytic theory of differential equations to call equation (1.1) an equation with a regular singularity. We first consider the simplest equation dU
dz
= Ao U
(1.7)
z·
It is not difficult to verify by a direct substitution that the operator function U(z)
=
eAo1uC
=
zA,C,
(1.8)
where C is an arbitrary operator, is a solution of (1.7) for z =F O. This solution is not single valued; it is multiplied by the operator eZ"jA, when one makes a complete circuit about the point Zo = 0 in the positive direction. Any solution of equation (1.7) that is single valued in a simply connected domain not containing the point Zo = 0 in its interior can be represented in the form (1.8) under a suitable choice of C and a single valued branch of the logarithm. For this reason the function zA, and, more generally, any solution of equation (1.4) having the analogous property will be called afundamental solution. 3. Case when the spectrum of the principal part does not contain pairs of points with integral differences. We pass to the consideration of the equation with operator (1.6) zdU -- = (
dz
I: AkZk 00
k=O
)
U,
(1.9)
1.
253
EQUATION WITH A REGULAR SINGULARITY
Izl
assuming that the series in the right side converges in a disk 0 ~ < p. We first suppose that the operator Ao satisfies the following condition: there does not exist a pair of points in the spectrum of Ao whose difference is equal to a natural number:
(n
A-W:f=n
1,2,3,.··; A, ,u E I1(Ao))·
=
(1.10)
By analogy with formula (l.8) we will seek a solution of equation (1.9) in the form (1.11) Formally substituting the expansion (1.11) into equation (1.9) and equating the coefficients of like powers of z, we obtain the system of recursion relations
AoUo - UoAo
=
0,
m
mUm - (AoUm - UmAo)
=
L:
AkUm- k
(m = 1,2,.··).
(1.12)
k=1
We put Uo = 1. To successively determine the operators Um (m > 0) we apply formula (1.3.10). We write (1.12) in the form (1.13)
(m = 1,2,.··),
where Fm = L:;=I AkUm- k. From condition (1.10) it follows that the spectra of the operators Ao and Ao -mI (m = 1,2,.··) do not intersect. Therefore the solution of equation (1.13) has the form (see (1.3.10))
Um = 41 2 n: = __ ~
4n:
§ § (Ao - mI - AI)-IFm(Ao F
,
A_
FCm) 0
§ § F, F,
,uI)-1 dA d
/I
r
l4o=---AI)-1 Fm (Ao - ,uI)-1 dA d,u, A + m - ,u
ro
,u (1.14)
where and rcim) are smooth contours surrounding the spectra of the operators Ao and Ao - mI respectively, the contour rcim) being a parallel displacement of the contour In order to prove the convergence of the series in (1.11) we estimate the norms of the operators Urn. Formula (1.14) readily implies that the following estimate is valid for sufficiently large m:
roo
We note in addition that, under our assumptions, for any PI < P lim IIAkllp~ = 0, k~oo
254
VI. LINEAR EQUATIONS IN THE COMPLEX PLANE
and hence
Since
~
max l"::s"::m-l
we have, beginning with some m
1
{II Usllpn c • -m
m
L; IIAkllp~, k=1
= mo,
I Urn I p;" ~ max {II Usllpn and consequently the sequence I Urn lip;" is bounded for any PI < p. Using this fact for some P2 (PI < P2 < p), we see that in each disk Iz I ~ PI « p) the series in l~s~m-l
(1.11) is majorized by a convergent geometric series and is therefore absolutely convergent. Thus the following theorem is valid.
THEOREM 1.1. If the operator Ao satisfies condition (1.10) and the series L;;;" AkZk converges in a disk < p, equation (1.9) has the solution
Izl
U(z) =
Cfo UkZk ) ZAo = B(z)zAo
(Uo = B(O) = I),
(1.11)
the series L;;;"UkZk converging in the same disk.
REMARK 1.1. Let G p be an open disk of radius p cut along some radius. By choosing a branch of ZA o that is single valued in Gp , we can obtain from (1.11) a single valued solution of equation (1.9). We assume that the boundary of Gp contains the origin. Then any solution that is single valued in this domain can be represented in the form V(z) = U(z)C, where C is a constant operator, i.e. U(z) is a fundamental solution. In fact, since B(O) = I, for sufficiently small Zo the operator B(zo) has a bounded inverse, and therefore the operator U-l (zo)
=
ZOAo B-1 (zo)
exists. In the simply connected domain Gp the operator U(z, zo)= U(Z)U-l(ZO) satisfies the condition U(zo, zo) = I and the differential equation (1.9), which does not have any singularities in this domain. Therefore V(z) = U(z, zo)V(zo) = U(Z)U-l(ZO)V(zo) = U(z)C.
2.
INTEGRAL DIFFERENCES OF EIGENVALUES
255
§ 2. Case of integral differences of eigenvalues 1. Solutions of minimal type of the inhomogeneous equation with a constant coefficient. In this section we consider equation (1.9) without the assumption (1.10) on the spectrum of Ao. We first collect some auxiliary facts. We consider the equation
+ fez),
dx jdz = Ax
(2.1)
under the assumption that A is an invertible operator. Suppose efk
E
'13; k
=
O,I,.··,n)
(2.2)
is a polynomial of degree n with vector coefficients. In this case there exists a solution n
x(z) =
L:
Xk Zk
(Xk
'13; k
E
k=O
=
O,I,.··,n)
(2.3)
of the same form. The coefficients Xk are most readily found by the method of undetermined coefficients. Substituting expressions (2.2) and (2.3) into (2.1) and equating coefficients oflike powers of z, we obtain the system of recursion relations AXn
+ In
=
0,
AXk-1 = kXk -
ik-l
(k = n, n - 1,.··,1),
which implies the equalities Xn= -A-lin, Xn-l = - A-I In-I
-
nA-2 fn.
(2.4) Xj
1
= - -.-, i·
. AJ-I
n
L:
s!A-s Is.
s=j
We recall that an entire function fez) is called a function of mininal type (more precisely, of minimal exponential type) if for any e > there exists a constant Ne such that the estimate Ilf(z) II ~ Neeel"1 holds throughout the complex plane. Suppose
°
00
fez)
= L: fk zk .
(2.5)
k=O
It is known (see Exercise 1.22) that this function is of minimal type if and only if lim (k! II fk 1I)Ilk
=
0.
(2.6)
k~oo
The above result concerning polynomial solutions admits the following generalization.
256
VI. LINEAR EQUATIONS IN THE COMPLEX PLANE
LEMMA 2.1. Equation (2.1) has a unique solution in the class of solutions of minimal type if the operator A has a bounded inverse and the vector functionf(z) is of minimal type. PROOF. We put in analogy with (2.4)
Xj
1
.
00
= - -.-, AJ-l 1:; s!A-SIs J.
(j = 0,1,. .. ).
(2.7)
s=j
For any s > 0 there exists by virtue of (2.6) an index jo such that s! Ills II ~ sS when s > jo. Therefore j!llXjl1
~
I;s! Ills II • IIA-llls-U-ll
S=J
whenj > jo, and hence
~im U!IIXjll}l/j = O. J~OO
Thus the function x(z) = 1:;;;" Xjzj is of minimal type. It follows from (2.7) that
(k = 1,2,.··).
(2.8)
Multiplying (2.8) by Zk-l and summing, we get that x(z) is a solution of equation
(2.1). For a proof of uniqueness it suffices to show that when A is invertible the homogeneous equation dx /dz = Ax cannot have nonzero solutions of minimal type. Suppose x(z) = 1:;k XkZk is a nonzero solution. Then Xk = Akxo/k! and Ilxoll = IIA-kAkxoll ~ IIA-lilk. Ilxkll • k! ~ skllA-lllk for arbitrarily large k and sufficiently small s, which implies Xo = O. LEMMA 2.2. Equation (2.1) has a solution of minimal type if the spectrum of A has an isolated poin(at A = 0 andf(z) is of minimal type. Two such solutions differ by a function whose values belong to the invariant subspace 58 0 of A corresponding to the point A = O. PROOF, Let Po be the projection corresponding to the point A = 0 of a(A). Equation (2.1) decomposes into the pair of equations
dYo/dz dYl/dz
= =
+ go(z), AlYl + gl(Z), AoYo
(2.9) (2.10)
2. INTEGRAL DIFFERENCES OF EIGENVALUES
257
where Yo = Pox, Yl = (/ - PO)X, go = Pof, gl = (/ - PO)f, Ao = APo, Al = A(/ - Po).
In the subspace 581 = (/ - Po)58 equation (2.10) satisfies the conditions of Lemma 2.1 and consequently has a solution Yl(Z) of minimal type. We consider equation (2.9) in the subspace 580 = Po58. In this equation the operator Ao has a spectrum consisting of only the point). = 0, while the function go(z) is of minimal type. Let us show that all of the solutions of this equation are also of minimal type. To this end we estimate the values of the derivatives of a solution of equation (2.9) at the origin. It is easily seen from (2.9) that
We note that under the above assumptions there exists for any c > 0 a constant N, such that
In fact the second estimate follows from (2.5) since go(z) is of minimal type. The first estimate can be deduced from the fact that the operator function (/ _ZAO)-1 is an entire function and the series ~;;ozkA~ representing it converges for any z. Further,
~ N,IIYo(O) II
• ck+l + N;(k + l)ek.
From this inequality one easily deduces the estimate lim {I II y&k+l) (0) II ~ c, k-->oo
which by virtue of the arbitrariness of c implies the equality lim { II ydk+l) (0) I }1/(k+1) = 0, k-->oo
proving the lemma. REMARK 2.1. It is not difficult to see that equation (2.1) has a polynomial solution x(z) if fez) is a polynomial and the dimension of the invariant subspace of Ao corresponding to the point). = 0 is finite. 2. Case when the integral differences correspond to isolated points of the spectrum of Ao. We consider the equation dU z -d Z
=
(00 ) ~ Akzk. U, k=O
(2.11)
VI.
258
LINEAR EQUATIONS IN THE COMPLEX PLANE
without assuming that condition (1.10) is satisfied. We note that an integer n for which there exist A, p
E
(j(Ao) such that
A-p=n
(2.12)
is a point of the spectrum of the transform (2.13)
WX= AoX - XAo
acting in the Banach space [f8] (p(W) does not contain differences of the points of (j(Ao); see § 3 below). We relax condition (1.10) and replace it by the following condition (a): each of the integers nl < nz < ... < np (p < (0) that are representable in the form nk = A - p for A, p E (j(Ao) is an isolated point of the spectrum of W. A solution of equation (2.11) will be sought in the form U(z) =
C~ Uiln
(2.14)
Z)Zk) ZAo
This expression is analogous to (1.11), although the coefficients here are no longer constants but functions of ~ = In z. Substituting the expansion (2.14) into equation (2.11) and equating coefficients of like powers of z, we obtain the system of relations Uk(lnz)(Ao
+ kI) +
Uk(inz) =
k
2: Ak-sUs(lnz)
s=o
(k
= 0,
1,.· .),
(2.15)
which is more conveniently written in the form Uk(~)
=
AOUk(~)
-
Uk(~)(Ao
+ kI) + Fk(~)'
where ~ = In z, Fk(~) = 2:~:::6 Ak-sUs(~). The relation Uo(~)
=
AoUo(~)
-
Uo(~)Ao
corresponding to k = 0 is satisfied if one takes Uo(~) = l. For 0 < k < nl we can take Uk(~) = Uk, i.e. we can assume that not depend on ~. Equation (2.15) then takes the form - AOUk + Uk(Ao
+ kJ)
Uk(~)
does
k-l
=
2: Ak-sUs
s=o
and can be investigated in exactly the same way as under condition (1.10). We next consider equation (2.15) for k = nl' Going over to the space [f8], it can be written in the form (setting X(~) = UnJ~)) X'(~) =
(W - nlI)X(~)
where Wis the transform (2.13) and Fn ,
E
[f8].
+ Fn "
2.
259
INTEGRAL DIFFERENCES OF EIGENVALUES
This equation has a solution X(~) which is a function of minimal type, i.e. an entire function satisfying the condition (2.16) for every c > O. For the sake of definiteness (although not out of necessity) we will assume that it satisfies the condition Pn,X(O) = 0,
(2.17)
where P n , is the spectral projection on the invariant subspace of ~ corresponding to the point nl of its spectrum. Under this condition a solution in the class of functions of minimal type is unique. For larger k we obtain an equation of the form (2.18) where the function Fk(O is of minimal type. The operator ~ - kI here has at the origin either a regular point (this is always true for k > np) or an isolated point of its spectrum. In either case equation (2.18) has solutions of minimal type. We can make the process of choosing such solutions completely determinate if we require that conditions of type (2.17) be satisfied. We have thus indicated a method for successively constructing the coefficients of the series (2.14). We will consider this series in a disk Go of radius R < P cut, for example, along the radius (- R, 0). We choose the branch of the logarithm that varies for z E Go in the halfstrip Do
=
glRe~
< InR,
-
7r < In~ <7r}.
From condition (2.16) it follows that the function Un(ln z)zn satisfies the condition limz~o II Un(ln z)zn II = O. In fact,
=
Un(~)en~
II Un(ln z)znll ~ Ceenlnlzl+ev'ln'lzl+Jr' ~ C;e(n-Ze)ln1zl = c;lzln-ze. To prove the convergence of the series (2.14) we consider a domain G of the form {ziO < (J ~ Izl ~ R - 7r < arg z < 7r}. Each of the functions Un(ln z) is bounded in this domain. We consider the" series (2.14) on the circle Izl = R and suppose R < Rl < p. By virtue of the estimate II Un(lnz)znll ~
II Un(lnz) II • Rn
= (R/Rl)nll Un(lnz)II R
7
the convergence of (2.14) will follow if we prove the uniform boundedness of the II Un (In z) IIR~. We consider the functions Uk(~)R~ (k = 1,,··,m). Since each of them is of minimal type there exists a constant Ce such that
VI.
260
LINEAR EQUATIONS IN THE COMPLEX PLANE
(k = 1,2,.··,m).
=
Hence for Fm+1,s
1::~=0
(2.19)
Am+1-rUrs we have
~ CecS
m+1
1:: liAr II· R;:.
r=1
For sufficiently large m the operator (~ - m/)-1 is invertible and satisfies the estimate II(~ ml)-111 ~ blm. The solution of minimal growth of the equation
-
U';'+1 = (~ - (m
+ 1)/)Um+1 + Fm+1
is given by a formula ofform (2.7): 00
Um+1,ss!
= -
(~
- (m + 1)/)s-1 1::
r!(~
- (m
r=s
+ 1)1)-rFm+1,r,
which implies the estimate
I Um+l.Ss! IIR~+1 m+1 ~ C 1:: IIArIIR;:. r=l
00
1::
e
£ IIArIIR;:
= Cem
~
I I
Since Ar R;:
-+
r=1 b
m
+
0 as r
m+1 1
r~1
-+ 00,
csll(~
(m
+ 1)1)-111 + 1)1)-111
+
-(S-1)
(m 1-
cll(~
-
1
(m
+
1)1)-111'
for sufficiently large m
=
I Uk(')R~11 ~ s=O I: I Ukss! I =
(m
-
IIArIIR;: . Cecs
and thus estimate (2.19) holds for all k ~
-
1 - cll(~
1)1)-11I r
-
crll(~
r=s
Cee' " I
~
1, 2,.··. But then
1's.),
~
I: csl~ls s.
Ce s=o Ceee-J R'+n:' .
We state our result in the form of a theorem. THEOREM 2.1. Suppose the following condition (a) is satisfied: there exist integers that are representable in the form n = it - fl" where it, fl, E a(Ao), each of them being an isolated point of the spectrum of the transform ~X =
AoX - XAo.
(2.13)
3. FINITE-DIMENSIONAL CASE
Suppose, further, that the series equation
.EO' AkZk
z ddU Z
261
converges in a disk /z/ < p. Then the
=( £ AkZk) U
(2.11)
k=O
has for 0 < /z/ < p a solution of the form U(z) = ( I
+ k~l UkOnz)zk ) ZA, =
B(Z)ZA"
(2.14)
where the Uk are entire functions of' = In z satisfying for every c > 0 the condition (2.20)
If the integral points of the spectrum of the transform (2. I 3) are poles of the resolvent R ..(2t'), the functions Uk(,) are polynomials. The validity of the last assertion of the theorem readily follows from the arguments presented in the proof of the theorem if one takes into account Remark 2.1. REMARK 2.2. Condition (a) means that there exist isolated points Akj and }kj of O"(Ao) whose difference is an integer:
and that there are no other points in O"(Ao) with integral differences. REMARK 2.3. Condition (2.20) implies that limz--+oB(z) = I. It can be deduced from this relation in the same way as in Remark 1.1 that any solution V(z) of equation (2.11) that is single valued in a disk /z/ < P cut along some radius can be represented in the form V(z) = U(z)C, where U(z) is a solution obtained from (2.14) by choosing a single valued branch of the logarithm, i.e. U(z) is a fundamental solution of the equation.
§ 3. Finite-dimensional case (dim )8 < (0) In this section we discuss the calculations needed to determine a solution of equation (2.11) in the finite-dimensional case. We represent the operator Ao in the form of a Jordan decomposition (see Exercise 1.3): n
Ao =
.E (AjPj + Qj).
(3.1)
j=l
Here the Aj (j = 1,.··,n) are the eigenvalues of Ao, the Pj are the corresponding spectral projections and the Qj are the corresponding nilpotent operators satisfying the relations (j
= 1,.··,n).
(3.2)
262
VI.
LINEAR EQUATIONS IN THE COMPLEX PLANE
Everywhere below we will make use of not the finite dimensionality of Q3 but relations (3.1) and (3.2), and thus our arguments remain valid in the infinitedimensional case whenever Ao is an operator of algebraic type (Exercise 1.9). We first consider the case when condition (1.10) is satisfied, i.e. when Aj - Ak "# n ("# 0) (Aj, Ak E a(Ao» for any integer n. In this case a solution is representable in the form of the series (1.11). To successively calculate the coefficients we must solve the equations
UmAo - AoUm + mUm
=
Fm.
(1.13)
The solution of this equation can be represented in the form
Um
(~ -
= -
ml)-IFm,
where ~ is the transform (2.13) acting in the space [Q3]. Let ifhm(A, fJ.) = (A - fJ. - m)-I. Then it is not difficult to conclude (see § I.3) that (~
- m I)-I =
We make use of formula (1.0.5), viz.
'!'(A B)X = L; L; _1'I' /, r . . 'I! '2! 11.12 r),r2
r
"
Q'~' "
xr Q'! r, h"
or,+r'
( 3) 3.
Applying this formula to the case when A = B = Ao and
j
In the special case when all of the elementary divisors are simple (nj = 1 for 1"",n), i.e. when Ao = L;~ AjPj, this formula acquires the simpler form
=
(3.5) We proceed now to the more complicated case when condition (1.10) is violated. In the present case, when the whole spectrum a(Ao) consists of a finite number of points, the condition (a) used in Theorem 2.1 is automatically satisfied. A solution of equation (2.11) is now representable in the form of the series (2.14). To calculate the coefficient functions Uk(~)' which, as we already know, are polynomials in the present case, we must find the solutions of minimal type of the differential equations (2.18) in [Q3]. We have seen that when m < nh where nl is the least integer that is representable in the form of a difference of the points Aj, we can assume that Fm(~) = Fm =
3.
263
FINITE-DIMENSIONAL CASE
const and Um(,) = const. In this case Um(,) = - (2( - ml)-lFm and all of the calculations are carried out according to formula (3.4) given above. In the remaining calculations we can no longer assume that Um is a constant operator. Our procedure for solving equation (2.18) depends on whether or not the operator 2( - ml is invertible. In the first case, i.e. when m #- Aj - Ak (j, k = 1"" ,n; j #- k), we apply formulas (2.4), permitting us to construct from the polynomial
a solution
in the form of a polynomial of the same degree. By virtue of (2.4) the coefficients of this polynomial have the form
1
d.
Umk = - kT s"fk s!(2( - ml)-(s-Hl) Fms .
(3.6)
We now make use of formula (3.3) for the function pq,m(A, fl.)
= (A -
f1. - m)-q.
We get Pj , Qj; Fms Pj • Qj: (Aj, - Aj. - m)q+r,+r, .
Substituting this expression into (3.6), we find, finally, that
Umk
=
-
1:;
±
-k\ nj,El nf\_ IY, S!(s(- k ~,rl,+,r2)! . s=k j"j.=l r,=O r.=O S .rl·r2· x
Pj , Qj: Fms Pj. Qj: . (Aj, - Aj. - m)r,+r.+s-Hl
We now cons ide! the second case, when m
=
nk (k
=
(3.7)
I,···,p), i.e. when
Here a solution of equation (2.18) should be sought by the method indicated in the proof of Lemma 2.2. It is convenient to do this in the following way. We write out (2.18) in more detail: (3.8)
MUltiplying this equality from the left and from the right by projections Pi, and
264
VI.
LINEAR EQUATIONS IN THE COMPLEX PLANE
Pi, and making use of formula (3.1), we obtain for the operator Xii, the equation
+ P;,UmP;,
For those cases when (3.10)
a solution is found by means of the same formula (3.7). The sum of these solutions is a polynomial U:,. of the same degree as Fm with coefficients
U:"k
1
= -
d.
nh- 1 nj,-1
n
.E .E .E (s=k j"j,=1 r,=O r,=O
-k'.E •
lY,
s!(s - k + '1 + '2)! ( - k)' , , S
·'1·'2·
(3.11)
If on the other hand (3.10) is not satisfied, equation (3.9) takes the form
According to formula (II. 1. II ) the solution of this equation satisfying the condition X;,;,(O) = 0 has the form X;,;,(~) =
~
J&i,(~-r) Pi,FmCr)P;,e-Qi,(C-r)dr.
(3.12)
o
We note that expression (3.12) is a polynomial since
eQ;t =
P;
.E s=O
QS
_j
IS.
s!
The degree of this polynomial, as can be easily verified, is equal to dm + Pi, + P;, +1. Summing expressions (3.12) over all i1 and i2 for which A;, - Ai, = m, we obtain an operator Um which when added to U:,. gives us the desired solution of equation (3.8). We see in addition that this solution is a polynomial if Fm is a polynomial. Since the function Fn,(~) = F::, is a polynomial of zero degree, the functions Um(~) are also polynomials for all m. We note that the degrees of these polynomials are the same for m > np. The coefficients of the polynomials Urn are found by means of simple algebraic operations after reducing the matrix of the operator Ao to a Jordan normal form. We note that in the simplest case, when Qj = 0 (j = l,.··,n), formula (3.7) takes the form
4.
INTEGRAL DIFFERENCES OF SPECTRUM POINTS
265
while formula (3.12) takes the form I;
X M, =
JPi, Fm(-r)Pi, d-r. o
§ 4. Case of integral differences of points of the continuous spectrum of Ao 1. Construction of the solution. In this section the restrictions on the spectrum of Ao under which we consider equation (2.11) will be weakened stilI further. We will be able to consider equation (2.11) in certain cases when the continuous spectrum of Ao contains points with integral differences. We will say that a decomposition
(4.1) of the spectrum O"(Ao) into spectral sets is normal if the difference A - fJ- is not equal to an integer when A and fJ- belong to one and the same spectral set O"j (j = 1,.··,n). By an integral shift of a spectral set O"j we mean a transformation taking it into a set of the form O"j - k = {fJ- = A - klA E O"j}, where k is a nonzero integer. The shift by one unit to the left wilI be called the elementary shift. An integral shift can result in two or more of the components of the decomposition (4.1) intersecting. By taking the union of the intersecting components, we obtain a new decomposition with fewer components. Such a shift will be called a coalescing shift. We say that the decomposition (4.1) is stably normal if every succession of coalescing integral shifts of its spectral sets again leads to a normal decomposition. Clearly, in order for the decomposition (4.1) to be stably normal it is necessary (but not sufficient) that the set of differences Ajk = {it = Aj - AklAj E O"j, Ak E O"k}
for each fixed pair of indicesj, k (j =1= k) contain not more than one integer. Thus there exists at most one integral shift of a spectral set O"k that coalesces it with another spectral set O"j (j, k = 1,.··, n; j =1= k). It follows from the definition that if O"(Ao) admits a stably normal decomposition, we can tran~form it by means of a finite number of integral shifts of its components into a set not containing pairs of points with integral differences (such a set will be said to be normal). We now prove an auxiliary proposition that plays an essential role in future constructions. LEMMA 4.1. Let O"(Ao) = 0"1 U 0"2 be a (not necessarily normal) decomposition of O"(Ao) into spectral sets and let Pk (k = 1, 2) be the corresponding spectral projections. The substitution
266
VI.
LINEAR EQUATIONS IN THE COMPLEX PLANE
(4.2) takes the equation dU z- = (CO .E AkZk dz k=O
)U
(Izl
< p)
(4.3)
(Izl < p)
(4.4)
into an equation
of the same form, the series in the right side of which converges in the same disk as the series in the right side of equation (4.3). The operator Bo satisfies the relation (4.5)
and leaves the subspace f8 2 = P2 f8 invariant. Its spectrum is the set (]'(Bo) = (]'l U «(]'2 1), which is obtained from (]'(Ao) by means of an elementary shift of the spectral set (]'2' PROOF.
Substituting (4.2) into (4.3), we obtain the equation
dY zP2Y + z(P2z + PI) ~d z
co
=.E
k=O
AkZk(P2Z + PI)Y
or, since
(4.6)
(z i= 0),
the equation Z
~~
=
(~2 + PI ) [k~O AkZk(P2Z + PI) -
J
zP2 Y.
Removing the parentheses and brackets, we obtain an equation of form (4.4) since the only term containing a negative power of z is equal to zero: P 2A oP l /z = A OP2P l /Z = O.
Relation (4.~) is verified by a direct calculation. From (4.5) it follows that BOP 2 = A OP2 - P2 = P2BoP2 and hence BoX E f8 2 for X E f8 2 (x = P2X). The last assertion of the lemma readily follows from the matrix representation of Bo, B_(Ao
o-
P2 Al
0)
Ao - I '
corresponding to the direct decomposition
4. INTEGRAL DIFFERENCES OF SPECTRUM POINTS
fB = fBI
267
+ fBz( = PlfB + P2fB).
REMARK 4.1. It is easy to see that the following formulas are valid: zP'
= eP,lnz = P2z + PI; (- zy, = - P2z + Ph
z-P' = P2/z
+ PI;
(- z)-P, = - P2/z
+ Pl'
(4.7a) (4.7b)
REMARK 4.2. Suppose there exists a decomposition 171 = 171 U171 of 171 into spectral sets of Ao such that 172 - 1 n17 1 = 0· Let i"{ denote the spectral projection of Bo corresponding to the spectral set 171' Then (4.8) For suppose Pz is the spectral projection of Bo corresponding to the set -1 U 171' Then i"{ P z = O. On the other hand, we have P2x E fB2 for any x E fB and, since both Pz and Bo leave fB2 invariant, PZP2x = P 2x. . Hence i"{P2x = i"{PZP2x = 0 for any x E fB. Lemma 4.1 directly implies the following result.
172
THEOREM 4.1. If the spectrum I7(Ao) admits a stably normal decomposition, every fundamental solution of equation (4.3) is representable in the form (4.9) where the Uk (k = 1,2, .. ·) and B are operators in [fB] determined by the equation, C E [fB] is an invertible operator and the series converges in the disk < p.
Izl
PROOF. There exists a finite sequence of elementary shifts of the components of the decomposition (4.1) that takes I7(Ao) into a normal set A. Let BgO) = Ao, BJl), ... ,Bgq) be the sequence of operators (4.5) corresponding to these shifts, so that (4.10)
where p~k) is the spectral projection of BJk) corresponding to the spectral set 172(B~k) being shifted under the recurrent transformation (4.2) and p?) = 1- pJk). Carrying out the series of transformations (4.2): yCk)
=
(P~k)Z
+ P{k))YCk+l)
(YCO) = U; k = 0,,,,, q - 1),
(4.11)
we arrive at the equation (4.12)
Since the spectrum of B~q) coincides by construction with the set A and hence
268
VI.
LINEAR EQUATIONS IN THE COMPLEX PLANE
does not contain pairs of points with integral differences, a fundamental solution of this equation has the form
(Yo(q)
= 1) (/z/ <
p).
By successively carrying out the transformations (4.11), we obtain the representation o U(z) = IT (P?)z + p?) ~ y}q)zj . zBSq)C. (4.13) 00
J=O
k=q
Removing the parentheses in (4.13), we get the required representation (4.9). The theorem is proved. 2. More exact analysis of the solution. The representation (4.9) is not sufficiently convenient for our investigation, and we therefore subject it to an additional transformation. In this connection we require the following auxiliary proposition. LEMMA
0"1 U 0"1,
4.2. Suppose that under the conditions of Lemma 4.1 the set where 0"1 and 0"1 are spectral sets of Ao such that
0"2 - 1 n O"{
* 0,
0"1 =
0"2 - 1 n 0"1 = 0·
There exists an invertible operator C E [58] such that the following represention is valid: (4.14)
where ff(~) (~ = In z) is an entire function of exponential type with values in [58]. The type of this function coincides with the spectral radius of the restriction of the transform \!{ - 1 to its invariant subspace
5821 = {P2XPl'/X E [58]}. PROOF.
The operator
S(~)
dS /d~
= ZBoC =
BoS
=
eBoCC satisfies the equation
=
(Ao - P2
+ P2 A1P1)S
(4.15)
and the condition S(O) = C. From the formulas in § 11.1 we see that S satisfies the integral equation
S
C
= e(Ao-P,)CC + Se(Ao-P,) (C-t) P2A 1P1S dt.
(4.16)
o
Multiplying this equality from the left by PI and taking into account the equality P1P2 = 0 and (4.7), we get
PIS = e(Ao-P,)CP1C = eAoC(P2 /z
+ PI) PIC =
eAocp1C.
Substituting this expression into (4.16), we obtain the representation
4.
INTEGRAL DIFFERENCES OF SPECTRUM POINTS
269
~
s = e(A,-P,)~C + Je(A,-P,)(~-t)P2AIPleAot dte.
(4.17)
o
Multiplying this equality from the right by Z-A, ZBoCz-A, =
e(Ao-P')~Ce-A,C
= e-A,c,
we find that
~
+ Je(A,-P,) (C-t) P 2A 1P 1eA.t dtCe-A,c.
(4.18)
o
Let P{ and P'{ denote the spectral projections of Ao corresponding to the spectral sets a{ and aI' We will seek the operator C in the form C = 1+ P 2 X 1 P'{. In this case P'{C = PI and equality (4.18) takes the form ZB'CZ-A, = e-P'C
+ e(A,-P')~P2XP'{e-A,C C
+ Je(A,-P,) (~-t) P2AIPle-Ao(~-t) dt o
P
= _2
z
+
P'{
+ Jce(A,-P,) (~-t) P 2A 1P{ e-A,(~-t) dt
(4.19)
0
c
+ e(A,-P,)CP2XP,{ e-A,e + Je(A,-P,)('-t) P 2A 1P,{ e-Ao(~-t) dt. o
We now show that the operator X can be chosen in such a way that the sum R of the last two summands in (4.19) becomes a constant. To this end we consider the Banach space [)B] and the transform 2(X = (Ao - P 2)X - XAo
acting in it. From (II. 1.9) we have the representation
which permits us to write R in the form
R
= eC~ P 2XP'{
c
+ Je(H)~P2AIP'{ dt. o
Let [)BEL denote the subspace of [)B] consisting of operators of the form PzXP'{ (X E [)B]). It is e1!sily seen that this subspace is invariant under the transform 2(.
n
From (1.3.10) it follows by virtue of the condition (az - 1) aI = 0 that its restriction 2(12 (which coincides with the restriction of 2( - J) to this subspace is invertible. Therefore ~
c
o
0
Je(H)~P2AIPl dt = Je(C-t)~"P2AIP'{ dt = -
e(C-t)~,,2(l:/P2AIP'{13=e'~"2f121P2AIP'{ - 2(121P2AIP,{,
270
VI.
LINEAR EQUATIONS IN THE COMPLEX PLANE
Setting X = - 2(1:/P2AIP]" we find that R = As a result, zBoCZ-Ao = P2/z + PI + 3"(~), 3"(~) = - 2(l2.1P2AIP],
2(l2.1P2AIP],.
-
,
+ Se C(-t)21P2A 1P{ dt.
(4.20)
o
It is easily seen that the exponential type of the entire function with the exponential type of the entire function C
S e CC -
o
2(n
co
dt
t)21"
=
r: 12 n=o(n+l)!
3"(~)
coincides
~n+1,
which can be calculated by means of the formula (see Exercise 1.22) v = lim 112(~2/(n n--+oo
REMARK
4.3. If (11 = 0, i.e.
=
(11
+
(11,
l)lll/n = lim 112(~2111/n. n-oo
and hence
(12n(11 =
0,
(4.21)
it follows from (4.20) that 3"(~) =
-
2(l2.1 P2AP1 = const.
Thus in this case ~Q~=~~+~+~~,
~~
where K E [)8] is a constant. We note that condition (4.21) is equivalent to the fact that the elementary shift in question is not a coalescing one. REMARK 4.4. If each of the sets (12 and (11 consists of a single point, the spectrum of the restriction of the transform 2( to the invariant subspace [)8121 = {P2XP{ IX E [)8]} consists of only zero. Therefore 3"(~) is in this case a function of minimal type. If in addition the subspaces P2)8 (c )8) and P{)8 (c )8) are finite dimensional, the subspace [)8121 (c [)8]) is also finite dimensional. An operator in a finitedimensional space with simply zero for a spectrum is nilpotent. This implies that the entire function 3"(s) degenerates into an operator polynomial. REMARK 4.5. Suppose the set (12 also decomposes into spectral sets of Ao: (12 = (1z U (1i{, such that (1i{ - 1 n (11 = 0, and suppose P z and Pi{ are the corresponding spectral projections of Ao. Let 2(' denote the transform 2('X = (Am - P2)X - XA 02 ,
(4.23)
where AOl and A02 are the restrictions of Ao to the invariant subspaces P{)8 and p z)8·
By arguing in the same way as in the proof of the lemma, we can establish the existence of an operator C such that the type of the function 3"(~) coincides with the spectral radius of the transform 2('. We are now ready to prove the following theorem.
4. INTEGRAL DIFFERENCES OF SPECTRUM POINTS
271
THEOREM 4.2. If the spectrum o'(Ao) admits a stably normal decomposition (4.1), afundamental solution of equation (4.3) is representable in the form U(z)
= [
1+ Fo(z)
+ ktl Fk(z)ffk(ln z)] ZA"
Izl
where Fk(z) = 'L.';Fkj Zj (k = O,I,.··,r; < p) isan analytic operator function in the disk < p and the ff k(~) are entire functions of exponential type. The number r coincides with the number of coalescing shifts while the type of each of the functions ff k(~) can be determined from the spectral sets that coalesce under these shifts with the use of Remark 4.5.
Izl
PROOF. We will make use of the notation introduced in the proof of Theorem 4.1. We show below that there exists a sequence of transformations (4.2) of the original equation (4.3) such that the following condition is satisfied at each step: p?+1) p~k)
=
°
(4.24)
(k = O,I,. .. ,q - 1).
We recall that p~k) is the spectral projection of BJk) corresponding to the spectral set (J~k) being shifted by an elementary transformation under the recurrent substitution (yCO)(z) = U(z); k = O,I,.··,q - 1), (4.11) p?) = I - p~k) is the complementary projection and hence NHl) is the spectral projection of BJH1l corresponding to the spectral set (J?+1) that remains fixed under the (k + l)th transformation. Since the operators B6k ) and BJHl) are connected by relation (4.10) it follows from Remark 4.2 that (4.24) will be satisfied if (J~Hl) ~ (J~k) (k = O,I,.··,q - 1) or, equivalently, (k = O,I,.··,q - 1).
(4.25)
We put aside for the present a proof of the existence of a sequence of elementary shifts satisfying condition (4.25) and use relations (4.24) to transform expression (4.13) .We write this expression in the form U(z) =
k=V-l (P?)z + P{k) [j~O Y?)zj ] [~: (ZB~k+l) CkrB~k»
] ZA"
where the Ck are 9perator constants chosen in the manner indicated in Lemma 4.2. We successively remove the parentheses in this expression. Let UsCz)
=
}t
k-q-l
Then US-1(z) = (P?-l)Z =
(P?-ll z
(P?)z
+
+ p?-l)
p?)
[~ y/q)zj][q~l (zB~k+l) CkrB&k»]. .-0
k-s
Us(z)zB~') CS_lrB~S-l)
+ p?-l)UsCZ) ( P~:-l) + pi H ) + P?~llffs(lnz)P?-1l), (4.26)
VI.
272
LINEAR EQUATIONS IN THE COMPLEX PLANE
where ,P;-,(O is the entire function described in Lemma 4.2. For s = q - 1 we have
.1::
Uq- 1(z) =
.1::
y/q)zj = 1+
J=O
y}q)zj = 1+ FoCq)(z),
J=1
where F!/)(z) is an operator function that is analytic in the disk =
Izl
< p (FoCq)(O)
0).
Let us assume that we have already established the representation Us(z)
=
1+ FoCs+1)(z)
1: F~S+I)(z).?k'+1)(lnz),
+
(4.27)
k=1
in which the entire functions .?ks+1) satisfy the relations .?~S+1) (In z) = .?kS+1) (In z)P~s) .
(4.28)
We will show that an analytic representation (4.27) with the same (one more than the same) number of terms is valid for Us - 1(z) if the shift corresponding to the transformation is not (is) a coalescing shift. The theorem will then follow since U(z) = UO(Z)ZA o• Substituting (4.27) into (4.26) and using (4.28) and (4.24), we get Us- 1(z)
=
(Pi'-I)Z
= I
+ p~S-l)( 1+
Fd S+1)(Z)
+ P~s-I) +
+ k~/~S+1)(z).?kS+1)(lnZ)J
X
[P?-I) /z
+
PiS-I) Fg+1) Pi'-ll
+
(P?-ll z
+
Pi'-ll( Fd s+1)(z)
+
(pCS-l)Z
+
pCS-l») ~ FCs+1)Zk-lpCS-l) 1 "-' Ok 2 k=2
2
pis-I) .?s(lnz)p~s-I)]
+ k~1 F~S+ll(Z).?kS+1)(1nz)Jpi'-I)
We put Fks)(z)
=
(Eis-I)z
+ p~s-I»)F~S+1)(z)
and
.?~s)(~) = .?~s+ll(~)Pi'-I)
(k
=
1,2,.··,rs)
(rS+1
=
rs
and, for the case of a coalescing shift, FCs)
r'+1
=
[(pCS-l) z 2
+
pCs-ll)FCS+l)(Z) 1
0
+
pCS-l) Z]pCS-l) 2
2
0 0 \
Fds)(z)
= (P?-I)Z + Pi'-I») Fd S+1) (Z)Pi'-l) + ~2 Fci%+llzk-lPi'-ll );
.?;~L(~) = .?s(~)pi'-ll.
(
+
1);
4.
INTEGRAL DIFFERENCES OF SPECTRUM POINTS
273
In the case of a noncoalescing shift the operator Fs«() is a constant and the corresponding summands should be attached to F~s) (z); here rS+l = rs. As a result, we obtain the formulas
condition (4.28) being satisfied by construction:
$>kS)«() = $>kS)«()Pi'-ll . If we show that (4.29) we will have proved formula (4.27) by induction. In order to prove (4.29) we note that the function Ys-1(z)
=
US_ 1(Z)z B6,-lJ
(4.30)
satisfies the equation z
dYS- 1 _ ~ B(S-l) kY d - t....l k Z s-l' Z
(4.31)
k=O
Substituting (4.30) into (4.31) and separating out the terms not depending on z, we see (cf. (1.12)) that US-1.0 = 1+ Pi'-l) FM-l) p~s-ll commutes with the operator BJS-l) and hence with its spectral projection pi'-ll. But then Pis-ll FM-l) p~S-l)
=
Pi'-1)(Us-1.0 - I)
=
Pi'-l) FM-l) p~s-l) pis-I)
=
(US-1.0 - I)Pi'-l) =
O.
Thus the proof of the theorem will be complete if we show that it is possible to construct a system of shifts satisfying relation (4.25). This can be done by regrouping the spectral sets of the decomposition (4.1) into a new decomposition in the following manner. We will say that two spectral sets (Jjl and (Jj, are comparable «(Jjl '" (Jj,) if there exists an integer kid, E Aid,' We partition the totality of spectral sets into minimal classes containing together with each spectral set all of the sets that are comparable with it. Two sets (J; and (Jj belong to one and the same class if there exists a chain In this case we put kif = k,'j, + kj,j, + ... + kj,_d' From the stable normality of the decomposition it follows that the number kij is uniquely defined. Arbitrarily choosing a set (Jj in a given class, we assign to each set (Js of this class the index as
=
kjs -
min kjp.
'"
VI.
274
LINEAR EQUATIONS IN THE COMPLEX PLANE
Finally, we let rik denote the union of the sets of index k from all of the classes (ifthey exist). As a result, we obtain the decomposition (J(Bo) = rio U rij, U ... U rij,'
The following sequence of elementary shifts has the required properties: we first shift the set rij,; after jp - jp-l elementary shifts we adjoin the set (Jj'_l to it and shift the union rij'_l U rij" and so on. The theorem is proved. COROLLARY 4.1. If the spectrum (J(Ao) admits a stably normal decomposition (4.1), a fundamental solution of equation (4.3) is representable in the form
+ k~l Uk(ln Z)Zk) zAo,
U(z) = ( I
where the Uk «(,) are entire functions of exponential type. Let nj (j = 1,.··,p) denote the natural numbers belonging to the spectrum of the transform 2( and let (Jj denote the connected spectral sets containing the points nj. Let )) ~ 0 denote the least number having the property that for any e > 0 the disk 1,1 - njl < )) + e covers the set (Jj. All of the functions Uk «(,) (k = 1,2"", 00) are of exponential type at most )).
In particular, if nh"', np are isolated points of the spectrum of 2(, all of the functions Ui(,) are of minimal type (cf. Theorem 2.1). EXERCISES 1. Suppose A(z) has a pole of order k at z = O. Obtain the following estimates (Hille [1]) for the behavior of the solution of problem (1.1) in a neighborhood of the origin: a) for k = 1
IIxoIlB- 1ICI M
~
Ilx(C)1I
~
IlxoIiBICI- M ,
where M = sup "'
>
IIxoIIB-1e-",,-kICk-ll
~
Ilx(C)1I
~
IlxoIIBe",,-klck-ll.
2. Suppose that condition (a) of § 2.2 is satisfied and, in addition, that the invariant subspaces corresponding to the integers that are points of the spectrum of the transform \}(X = AoX - XAo are finite dimensional. We denote the sum of their dimensions by N. Show that a solution of equation (2.11) ofform (2.14) can be obtained by putting U(z) =
.
ON
hm~w(C,7]), ~-o
u7]
(0.1)
where w(C,7]) = L.;:~o a m(1) CAo+cm+~)I and the functions am (7]) are regular in a neighborhood of = 0 (Hille [1]). Hint. In the equations for the coefficients resulting from the substitution of the series (0.1) into the equation, put a m(7]) = 7]N-k. b m(7]), where k m is the sum of the dimensions of the invariant subspaces corresponding to the integral points of O"(Ao) not exceeding m. 7]
NOTES
275
3. Show that with the use of thesubstitutionx(~)=1(~)w(~), where 1(~) is regular in a neighborhood of the origin, it is possible to reduce equation (1.1) to the form ~w'(~) =P(~)w(C), where P(C) is a polynomial containing only those terms whose exponents are points of the spectrum of 2[. NOTES The theory of equations with a regular singularity was constructed in the finite-dimensional case (for a system of equations of first order and for a single equation of higher order) by Fuchs [1] and Frobenius [1]. This case has been extensively studied; a presentation of it can be found, for example, in the books of Gantmaher [1] and Coddington and Levinson [1]. The more complicated variant when the operator Ao has eigenvalues with integral differences is usually considered with the use of the Jordan normal form of the matrix Ao. Another method, which was applied by Frobenius in the case of an equation of higher order to obtain a definitive expression for the solution, requires differentiating up to an order equal to the sum of the multiplicities of the corresponding eigenvalues. E. Hille [1] was the first to consider equation (1.1) in the infinite-dimensional case. His approach is followed in § 1. In the case when the spectrum I1(Ao) has numbers with integral differences, Hille modifies the Frobenius method. He is able to do this only when the corresponding points of the spectrum are isolated and have finite multiplicities. In § 2 we apply another method (Ju. L. Daleckii and I. K. Korobkova [1]), which is based on the use of double logarithm-power series and does not require finite multiplicities of the eigenvalues. The results presented in § 4 are due to P. A. Svarcman [1]. They were obtained with the use of an extension of the method applied in the finite-dimensional case by Coddington and Levinson [1]. Their presentation has been slightly altered. The authors wish to express their deep gratitude to P. A. Svarcman for his friendly criticism and help in the presentation of this chapter.
CHAPTER
Vll
NONLINEAR EQUATIONS In this chapter we study nonlinear equations that, as a rule, differ little in some well defined sense from linear equations. We present (in § 1) only the simplest theorems for the existence and uniqueness of solutions of local and global type since the more refined results are not used in this book (some of the more refined theorems are indicated in the Exercises). Also in § 1 we introduce basic stability concepts and prove a general theorem on the stability of the negativeness property of the Bohl exponent of a nonlinear equation under nonlinear perturbations. This result is then twice repeated in special cases: in § 2 ( § 3) for nonlinear perturbations of a stationary (nonstationary) linear equation, in order to illustrate other methods and to obtain more exact estimates. The result of § 2 generalizes a classical stability theorem; here we also cite different generalizations of a theorem on the instability of the zero solution of an equation with a stationary principal linear part. The proofs make essential use of the technique developed in Chapter II of renorming a Banach space (with definite and indefinite norms). In § 3 we consider an equation with a nonstationary principal part and we elucidate the important role of the negativeness of the Bohl exponent of a linear approximation of this equation. In § 4 we proceed to a consideration of the nonlinear equation dxfdt = A(t)x
+ F(t, x)
with an exponentially dichotomic principal linear part and a sufficiently small nonlinear addition satisfying a Lipschitz condition with a small constant. For the sake of simplicity we discuss only the case when the principal part is stationary (A(t) == A); here the Green operator function introduced in Chapter II plays an essential role. Since an analogous function also exists for a nonstationary e-dichotomic equation, the majority of results, as noted in the appropriate places, easily carryover to the general case. The main result consists in a proof of the existence of a solution xo(t) lying in a small ball for all t EO (-00,00). We also investigate conditions for the periodicity and almost periodicity of this solution. In addition, we prove the existence in a small ball of two initial manifolds that are canonically homeomorphic to neighborhoods of invariant subspaces ~ + and ~ _ of A and have the property that solutions initially in them tend to xo(t) as t ---> -00 and t ---> 00 respectively. We note that under the condition F(t, 0) = 0 the results of § 2 imply that when xo(t) == 0 these solutions must leave a small ball in the direction opposite to that indicated above (as t ---> 00 and t ---> - 0 0 respectively). The more complicated case when the spectrum of A contains points lying on the imaginary axis is considered in §§5 and 6 (an instability theorem for this case is given in § 2). Under certain conditions the equation can be decomposed so that one part of it, roughly speaking, determines integral manifolds of a special form of the equation, while the other part (corresponding to the points of the spectrum lying on the imaginary axis) determines the behavior of the trajectories on these manifolds. In § 5 we prove that such a manifold exists in an arbitrarily thin hypercylinder for all t EO ( - 00,00), and in § 6 we determine conditions for its stability. Most of the results of
276
1. EXISTENCE OF SOLUTIONS
277
§§5 and 6 can be extended to equations with a variable exponentially dichotomic principal linear part in the case when the space is a Hilbert space. In § 7 we present two variants of the so-called averaging principle for nonlinear equations. One of them concerns the behavior of solutions over a finite time interval while the other concerns their behavior over an infinite time interval.
§ 1. Existence of solutions. Basic stability concepts 1. Existence theorems.!) In this chapter we consider in a Banach space ~ nonlinear differential equations of the form
dx/dt = f(t, x),
(1.1)
where f(t, x) is a given function with values in ~ of a real variable t and a variable x E ~. As a rule, we will assume thatf(t, x) is continuous in t. We will be interested in various questions connected with the behavior of the solutions of equation (1.1) as t -+ 00. There first of all arises the question of the existence of solutions of this equation. This question is thoroughly examined in the literature. We give only the simplest result, which generalizes the classical theorem of Picard. The reader can find various tests for the existence of solutions, their uniqueness and extendability to an infinite interval in the work of M. A. Krasnosel'skii and S. G. Krein [2], for example (see Exercises 1-3). In the sequel, when studying the behavior of solutions at infinity, we will usually assume their existence a priori without binding ourselves to additional conditions. THEOREM 1.1 (LOCAL). Suppose there exists a neighborhood of a point (to, xo) in which the function f(t, x) is continuous in t and satisfies the Lipschitz condition
I f(t, xz) -
f(t, Xl) I ~ M
IIX2 - XIII
(1.2)
for some finite positive constant M. Then there exists a neighborhood of to in which equation (1.1) has a unique solution x = p(t) satisfying the condition (1.3) The proof is based on the contraction principle. It immediately follows from the hypothesis of the theorem that there exist Xo ~ 7J the positive constants c and 7J such that when It - tol ~ c and functionf(t, x) is continuous, satisfies (1.2) and is bounded:
Ilx -
Ilf(t, x) I ~
MI
<
00.
I
(1.4)
Let 0 = min (c, 7J / M I ) and let Ca(~) denote the Banach space of continuous functions x(t) that are defined for it - tol ~ 0, take their values in ~ and have the norm 1) The authors wish to acknowledge that they are indebted to the translator for an improvement in the presentation of this subsection
VII.
278
NONLINEAR EQUATIONS
Illxlll
=
SUp
It-t,l~o
Ilx(t)ll·
We consider in this space the closed ball B~(xo) = {x
E
xolll
Co 1IIIx -
~ 7J}.
The boundedness condition (1.4) implies that the operator t
(Tx)(t)
== Xo + JI(-r, x(-r»)d-r
(1.5)
t,
maps B~ into itself, since II(Tx)(t) - xoll ~ oMl ~ 7J for x(t) E B~. Further, when Xl> X2 E B~ the Lipschitz condition (1.2) implies the estimates II( Tx2)(t) - (TX1)(t)
I
~
t
JII/(-r, X2(-r») t,
~ M
t
J Il x2(-r) t,
- I(-r, Xl(-r») Ild-r xl(-r)lld-r ~ M(t - to)lll x 2 -
xliii
which in turn imply the estimates II( T2X2)(t) - (T2Xl)(t)
I
t
JI (TX2)(-r) -
~ M
(TX1)(-r) Ild-r
t,
from which we induce that
and hence that IIIrnx2 -
rnxllll
(oM)n
~ -n-!-lll x2 -
xliii·
(1.6)
It follows that the operator rn is a contraction in B~ for sufficiently large n. Thus, by the contraction principle there exists a unique solution x(t) E B~ of the integral equatiQJl t
x(t) = Xo
+ JI(-r, x(-r»)d-r,
(1.7)
t,
which is easily seen to be equivalent to the Cauchy problem (Ll), (1.3) for x(t) E B~. The theorem is proved. REMARK 1.1. Theorem 1.1 asserts the existence of solutions only in a certain neighborhood of the point to. But having constructed a solution in the interval [to - 0, to + 0], we can attempt to extend it further. It is obvious that we will be
1.
EXISTENCE OF SOLUTIONS
279
able to continue such a procedure indefinitely if, for example, conditions (1.2) and (l.4) are satisfied for all t and x E 18 with one and the same constants M and MI' In particular, if conditions (1.2) and (1.4) are satisfied for t E [a, (0), Xo ~ and the solution x(t) of equation (1.1) is known to lie in some ball Ilx(t) - Xo ~ < 7j, it can be extended indefinitely as t ~ 00. If we impose requirements of a global character on J(t, x), we can achieve unlimited extendability of the solutions without a priori assumptions on their behavior.
Ilx - I r; I
r;o
THEOREM 1.2 (GLOBAL). Suppose there exists a domain [a, b] x 18 on which the Junction J(t, x) is continuous in t and satisfies the Lipschitz condition (1.2). Then . Jor any (to, xo) E [a, b] x 18 the Cauchy problem (1.1), (1.3) has a unique solution x = ¢(t) defined on [a, b].
The proof is analogous to the proof of Theorem 1.1 and therefore need not be given. It suffices to note (i) that the hypothesis of the theorem implies the boundedness of J(t, x) on [a, b] x S, where S is an arbitrary compact subset of 18, and (ii) that the role of B~ is played by the Banach space C(l8) of continuous functions x(t) that are defined on [a, b], take their values in 18 and have the norm
Illxlll
=
sup
tE[a,b]
Ilx(t)ll·
2. Stability. Variational equation. We extend to nonlinear equations some of the definitions introduces in §§3 and 4 of Chapter III. We consider in a Banach space 18 the differential equation dxJdt = J(t, x)
(to
~
t < (0),
(1.1)
under the assumption that the function J(t, x) with values in 18 is defined for t E [to, (0) and x in some subset of 18 (possibly coinciding with the whole space). A solution x = ¢(t) of equation (1.1) on [to, (0) is said to be stable (according to Ljapunov) if for any c > 0 and any tl ~ to there exists a 0 > 0 such that every other solution x = 1'(t) defined in a neighborhood of tl and satisfying the inequality I ¢(tl) - 1'(tl) II < 0 exists for all t ~ tl and satisfies the inequality 1I¢(t) - 1'(t) II < c (t ~ tl)' A solution ¢(tns said to be uniformly stable if the constant 0 can be chosen independently of tl ~ to. Finally, a solution ¢(t) is said to be asymptotically stable if it is stable and for any tl ~ to there exists a 00 > 0 such that the inequality II ¢(tl) - 1'(tl) I < 00 implies lim 1I¢(t) -
t __ +oo
sHt)1I
=
o.
The definition of a stable solution is consistent with the definition given in § 3
VII.
280
NONLINEAR EQUATIONS
of Chapter III in the sense that if the linear equation dx Idt = A(t)x (to ~ t < (0) is (right) stable, every solution of it is stable. In fact, for any pair of solutions X1(t), X2(t) of this equation we have
where Ntl = SUPt II U(t, t 1) I < 00, and it suffices to take;; = e INt" An investigation of the stability of a solution x = 1>(t) of equation (1.1) can always be reduced to an investigation of the stability of the zero solution of an auxiliary equation. In fact, setting yet) = x(t) - 1>(t), we obtain for yet) the equation (1.8)
dYldt = 1/f(t, y),
where 1/f(t, y) = /(t, Y + 1>(t)) - /(t, 1>(t)). We note that 1/f(t, 0) = O. Here the solution x = 1>(t) of equation (1.1) goes over into the zero solution y == 0 of equation (1.8) while any other solution x = ¢(t) goes over into the difference y = 1>(t) - ¢(t). Equation (1.8) can be reduced to a more convenient form for purposes of study if the function/(t, x) is continuously differentiable with respect to x in a neighborhood of the point x - 1>(t). One can then write
+ 1>(t))
1/f(t, y) = /(t, Y
- /(t, 1>(t)) = A(t)y
+ F(t, y),
where A(t) = /~(t, 1>(t)) and IIF(t, y)11 ~ et(t)llyll, with limy~oet(Y) = O. It sometimes turns out that this convergence is uniform in t, i.e. lim e(y) = O.
(1.9)
y~o
Such a situation holds, for example, if the function/(t, x) has a bounded second derivative or, more generally, if 11/;(t,1>(t)
+ h)
- /~(t, 1>(t)) I ~ e Ilh II·
This follows from the obvious representation (see (1.9.7)) /(t, y
+ 1>(t))
- /(t, 1>(t)) - /;(t, 1>(t))y
1
=
J[/;(t, 1>(t) + ~y)
o
- /;(t, 1>(t))]d~ • y.
Thus equation (1.8) can be written in the form dYldt
=
A(t)y
+ F(t,y).
(1.10)
Such an equation, when condition (1.9) is satisfied, is sometimes said to be quasilinear.
The linear equation dYldt
=
A(t)y
(A(t) = /~(t, 1>(t))
(1.11)
1. EXISTENCE OF SOLUTIONS
281
which plays an especially important role in the investigation of equation (1.10), is called the variational equation of equation (1.1) corresponding to the solution x = ifJ(t). We note that if condition (1.9) is satisfied then for any q > 0 there exists a p > 0 such that
iBlllxll
IIF(t, y)11 ~ q11Y11 ~ p}.
(1.12)
in the ball Bp = {x E We will sometimes proceed directly from equation (1.10) under the assumption that condition (1.12) is satisfied with a sufficiently small fixed q > 0 (this condition is weaker than (1.9»). We consider some important special cases that are frequently encountered in the applications. An equation
dx/dt
(1.13)
f(x),
=
whose right side does not depend explicitly on t, is said to be autonomous. Clearly, if x = ifJ(t) is a solution of equation (1.13), any shifted function x = ifJ(t - to) is also a solution of it. Let Xo be a critical point off(x):f(xo) = O. In this case equation (1.13) has the stationary solution x(t) == xo, while the variational equation (1.11) turns out to be a stationary linear equation dx/dt = Ax with the operator
A = f'(xo).
(1.14)
We obtain another important special case when the function f(t, x) is periodic in t for all values of x: f(t + T, x) = f(t, x). Suppose x = ifJ(t) is a periodic solution of such an equation (with the same period): ifJ(t + T) = ifJ(t). In this case the operator function A(t) = f~(t, ifJ(t») is obviously also T-periodic. In particular, if an autonomous equation has a periodic solution y = ifJ(t), its derivative dy/dt = ifJ'(t) satisfies the variational equation. In fact, differentiating the identity difJ(t) /dt = f(ifJ(t»), we obtain the relation
d~?) = f'(ifJ(t»)ifJ'(t) =
A(t)ifJ'(t).
3. General theorem on the stability of the Bohl exponent. Inasmuch as it was shown above that it suffices to carry out an investigation of stability for the zero solution, we will frequently consider the equation dx/dt
=
f(t, x)
(t
~
0)
(1.1)
under the additional condition
f(t, 0)
=
o.
(1.15)
282
VII. NONLINEAR EQUATIONS
We will say that an equation (1.1) satisfying condition (1.15) has property N, p) (- 00 < )) < + 00, N > 0, p > 0) if every solution x(t) of it for which Ilx(to) II ;;:;; p at some moment to satisfies the estimate
~()),
(1.16) for all t > 'r ~ to for which the solution x(t) is defined. We generalize a definition of § III.4.2 by calling the greatest lower bound of numbers A = -)) for which there exist numbers pv and Nv such that equation (1.1) has property ~()), N v, pv) the (upper) Bohl exponent at zero of this equation and denote it by KB. REMARK 1.2. If the conditions of the local existence theorem are satisfied in a neighborhood of zero and the Bohl exponent at zero of the equation is negative, then the solutions whose initial values lie in a sufficiently small neighborhood of zero can be extended indefinitely (see Remark 1.1). Since, furthermore, these solutions tend to zero as t --+ 00, the zero solution in the case in question is uniformly and asymptotically stable. Because of this fact we will speak below without further ado of the negativeness of the Bohl exponent implying stability. By virtue of the above definition, for each)) > KB the solutions of equation (I. 1) that are contained in the ball B p, = < pv} at some moment of time to satisfy the condition
{xlllxli
(1.17)
It is not difficult to see that for a linear equation the Bohl exponent at zero coincides with the upper Bohl exponent of the equation. We now see that, just as in the linear case, the Bohl exponent is stable with respect to small perturbations of the equation. This fact is implied by the following result.
THEOREM 1.3. Suppose that a function f(t, x) for t E [0, 00) and I x I ;;:;; r is continuous in t and satisfies condition (1.15) and the Lipschitz condition
1[J(t, X2)
- f(t, Xl) II ;;:;; LllX2 -
xtll·
Suppose further ,,that equation (1.1) has property ~()), N, p) for some)) > 0 and p < r. Then for any N1 > Nand))l < )) there exists a number q > 0 such that if a function g(t, x) satisfies the inequality
(t
E
[0,
oo);llxll ; :; r),
(1.18)
the equation dx/dt
=
f(t, x)
has property .?l())t> Nt> P1) for some P1 > O.
+ g(t, x)
(1.19)
1.
283
EXISTENCE OF SOLUTIONS
PROOF. It suffices to show that for some h > 0 every solution of equation (1.19) that is contained in a ball Eo at some moment to satisfies the estimates
Ilx(z- + s)11 ~ N1e-v,s Ilx(z-)II Ilx(z- + h) I ~ e-v,h Ilx(z-) I
(0
~
s
~
h),
(1.20)
for z- ~ to (provided this solution is defined at the corresponding points). For in this case the inequality
Ilx(z- + nh)11 is valid for any z-
~
~ e-v,nhllx(z-)
I
to and any integer n and, finally,
Ilx(t)II
Ilx(z- + nh + s)11 ~ N1e-v,Sllx(z- + nh)11 ~ N1e-v, (S+nh) Ilx(z-) I = N1e-v, Ilx(z-) I (t = z- + nh + s). =
(H)
for any t > z- ~ to We consider a solution yet) of equation (1.1) satisfying the condition Ily(to) I ~ 00, where
00 = min {p /N, p}.
(1.21)
Since it is subject to inequality (1.16), in particular,
I y(t) I
~
Nlly(to)11
<
c.
From what was said in Remark 1.2 it follows that the solution yet) is defined for all t E [to, (jJ). We now choose an h > 0 so that (1.22) and consider on the interval [to, to + h] a solution x(t) of equation (Ll9) satisfying the condition x(to) = Xo E Eo (for those values of t at which this solution is defined). From the relation x(t)
t
t
to
to
= x(to) + Jf(s, x(s»)ds + Jg(s, x(s»)ds
(1.23)
we obtain by virtue of (Ll8) the estimate Ilx(t) I ~
t
IIxoll + (L + q) Jt, IIx(s)llds,
which implies by virtue of Corollary III.2.1 that IIx(t)1I ~ Ilxolle(L+q)h.
(1.24)
Suppose now yet) is a solution of equation (Ll) satisfying the same condition y(to) = Xo. From (1.23) and the equation t
yet) = Xo
+ J(s, y(s»)ds t,
VII. NONLINEAR EQUATIONS
284
we deduce, using (1.24), the estimate Ilx(t) - yet) I ~ L
t
J Ilx(s) -
yes) lids
+ qh Ilxo Ile(L+q)h,
to
which implies by virtue of Corollary III.2.1 again that Ilx(t) - y(t) I ~ qh llxo\\e(2L+q)h ~ qhe(2L+q+v,)he-v,(t-to) Ilxoll· Thus for those values of t have the estimate
E
Ilx(t) I ~ ~
[to, to
+ h) at which the solution x(t) is defined
I
I
I
yet) + IIx(t) - yet) [N + qhe(2L+q+v,)h] e-v,(t-to)
we
Ilxoll.
We now subject the number q (> 0) to the condition qhe(2L+q+v,)h
= min(NI
-
N, 'rj).
(1.25)
Then (to
~
t < to
+ h).
(1.26)
On the other hand, taking into account (1.22) and (1.16), we get IIx(to
+ h) I
~
I y(to + h) I + Ilx(to + h) -
~ [1 - 'rj
y(to + h) I + qhe(2L+q+v,)h]e- v,h IIx(to) II,
which implies by virtue of (1.25) that IIx(to
+ h) I
~ e-v,hllx(to)ll·
(1.27)
Since all of the arguments can be repeated with the point to replaced by the points to + h, to + 2h,.··, we can assert that estimate (1.26) is satisfied for all t ~ to at which the solution x(t) is defined. We now choose 0 = min (ooINI> (0). It then follows from (1.26) for Ilx(to) I < 0 that IIx(?:") I ~ Nlo ~ 00 ('L" ~ to) and hence that all of the above estimates can be constructed with to replaced by 'L". In this connection (1.26) and (1.27) go over into estimates (1.20). Thus the theorem is proved, it following from (1.21) that 0= min {p, pIN, pI NNd,
and the number q being determined from (1.25) and (1.22). COROLLARY 1.1. If the Bohl exponent at zero of equation (1.1) is negative, then the Bohl exponent of the perturbed equation (1.19) for sufficiently small q > 0 is also negative. REMARK 1.3. Suppose the function get, x) satisfies the condition
2.
STABILITY: STATIONARY PRINCIPAL PART
lim Ilg(t, x) ,,~o
Ilxll
I
285
=0
uniformly in t. In this case we will briefly write g(t, x) = o(llxll), bearing in mind that the convergence is uniform in t. Then for any q > 0, no matter how small, we can find a neighborhood of zero in which condition (1.18) is satisfied. Consequently, for a sufficiently small Pl > 0 equation (1.19) will have property &6'(Vl' Nl , Pl) with Vl arbitrarily close to v. Hence in this case the Bohl exponents at zero of equations (1.1) and (1. 19) coincide.
§ 2. Stability and instability of the solutions of the nonlinear equation with a stationary principal part 1. Stability. Suppose the stationary linear equation dx/dt
= Ax
(2.1)
has a negative Bohl exponent /CB and the function F(t, x) is subject to the estimate IIF(t, x)11 ~ qllxll
(t ~ 0;
Ilxll
~ p).
(2.2)
Then from Theorem 1.3 it follows that the equation dx /dt
= Ax + F(t, x)
(2.3)
has a negative Bohl exponent at zero whenever q > 0 is sufficiently small. We will independently obtain this same result in order to illustrate a method (a natural generalization of the so-called second method of Ljapunov) which will be used later to prove an instability theorem. The idea underlying this method was presented in Chapter II. We recall some of the geometric observations made there. If the spectrum a(A) lies in the interior of the left halfplane, the vector field Ax (of tangent vectors to the integral curves of equation (2.1)) at each point x E}B after an appropriate renorming of the space }B turns out to be directed toward the interior of the sphere centered at the origin which passes through the point x. But if a(A) contains a component in the right halfplane there exist points x E }B at which the vector Ax is directed toward the exterior of this sphere. It is obvious that a small perturbation of the equation cannot produce a large change in the ch~racter of the integral curves. We can therefore anticipate that under certain conditions the question of stability of the zero solution of equation (2.3) under a "not too large" perturbation FCt, x) can be resolved by the spectrum of the linear approximation of this equation. These general observations receive a rigorous formulation below. THEOREM 2.1. Suppose the spectrum of the operator A lies in the interior of the left halfplane, so that IleAl ~ Noe-v,t, and suppose condition (2.2) is satisfied for q < vo/No.
I
VII.
286
NONLINEAR EQUATIONS
Then equation (2.3) has a negative Bohl exponent at zero.
I
Consider a solution x(t) of equation (2.3) satisfying a condition Ilx(to) P (the number Po will be chosen below) and let to ~ t < T be an interval in which Ilx(t)11 < p. We introduce in the space )8 a new norm PROOF.
< po
~
co
I x llA
As was shown in Theorem 1.4.2, M = No/vo. We put yet) = rAtx(t). Then
=
J I eAtxlldt. o
mllxll
~ IIxllA ~ Mllxll, where m
dt/dy = rAt (dx/dt - Ax) =
e~AtF(t,
=
l/IIAII and
x)
and co
co
o
t
Ilx(t)IIA = J lI eA (r+t)y(t)lI dZ' = J lI eAry(t)lI dZ'. Let us calculate the divided difference I x(t
+ h) IIA
Ilx(t) IIA
-
h
7 lIeAry(t + h)hI -
= _ ~ ttlleAry(t)lIdZ' + h t
lI eAry(t)1I d'r.
t+h
Denoting the first summand of the right side by .P1(h) and the second by .P2 (h), we estimate them, assuming that to ~ t < T. We first note that
In addition, setting Z' .P2(h) ~
=
s
+ t + h in the integral below, we get
7 lIeAr[y(t + hh) -
r
y(t)] I dZ'
Hh
~
I eAseA(t+h) yet
+ hh -
yet) II ds
=
II eA(t+h) y(t
+ hh -
yet)
t
~ Mil eA(t+h) y(t + hh - yet) II ~ Mil eAh II II eAt yet + hh - y(t)~. The last estimate implies the inequality lim .P2(h) ~ MlleAtY'(t) II h~O
=
M IIF(l, x) I ~qMllxll.
(2.5)
2.
STABILITY: STATIONARY PRINCIPAL PART
287
I
Ik Ilx(t + h) IIA - Ilx(t) IIA ,
Thus for the right first derivative of the norm x(t)
d+ Ilx(t) IIA = lim ~
h
h-+O
which exists, as was shown in Chapter I (see Exercises 1.17 and 1.18), we obtain from (2.4) and (2.5) the estimate
d+llx(t)IIA/dt ~ - vllx(t)IIA' qM)/m > 0 since qM < (vo/No)(No/vo)
where v = (1 From inequality (2.6) we obtain the estimate
(2.6)
1.
=
(to ~ .. ~ t < T),
and, finally, after returning to the original norm, we get
Ilx(t) I M /m.
~
Ne-"(I-T) Ilx(..)I
(to
~
..
~ t
< T),
where N = We put po = min(p, p /N). Then Tcan be taken arbitrarily large. For suppose Ilx(T) = p. Since we can pass to the limit in (2.7) for obtain the contradictory inequality
I
p
= Ilx(T) I
~
Ne-"(T-t,) Ilx(to) I <
(2.7)
t ~ T, we
p.
The theorem is proved. 2. Conditions for the absence of stability. We now consider equation (2.3) under the assumption that the spectrum (l(A) contains points lying in the interior of the right halfplane. This will lead us to a generalization of the classica1 Ljapunov instability theorem. We begin by assuming that we have a decomposition (l(A)
= (l+(A) U (l-(A),
(2.8)
and we let P + and P _ denote the corresponding spectral projections. Thus we assume that the spectrum of A does not intersect the imaginary axis and that (l +(A) is not empty. As in the preceding subsection, we assume that
(11xll
IIF(t, x)11 ~ qllxll
~ p; t ~ to)·
(2.2)
The subsequent arguments are analogous to those in the proof of Theorem 2.1. We introduce in)8 the norm 00
IlxiiA = IIp+xl\A+ IIp_xllA =
00
J Ile-Arp+xlld.. + 0J IleArP-xlld... o
(2.9)
It was shown in § I.4.5b) that this norm is equivalent to the original one:
mllxll ~ IlxiiA ~ Mllxll· In conjunction with (2.9) we consider the indefinite norm
288
VII.
NONLINEAR EQUATIONS
Let x(t) be a solution of equation (2.3) and let [to, T] be an interval on which it satisfies the condition I'X(t) II < p (such an interval exists by virtu. e of the continuity of a solution if IIx(to) I < p). Settingy(t) = e-Atx(t), we get as before y'(t) = e-AtF(t, x). In this connection IIp+x(t)IIA =
00
00
o
~
f lIe-A(r-l)p+y(t)lIdz- = f lIe-Arp+y(t)lIdz-.
Let us estimate the right derivative of the function lip +x(t) IIA. We write IIP+x(t
+h)lI~
-
IIp+x(t)IIA = f£l(h)
+ f£z(h),
where
and f£z(h) =
r
lIe-Arp+y(t
+ h)~ -
IIrArp+y(t)1I dz-.
-I
Here lim I f£l(h) I = lIeAtP+y(t) I = IIp+x(t)1I
(2.10)
h~O
and
since If£z(h)
I~ =
I
II e- ArP + y(t
JII
+ h~ -
e-Asp + eAt y(t
y(t) II dz-
+ h~ -
y(t) II ds
=
II eAtp + y(t
+ h~ -
y(t)
From (2.10) and (2.11) we obtain the estimate d+IIP+xIIA = lim IIp+x(t dt h~O
;;:; lim If£l(h) h~O
+ h)IIA -
I-
IIp+x(t)IIA
dt
lim If£z(h)
h~O
1 ;;:; M lip +x(t) IIA -
I ;;:;
lip+xll -
lip +F(t, x(t) IIA'
lip +F(t, x) IIA
t
2.
289
STABILITY: STATIONARY PRINCIPAL PART
which reduces to the inequality
lip+X(t) IIA - lip+x(r) IIA ~ J{~ lip+x(s) IIA - lip+F(s, xes)) IIA } ds.
(2.12)
Analogously, we deduce the inequality d+llp-x(t)IIA dt
=
lim IIp-x(t
+ h)I!A - IIp-x(t)IIA
h~O
~
h
- lip-xII
+
lip_F(t, x(t)) IIA
~ - (l/M)llp-x(t)IIA
+
IIp-F(t, x(t))IIA
and the estimate
I P_x(t) IIA - IIp-x(z-)IIA ~ ~ IIp_x(s)IIA +
I {-
(2.13) IIp_F(s,x(s))IIA }ds.
Subtracting (2.13) from (2.12), we obtain the estimate <x(t) A ~ {x(z-) A + p(t, z-),
(2.14)
where pet, z-)
f {~ Ilx(s)IIA -
=
IIF(s, xes)) IIA } ds.
We note that it follows from (2.2) and the equivalence of the two norms being considered by us that for sufficiently small q (q < qo = m/M) p(t, z-) ~ a
t
J Ilx(s) I Ads,
(2.15)
T
where a is a positive constant. Finally, from (2.14) and (2.15) we obtain the estimate {X(t)A - {X(Z-)))A ~ a
t
J Ilx(s)IIA ds
(q
~
qo; to
~
z-
~
t < T),
(2.16)
which will be used, in subsequent arguments. Inequality (2.16) means that the indefinite norm «X(t)))A of a solution x(t) increases with increasing t when the condition Ilx(t) I < p is satisfied. THEOREM 2.2. Suppose the spectrum O"(A) does not intersect the imaginary axis and the spectral set 0" +(A) is not empty. There exists a number qo depending only on the operator A such that if the function F(t, x) satisfies condition (2.2)for q ~ qo, the zero solution of the differential equation (2.3) is unstable for t -+ + 00.
290
VII.
NONLINEAR EQUATIONS
Moreover, every nonzero solution x(t) of this equation for which ((x(tO»))A leaves any ball I x II ;£
PI
~
(2.17)
0
of radius PI < P at some t > to·
Proof. From (2.17) and (2.16) it follows that the strict inequality r = ((x(tO»))A> 0 holds for to > to. From the inequalities (2.16) and IIx(t)IIA ~ ((X(t»))A we obtain the inequality I
IIx(t) IIA ~ a
J IIx(s) IIA ds + r, 1'0
which on the basis of Corollary III.2.1 reduces to the estimate (2.18) since the function u(t) = rea(I-t;) satisfies the equation I
u(t)
=
a
J u(s)ds + r·
I',
For sufficiently large t the required inequality IIx(t)II ~
(2.19)
PI
follows from (2.19) provided PI < p. We note, finally, that if P -x(to) = 0, condition (2.17) is automatically satisfied provided x(to) 'I O. Thus we can find a solution with an arbitrarily small initial value I x(to) II satisfying condition (2.19) for sufficiently large t. This implies the instability of the zero solution of equation (2.3). The theorem is proved. REMARK 2.1. If the condition IIF(t, x) I ;£ qllxll is satisfied for - 00 < t;£ to and the spectral set IT_(A) is not empty, a.solution x(t) satisfying the condition (2.20) leaves a ball IIxil ;£ PI < pas t -+ - 00. In the sequel we must consider the case when the function F(t, x) is defined on the real line and condition (2.17) is satisfied for all t. In this case every nonzero solution of equation (2.3) leaves any ball IIxil ;£ PI < p. 3. Another type of instability condition. The condition of Thorem 2.2 concerning F(t, x) is satisfied in every case if IlF(t, x) II ;£ qllxll1+p
(p> 0;
/lxll
;£p; t ~to).
(2.21)
In fact, estimate (2.21) in a sufficiently small ball implies estimate (2.2) with a sufficiently small q.
2.
291
STABILITY: STATIONARY PRINCIPAL PART
It turns out that we can deduce from (2.21) a result that in a certain sense is more general, the assumption that O"(A) does not intersect the imaginary axis having been dropped. THEOREM 2.3. If the spectrum O"(A) of A contains points lying in the interior of the right halfplane and condition (2.21) is satisfied, the zero solution of equation (2.3) is unstable. PROOF.
Re A (A
Let A = a + (3i be a point of O"(A) having a maximal real part: a (A)). Then for any 1) > 0
~
EO"
(1)
> 0; t
~
z-).
Moreover, as follows from Lemma 1.4.1, for each T > 0 and a sufficiently small 0 there exists a vector ~ = ~iJ,T (II~II = 1) such that
o>
ea(t-t')(l - 0) ~
I eM-I,); I
~ ea(t-to)(l
+ 0)
(0 ~ t - to ~ T).
Let x(t) (t ~ to) be the solution of equation (2.3) satisfying the condition x(to) = Xo. We write it in the form x(t) = X1(t) + xz(t), where I
X1(t)
= eA(I-I,)xo; xz(t) = JeA(T-I,)F(z-, x(z-))dr:. I,
Let Xo = e~iJ,T and let R be a number satisfying the condition 1 + 0 < R < min (2, 1 + 0 + 01), where the number 01 > 0 will be chosen later on. By virtue of the continuity of x(t) there exists an interval (0 ~ t - to ~ T1) on which
I x(t) I
~ eRea(t-t,) .
(2.22)
We estimate both parts of the solution on this interval, assuming that 1)
< ap, eReaT,
~
p.
Under these conditions we have Ilxz(t)
I
I
~
J IleA(I-T) 1IIIF(z-, x(z-)) IldzI, I
~ N~q
J e(a+~)(I-T) Ilx(z-)II1+PdzI, I
~ (eR)1+PN~q
J e(a+~)(t-T)+a(P+1)(T-I,) dz10
and
(2.23)
292
VII.
NONLINEAR EQUATIONS
Il x1(t) II ~ (1
Therefore, provided T1
~
+ o)cea(t-t,).
T,
Ilx(t) II ~ Il x 1(t) II
+ Ilxz(t) II ~ [ 1 + 0 + NrR
(0 < t - to
cPR1+Peap (t-t,) ] celt-to)
ap - r;
~ T1).
If it turns out in this connection that
1+
() +
N~q
ap - r;
cPR1+PeapT, < R,
then the continuity of x(t) implies that condition (2.22) is also satisfied on a larger interval than [to, to + T1]. Therefore inequality (2.22) is always satisfied on the interval [to, to + T1], where T1 satisfies the equation 1
+0+
N~q
ap - r;
cPR1+PeapT,
=
R,
which can be rewritten in the form cea T, = _1_ [(R - 1 - (})(ap - r;) J1IP. R N~qR
(2.24)
For sufficiently small c this equation always has a unique solution T 1(c) > 0, which when 0, Rand r; are given depends only on c and not on T. Thus T can be chosen arbitrarily and we put T = T1(C). . We now note that (2.24) implies cReaT,
=
(ap -
r; )l1P (R _ 1 _ o)llP
N~qR
::S;
-
ollP 1
(ap -
r; )lIP.
N~q
Setting 01 ~ N~qpP I(ap - r;), we ensure the fulfilment of inequality (2.23) and thereby the validity of all of the arguments constructed above. Thus inequality (2.22) is satisfied on the interval 0 ~ t - to ~ Tb where T1(c) is a solution of equation (2.24). But then Ilx(to + T1) II ~ II x 1(to + T1) II .
~ (1 - o)ceaT, =
[1 - 0 -
- Ilxz(to + T1) II N~q
ap - r;
N~q
ap - r;
cPR1+PeapT'ceaT,
R1+PcPeaPT, ] ceaT,
_ _ [(R - 1 - (})(ap - r;)J 1IP _ - (2 R) N~qR1+P -
C
>
o.
293
3. NONSTATIONARY PRINCIPAL PART
Ilxoll
This estimate implies the instability of x(t), since E = can be taken arbitrarily small and yet there still exists a value T1(E) for which Ilx(to + T1) ~ c. 4. Stability of a stationary solution of the autonomous equation. We consider the autonomous nonlinear equation dx
dt = f(x)
(to
~
I
(2.25)
t < 00).
If Xo is a critical point:f(xo) = 0, in a neighborhood of which the functionf(x) is continuously differentiable, the difference y(t) == x(t) - Xo will satisfy the quasi linear equation dyJdt = Ay
+ F(y).
(2.26)
On the basis of what has been said above we can conclude that the behavior of a small solution of this equation is determined by the spectrum of the operator A = f'(xo). If the spectrum (J(A) lies in the interior of the left halfplane, the zero solution of equation (2.26), and hence also the stationary solution x == Xo of equation (2.25), is uniformly and asymptotically stable. But if the points of (J(A) also lie in the interior of the right halfplane, we can assert that this solution is unstable when either (i) (J(A) does not intersect the imaginary axis or (ii) a condition of type (2.21) is satisfied (it is satisfied, for example, if the function f(x) is twice differentiable).
§ 3. The nonlinear equation with a nonstationary principal part 1. Another variation of the theorem on the stability of the Bohl exponent. In this section we consider the question of stability of the solutions of the nonlinear equation dx
dt = A(t)x + F(t, x),
(3.1)
without assuming that its principal part is stationary. As indicated in § 1, it suffices to study conditions for the stability of the zero solution of equation (3.1). Thus we can assume that F(t,O)
= O.
(3.2)
From Corollary 1.1 to Theorem 1.3 it follows that a small addition F(t, x) does not violate the property of negativeness of the Bohl exponent of the equation. We now give an independent proof of this fact, following the same idea as in the proof of Theorem III.4.6, in order to obtain more exact values of the constants entering into the estimates. Also we replace condition (1.18) by the analogous integral condition
IIF(t, x) I
~ r;(t)llxll
(t ~ to;
Ilxll
~ p),
(3.3)
VII. NONLINEAR EQUATIONS
294 where
1
t-+
Co
-7:0 Jt 7J(7:)d7: ;£ q.
(3.4)
An analogous generalization could have been made in Theorem 1.3 (see Exercise 5). THEOREM 3.1. Suppose the equation
dxldt = A(t)x
(to ;£ t < <Xl)
(3.5)
has property ~(J.!, N) (J.! > 0) while the function F(t, x) satisfies condition (3.3), (3.4) for a fixed 7:0' Then when q < J.! IN equation (3.1) has property ~(J.!t, Nt, Po) for some po > 0, Nl = Ne Nco andJ.!l = J.! - Nq > O.
{xlllxli
PROOF. Suppose po < min (pINl , p), Xc E Bpo = ;£ Po} and x(t) is the solution of equation (3.1) satisfying the condition x(7:) = XC' We can always find aT> 0 such that < p ;£ t < + In the indicated interval x(t) can be written in the form
Ilx(t)11
(7:
7:
T).
t
x(t) = U(t,7:)xc
(7: ;£ t < 7: + T).
+ J U(t, s)F(s, x(s»ds
The evolution operator U(t, estimate
(3.6)
7:) of equation (3.5) is by assumption subject to the (t
~
7:).
Using this inequality and inequality (3.3), we obtain from (3.7) the integral inequality
Ilx(t)11
;£ N~lxclle-v(t-c)
t
+ J Ne-v(t-s)r;(s)llx(s)llds c
(7: ;£ t < 7: +
T),
which implies (see Corollary 111.2.2) that on the interval 0 ;£ t -
Ilx(t)II
;£ Nlllxclle-(v-Nq)(t-c)
From estimate (3.7) and the condition
Ilx(t) I
;£ pe-(v-Nq) (t-c)
Ilxcll
7: < T
(Nl = Ne Nco ).
(3.7)
;£ po we obtain the inequality
(0 ;£ t -
7: <
T),
from which it is not difficult to deduce the fact that the quantity T can be chosen arbitrarily large. Indeed, there would otherwise exist a number T > 0 such that + = p, which contradicts inequality (3.7). Thus inequality (3.7) is satisfied for any t and (t ~ which proves the theorem.
Ilx(7: T)II
7:
7:),
3.
295
NONSTATIONARY PRINCIPAL PART
COROLLARY. For sufficiently small q > 0 the zero solution of equation (3.1) is uniformly and asymptotically stable if equation (3.5) has a negative Bohl exponent.
(Here it is assumed that the perturbed equation satisfies the conditions of the local existence theorem (see Remark 1.2).) 2. Stability of the zero solution of the quasilinear equation. If equation (3.1) is quasilinear, i.e. IIF(t, x)11 = o(llxll), condition (3.4) is satisfied for arbitrarily small q in a sufficiently small neighborhood of x = O. Thus the quasilinear equation (3.1) has property £6>(Vh Nh P1) with V1 arbitrarily close to v; in particular, its zero solution is uniformly stable. It turns out that this result has a converse. THEOREM 3.2. In order for the zero solution of equation (3.1) to be uniformly stable for any perturbation F(t, x) satisfying the quasilinearity condition it is necessary and sufficient that the corresponding unperturbed equation have a negative Bohl exponent.
The sufficiency of the assertion follows directly from the above arguments. We prove the necessity by considering only one perturbed equation (3.1) with F(t, x) = xllxllp for some fixed p > 1. Let Bp = {xlllxli < p} be a ball having the property that every solution x(t) of the equation under consideration with Xo = x(to) E Bp is indefinitely extendable for t > to. Each such solution can be represented in the form x(t) = U(t, to)y(t), where U(t, to) is the evolution operator of equation (3.5). From (3.1) we obtain the following equation for y(t); PROOF.
y'(t) = ¢(t)y(t)
(¢(t) =
It follows from (3.8) that yet)
¢(t) = exp
~ to).
(3.8)
(> 1 for t > to),
(3.9)
I U(t, to)y(t) lip; t
= ¢(t)xo, where
(1. ¢(s)ds)
and therefore by virtue of (3.8) ¢'(t)
=
¢P+1(t) I U(t, to)xo lip·
(3.10)
From this result, taking into account that ¢(to) = 1, we get ¢'(7:) 1 f. II U(7:, to)xo lip d7: = f ¢P+1(7:) d7: ?:. p t
t
(xo E £6> p).
(3.11)
But this estimate implies by virtue of Theorem III.6.2 and Remark III.6.3 that the Bohl exponent ICB of equation (3.5) is negative under the additional assumption of integral boundedness of the coefficient A(t). In the general case, according to the same Theorem III.6.3, it remains for us to show that ICB < 00. To this end we make use of the fact that uniform stability must
296
VII.
NONLINEAR EQUATIONS
hold for the perturbed equation under consideration. We choose an c > 0 and find a 0 such that Ilx(to) < 0 implies < c independently of the choice of to· Then for < min (0, p) we will have
Ilxoll
I
Ilx(t) I
Ilx(t) I
= ¢(t) I U(t, to)xo I <
c
(t ~ to),
which implies and hence ICB ~ o. 3. Stability oj a periodic solution. An important class of equations of type (3.1) is the set of equations of this type with periodic coefficients. We have already noted in § 1.3 that if pet) is a T-periodic solution of the equation dx fdt = J(t, x)
(3.12)
with a T-periodic continuously differentiable function J(t, x), the difference yet) = x(t) - pet) satisfies the quasilinear equation dy fdt
= A(t)y + F(t, y),
(3.13)
where A(t) = J;(t, pet)). If the variational equation dy fdt = A(y)t in this connection has a negative Bohl exponent (i.e. the spectrum of its monodromy operator lies in the interior of the unit disk), equation (3.12) will also have a negative Bohl exponent at zero. Thus in this case the periodic solution pet) is uniformly and asymptotically stable. A different situation arises when equation (3.12) is autonomous: J(t, x) == J(x). In this case, as we saw in § 1, the variational equation corresponding to a periodic solution pet) must have a periodic solution, and hence the spectrum of its monodromy operator intersects the unit circle. We return again to this case later.
§ 4. The nonlinear equation with a principal part the spectrum of which does not intersect the imaginary axis 1. Solutions bounded on the real line. In this section we will consider the behavior of the solutions of the nonlinear equation
dx fdt
= A(t)x + F(t, x),
(4.1)
under the assumption that the spectrum of the operator A(t) in its principal part dx fdt = A(t)x
(4.2)
does not intersect the imaginary axis. We will not as a rule assume that F(t, 0) = 0 and we will see that the properties of equation (4.1) resemble the properties of the inhomogeneous linear equation dx fdt
= A(t)x + J(t)
(4.3)
4.
SPECTRUM NOT INTERSECTING THE IMAGINARY AXIS
297
We assume here as everywhere else in this chapter that the function F(t, x) is continuous in t. In order to simplify the presentation we will consider below the more restricted equation dx/dt = Ax
+ F(t, x)
(4.4)
with a stationary principal part such that O"(A) =
0" +(A)
U0" _(A),
(4.5)
where either of the sets 0"±(A) may be empty. But it is not difficult to note that only the possession by equation (4.3) of a principal Green function (see Chapter IV), which is subject to the usual estimates, is of significance in the proofs constructed below. The passage to the more general case will be effected in special remarks. The following terminology will be used in the sequel. A function F(t, x) that is defined and continuous in t on the product space (- 00,00) x B p, where Bp = {xllix ~ p}, is of class (M, q, p) if on the indicated set it is subject to the condition
I
IIF(t, x) I
~ M
(4.6)
and the Lipschitz condition IIF(t,
F(t, Xl)
X2) -
I
~
qllx2 -
xd·
(4.7)
A function satisfying the above conditions for (t, x) E (- 00, 00) x 58 is of class (M, q). Finally, the symbols (M, q, p)± and (M, qh will be used to denote the function classes resulting from the above classes when the real line (- 00, 00) is replaced by a right halftine (a, 00) or a left halftine ( - 00, a) respectively. We first consider equation (4.4) on the real line. THEOREM 4.1. Suppose the operator A satisfies condition (4.5). For any p > 0 there exist a constant M > 0 depending only on A and p and a constant q > 0 depending only on A such that if F(t, x) E (M, q, p), equation (4.4) has one and only one solution x(t) that remains for all t in the ball Bp:
sup
Ilx(t)
-00<1<00
I
~ p.
PROOF. Suppose x(t) is a solution of equation (4.4) that remains in Bp. Then the function F(t, x(t») is bounded on the real line and by virtue of Theorem 11.4.1 a solution of equation (4.4) can be represented in the form 00
. x(t)
=
J GA(t -
r)FCr, x(z»)dr,
where GA is the principal Green function of the operator A. This function is subject to an estimate of the form
(4.8)
VII.
298
NONLINEAR EQUATIONS
where Nand)) are positive constants. Thus the considered solution of equation (4.4) satisfies the integral equation (4.8). Conversely, on the basis of the same theorem every solution of the integral equation (4.8) that remains in Bp satisfies equation (4.4). Consequently, for the considered solutions x(t) that are defined on the real line and remain in B p, equations (4.8) and (4.4) can be regarded as equivalent. For a proof of the existence and uniqueness of the solutions of equation (4.8) we consider in the space G(lS) the ball Bp consisting of the functions x(t) (- 00 < t < (0) satisfying the condition
Illxlll
sup
=
-00<1<00
Ilx(t)11 ~ p.
For sufficiently small M > 0 the transformation 00
yet) = (Tx)(t) =
I
GA(t - s)F(s, x(s))ds
acts in Bp and is a contraction if q is sufficiently small. In fact, IIIF( • ,xC • )) ~ Mfor x(t) E B p, and therefore
II
Illy I I ~ M if M
~
7IIGA(t) I dt ~
-00
M
2:
~
p,
W(2N. Further, 00
IIITx2 - Txtlll ~ q sup 1
I
IIGA(t -
s)II·llxz(s) -
xl(s)llds
-00
2N ~ q-))-lll x 2 -
Xliii·
If q < )) (2N, the transformation T is a contraction, On the basis of the contraction principle we conclude that there exists one and only one solution of equation (4.8), and hence of equation (4.4), in Bp. The theorem is proved. REMARK 4.1. The frequently encountered equation dx (dt = Ax
+ get) + F(x)
(4.9)
Illglll
with a bounded continuous function g(t) satisfying the condition < p))(2N and an autonomous function F(x) satisfying conditions (4.6) and (4.7) is a special case of (4.4). REMARK 4.2. If the condition F(t, 0) = 0 is satisfied and consequently the estimate IIF(t, ~ holds, the only bounded solution on the realline of equation (4.4) is the trivial solution x(t) == o. Any other solution leaves every ball B p, (Pl < p) either as t -> +00 or as t -> -00.
x) I
qllxll
299
4. SPECTRUM NOT INTERSECTING THE IMAGINARY AXIS
REMARK 4.3. Theorem 4.1 can be extended verbatim to equation (4.1) if equation (4.2) is e-dichotomic. 2. Almost periodic and periodic solutions. Theorem 4.1 can be sharpened if one assumes that the function F(t, x) is not only bounded but almost periodic in t for each x E Bp. LEMMA 4.1. Suppose F(t, x) is an almost periodic function in tfor each x E Bp and satisfies condition (4.7). Then the function F(t, x(t)) is almost periodic if x(t) is an almost periodic function all of whose values belong to Bp. PROOF. We consider a sequence of translates F(t + 'r n , x), Let D be a compact set in Bp. We will prove that {'rn} contains a subsequence {'rn.} such that the sequence of functions F(t + 'r n ., x) converges uniformly on D. To this end, using the diagonal method, we choose {'rn.} so that the sequence {F(t + 'r n., x)} converges at each x of a countable subset DI that is dense in D. Suppose XED and Xl E D I . From the inequality
IIF(t + 'rn" x) -
F(t + 'r n" x)11 + 'r n" x) - F(t + 'r n" xI)11 + IIF(t + 'rn., Xl) - F(t + 'rn" Xl) I + IIF(t + 'rn" Xl) - F(t + 'r n" x) I ~ 2qllx - XIII + IIF(t + 'rn" Xl) -F(t + 'rn"
~ IIF(t
it follows that for any $ > 0 there exists a natural number N(xb IIF(t
+ 'rn"
x) - F(t
+ 'rn" x)11 <
$
Xl) $)
I
such that
(nk' nj ~ N)
(4.10)
for all x E U(Xb $) = {xlllx - xtll < cj4q}. The open balls U(Xb $) (Xl E D I) cover D. Choosing from among them a finite covering and taking as N the greatest of the N(XI' $) corresponding to the members of this covering, we obtain inequality (4.10) for all XED. This proves that the subsequence {F(t + 'r n" x)} converges uniformly on D. We now choose from the sequence {'rn} a subsequence {'rn.} for which the sequence {x(t + 'rn.)} converges and for which the sequence {F(t + 'r n" x)} converges uniformly on the closure D of the set of values of x(t) (this set is compact by virtue of the almost periodicity of x(t)). We then obtain the estimate IIF(t
+ 'rn., x(t + 'rn.)) - F(t + 'rn" x(t + 'rn)) I + 'rn., x(t + 'rn.)) - F(t + 'rnJ, x(t + 'rn.)) " + IIF(t + 'rn" x(t + 'rn.)) - F(t + 'rnJ , x(t + 'rn)) I maxllF(t + 'rn" x) - F(t + 'rn" x) I + qllx(t + 'rn,)
~ IIF(t ~
D
which implies the convergence of the sequence {F(t
+ 'rn., x(t +
(4.11) -
x(t
'rn.))}'
+ 'rn)ll,
300
VII. NONLINEAR EQUATIONS
We have proved the precompactness of the family F(t the almost periodicity of the function F(t, x(t»).
+ 7:, x(t + 7:») and hence
THEOREM 4.2. Suppose the spectrum (T(A) does not intersect the imaginary axis while the function F(t, x) is almost periodic for each x E Bp and of class (M, q, p) with the constants M and q restricted as in the proof of Theorem 4.1. Then the unique bounded solution of equation (4.4) whose existence was proved in Theorem 4.1 is almost periodic. PROOF. The set fJp of almost periodic functions lying in the ball Bp is closed in this ball. From the proof of Theorem II.4.4 and Lemma 4.1 it follows that the contraction T (see the proof of Theorem 4.1) leaves fJ p invariant. Therefore the unique fixed point of Tin Bp lies in fJ p. We briefly discuss the question of periodic solutions of equation (4.4) for the case when the function F(t,x) is continuous and periodic in t: F(t
+ 2n, x) =
F(t, x)
If the conditions of Theorem 4.1 are satisfied, there exists exactly one bounded solution x(t) of this equation. It is not difficult to verify that it is a periodic function. In fact, the function x(t + 2n) satisfies the same equation, and, by v1rtue of the uniqueness, x(t) == x(t + 2n} All of what has been said applies in particular to equation (4.9) when the function get) is almost periodic or periodic. REMARK 4.4. The results of this and the preceding subsection can be extended without difficulty to equations of the form dx fdt = A(t)x + F(t, x) with a periodic operator function A(t) if the unperturbed equation is reducible. 3. Solutions bounded on a halfline. Let x(t) be a solution of equation (4.4) that remains in the ball Bp = ~ p} for t > to. From the results of § II.4.3. it follows that x(t) satisfies the integral equation
{xlllxli
00
x(t)
=
eACHo)yo
+ S GA(t
- s)F(s, x(s»)ds,
(4.12)
to
where Yo = P ~x(to). The converse is also true: a solution of the integral equation (4.12) satisfies the differential equation (4.4) for t > to. The investigation of equation (4.12) is carried out by the same methods as the above. THEOREM 4.3. Suppose that the spectrum (T(A) does not intersect the imaginary ~ Ne~"lti (v> 0, N ~ 1) is satisfied. axis and hence that the estimate For any p > PI > 0 there exist positive numbers M, q and I depending only on N, v, p and PI such that if F(t, x) E (M, q, p)+, there corresponds to each Yo E 1lL n
IIGA(t)II
4.
301
SPECTRUM NOT INTERSECTING THE IMAGINARY AXIS
B p/ZN one and only one solution of equation (4.4) satisfying the conditions P _x(to) = Yo and sup to+/:"'tllx(t) I ~ Pl· The following estimate is valid for any two solutions corresponding to different values of Yo: (4.13)
where cp. is a positive constant not depending on to while f1. arbitrarily close to J) by decreasing the number q.Z)
E
(0,
J))
and can be made
PROOF. The first part of the theorem is proved with the use of the contraction principle by foIlowing the proof of Theorem 4.1. The additional condition IIYol1 ~ p/2N is imposed to ensure that the initial portion of the trajectory x(t) does not leave the balI Bp. When t - to > I the first term in (4.12) becomes sufficiently smaIl and the trajectory of the solution finds itself in BpI. Let XI(t) and xz(t) be a pair of solutions. From (4.12) we deduce the inequality 00
Ilxz(t) - XI(t) I ~
Ilyzo -
YlOIINe-v(t-t,)
+ S Ne-vlt-slqllxz(s)
- xI(s)llds,
t,
which implies by virtue of Corollary III .2.3 that
Ilxz(t) -
xI(t)!1
~
¢(t)llyzo - YlOll,
where
¢(t) =
J)
+
vv22NJ)_ 2qNJ)
e-(t-t,).Jv'-ZqNv,
and this implies the proof of the second part of the theorem. REMARK 4.5. To each vector Yo E ~L with a sufficiently small norm there corresponds a unique bounded trajectory x(t) (t ~ to) such that Yo = P _x(to). Thus the initial points x(to) of the trajectories that are bounded for t -> + 00 form in B p/ ZN a manifold IJJI+ of dimension equal to the dimension of ~L. In fact, to each point Yo E 58_ Bp/ZN there corresponds one and only one point x(to) such that Yo = P _x(to) and from which there emanates a trajectory that is bounded for
n
t -> 00.
The manifold IJJI+ is conveniently imagined as a "surface" in the Cartesian product of the subspaces P +58 and P -58. The "equation" of this surface, i.e. the relation between Yo = P -x(to) and P +x(to), is implicit in (4.12). If x(to) E Bp/ZN\IJJI+, the solution x(t) leaves any ball BpI (PI < p) for sufficiently large t. When the spectral set(i +(A) is empty, the space 58 coincides with 58- and therefore all of the solutions emanating from Bp/ZN terminate in BpI for t -> 00. By virtue of (4.13) each of them is uniformly and asymptoticaIly stable. 2)
Hence f1. can be made arbitrarily close to the distance from a(A) to the imaginary axis.
VII. NONLINEAR EQUATIONS
302
On the other hand, if (J' _(A) is empty, the manifold 9](+ becomes a single point x, for which P -x = O. In this case there is only one trajectory that is bounded for t > to. It is obtained when Yo = o. Analogous results hold for a left halfline, the role of 58_ being played by 58+. The initial values x(to) of the solutions that remain in Bp as t --+ - 00 fill out a manifold 9](- situated in B p/ 2N• Each point x of this manifold is uniquely determined by its projection y = P +x, and therefore the "dimension" of 9](- coincides with the dimension of 58+. We again consider equation (4.4) on the real line. From Theorems 4.1 and 4.3 and the observations made in Remark 4.5 we obtain the following result. THEOREM 4.4. Suppose the spectrum (J'(A) does not intersect the imaginary axis. Then for any p > 0 there exist M and q depending only on A and p such that if F(t, x) E (M, q, p), a sufficiently small neighborhood of the zero element of 58 will contain manifolds 9](- and 9](+ with the following properties. a) The manifold IJJ(± is homeomorphic to a neighborhood of zero of the subspace 58±. b) The manifolds 9](+ and 9](- have exactly one point z in common. c) The solution xo(t) of equation (4.4) satisfying the condition xo(to) = z is bounded on the realline. d) The solutions x(t) of equation (4.4) satisfying the condition x(to) E 9](+ (9](-) exponentially approach xo(t) as t --+ + 00 (t --+ - (0) and exponentially recede from it as t tends indefinitely far in the opposite direction. REMARK 4.6. If one of the manifolds 9](± reduces to a single point belonging to xo(t), this solution is uniformly and asymptotically stable over the corresponding halfline. But if the dimensions of both 9](+ and 9](- are not equal to zero, we say that this solution is conditionally stable. REMARK 4.7. The above results extend without difficulty to equation (4.1) in the case when its nonstationary principal part admits an e-dichotomy of the solutions. 4. Orbital stability of a periodic solutions of the autonomous equation. We consider the autonomous equation (4.14)
dx/dt = f(x)
with a continuously differentiable function f(x) and having a periodic solution ¢(t) = ¢(t + T) (- 00 < t < 00). We have already indicated that in this case the variational equation dx/dt
= A(t)x
(A(t)
= !'(¢(t»)
(4.15)
has the periodic solution x(t)
= ¢'(t)
(4.16)
and hence (we assume that ¢(t) ¢ const) the spectrum of the monodromy operator U(T) of this equation contains the point A = 1.
4.
SPECTRUM NOT INTERSECTING THE IMAGINARY AXIS
303
We assume that this point is an isolated simple eigenvalue while the rest of the spectrum of U(T) lies in the interior of the unit disk and does not surround the origin. In this case there exists a periodic operator function Q(t) that reduces (4.15) to the form dx /dt = Ax, where A is a constant operator with zero as an isolated simple eigenvalue and the rest of its spectrum (1 -(A) lying in the interior of the left halfplane. We take to as the initial point and assume that Q(to) = I. The change of variable
+
x(t) = rp(t)
Q(t)z(t)
(4.17)
takes (4.14) into the quasilinear equation dz/dt
= Az + F(t, z)
(F(t, 0) = 0),
(4.18)
the function F(t, z) satisfying in a sufficiently small neighborhood of zero the Lipschitz condition IIF(t, zz) - F(t,
Zl)
I
~ q
Ilzz - ztll
with a sufficiently small constant q > O. We denote by Po and P _ the spectral projections of A corresponding to the parts (10 = {O} and (1- of its spectrum. Let - )) = SUP'<EIJ_ Re A. The substitution
z = e- (4z) (I-I,) z converts equation (4.18) into the equation
~:
=
(A + ~ I) z + F(t, z)
(4.19)
where the function F(t, z) = e(v/Z) (I-I,) F{t, e- (v/Z) (I-I,) z) has the same properties as F(t, z). But the principal part of this equation is e-dichotomic and we can therefore apply the results of the preceding subsections to it. Since the only solution of equation (4.19) that is bounded on the real line is the zero solution, the solutions issuing from the manifold 9)1+ (described in Remark 4.5) tend to zero as t -+00. Returning to equation (4.18), we obtain the following estimate for any solution of it satisfying the condition z{to) E 9)1+ : (4.20) where f1. < )) but can be made arbitrarily close to )) by requiring that z{to) lie in a sufficiently small neighborhood of zero. These solutionsz(t) are determined from an equation of type (4.12), which, after one returns from (4.19) to (4.18), takes the form co
z(t)
=
eA(I-I,) P _z(to)
+ S Git I,
where
s)F(s, z(s)ds,
(4.21)
304
VII.
NONLINEAR EQUATIONS
G (){ eAtp_ At _ e-Atpo
=
Po
_
fort> 0, for t < O.
We note that although IIGit - s)11 = Ilpoll (s> t) does not decrease as S-Hy), expression (4.21) is applicable to the solutions under consideration, since by virtue of estimate (4.20) IIF(s, z(s))
I
;:£ q Ilz(s)
I
;:£ cl'qe-l'(s-to) lip _z(to) II·
After this, from (4.21) we obtain the estimate Ilz(to) - P-z(fo) -z(to)
lip
I
I
<
Ilpoll
C
=
I'
fJ.
q,
which shows that in a sufficiently small neighborhood of zero the manifold 9)1+ described by the points z(to) lies outside a cone whose axis of rotation is the onedimensional subspace 5Eo = Po5E. We note that 9)1+ is projected in a one-to-one manner onto a neighborhood of zero in the subspace 5E- = P-5E and therefore divides a sufficiently small ball with center at zero in the space 5E. We proceed, finally, to a consideration of the solutions of equation (4.14). The periodic solution x(t) of equation (4.15) has the form x(t)
=
Q(t)eA(t-to) Pox(to)
=
Q(t)Pox(to),
and by virtue of (4.16) ¢'(to) = x(to)
=
Pox(to) E 5E o.
Thus the periodic solution ¢(t) is tangent to the one-dimensional space 5Eoatt = to. On the other hand, the points (4.22) with z(to) E 9)1+ constitute a surface 9)1i that is obtained from 9)1+ by means of a parallel translation into the point ¢(to), lies outside a cone with axis 5Eo and divides a sufficiently small ball Ba(¢(to)) with center at this point. Since the trajectory of ¢(t) in the vicinity of to obviously passes from one nappe of the cone to the other, for sufficiently small any trajectory passing through the ball under consideration also penetrates, by virtue of the continuity of I(x) , each of the nappes of the cone and consequently necessarily intersects the manifold 9)1i at some instant t 1• Thus, o) - ¢(to) < if at some time t= to there exists a t1 such that X(t1)
a
Ilx(t
I a
E 9)1i.
Let X1(t) = x(t - to + t1)' Since equation (4.14) is autonomous, X1(t) is also a solution of it, this solution now intersecting 9)1i when t = to. The corresponding solution Zl(t) of equation (4.18) has the property Zl(tO) = X1(tO) - ¢(to)
and is therefore subject to the estimate
E 9)1+
5. STABLE INTEGRAL MANIFOLDS
305
II Zl(t) II ~ c!'e-!'(t-t,) lip -ZI(tO) II· Hence by virtue of (4.17) we conclude that Ilxl(t) - cp(t) II ~ c!,e-!'(t-t,) , where C!' is a certain constant, or, finally, that Ilx(t -(tl - to)) - cp(t)11 ~ c!,e-!'(t-t,).
(4.23)
Inequality (4.23) does not mean that x(t) and cp(t) approach each other as t-HfJ, and thus it does not imply the stability of cp(t). But this inequality does imply that if a solution x(t) finds itself at some moment of time in a sufficiently small neighborhood of the set 2 = {cp(t)IO ~ t ~ T}, viz. the trajectory of the periodic solution cp(t), it will subsequently approach this set as t -+ 00. The property described above is called orbital asymptotic stability. The shift in time 'Z'o = tl - to after which the solutions asymptotically approach each other is called the asymptotic phase. The arguments presented above lead to the following result. THEOREM 4.5. If the autonomous equation has a periodic solution (not identically constant) to which there corresponds a simple isolated eigenvalue of the monodromy operator of the variational equation while the rest of the spectrum of this operator is in the interior of the unit disk and does not surround the origin, the indicated periodic solution is orbitally asymptotically stable and, moreover, each solution that is sufficiently close to its trajectory has an asymptotic phase.
§ 5. Stable integral manifolds 1. Derivation of an integral equation for manffolds. In this section the equation dx/dt = Ax
+ F(t, x)
(5.1)
will be considered under less rigid restrictions concerning the operator A. We will no longer assume that the spectrum O'(A) does not intersect the imaginary axis. This creates a more complicated situation. We assume that the spectrum of A can be divided into two spectral sets O'(A) = O'I(A) U O'z(A) such that for some a > 0 I ReAl> a I ReAl < a
for A E O'z(A), for A E O'I(A).
Thus in our case the equation dx /dt = Ax admits an exponential splitting of its solutions of generally third order. We will say in this connection that the spectrum O'(A) admits an a-separation. As usual, Ph P z, )81 and )8z will denote the spectral projections and invariant subspaces of A corresponding to the spectral sets O'I(A) and O'z(A).
VII.
306
NONLINEAR EQUATIONS
We will assume that F(t, x) is of class (M, q), i.e. it is continuous and subject to the estimates IIF(t, x)11 ~ M,
(5.2)
IIF(t, X(2)) - F(t, X(l)) II ~ qllx(2) - x(1) II
(5.3)
. on the product space (- 00, (0) x )8. The purpose of the following discussion is the construction of certain special integral manifolds for equation (5.1). When the spec.trum of A does not intersect the imaginary axis, these manifolds convert into the individual trajectories studied in the preceding section. In order to explain the essence of the matter we consider as an elementary example the system of differential equations
I
dXl/dt = 0, dx1fdt = Xl, dX2/dt = - X2,
or in vector form, setting X
=
(xl, xl', X2),
d~=
d
(0 ° 0) 10
Ox
° °-1
(= Ax).
°
The spectrum (l(A) in this case consists of the two points ).0 = and ).1 = - 1, and the three-dimensional space )8 decomposes into the direct sum )8 = )81 )82 of the two-dimensional space )81 consisting of the vectors (xl, xl', 0) and the onedimensional space )82 consisting of the vectors (0, 0, X2). The solution of the system has, as can easily be seen, the form
+
It readily follows from this formula that the zero solution of the system, as well as any other solution initially in )81, is unstable. But the whole set of solutions that are initially in )81 (and, incidentally, remain in )81) is stable in the sense that the distance between any other solution and )81 tends to zero as t --+ + 00. We can say that this set of solutions forms a stable integral manifold )81. An analogous situation exists, as we will see below, for more general nonlinear equations. The individual solutions of such equations will not as a rule be stable or even conditionally stable because of the presence of a "critical" part of the spectrum lying on the imaginary axis. At the same time certain sets of solutions, which in general are no longer linear, can turn out to be stable (or conditionally stable) in the sense that near solutions will be "attracted" to them. The "dimension" of such a stable manifold of solutions always coincides with the multiplicity of the critical
5.
STABLE INTEGRAL MANIFOLDS
307
part of the spectrum of the coefficient of the principal linear part of the equation (and if the critical part is absent, the manifold reduces to a single trajectory). Another example of this type of stable integral manifold is the orbit of a periodic solution of the autonomous equation considered in the preceding section. In this section we deduce a functional equation for integral manifolds of a specific form and prove their existence, while in the next section we establish the stability of such a manifold under certain assumptions on the equation. We note that after finding such a manifold, in order to determine the solutions composing it, we must consider, as will be seen, a differential equation in a space of dimension equal to the multiplicity of the critical part of the spectrum, i.e. less than the dimension of the initial phase space. In particular, this dimension can be finite. We proceed to more precise definitions. By an integral manifold of equation (5.1) we will mean a set WI composed of the trajectories of this equation in the product space (- 00, (0) x Q3. Strictly speaking, we depart here from the usual meaning of the term "trajectory" as the set of values of a solution x(t) (- 00 < t < (0) in the phase space Q3 and think of it rather as the set of points (t, x(t») of the space (- 00, (0) x Q3. The component x E Q3 of an element (t, x) will be called its spatial projection. From this point of view a trajectory in the usual sense of the word is the spatial projection of the trajectory being considered by us. Thus, by definition, through each point of an integral manifold WI there passes a trajectory x(t), and each trajectory having at least one point in WI has all of its points in WI. We will consider the integral manifolds of equation (5.1) that can be described by equations of the form
(t E (-00, (0); Xl E Q31) (5.4) = ¢(t, Xl) in the "coordinates" t, Xl = PIX and Xz = Pzx. Here we will assume that the function ¢ is continuous in t and satisfies a LipXz
schitz condition in Xl with a constant not depending on t. The set of all such functions is a linear space, which we denote by L. Let L1J denote the subset of L consisting of the functions satisfying the condition ., !I¢(t, x?) - ¢(t, x?) (x?l, x?) E Q31;
II
~ 7j I x?) - x?) II
7j = const > 0)
(5.5)
(the functions of class L1J)' The integral manifold WI described by equation (5.4) is called a (p,7j)-manifold if ¢ E L1J and satisfies the condition
(t
E (-00,
(0);
Xl E Q31)'
(5.6)
Thus the spatial projection of a (p, 7j)-manifold is completely contained in the
VII.
308
NONLINEAR EQUATIONS
cylinder Up of 1S whose base is the ball Bp(1S z) in 1Sz and whose "axis" is the subspace 1S1. If a trajectory (t, x(t) lies on the considered manifold and cf;(t) = PIX(t) is its projection onto the subspace 1S1o the whole trajectory can be expressed in terms of the function cf;(t) by means of the formula X(t)
= cf;(t) + ¢(t, cf;(t).
(5.7)
For constructing the manifolds of equation (5.1) that are of interest to us it is convenient to write this equation in the form of a system of two equations in the subspaces 1S1 and 1S z. Multiplying (5.1) from the left by the projections PI and P z respectively, we obtain the required system dXI = f dt
l dt d.x2
=
AIXI
+ FI (t, Xl + X2),
A ZX2 + F2(t,
Xl
(5.8a)
+ X2),
(5.8b)
where Xk = Pkx, Fk = PkF, Ak = PkA (k = 1, 2). Let us derive an equation which the function ¢(t, Xl) defining a (p, 1])-manifold 9J1 must satisfy. If X = x(t) is a trajectory lying on this manifold, the functions Xl = cf;(t) = PIX(t) and X2 = ¢(t, cf;(t) must satisfy system (5.8). Here equation (5.8a) takes the form dcf;/dt
=
Alcf;
+ FI(t, cf; + ¢(t, cf;)).
(5.9)
Conditions (5.3) and (5.5) imply the condition II FI(t, cf;"
+ ¢(t, cf;")
- FI(t, cf;'
+ ¢(t, cf;')
II ~ q(1
Therefore equation (5.9) satisfies for each given ¢ E 1S1 a unique solution cf;(t)
1.2 and has for each XlO XlO at t = -c:
E
=
+ 1])IIcf;"
- cf;'II·
(5.10)
L the conditions of Theorem Wet, -c, xlOl¢) that is equal to
(5.11) We note tha! when t, -c and XlO are fixed the function Wet, -c, xlOl¢) is with respect to ¢ a nonlinear operator defined in the space L and taking its values in 1S1. On the other hand the function X2(t) = ¢(t, cf;(t)) is a bounded solution on the real line of the equation dXz/dt
= A2X2 + F2(t, cj;(t) + X2)
(5.12)
and must therefore satisfy the integral equation 00
xz(-c)
=
J G2(-c -
t)F2(t, cf;(t)
+ xz(t)dt,
(5.13)
5.
309
STABLE INTEGRAL MANIFOLDS
where Gz(t) is a principal Green function of the operator A z in the subspace ~z. We now choose for a fixed 'C the solution ¢(t) of equation (5.9) that is equal to a fixed element Xl at t = 'C: ¢(t) = Wet, 'C, XI!¢). From (5.13) we then obtain the integrofunctional equation 00
¢('C, Xl) =
S Gz('C
- t)Fz(t, Wet,
'C,
XI!¢)
+ ¢(t, Wet, 'C, xd¢)))
dt
(5.14) ('C E ( - 00,
<Xl); Xl
E ~l)
for the function ¢(t, Xl). We emphasize that this equation is not simply an integral equation, since ¢ enters into it not only as a function but as an element of the space L. We have thus shown that the function ¢(t, Xl) defining a (p, 7J)-manifold of equation (5.1) must satisfy equation (5.14). On the other hand, a solution ¢(t, Xl) of equation (5.14) satisfying conditions (5.5) and (5.6) defines a (p, 7J)-manifold of equation (5.1). In fact, for any point (XlO, Xzo, to) lying on 9)(, i.e. satisfying the relation Xzo = ¢(to, XlO), equation (5.9) has a solution Xl =
¢(t) = W(t, to, XlO!¢)
with the property Xl (to) = XlO. It accordingly follows from (5.14) and the obvious relation Wet,
'C,
W('C, to, XlO!¢)!¢) = Wet, to, XlO!¢)
that the function Xz = ¢(t, ¢(t)) is a solution of equation (5.12). This result can be formulated in the following manner. LEMMA 5.1. A (p, 7J)-manifold of the differential equation (5.1) is described by a solution ¢ of the integrofunctional equation (5.14) that is of class L~ and satisfies condition (5.6). The trajectories lying on the manifold are determined by the solutions of the differential equation (5.9).
It will be proved below that for given p > 0 and 7J > 0, when M and q are sufficiently small, equation (5.14) has one and only one solution possessing the necessary properties, and that the integral manifold corresponding to it consists of all of the trajectories of equation (5.1) that lie in the cylinder Ilxzll ;£ p. 2. Proof of the existence of (p, 7J)-manifolds. Let us first prove an auxiliary proposition. We note that if the operator A admits an a-separation, there exists a > 0 for which the spectral set O"I(A) lies in the strip IRe AI < a and hence
a
a
IleM11 ;£
Nae(a-alltl,
(5.15)
where Al = PIA. On the other hand, it can be assumed that the spectral set O"z(A) lies in the domain Re A! > a + and hence that the estimate
!
a
310
VII. NONLINEAR EQUATIONS ~ Noe- (aH) III
II G2(t) II
(5.16)
is valid for the Green function G 2(t). LEMMA 5.2. The operator //f(t, -r, xllqS) representing a solution of equation (5.9) satisfies the conditions
1) II//f(t, -r, xi 2)lqS(2)) - //f(t, -r, X?) IqS(l)) I
~ No {lIxiZ)
- XiI) lIe(a-o+~.) II-TI
(5.17)
i
+ q I e(a-0+~')II-Sls~pIW2)(S, Xl) (qS(k) where (3q
-->
0 as q
-->
E
- qS(l)(s, Xl) lids
L7J;
X?)
E )81;
I} k = 1,2),
0, and
PROOF. Let xil)(t)
=
//f(t, -r, x?) WI)), Xl(t)
=
x?)(t)
=
//f(t, -r, X?) W2)),
//f(t, -r, xiI) W2)).
These functions satisfy the integral equations 1
eA,(H)x?)
+ S eA,(I-S)Fl(s, x?)(s) + qS(l)(S, x?)(s)) ds,
x?)(t) = eA,(H)xi2)
+ S eA,(I-S)Fl(s, xi 2)(s) + qS(2)(S, x{Z)(s))) ds,
xiI)(t)
=
(5.19)
1
(5.20)
T
1
Xl(t)
=
eA,(H)xi l )
+ S eA,(I-S)Fl(s, Xl(S) + qS(2)(S, Xl(S))) ds.
(5.21)
T
From (5.19) and (5.21), using estimate (5.15) and the Lipschitz conditions (5.3) and (5.10) for the functions F and qS(l), qS(2), we obtain the estimate iixil)(t) - xl(t)1I ~
1
S IIeA,(I-s) II{ IIFl(s, x?)(s) + qS(l)(S, x?)(s)) +
T
IIFl(s, Xl(S)
+ qS(l)(s, Xl(S)))
- Fl(s, Xl(S) - Fl(s, Xl(S)
1
~ Noq(1
+ r;) Se(a-o) (I-s) IIxiI)(s)
- Xl(S) lids
T
1
+ Noq Se(a-o) (I-S) T
sup IWl)(s, Xl) - qS(2)(S, Xl) lids. x,
+ qS(1)(s, Xl(S)) I + qS(2)(S, Xl(S)) II} ds
S.
311
STABLE INTEGRAL MANIFOLDS ~ 'C
It can now be deduced from Remark III.2.1 that when t
Ilxil)(t) -
XI(t)
I
~ u (t),
where u(t) is a solution of the integral equation
=
u(t)
Naq(1
+ r;)
t
S e(a-a) (t-s)u(s)ds T
t
+ NaqS e(a-a)(t-s)supll¢(l)(s, Xl)
-
¢(Z)(s,
Xl) I ds,
(t-S)supll¢(1)(s, Xl) -
¢(Z)(s,
Xl) I ds.
T
X,
which,as can easily be verified, has the form u(t)
=
t
Naq S e[N,q(1+~)+a-aJ T
X,
Thus
t
~ Naq S e(a- H fl,)(t-S)supll¢(1)(s,
Xl) -
¢(Z)(s,
where the constant f3q = Naq(1 + r;) tends to zero as q Further, (S.20) and (S.21) imply
IlxI(t) -
xiZ)(t)
I
Xl) lids,
(S.22)
x,
T
~
o.
I I XiI) - xiZ) I t + S IleA,(t-S) I I FI(s, XI(S) + ¢(Z)(s, XI(S»))
~ IleA,(t-T)
T
- FI(s, xiZ)(s) ~ Noe(a-o) (t-T)
+ Noq
Ilx?) -
xi Z)
I
t
(1
+ ¢(Z)(s, x?)(s») lids
+ r;) S e(a-a) (t-s) IlxI(s) -
I
x{2)(s) ds.
T
Using Lemma II1.2.1 once again, we readily deduce from this inequality the estimate
IIW(t, 'C, x?) I¢
Wet,
'C,
xi Z)
I¢(Z» I
~ Nallx?) - xi Z) Ile(a-H!3,)(t-T).
(S.23)
Combining (S.22) and (S.23), we obtain the required estimate (S.17) for t ~ 'C. An analogous derivation exists for t ~ 'C. Inequality (S.18) follows directly from (S.2) and (S.lS). This lemma can be used to establish the existence of the integral manifolds that are of interest to us.
312
VII.
NONLINEAR EQUATIONS
THEOREM 5.1. Suppose the spectrum l1(A) admits an a-separation. For any r; > 0 and p > 0 there exist constants M > 0 and q > 0 depending only on r;, p and the operator A such that if F(t, x) E (M, q) equation (5.1) has a unique (p, r;)-manifold m. This manifold consists of all of the solutions of equation (5.1) that lie in the cylinder Dp: Ilp2x(t) I ~ p (- CIJ < t < CIJ). PROOF. We consider, in the Banach space of functions (t, x) E ~2 that are continuous and bounded on (- 00, CIJ) X ~l with norm III III = SUPt...J(t, Xl) II, the closed subset St(p, r;) consisting of the functions of class L'fj that satisfy condition (5.6). We will show with the use of the contraction principle that for FE (M, q) with sufficiently small q and M the operator S: ~ ;p defined by the formula
00
J G2(7: -
(fi(7:, Xl) =
t)F2(t, Wet,
7:, XII(t, Wet, 7:, Xl I») dt,
acts in the space St(p, r;) and has precisely one fixed point in it. We first note that (5.2) and (5.16) imply the inequality
00
2MN
11(fi(7:, Xl) I ~ MNo 100 e-(aH)ltl dt = a + a' which shows that the function (fi satisfies condition (5.6) for sufficiently small M >
O. Then, using inequalities (5.16) and (5.10) and Lemma 5.2, we obtain the estimate
11(fi(7:, x?») - (fi(7:, xi2») I 00
~ No
J e-(aH)'H'IIFI(t, Wet, 7:, x?) I
7:, x?) IqI) + (t, Wet, 7: x?) 1
00
~ Noq(1
+ r;) J r(aH)'r- t' IIW(t,7:,x?)I
~ N~q(1
+r;)llx?) - x{2)11
00
J e-(aH)IHI+(a-O+~.)IHI dt
= 2q(1 + r;)N~ Ilx(l) _ X(2) II. 20 _
~q
I
I
By taking q > 0 sufficiently small, the coefficient in the right side of this inequality can be made less than r;. This proves that the function (fi satisfies condition (5.5) and hence that the operator S acts in the space St(p, r;). In regard to a proof of the first assertion of the theorem it remains to show that the operator S is a contraction in this space. From Lemma 5.2 it follows that
5.
313
STABLE INTEGRAL MANIFOLDS
co
~ No
Je-CaHJI<-tIIlFI(t, W(t, Z", xllpz) + Pz(t, Wet, Z", xlIPz») - FI(t, W(t,
Z",
xllpl) + plt, Z", Xl IPI))) I dt
co
~
J e- CaH ) IT-t (RI + Rz)dt,
Noq
1
-co
where
and
Xllpl» I ~ IIpz(t, w(t, Z", x2Ip2» - PI(t, Wet, Z", xllpz» I + IIpI(t, w(t, Z", xllp2» - PI(t, w(t, Z", Xllpl» I ~ IIIp2 - PI I I + 7JII W(t, Z", xllp2) - Wet, Z", xllpl) I eCa-HfJ,J It-
R2 = IIp2(t, Wet,
Z",
Xllp2» - PI(t, Wet,
Z",
Using these estimates, we obtain after an integration the inequality 111
-
P2 -
PI I I -
2Naq [
~ a +0
I
+
q(1 + 7J)(a + 0) ] (20 - f3q)(a - 0 + f3q)
I I P2 - PI I I ,
which shows that the operator S is indeed a contraction for sufficiently small q> O. It remains to prove that the constructed manifold contains every solution x(t) of equation (5.1) that satisfies the condition IIp2 x(t)1I ~ p (-00 < t < (0). Suppose x(t) is a solution with this property and let cf;(t) = PIX(t), X(t) = P2x(t). These functions satisfy the system
+ FI(t, cf;(t) + X(t»), = J G(t - s)F2(s, cf;(s) + Xes») ds,
dcf;Jdt = AIcf; 00
X(t)
cf;(t)lt=t, = cf;(to). It is not difficult to show by the same method as was used above that this system has a unique solution satisfying the condition IIX(t) ~ p (- 00 < t < 00). On the other hand, the same system is satisfied by the functions cf;1(t) and XI(t), where cf;l(t) is a solution of equation (5.9) under the condition cf;1(tO) = cf;(to) while XI(t) = pet, cf;l(t»). Therefore cf;(t) cf;1(t) and X(t) XI(t) pet, cf;(t»); but this means that the solution under consideration lies on our manifold. The theorem is proved.
I
=
=
=
r~E~:nzo DE irr/~:ST'C;'~'~-~'~ I
314
VII.
NONLINEAR EQUAnONS
REMARK 5.1. The result obtained in Theorem 5.1 bears a global character. We were able to obtain this result because we assumed that the condition FE (M, q) is satisfied on the whole space. But usually the function F(t, x) is defined for x E B p, where p (> 0) is sufficiently small, and satisfies such a condition on (- 00, 00) x Bp. It is not difficult in this case to extend F(t, x) onto the whole space ~ with preservation of the constants M and q. This can be done, for example, by letting F(t, x) be constant on the extensions of the radii of the ball Bp. After this one can assert on the basis of Theorem 5.1 that there exists a manifold 9R which will serve as a "local" manifold for the original problem. By solving equation (5.9) on this manifold, we can observe the component Xl = PIX of the solution and hence the whole solution as long as it remains in the ball Bp. Such a "local" integral manifold exists, in particular, if the function F(t, x) satisfies the quasilinearity condition
IIF(t, x)
I ~ c(x)llxll
(c(x)
~
0 as x
~
0)
and the condition (5.24) REMARK 5.2. Under certain conditions the results of this (and the following) section carryover to equations of the form
dx/dt = A(t)x
+ F(t, x).
(5.25)
Suppose, for example, that the equation dx/dt
= A(t)x
(5.26)
is reducible to a stationary equation dx fdl = Cx with an operator C whose spectrum admits an a-separation. Then equation (5.25) can be transformed by means of a change of variable into the equation studied above, and this permits one, as can easily be seen, to transfer the results obtained above to equation (5.25). This is true, in particular, for a periodic reducible equation (5.26) with a monodromy operator whose spectrum admits an a-separation. A more general case is when equation (5.25) admits a third order exponential splitting of its solutions, the Bohl spectrum of this equation admitting an a-separation. In this case Theorem 5.1 remains valid at least when the phase space ~ is a Hilbert space .p.3) In fact, by virtue of the results of § IV.6 the equation can in this case be reduced by means of a kinematic similarity transformation to a form analogous to (5.8), with variable operators AI(I) and AzCt). The second of these equations turns out to be e-dichotomic, a Green function exists for it, and therefore all of the subsequent calculations will be valid. 3)
We do not know the extent to which this assumption is essential.
6.
315
BOUNDED INTEGRAL MANIFOLDS
REMARK 5.3. In the applications one frequently encounters the equation dx/dt
=
Ax
+ sF(t, x)
with a small parameter s and a bounded function F(t, x) satisfying a Lipschitz condition. The results presented above are applicable to equations of this type since for any M > 0 and q > 0, no matter how small, the condition F(t, x) E (M, q) will be satisfied if s is sufficiently small. An analogous remark can be made in regard to equations of the form dx/dt
=
Ax
+ sF(t, x) + Fl(t, x),
where Fl(t, x) is quasilinear and satisfies condition (5.24).
§ 6. Integral manifolds bounded for t
---+
+
00
(or t
---+ -
(0)
In this section we consider the integral manifolds of equation (5.1) that are composed of trajectories bounded for t tending to infinity in one direction only. The integral manifold of the differential equation (5.1) defined by the equation will be called a (p, r;, + )-manifold if there exists a number t such that when t > t the function 1>(t, Xl) is of class L~ and satisfies the inequality 111>(t, Xl) I ~ p. The (p, r;, - )-manifolds composed of trajectories with the component X2 bounded on a negative halfline are defined analogously. In order to construct these manifolds we consider in more detail the separation of the spectrum of the operator A. Let 0'2. +(A) and 0'2. _(A) be the spectral sets of A consisting respectively of the points lying to the right of the straight line Re A = a and to the left of the straight line Re A = - a. Thus 0'2(A)
=
0'2. +(A) U0'2. _(A).
We again consider a principal Green function G2(t) of the operator A2 = P2A in the subspace 18 2• By carrying out arguments analogous to those that led us to equation (5.14) and using Theorem 4.3, we can show that the (p, r;, + )-manifolds are described by functions X2
=
1>(t, XI. y)
satisfying the integral equation
1>(1:, XI. y) = eA(~-lo)y +
(6.1) 00
J Gz('1: -
t) F2(t, Wet,
The function Xl
= W(t,
'1:,
xll1» + 1>(t, Wet, '1:, Xl 11», y))dt (to
10
'1:,
xlOl1»
~ '1:
<(0).
here is the solution of the differential equation
VII.
316
NONLINEAR EQUATIONS
= AlXl + Fl{t, Xl + ¢(t, Xb y») which satisfies the condition Xl(Z) = XIO'
(6.2)
dXl/dt
LEMMA
6.1. Suppose
11¢{t, xi2), y(2») - ¢(t, xil), y(1) II ~ r;11 x?) - x?) II
+ Ke- vCHo ) Ily(2) - yCl) II
(6.3)
a>).i > 0).
(xil),xi 2) E){3l; yCl\y(2) E){32,-; t ~ to; a -
= ¢(t, Xb yCl)) and ¢(2) = ¢(t, Xby(2»
Then/or ¢(l)
IIW{t, -r, x?) j¢C2») - W{t, -r, x?) 1¢(1)) II
~
No { Ilx?) - xiI) Ile Ca - H !3.) It-rl
+ PROOF.
(6.4)
Kq Il y (2) - y(1) II [e-vCr-to)+Ca-Hfl,) It-rl a - a + ).i + f3q
_
e-VCt-to)]}.
We have
/ieCa-Hfl)lt-sls~pll¢{s,xbyC2» ~ ! ~
-
- ¢(S,Xl,yCl))II/dS
K\
e Ca - H !3,)It-sI-vCs-to) ds/ll y (2) - yCl) II
a -
a + ).i + f3q
K
{e-vCr-to) +Ca-o+fl.) It-rl
_
e-v(t-to)} Ily(2) _ yCl) II.
Substituting this result into inequality (5.17), we obtain the required estimate. Let Sf(p, R, '1}, K, ).i, i) denote the space of functions ¢(t, Xb y) (t E [to, 00); Ily II ~ R; Xl E ~h), satisfying condition (6.3) and the condition
11¢(t, Xb y) II ~ p
(t ~ i > to)
(6.5)
and with the norm
We will investigate in this space the operator S: ¢
-t
~ defined by the equality
(6.6) co
= eACr-to)y + LEMMA
).i <
J G2(-r to
t)F2{t, W(t, -r, xll¢)
'1} are arbitrary positive numbers, K > No and There exist numbers qo > 0, Mo > 0 and tl > to having the property
6.2. Suppose R, p and
a-a.
+ ¢(t, W(t, -r, xll¢), y))dt.
317
6. BOUNDED INTEGRAL MANIFOLDS
that the operator t> tl.
S acts in the space step, R,
'1),
K,
].i,
t) when q ;;:; qo, M ;;:; Mo and
PROOF. Inequalities (5.2) and (5.17) imply the inequality
11q5(r, x,y) I ;;:; Noe-(a-o)(t-lo)R + a ~ 0 which shows that when M
(zo
~
i;
IIYII ;;:; R; Xl E 581),
> 0 is sufficiently small and i is sufficiently large
III~III
=
_sup
<~I;x,;lIyll~R
11~(zo, XhY) I
;;:; p.
Let us verify the preservation of condition (6.3) .We have
xiI), y(1) I ;;:; Ile (<-I )(y<2) - y(1) I + SIIGz(zo - t)IIIIF2(t, W(t, zo, xi2)1¢(2» + ¢(t, W(t, zo, xi 2)1¢2), y(2») - F2(t, W(t, ZO, xiI) I¢(I) + ¢(t, W(t, zo, xiI) I¢(I», y(I)) I dt ;;:; Noe-(a-o)(<-Io) Ily(2) - y(I) I + qNo Se-(aH) 1<-IIIIW(t, zo, x?) I¢(Z» - W(t, zo, x?) I¢(I» I
11~(zo, xi2l, y(2» - ~(zo,
A
o
10
co
10
11¢(t, W(t, zo, xi2)/¢<2», y(2» - ¢(t, W(t, zo, xiI) I¢(I», y(I) I dt ;;:; Noe-(a-o) Ily(2) - y(I) I + q(l + '1)No Se-(aH) 1<-IIIIW(t, ZO, x{Z) 1¢(2) - W(t, zo, x?) 1¢(1) I dt +
(c-1o)
co
10 co
+ qNoK Se-(aH)I<-lle-v(I-lo) dt·
Ily(2) - y(I)II·
10
Estimating the second summand with the use of Lemma 6.1 and calculating the integrals in this inequality, we obtain the estimate 11~(t,
X(2), y(2) -
;;:; 2q(l
+ '1)N~
~(t,
x?), y(1)
I
I XI(2) - xI I + (1)
~-~
lIy(2) - y(I) I {No
+
2( + 0) + qNoK ~--,--a~-,----cc~+~-~
2Kq2(1 + '1)NJ }e-V(I-Io). (n - (3 + ].i + (3q) (20 - (3q)
It follows from this estimate that the function ~ satisfies condition (6.3) when q > 0 is sufficiently smalL The lemma is proved. THEOREM 6.1. Suppose the spectrum (l(A) admits an a-separation. For any positive constants '1), p, R, ].i < ex and sufficiently large K there exist positive numbers I, M and q depending only on these constants and the operator A such that if F(t, x) E
VII.
318
NONLINEAR EQUATIONS
(M, q, +), there corresponds to each y E 18 2• _ (IIYII ;;:; R) one and only one (p, 7J)manifold 9Jl:'r; (y) of the differential equation (6.1), which is defined by the equation X2
¢(t, XI. y)
=
(Xl
E
181 ; y
E
182.-; X2
E
182 ; to ;;:; t < (0),
where the function ¢ satisfies the conditions 1) P2.-¢(tO,
2)
XI.
y)
=
y;
11¢(t, Xl, y)11 ;;:; pfor It -
tol ~ I,
IIYII ;;:; R;
(6.7)
In particular, all of the manifolds 9Jl:'iy) exponentially approach each other as t
~
+
00.
The manifold 9Jl:' iY) consists of all of the trajectories of the differential equation (6.1) that satisfy the conditions (t - to ~ I). PROOF. It follows from Lemma 6.2 that the operator S acts in the space !£'(p, R, r;, K, ).I, to + I) when M> 0 is sufficiently small and I> 0 is sufficiently large.
Let us show that it is a contraction in this space. Using Lemma 5.2, we get that
OJ
;;:; No
S e-(a+O)i T- ti IIF1(t, W(t, ZO,X1!¢C2») + ¢(2)(t, W(t, zo, x11¢(2»), y))
(6.8)
t,
OJ
;;:; Noq Se- (a+O) iT- t i(R1 + R 2)dt, t,
where R1
W(t, zo, X1!¢C1») I e(a-o+(J,) it-Ti ;;:; q a + (3q ~~~ 11¢(2)(s, XI. y) - ¢(1)(s, Xl, y) II, =
IIW(t, zo, X1!¢C2») -
a
R2
y)11 ¢(1)(t, W(t, zo, X11¢(2»), y)11
=
1I¢(2)(t, W(t, zo, X1!¢C2»), y) - ¢(1)(t, W(t, zo, X11¢(1»),
~
IlqI(2)(t, W(t, zo, X11¢(2»), y) +
;;:;
11¢(1)(t, Wet, zo, X1!¢C2»), y) - ¢(1)(t, W(t, zo,x11¢(1)), y)11 e(a-o+fJ,) it-Ti } SUpl _ !¢C2)(S, XI. y) - ¢(l)(s, XI. y) I { 1 + qr; a - + (3q .
a
Substituting these estimates into (6.8) and integrating, we obtain the required result for sufficiently small q > 0 in exactly the same way as in the proof of Theorem 5.1.
7.
319
AVERAGING PRINCIPLE
The proof of the last assertion of the theorem is carried out in exactly the same way as in Theorem 5.1. § 7. Averaging principle 1. A veraging principle for a finite interval. We consider in a Banach space )8 the differential equation
(7.1)
dx/dt = cf(t, x)
containing a small parameter c. We assume that on the set [0, (0) x B, where B is a ball in )8, the functionf(t, x) is bounded, continuous in t and satisfies the Lipschitz condition (XI. X2 E
B)
(7.2)
with a constant not depending on t. We assume further that the mean value off(t, x) with respect to t: fo(x) = lim T-'too
1
T
T
S f(t, x)dt
(7.3)
0
exists for each x E B. Under this assumption it is expedient to consider along with equation (7.1) the equation (7.4)
dx/dt = cfo(x).
It follows from (7.3) that the functionfo(x) also satisfies condition (7.2) in B.
Let y(t) (0;:;:; t ;:;:; To) be a vector function taking values in B that satisfies the equation dy
dt =fo(y)
(7.5)
Then, as can easily be seen, the function xo(t; c) = y(ct) will be a solution of equation (7.4) on the interval t ;:;:; To/c. We will establish below a connection between xo(t; c) and the solution x(t; c) of equation (7.1) that satisfies the condition
°;:;:;
x(o; c) = yeO) (= xo(O; c)).
(7.6)
More precisely, we will prove the following assertion. THEOREM 7.1. Suppose that on the set [0, (0) x B the function f(t, x) is bounded, continuous in t and satisfies the Lipschitz condition (7.2). Suppose further that its mean value (7.3) exists for each x E B. If yet) is a solution of equation (7.5) that takes the interval [0, To] into B, there exists for any r; > an co > such that when < c < co the solution x(t; c) of equation (7.1) satisfying condition (7.6) is subject to the estimate
°
°
°
320
VII.
NONLINEAR EQUATIONS
Ilx(t; e) - y(et) II ~
(0
1)
~
t
~
Tole).
Theorem 7.1 is obtained as a corollary of a general theorem on the continuous dependence of solutions of differential equations on a parameter. To this end we consider the more general (than (7.1») equation dx d7: = /(7:, x; e)
(0
~
7:
~
To; 0
~
e
~
eo)
(7.7)
with a function/(7:, x; e) satisfying the Lipschitz condition
xI/!
II/(7:,X2; e) - /(7:, Xl; e) II ~ CllX2 -
(7.8)
and the condition '!O
To
0
0
lim J/(7:, x; e)d7: = c~o
J/(7:, x; O)d7:
(7.9)
which is called the condition of integral continuity at e = O. Equation (7.7) can be converted into (7.1) by putting /(7:, x; e) = {ffc«7: 1)0$, x) e ~ 00, ox, e- ,
and making the change of variable 7: = et. Condition (7.9) is satisfied in this case since it takes the form lim
) J/ (7:-, X d7: = 7:ofo(x) e . T,
c~o 0
or, equivalently, lim -
e
c~o 7:0
T,1e
J0 /(t, x)dt = fo(x),
which is equivalent to (7.3). Let us prove an auxiliary assertion. LEMMA 7.1. Suppose x(7:) is a continuous/unction on [0, Tol. Then (7.8) and (7.9) imply the relation
PROOF.
T,
T,
lim
J /(7:, x(7:); e)d7: = J /(7:, x(7:); O)d7:.
e~O
0
0
From (7.9) it follows that for any 7:10 7:2 TZ
Tz
"'1
orr
lim e~O
and hence that for any 0
~
E
(7.10)
[0, Tol
J/(7:, x; e)d7: = J/(7:, x; O)d7:, 7:1 < ... <
'Z"n-l
<
'Z"n
= To,
Xk E
B (k
= 1,2,.··, n)
7. n k=l
c---)Q
n
Tot
lim 1:;
f
321
AVERAGING PRINCIPLE
Tot
f k=l
f(-r, Xk; c)d7: = 1:;
!'.t-l
f(7:, Xk; O)d7:.
1"1--1
If we introduce the step function
(7:k-1
~
7: < 7:k)
(k
= 1,2,. ··,n),
we can write the latter equality in the form To
lim
f f(7:, x(7:); c)d7:
e~O
0
To
=
f f(7:, x(7:); O)d7:.
0
This proves relation (7.10) for step functions. Consider a sequence {Xn(7:)} of step functions that converge to x(7:) uniformly on [0, To]. Using the Lipschitz condition for the functionf(7:, x; c), we obtain the estimate II f'f(7:, x(7:); c)d7: - r[(7:, x(7:); O)d7:
~
/I
To
f Ilf(7:,x(7:);c) o
-f(7:,Xn(7:);c)lld7:
To
+f
o
Ilf(7:,X(7:); 0) -f(7:,Xn(7:);O)lld7:
~ 2To sup·IIX(7:) - Xn(7:) II
+I
['[f(7:, Xn(7:); c) - f(7:, xi7:); 0)] d7: II,
which immediately implies the required assertion. We will now prove the theorem on continuous dependence on a parameter in the form needed for our purposes. THEOREM 7.2. Suppose that for 7: E [0, To], x E Band 0 ~ c < C1 the function f(7:, x; c) is defined, bounded, piecewise continuous in 7:, integrally continuous at c = 0 and satisfies the Lipschitz condition (7.8). Suppose further that the equation
dx/d7: = f(7:, x; c)
(7.11)
has a solution x(7:; 0) for c = 0 that satisfies the condition x(7:; 0) E B (0 ~ 7: ~ To). Then for any 1) > 0 there exists an co (0 < co < c1) such that every solution x(7:; c) of equation (7.11) for 0 < c < co that is defined on the interval [0, To] and satisfies the condition x(O; c) = x(O; 0) (= xo) is subject to the estimate
VII.
322
NONLINEAR EQUATIONS
(7.12)
Ilx(-r; c) - x(r; 0) II < r; PROOF.
The functions x(r; c) and x(r; 0) satisfy the integral equations T
x(r; c) = Xo
+ Jf(s,
x(r; 0) = Xo
+ Jf(s, x(s; 0); O)ds.
o
x(s; c); c)ds
and <
o
Using the Lipschitz condition (7.8), we obtain the estimate
Ilx(r; c) - x(r; 0)11 = II
~
I
[f(s, x(s; c); c) - f(s, x(s; 0); O)]ds II
<
J Ilf(s, x(s; c); c)
o
- f(s, x(s; 0); c) lids
I
+ II [f(s, x(s; 0); c) - f(s, x(s; 0); O)]ds II ~ c
<
J Ilx(s; c) -
o
x(s; 0) lids
+ ¢(c),
where
¢(c) = II T[f(S, x(s; 0); c) - f(s, x(s; 0); O)]ds II· On the basis of Corollary III.2.1 we can now conclude that
Ilx(r;c) - x(r;O)11 ~ ec<¢(c), and it remains only to note that lime~o ¢(c) = 0 by virtue of the integral continuity condition (7.9) and Lemma 7.1. The theorem is proved. Theorem 7.1 immediately follows from what has been proved, as follows from the arguments given at the beginning of the present subsection. 2. Averaging I?rinciplefor the real line. We can obtain more complete results if we make additional assumptions on the right side of the averaged equation. Suppose again 1
fo(x) = lim T T-)oo
T
J0 f(t, x) dt.
Let us assume that there exists a point Xo
E ~
fo(xo) = O.
at which (7.13)
7.
323
AVERAGING PRINCIPLE
In this case the differential equation dx/dt = cfo(x)
(7.14)
has the stationary solution xo(t) == Xo. We will see that under certain assumptions the equation dx/dt = cf(t, x)
(7.15)
has for sufficiently small c > 0 a bounded solution on the real line that does not leave a small neighborhood of the point Xo. We require that the following conditions be satisfied. (C l ) The functionf(t, x) is defined and bounded for t E ( - 00, (0) and x E Bp(xo), where Bp(xo) is the ball ofradius p > 0 with center at Xo. The limit 1
fo(x) = lim
T
t+T
Jt f(s, x)ds
T--too
(7.16)
exists uniformly in both t E ( - 00, (0) and x E Bp(xo), the functions f;(t, x) and fo(x) being bounded for t E ( - 00, 00) and x E Bp(xo). (C 2) The vector functions f(t, x) and fo(x) have in Bp(xo) continuous bounded derivatives with respect to x of first and second order. (C3) Relation (7.16) can be termwise differentiated twice, i.e. the relation fJk)(X) = lim T~oo
1
T
t+T
Sf~kl(s, x)ds
(k
t
=
1,2)
(7.17)
holds uniformly in t and x. We recall that, by definition, for each x E Bp(xo) the expressionfo(x) represents an operator from [~] satisfying the condition fo(x
+ h)
+ R(x, h)
- fo(x) = fo(x)h
(x, x
+ hE Bp),
(7.18)
where IIR(x, h) II =
Let A
= fo(xo)
and S(h)
= R(xo, h).
fo(xo
(J
(7.19)
Then by virtue of (7.13) and (7.18)
+ h) =
We note that in the ball II h II ~ condition
o(llhll)·
Ah
+ S(h).
(7.20)
the vector function S(h) satisfies the Lipschitz «7.21)
where lim II~O
elI =
O.
(7.22)
VII.
NONLINEAR EQUATIONS
S'(h)
= fo(xo + h) - fo(xo),
324
For (7.20) implies (7.23)
and by virtue of the continuity offo the quantity C"
= sup
iifo(xo
IihIlC"
+ h)
(7.24)
- fo(xo)ii
satisfies condition (7.22). Inasmuch as 1
S(h2) - S(hI) =
fS'C hI + S(h2 -
hI) )(h2 - hI)ds,
o
we obtain (7.21). In order to investigate equation (7.15) we will transform it into a form permitting an application of the results of § 4. We first rewrite it with the use of (7.20) in the form dh/dt
= cAh + cg(t, h),
(7.25)
where h = x - Xo and g(t, h)
= f(t, Xo + h) - fo(xo + h) + S(h).
In equation (7.25) we will make the change of variable h = z - w(t, z; c),
(7.26)
where co
u(t, z; c)
f V(s + t, z)e-esds
=
(7.27)
(c> 0)
o
is the Laplace transform of the function V(t, z)
= f(t, Xo + z) - fo(xo + z).
(7.28)
We will use the following well-known proposition to show that the transformation (7.26) has an inverse. 7.2. Suppose a function ifJ(s; J.) (0 ~ s < Cf) taking values in a Banach space and depending on an abstract parameter J. E A satisfies the condition
LEMMA ~
(M =) sup{11
+1
ifJ(s; J.)ds
I I t ~ 0; J.
and, uniformly in J. E A, the condition
lim ~
T~oo T
Then the relation
JifJ(s; J.)ds = O.
0
E
A} <
00
7.
325
AVERAGING PRINCIPLE 00
holds uniformly in ). PROOF.
p
E
lim p
f ¢(s; ).)e-Psds =
P~O
0
0
A.
Integrating by parts, we obtain the relation
J
e-Ps¢(s; )')ds
=p
r
e-psd[
f ¢(t; )')dt ]
0 0 0
=p
~
{ (se- PS )
= p2
J¢(t; )')dt \: + p I[ 1¢(t; )')dt ] e-Psds }
(7.29)
I[1
¢(t; )')dt ] e-psds.
For any 0 > 0 there exists aT> 0 such that when s
I
+1
¢(s; )')ds /I
~
T
~ o.
Therefore (7.29) implies the estimate
k
[¢(s; ).)e-Psds
= I p2 { Is [ ~
I
+I
M [1 - e-PT(I
¢(t; )')dl ] e-Psds
+ Jse- PS [
~
I
¢(t; )')dt
J
ds
}II
+ pT)] + oe-PT(l + pT),
which by virtue of the arbitrariness of 0 > 0 proves the required assertion. LEMMA
7.3. The following relations hold uniformly in 1 E ( - 00,
00)
and Z E Bp:
lim eU(t, z; e) = 0,
(7.30)
lim eu;(t, z; e) = O.
(7.31)
dO dO
For by virtue of (7.16) (7.17) and (7.28) the bounded functions V(s + I, z) and V;(s + t, z) satisfy the conditions of Lemma 7.2, the role of the parameter). being played by the pair (I, z). From (7.31) it follows that when 0 is sufficiently small the operator 1 -eu;(/, z; e) (t E ( - 00, (0), Z E Bp) has a bounded inverse, and hence the mapping (7.26) is invertible. In addition, from (7.30) it follows that for any preas0 and an r; 0 such that < Po when (0 <) signed po E (0, p] there exist an < and = 1). Thus the change of variable (7.26) can be carried out in a sufficiently small neighborhood of the origin and for sufficiently small e > O.
e>
e eo
Ilzll
eo>
>
Ilhll
326
VII.
NONLINEAR EQUATIONS
Representing the function u in the form 00
u(t, z; c)
=
eel
JV(s, z)e.-esds,
I
we find that
uf(t, z; c) = cu(t, z; c) - V(t, z),
(7.32)
and consequently
dh
dt
=
dz
,dz
z
V
dt - cU. dt - c u + c .
(7.33)
Using relations (7.26), (7.28) and (7.33), we reduce equation (7.25) to the form
(/ - cu;)dzJdt = c[f(t, Xo + h; c) - f(t, Xo + z; c)]
+ c/o(xo + z) + czu(t, z; c) = c(Az + S(z») + cZu(t, z; c) + c[f(t, Xo + z - cU, c) - f(t, Xo + z; c)] or, after multiplying by the bounded operator (/ - cU;)-l and introducing the "slow" time 7: = ct as the independent variable, to the form
dzJdt
= Az + F(z, 7:; c),
(7.34)
where, as is easily seen, F(z, 7: ; c) has a bounded first derivative F: (z, 7: ; c) and satisfies the relation
F(z, 7:; c) - S(z)
=
O(c).
uniformly in 7: E ( - 00, (0) and z E Bp. Hence for any M > 0 and f.J. > 0 there exist an co and a c < co and Ilzll < /30 we have
IIF(z, 7:; c) II ~ M
/30 such that
when (0 <) (7.35)
and (7.36) Suppose now that the spectrum of the operator A = fo(xo) does not intersect the imaginary axis~ By virtue of the results of § 4 we can conclude that there exists for sufficiently small po > 0 an co > 0 such that when (0 <) c < co equation (7.34) has a unique solution zo(7:) (- 00 < 7: < (0) whose trajectory lies entirely in the ball Ilz(7:) II ~ po. This solution is almost periodic if the function F(z, 7:; c) is almost periodic. Moreover, there exists an initial manifold 1JJ1~ with a one-to-one projection onto a neighborhood of the origin in the subspace)8_ = P -)8 (P _ is a spectral projection of the operator A) such that when z(7:o) E 1JJ1~ the corresponding solution z(7:) of equation (7.34) satisfies the condition
327
EXERCISES
(7.37) All other solutions leave any ball of radius less than pa as 7: --+ 00. If we return to the solutions of equation (7.15) by means of the inverse of the transformation (7.26), we obtain the following result. THEOREM 7.3. Suppose conditions (Cl ), (Cz), (C3) and (7.13) are satisfied and the spectrum of the operator A = fO(xa) does not intersect the imaginary axis. Then for sufficiently small pa > 0 there exists an ea> 0 such that when (0 <) e < ea equation (7.15) has a unique solution xa(t) satisfying the condition
supllxa(t) - Xall < pa· t
There exists an initial manifold WL (with a one-to-one projection onto a neighborhood of the point P -Xa in the subspace )8- = P _ )8)4) such that when x(ta) E WL IIX(t) - xa(t) II ~ ce-v(t-t,) Ilx(ta) - xa(ta) I
(t ~ ta).
(7.38)
If x(ta) E 9JL,
the solution x(t) leaves any ball Bp(xa) with p < pa as t --+ 00. The solution xa(t) is almost periodic (periodic) if the functions f(t, z) and f~(t, z) are almost periodic (periodic). REMARK 7.1. If the whole spectrum of the operator A lies in the interior of the left halfplane, the manifold 9JL coincides with a neighborhood of the point Xa in the space )8. Formula (7.38) shows that in this case the solution xa(t) of equation (7.15) is uniformly and asymptotically stable. If part of the spectrum (I(A) (but not all of it) is located in the right halfplane, this solution is only conditionally stable.
EXERCISES We cite certain more precise existence and uniqueness theorems for equation (l.I). The following propositions 1-3 were proved by M. A. Krasnosel'skii and S. G. Krein [2]. l. Suppose the/unction/(t, x) is representable in the/orm
/(t, x) = fr(t, x)
+ fz(t, x)
(11x - xoll
~
r;
It - tol
~
a),
where fi(t, x) is completely continuous andfz(t, x) satisfies the condition
IIfz(t, x) - fz(t,y)1I 1/ 0
~
K(t)llx - yll.
< h ~ a is such that to+h
f K(t)dt< 1
to-h
and
sup II fi(t x)1I h{ 11%-XoI~'" It-tol~a
+
II.~~,PII~_'" • • =.
lIfz(t, x)11 }~ r,
It-tol~a
') The property indicated in parentheses ofthe manifold WLgenerally requires further explanation.
328
VII.
NONLINEAR EQUATIONS
there exists a solution of equation (1.1) that is defined on the segment [to - h, to + h] and satisfies the initial condition x(to) = xo. 2. Suppose Ilf(t, x) II ~ L(I/i(x»¢(t), where I/i(x) is a nonnegative functional on ~ satisfying a Lipschitz condition and such that limllxll_=I/i(X) =00, L(u) is continuous, nonnegative and such that f=[L(u)]-ldu = 00, and ¢(t) is nonnegative and integrable on each finite interval. Then every integral curve of equation (1.1) can be extended indefinitely far to the right. 3. Suppose x = XI(t) and x = X2(t) are two solutions of equation (1.1) satisfying one and the same initial condition x(to) = Xo and lying in the ballllx - xoll ~ r. Suppose IIf(t, x) - f(t, y)1I
~
L(I/i(y - xȢ(t)
with fo[L(u)-I]du = 00. Then these solutions coincide: XI(t) = X2(t). 4. Consider in the finite-dimensional Euclidean phase space Rn the autonomous equation
(0.1)
dx/dt =f(x)
with a continuously differentiable functionf(x). This equation induces on the differentiable functions in Rn with compact support a linear (unbounded) operator Arp(x) =
:r rp(x(t» Lo'
where x(t) is the solution of equation (0.1) that is initially (when t = 0) at the point x. a) Show that the operator iA is symmetric') in the Hilbert space .!i'2(Rn) precisely when (0.2)
trf'(x) = O.
(We recall that when x is fixed the derivativef'(x) is a linear operator in Rn.) Hint. Make use of the Gauss-Ostrogradskii formula. b) The following nontrivial test for the indefinite extendability of solutions is due to A. Ja. Povzner [1]. In order for a system (0.1) satisfying condition (0.2) to have a solution in the large for t> 0 (t <0) (in the sense that each point xo, except for the points of a set of zero measure, does not go to infinity within a finite period of time) it is necessary and sufficient that the deficiency index in the upper (lower) halfplane of the closure of the operator iA be equal to zero. The cited paper of A. Ja. Povzner also contains a number of generalizations of this result. 5. Extend Theorem 1.3 to the case of the differential equation (1.19):
~~
= f(t, x)
+ g(t, x)
(O~
t< 00; IIxll
(0.3)
in which the functionsf(t, x) andg(t, x) respectively satisfy the Lipschitz condition IIf(t, xz) - f(t, XI) II
and the condition
~
L(t)lIx2 - xdl,
(0.4)
~
Ilg(t, x) II
~
(0.5)
q(t)lIxll
with integrally bounded L(t) and q(t): 1
sup <
'r
t+r
f L(s)ds
t
<
00;
1
t+r
'r
t
sup t
f q(s)ds
<
00
(0.6)
(see V. E. Germaidze [1]). 6. Extend to nonlinear equations the test for negativeness of the Bohl exponent contained in Theorem III.6.1. ') I.e. (iArp, ¢)=(rp, iA¢) for any pair of differentiable functions
rp and ¢ with compact support.
329
EXERCISES
7. We will say that the differential equation dxfdt
= I(t, x)
(0.7)
(0 ~ t
has property O,(N, rp), where rp(t) (0 ~ t < 00) is a given continuous function (B. A. Scerbakov [1]), if every solution x(t) of this equation that is in the ball of radius c 0 at some initial moment to ~ 0 (lIx(to)II c) is subject for all t ~ to for which it exists to the estimate
>
<
Ilx(t)lI~ N Ilx(to)lIrp(t -
(0.8)
to).
a) Show that equation (0.7) has a negative Bohl exponent at zero if it has property O,(N, rp) for a continuous rp(t) tending to zero at infinity: Iimt_= rp(t) = O. Hint. Make use of the result of the preceding exercise. b) Suppose equation (0.7) satisfies condition (0.4) with a bounded L(t) and has property O,(N,rp) for some c 0, some 0 and rp(t) -> 0 as t -> 00. Suppose, further, ¢(t) (0 ~ t 00) is a continuous bounded function satisfying the conditions 1) rp(t)l¢(t) -> 0 as t -> 00,
>
N>
2) rp(t) ~ ¢(t) (0 ~ t 3) ¢(tl ) ¢(tz) ~ ¢(tl
<
< 00),
+ tz) (0 ~ t l ; tz < 00).
Show (B. A. Scerbakov [1]) that for any M> N there exist numbers 0 for any functiong(t, x) satisfying the condition
Ilg(t, x) II
~
qllxll
(11xll
> 0 and q > 0 such that
< r; 0 ~ t < 00),
equation (0.3) has property Oo(M, ¢). c) Extend the latter result to the case of integrally bounded L(t) and q(t) (see (0.6». 8. Show that the origin of the two-dimensional phase space of the nonlinear system
{ d~ddt= -a~1 (l12
(0.9)
is unstable (0. Perron [2], page 706), whereas a system of first approximation to it is stable (see § 111.4.5). Compare this result with Theorem 3.2. Hint. Verify that system (0.9) has the general solution
= C1e-
~I {
al ,
~z = eCsin In I-Za)1
(Cz + Cr
and investigate the values of ~ = {~klI at t = exp (2n 9. Consider the quasilinear equation dxfdt
= A(t)x + F(t, x)
ie'
sin In
+ t)n-
(F(t, x)
'd" ),
(n = 1,2,.··).
= 0 (1Ixll»
(0.10)
with a bounded operator function A(t), and along with it the equation dyfdt
= A(t)y + F(t, y) + $(t, y),
(0.11)
<
<
where $(t, y) is a bounded continuous function: 1I$(t, y)11 ~ M (0 ~ t 00; IIxll r). a) Show (P. Bohl [2]) that the equation x = A(t)x has a negative upper Bohl exponent precisely 0 such that for M fJ. any solutions x(t) and y(t) when there exist numbers II 0, 0 0 and k of equations (0.10) and (0.11) that are initially at one and the same point of the ball Bp = (xlllxll oj : x(O) = y(O) E B p, satisfy on their whole interval of existence the inequality Ilx(t) - y(t) II kfJ.. b) Extend this result to the case of integrally bounded A(t) and $(t, y). Bohl's result implies that equation (0.10) with a negative upper Bohl exponent at zero has the following property.
>
>
>
<
<
<
VII.
330
NONLINEAR EQUATIONS
>
>
<
> <
<
For any e 0 there exist numbers fJ. 0 and jj 0 such that when Ilxoll jj and M fJ. the solutions of equation (0.11) satisfy the estimate I y(t) If e. This property was subsequently rediscovered under the name, stability of the zero solution of equation (0.10) under constantly acting perturbations (see I. G. Malkin [5]). For a generalization of the property of stability under constantly acting perturbations see V. E. Germaidze and N. N. Krasovskii [1). 10. Prove the following theorem (D. L. Kucer [2]). Suppose the equation x = A(t)x has property 86'(v o, No) and IIF(t, x) II
with SO' f P(t)dt
<
00
for some p
(II xii
~f(t)llxll
~
r;
t ~
0),
> 1. Then for any e > 0 there exists an N> 0 for which the equation dx/dt
= A(t)x + F(t, x)
(0.12)
has property 88(vo ~ e, N). If p = 1, the assertion is true for e = O. 11. Prove the following theorem (D. L. Kucer [2]). Suppose the equation condition sup
O~'t'~t
I U(t, ,,)11
>
x=
A(t)x satisfies the
00
and IIF(t, x)11 ~f(t)llxll (11xll ~ r; t ~ 0),
where
fo fP(t)dt <
00.
N>
Then the solutions of equation (0.12) are subject to the estimate IIx(t) I ~ Nllx(to)11 for some O. 12. Prove the following theorem (D. L. Kucer [2]). Let m 1 and p 1. In order for the zero solution of equation (0.12) to be uniformly stable for any F(t, x) satisfying the condition
>
IIF(t, x)11
~f(t)llxlim
(11xll
~
r; t
>
~
0),
<
where SO' fP(t)dt 00, it is necessary and sufficient that the Bohl exponent of the equation x = A(t)x be negative. 13. Prove a theorem analogous to the preceding one, assuming m 1, p = 1 and replacing the requirement that the Bohl exponent be negative by the requirement sup 0,,;,";'<00 I U(t, ,,)11 00. 14. Consider in a Hilbert phase space H the differential equation
>
dx/dt
= f(t, x)
(~oo<
<
t
(0.13)
where f(t, x) is a continuous function of t and a continuously differentiable function of x for ~oo
~
aI
(a =const
> 0; t E
[~oo,
00);
x E.)5)
equation (0.13) has the convergence property, i.e. it has a unique solution x(t) that is bounded on the real line and any other solution of this equation is indefinitely extendable to the right and exponentially approaches x(t). We note that the indicated conditions are satisfied iff(t, x) = f(t) + g(x), wheref(t) is a bounded function and Re g'(x) «0 uniformly in x. Hint. Show that there exists an R 0 such that every solution of equation (0.13) ends up in the R as t--> 00. Show that any two solutions Xj(t) and X2(t) that are in this ball at some ball IIxll moment to are subject to the estimate
<
>
331
EXERCISES (t~
-r).
(0.14)
To prove the existence of a solution bounded on the real line consider the sequence (xit») of solutions satisfying the conditions xi - k) = 0 (k = 1,2,.··) and show, using (0.14), that (xiO») is a Cauchy sequence in !P. The desired solution x(t) is then determined by the condition x(O) = limk_ooxiO).
15. Let A be an operator in a finite-dimensional space whose spectrum does not intersect the imaginary axis, let f(x) be a function satisfying the Lipschitz condition with a sufficiently small constant in a neighborhood of the origin and suppose f(O) = O. Then there exists in a neighborhood of the origin a homeomorphism y = r[J(x) that transforms the solutions of the equation dx/dt
= Ax + f(x)
(0.15)
into the solutions of the equation dx/dt = Ax. (D. M. Grobman; see the book [1] by B. F. Bylov et al.) Hint. Extend the functionf(x) to the whole space with preservation of the norm supremum and Lipschitz constant in such a way that it vanishes outside some sphere. If Xo EO !P, consider the solution x(xo, t) of equation (0.15) satisfying the condition x(xo, 0) = Xo and put (see (4.8» r[J(xo)
16. Suppose the equation dx/dt tion A(t) is almost periodic. Consider the equation
= Xo -
L GA( --r)f(x(xo, -r»d-r.
= A(t)x is e-dichotomic on the real line and the operator funcdx/dt
= A(t)x + F(t, x),
where F(t, x) satisfies the same conditions as in Theorem 4.2. Generalize this theorem by proving the existence of an almost periodic solution of this equation. Show that the frequency module of this solution is contained in the minimal module containing the spectra of the almost periodic functions A(t) and F(t, x) (assuming here that the module of F(t, x) does not depend on x). Hint. Make use of the results of § 4.2 and Exercise ILl 1. 17. a) Generalize Theorems 4.1, 4.3, 5.1 and 6.1, replacing the condition F(t, x) EO (M, q) by the conditions IIF(t, x) II ~ M(t),
IIF(t, X(2) - F(t, xCI)11 ~ q(t)lIxC2 ) - xU)11
(0.16)
with integrally bounded functions M(t) and q(t). b) Show that if conditions (0.16) are satisfied with
-I
M(t)dt
= f1.
and
Lq(t)dt = k < co,
and, in addition, sup,lIeA 'P I IIO and k>O. It can be assumed here that conditions (0.16) are satisfied only in some ball. " An analogous assertion holds with respect to Theorem 6.1. c) Extend this result to the case when 00
f tnM(t)dtco. 18. Consider in the two-dimensional phase space R2 the autonomous equation dx/dt = f(x). Assume that it has a periodic solution x(t) = x(t + T). The following test is due to Poincare. The solution x(t) is orbitally asymptotically stable whenever
332
VII.
NONLINEAR EQUATIONS T
f tr f'(x(/»dl
o
<00.
(0.17)
Hint. Derive, by making use of Jacobi's well-known formula for the determinant of a fundamental matrix of a system of differential equations, the formula
tr In U(T)
= ~ [
trf'(x(/»dl,
(0.18)
where U(T) is the monodromy operator for the variational equation corresponding to the solution X(/). From this formula it follows by virtue of (0.17) that the only eigenvalue of the monodromy operator that is different from A = 1 lies in the interior of the unit disk.
We note that in the multidimensional (and infinite-dimensional) case: n>2, when the spectrum of the monodromy operator for the variational equation contains points lying in both the interior and exterior of the unit disk, some trajectories will approach a periodic motion for 1->00 and some, for 1->-00. In this connection there can arise a complicated phenomenon discovered by H. Poincare, the so-called homoclinic motion: a trajectory that is asymptotic to a periodic motion for both 1->00 and 1->-00. A deep study of this phenomenon in the multidimensional case has recently been made by Ju. I. Neimark in the paper [1] (which contains a bibliography of the literature on this question). NOTES The existence and uniqueness theorems of § 1 are contained in various textbooks and have been well known for a long time. The result of Theorem 1.3, viz. the stability of the Bohl exponent of a nonlinear equation under nonlinear perturbations, is essentially contained in the works of Bohl (see the Notes to Chapter III and Exercise 9). The proof given by us is based on a method applied by B. A. Scerbakov [1] in a more general situation (see Exercise 5 and the comments to it). Theorems 2.1 and 3.1 (see M. G. Krein [2] and the "Lectures") constitute more special (but somewhat more exact) results. The method employed in Theorem 3.1 makes use of integral inequalities (a similar method was essentially used by Bohl; see the Notes to Chapter III). The method of proof of Theorem 2.1 is a generalization of a method of Ljapunov, as is also the method of proof of Theorem 2.2 (see the "Lectures"), which generalizes another well-known theorem of Ljapunov, viz. his instability theorem. In the theorems of §§2 and 3 mentioned above the perturbing term has first order smallness. The stability of the solutions of nonlinear equations under such an assumption on the perturbations was apparently first studied by O. Perron. The other instability theorem cited by us (Theorem 2.3) was proved by M. A. Rutman [1] in a more complicated way and under more complicated assumptions. Our proof, which is based on the same idea, was obtained by Ju. L. Daleckii after analyzing the original proof. Nonlinear perturbations oflinear e-dichotomic equations were first considered by P. Bohl (for the case when the original equation is stationary) and later by o. Perron (see the Notes to Chapter IV). Almost periodic solutions have been studied in the finite-dimensional case by many authors (G. I. Birjuk [1], N: N. Bogoljubov [1], N. N. Bogoljubov and Ju. A. Mitropol'skii [1] and B. P. Demidovic [2]). In this connection we have waived the usually imposed requirement that the almost periodicity of F(/, x) in 1 be uniform in x. Theorem 4.5 was obtained in the two-dimensional case by A. A. Andronov and A. A. Vitt [1]; a more exact finite-dimensional version due to E. A. Coddington and N. Levinson [1] (see also the book by B. P. Demidovic (3]) has been carried over in the present book to the infinite-dimensional case. The method of investigating multidimensional integral manifolds and their application to the substantiation and development of the averaging principle were proposed by N. N. Bogoljubov in his deep investigation [1]. The ideas of this investigation were used in all of his subsequent works.
NOTES
333
Various versions of results in the finite-dimensional case that are similar to those presented in §§ 5 and 6 have been given in the works of N. N. Bogoljubov and Ju. A. Mitropol'skii (N. N.
Bogoljubov [1] and N. N. Bogoljubov and Ju. A. Mitropol'skii [1]). Ju. A. Mitropol'skii and o. B. Lykova (Ju. A. Mitropol'skii [2] and Ju. A. Mitropol'skii and o. B. Lykova [1]) have carried over some of these results to the case of Hilbert space. The present account is of a more general character and in a number of places contains simpler proofs. It was obtained as a result of analyzing the above mentioned works and is due to Ju. L. Daleckii [6]. The material presented in Chapter IV has permitted us to carry over many of these results to the nonstationary case. The results presented in § 7 are a development of the well-known method of van der Pol. A theorem of the type of Theorem 7.1 was proved for systems with periodic coefficients by L. I. Mandel'stamm and N. D. Papaleksi [1], and for arbitrary systems of finite order by N. M. Krylov and N. N. Bogoljubov [1]. I. I. Gihman [1] noted that this theorem is a corollary of a special theorem on the continuous dependence of the solutions of a differential equation on a parameter. M. A. Krasnosel'skit and S. G. Krein [1] generalized this result and carried it over to the case of an equation in Hilbert space. There exists a number of other generalizations due to J. Kurzweil and Z. Vorel [1], H. A. Antosiewicz [1] and others. We have cited only the simplest version in order to avoid widening the circle of ideas considered in this book, and our presentation follows that of M. A. Krasnosel'skii and S. G. Krein. The results presented in § 7.2 concerning the estimation of the behavior of a solution over an infinite interval of time were obtained in the finite-dimensional case by N. N. Bogoljubov [1]. They were carried over to the case of Hilbert space by Z. Sircenko [1]. We have given a somewhat simpler proof. An extensive survey of the results concerning integral manifolds up to 1961 inclusively is contained in the report of N. N. Bogoljubov and Ju. A. Mitropol'skit [2], while a survey of more recent works can be found in the book of Ju. A. Mitropol'skii and O. B. Lykova [2]. For a survey of the results connected with the method of averaging see the book of Ju. A. Mitropol'skit [3].
CHAPTER
VIII
ASYMPTOTIC REPRESENTATION OF THE SOLUTIONS OF A LINEAR DIFFERENTIAL EQUATION WITH A LARGE PARAMETER In this chapter we find an asymptotic expansion of the solutions of a linear differential equation in the inverse powers of a large parameter appearing in the equation. In the simplest case, when the principal part of the coefficient of a finite-dimensional system has a spectrum consisting of eigenvalues whose multiplicity does not change over the whole interval of time, such expansions have been the object of extensive studies, beginning with the works of Birkhoff and Tamarkin. Usually these expansions are used to prove the completeness of the eigenfunctions of selfadjoint boundary problems. We do not cite such applications since they lie to the side of our main theme and, furthermore, have appeared in textbooks for some time now (see, for example, M. A. Naimark [1]).
Of greater relevance for us is the use of asymptotic methods for the approximate calculation of the monodromy operator of a periodic equation and for the determination of estimates for stability bands. Applications of this kind are partially touched upon in the examples of § 2 and in the Exercises at the end of the chapter. On the other hand, equations with a large parameter can be interpreted in an equivalent form as equations with slowly varying coefficients. In § 1 we present a method of asymptotically decomposing an equation with a large parameter in accordance with a decomposition of the spectrum of the principal part of its coefficient into nonintersecting spectral sets. In § 2, under the assumption that the spectrum of the principal part of the coefficient lies in the left halfplane, we estimate the error sustained in replacing the solution of an equation by its nth approximation. In § 3 we consider the more complicated case when, after going over to slow time, the coefficients of the equation can rapidly oscillate against a background of slowly varying amplitudes. In this case there can arise distinctive resonance phenomena analogous to parametric resonance.
§ I. Approximate decomposition of the equation I. Statement of the problem. We consider in a phase space ~ the differential equation
~~
=
AB(7:; A)¢
(0 ;2;
7:
;2; T),
(1.1)
in which A is a sufficiently large parameter. More precisely, we will assume that A varies in a domain llAo of the complex plane which lies outside the disk of radius Ao (IAI ~ Ao) and contains the ray A ~ Ao. We assume that the following representation holds:
B(7:; A)
= Bo(7:) + (l/A)B(1)(7:; A) 334
(A
E
ll)o)'
(1.2)
1.
APPROXIMATE DECOMPOSITION OF THE EQUATION
335
where the operators Bo('r) and B(l)('r; A) are continuous in '0 on the interval [0, T] and, in addition, ('0 E
[0, T]; A E il).,).
In the sequel some differentiability requirements will also be imposed on these operator functions. It will be assumed everywhere in this chapter that the spectrum O'(Bo('r)) decomposes into several spectral sets: n
O'(Bo('r))
=
U O'k('r), k=l
(1.3)
which do not intersect for any '0 E [0, T] and continuously depend on '0. The latter means that a smooth contour Fk separating a spectral set O'k('rO) from its complement in the spectrum O'(Bo('ro)) for some '00 E [0, T] has the same property for any other sufficiently close values of '0 E [0, T]. We recall that according to the formula
Pk('r) = - - 12. § (Bo('r) - p,l)-ldp, 71:1 r
(k
= 1,.· ·,n)
the spectral projections Pk('r) corresponding to the decomposition (1.3) are continuously differentiable in '0 as many times as the operator Bo('r). Equation (1.1) can be written in a different form if one introduces the new variable t = A'r, viz.
d¢/dt = [Bo('r)
+ £B(1)('rie)]¢
(£ = I/A).
(1.4)
In the case when the variable t has the meaning of time, the variable '0 = £t (for real £ > 0) is called the "slow time". The equations ofform (1.4) are called equations with slowly varying coefficients. When £ = equation (1.4) degenerates into the equation d¢ /dt = Bo(O)¢, whose evolution operator is easily calculated:
°
n
L; Pk(O)eB,(O) p. (0) (t-s). (1.5) k=l It is natural to expect that the basic structure of this expression is preserved for sufficiently small values of £ "# with the constant operators being replaced by slowly varying operators plus terms of order 0(£). We will therefore seek an approximate expression for the evolution operator of equation (1.4) in the form
U(t, s)
=
eB,(O)(t-s)
=
°
_
U(t,to;£)
n
=
L; [Pk('r) k=l
+ £Vk('r;£)]Ylt,to;£),
where the operator Yk(t, to; £) satisfies the equation
dYk(~/O; £)
=
[Pk('r)Bo('r)
+ £Ok('r; £)]Yk(t, to; £)
VIII.
336
EQUATION WITH A LARGE PARAMETER
and the condition (1.6) Returning to the original notation and using (1.6), we rewrite the indicated relations in the form D(7:,
7:0;
dYk('d:O; A)
A) =
~JI +
l
Vk(7:; A)] Yk(7:,
= A[Pk(7:)Bo(7:) + (7:,7:0 E
-t
7:0;
A),
Qk(7:; A)] Yk(7:,
7:0;
(1.7) A)
(1.8)
[0, T]; A Ell).,).
The operators Vk and Qk (k = 1,2,.··, n) are to be found. We will try to choose them so that condition (1.6) is satisfied, i.e. so that the range of Y k lies in the subspace ~h(7:) = Pk (7:)'iS (k = 1,.··, n). We assume that the operator B(l)(7:; A) has an asymptotic expansion in inverse powers of A, i.e. (1.9) where IIB(P+l)(7:; A) II ~ c.
(1.10)
We will seek the operator functions Vk and Qk in the form 1
T Vi7:; A) 1
P
= E/-s VkS(7:),
T Qk(7:; A) =
P
S~l A- sQks(7:)
(1.11 ) (1.12)
and attempt to choose the coefficients of these expansions so that the operator function (1.7) satisfies the equation dD(7:, , (7:,. A')U-( 7:,7:0,. A') d7:7:0; A) -_ AB
1 '" ( 7:,. A, ') + »'Vp
(1.13)
which differs from the equation. dUjdt
= AB(7:; A)U
for the desired evolution operator by a term of order O(A-P). Under certain conditions, which will be described below, a solution of equation (1.13) differs from the evolution operator of equation (1.1) by a quantity of order O(A-P).
Once we have indicated a method of determining the coefficients of the expansions (1.11) and (1.12), the approximate solution of the Cauchy problem for
1.
APPROXIMATE DECOMPOSITION OF THE EQUATION
.),)1
equation (1.1) will be reduced to the solution of equations (1.8) in the phase spaces ~M't"o). Such a reduction is called an approximate (asymptotic for . :l. -+ 00) decomposition of equation (1.1). 2. Some auxiliary propositions. LEMMA 1.1. Let Q('t", 't"o) be a differentiable operator with respect to '0 that "tracks" the subspaces IBi'r):
Pk('r)Q('r, Q('ro,
'00)
=
'00)
= l.
Q('r, 'ro)Pk('ro)
(k
= I, .. ,n),}
(1.14)
If the relation Qk('r; ).) holds identically in
'0,
=
Pk('t")Qk('r; ..:l.)Pk('r) + Q'('t", 'rO)Q-l('r, - Pk('r)Q'('r, 'rO)Q-l('t", 't"O)Pk('r)
'00)
(1.15)
the relation
Pi'ro)Yk('ro, 'ro;..:l.) = Yk('ro,
'00;
..:l.)Pl't"o) = Yl'ro, 't"o;..:l.)
(1.16)
implies the relation ('0 E [O,T]). (1.17) = Yk('r, '00; ).)Pk('ro) = Yk('r, 'ro;..:l.) PROOF. The substitution Yk('r, '00; ).) = Q('r, 't"o)Xk('r, '00;).) takes equation (1.8)
Pk('r)Yk('r,
'00;).)
into the equation dXk/d't" = Q-l('r, 't"o){..:l.Pk('r)Bo('r)
+ Qk('r,..:l.)
- Q'('r, 'rO)Q-l('r, 'ro)}Q('r, 'ro)Xk,
which by virtue of conditions (1.15) and (1.14) reduces to the form dXk/d't" = Pk('t"O)Q-l('t", 't"o){ ..:l.Pk('r)Bo('r)
+ Qk('r; ..:l.)
- Q'('r, 'rO)Q-l('r, 'ro)}Q('r, 't"O)Pk('t"O)Xk. It is not difficult to see that this equation is satisfied by not only Xk('r, '00; ).) but also the operators Xk('t", '00; ..:l.)Pk('ro) and Pk('rO)Xk('t", '00; )'). From (1.16) it follows that all three operators coincide for '0 = '00 and hence for all '0 E [0, T]. Inasmuch as .,.
Pk('r)Yk('r,
'00;).) =
Pk('r)Q('r, 'ro)Xk('r, 'ro;..:l.)
= Q('t", 'ro)Pk('rO)Xk('t", '00; ..:l.) and Yk('r,
'00;
equality (1.17) is proved.
..:l.)Pi't"o) = Q('r, 'ro)Xk('t", 't"o; ..:l.)Pk('ro),
338
VIII. EQUATION
WITH
A LARGE PARAMETER
REMARK 1.1. Condition (1.15) is obviously satisfied for an operator Ok(r; A) of form (1.12) if
Okl(r)
= Pk(r)Okl(r)Pk(r) + Q'(r, -ro)Q-l(r, -ro) - Pi-r)Q'(-r, -rO)Q-l(-r, 7:o)Pk(-r),
Oks(-r) = Pk(-r)Oks(-r)Pk(-r)
(s
~
2).
(1.18) (1.19)
REMARK 1.2. In the sequel it will be convenient to take as Q(-r, -ro) the conjugation operator constructed in § 111.1. It satisfies the differential equation
In this case equality (1.18) takes the form
Okl(-r)
= Pk(-r)Okl(-r)Pk(-r) +
n
~
P;(-r)Pj{-r).
(1.20)
j=l
Condition (1.16) will be satisfied, for example, if Yk(-ro, -ro; A) = Pk(-ro). From Lemma 1.1 and Remarks 1.1 and 1.2 it follows that if the operators Oks(-r) are selected so that they satisfy conditions (1.19) and (1.20), the operators Yk(-r, -ro; A) determined from equations (1.8) will satisfy condition (1.6) for all -r if they satisfy it for -r = -roo 3. Construction of the approximate solution. We proceed to calculate the coefficients of expansions (1.11) and (1.12). Differentiating (1.7) and using these expansions and (1.8), we find
1
=
n
p
T k'f1 s'f1 A- sV;sC-r) Yk(-r, -ro; A)
+ k~l
[I + S~l A-SVkS(-r)] [Pk(-r)Bo(-r) + S~l A-SOkS(-r)] Yi-r, -ro; A) (1.21)
On the other hand, the right side of equation (1.13), after dividing through by
A and using (1.9), (1.7) and (1.11), takes the form
1.
B(-I:; ).)0('1:, '1:0;).)
+ ).-(P+1)f/JP{'1:;).)
= [sto ),-sBs('1:) x
=
339
APPROXIMATE DECOMPOSITION OF THE EQUATION
+
EJ1 + ~1
).-(P+1)B(P+1)('1:; ).)]
).-SVkS('1:)] Yk('1:, '1:0;).)
+
).-(P+1)f/Jp('1:;).)
k~l {BO('1:) + St1 ).-s[Bs('1:) + BO('1:)Vks('1:)]
+ +
(l.22)
).-(P+l)B(P+1)('1:; ),)[1 + st1 ).-sVkS ('1:)]} Yi'1:, '1:0;).)
).-(P+1)f/Jp('1:; )').
The right sides of equalities (l.21) and (l.22) must coincide. By equating the coefficients of like powers of )., making use of relations (1.17) and discarding the common factor Yk('1:, '1:0; ).), we obtain the system of relations Vk1 ('1:)Bo('1:)Pk('1:)
+ Ok1(r')Pi'1:)=
Vk.('1:)Bo('1:)Pk('1:)
+
V';'S-l('1:)Pk('1:)
= B.('1:)Pk('1:)
+
BO('1:)Vk.('1:)Pk('1:)
B1('1:)Pk('1:)
+ Bo('1:)Vk1 ('1:)Pk('1:); s-l
+ Oks('1:)Pi'1:) + r;
j=l
Vk,s-l('1:)Okj{'1:)Pk('1:)
s-l
+ r;
(2~
B s-i'1:)Vkj{'1:)Pi'1:) j=l which is more suitably written in the form
s
~
BO('1:)Vk1 ('1:)Pi'1:) - Vk1 ('1:)Pk('1:)Bo('1:)
(1.23)
= Ok1('1:)Pi'1:) - B1('1:)Pk('1:); BO('1:)Vks ('1:)Pk('1:) - V ks ('1:)Pi'1:)Bo('1:) = OkS('1:)Pk('1:) - Tk.('1:)Pk('1:)
p),
(2
~
s
~
p),
(1.24)
where Tks('1:)
s-l
s-l
;=1
j=l
= r; Bs-i'1:) Vkj{'1:) + Bs('1:) - v,{, S-l('1:) - r; Vk. s-i'1:)Okj{'1:).
(1.25)
Moreover, a comparison of the terms of order O().-(P+1» in (1.21) and (1.22) leads to the equality f/Jp('1:, ).) n {
= k"f1 V£p{'1:)
zp
P
+ S=f+1 ).-(s-P-l) j=~_P [Vk,s-j{'1:)Ok;('1:) - B(P+1)('1:,
- Bs-j('1:)Vk;('1:)] (1.26)
).(1 + st1 ).-sVkS('1:)]} Yk('1:,
'1:0, ).).
VIII.
340
EQUATION WITH A LARGE PARAMETER
The system of equations (1.23)-(1.24) is recursive in nature inasmuch as the expression (1.25) for Tks(-r) contains only those operators Vkj{-r) and Qkj{-r) with subscripts j = 1,.··, s - 1 and the derivative V':'s-l(-r). We will now show how to successively determine all of the operators Qkj and Vkj (j = 1,.··,p; k = 1,.··,n) from these equations with the use offormula (1.3.14). It will be assumed in this connection that the operator function Bj{-r) (j = 1,. .. ,p) has strongly continuous derivatives up to order p + 1 - j inclusively. We begin by considering the equation of first approximation (1.23). Multiplying it from the left by the projection Pi-r), we obtain for the operator XR)(-r)
= Pj{-r)Vkl(-r)Pk(-r)
the equation BO(-r)Xj~l)(-r) - XN)(-r)Bo(-r)
(j = Pj(-r)[Qkl(-r) - B1(-r)]Pk(-r) This equation is satisfied for j = k if one puts
(1.27) =
1,2,. ··,n).
(1.28)
Xkr(-r) = 0
and Pk(-r)Qkl(-r)Pk(-r)
= Pl-r)B1(-r)Pk(-r).
(1.29)
Equalities (1.20) and (1.29) permit us to determine the operator Qk1(-r): n
Qkl(-r) = Pk(-r)B1(-r)Pk(-r)
+ L: P;(-r)P,(-r). ,=1
(1.30)
When j =f. k we have Y(-r) = Pi-r)Y(-r)Pk(-r), where Y(-r) denotes the right side of (1.27) and the projections Pi-r) and Pk(-r) correspond to disjoint spectral sets of Bo(-r). We can therefore apply formula (1.3.14), which gives us
XJ.Nr) _
_1_
§
§
- - 4,,2 F.«) F,«) ___1_
-
§
(Bo(-r) - vI)-IP/-r)[Qkl(-r) - B1(-r)]Pl-r)(Bo(-r) - f.JI)-1 dv df.J v - f.J
§
(1.31)
(Bo(-r) - vI)-IP/-r)[Pf(-r) - B1(-r)]Pk(-r)(Bo(-r) - f.JI)-1 dv df.J. 4,,2 F.«) F,«) v - f.J
Finally, we find the operators Vk1 (-r) by putting (1.32) As has already been mentioned the projections Pk(-r) have as many continuous derivatives as the operator Bo(-r). Therefore, as can be seen from formulas (1.30) and (1.31), the operators Qkl(-r) and Vk1 (-r) have continuous derivatives of order p. We consider the equations for second and higher approximations in a completely analogous manner. Multiplying (1.24) from the left by Pj{-r), we obtain for the
2.
ESTIMATE OF THE ERROR
341
operator the equation Bo('C)Pj{'C)Xj~)(-1:)
- xjt/('C)BO('C)Pk('C) = Pj('C) [OksC'C) - Tks('C)]Pk('C),
(1.33)
which is investigated in exactly the same way as (1.27). Whenj = k we put (1.34) and and get, using (1.19),
OksC'C) = Pi'C)TksC'C)Pk('C)
(s = 2,.··, p).
(1.35)
After this we find
XjZl('C)
_
f f
4n2 r.Cr) rjCr) and put, as before,
(Bo('C) - vI)-lPj{'C)Tk,('C)Pi'C)(Bo('C) - fLI)-l dv dfL (1.36) v - fL Vk,('C) = L; xj%)('C).
(1.37)
#k
We note that the operator function
Tk2('C)
=
B1('C)Vk1('C)
+ Bz('C)
- Vk1('C) - Vk1('C)Okl('C)
is continuously differentiable p - 1 times, and hence so also are the operators Vk1('C) and Ok2('C). It is not difficult to show by induction that the operators Vkr('C) and Okr('C) are continuously differentiable p + 1 - r times. For if this is true when r = s - 1, the operator Tks('C) is according to formula (1.25) continuously differentiable p - s + 1 times, and it follows from formulas (1.35) and (1.37) that this property is also possessed by the operators Vkr('C) and OkrC'C). In particular, th~, operator Vkp{'C) has a continuous derivative. Thus all of the coefficients of expansions (1.11) and (1.12) have been determined so that the operator function (1. 7) satisfies equation (1.13). In this connection, as follows from formula (1.26), the operator function Wp{'C; i\) is continuous in 'C for 'C E [0, T] and i\ E U".
§ 2. Estimate of the error 1. Basic lemmas. We proceed to estimate the error sustained in using the above method for constructing the evolution operator of equation (1.1).
342
VIII.
EQUATION WITH A LARGE PARAMETER
This estimate is obtained under certain additional assumptions connected with the location of the spectrum of the operator Boer). One of these additional assumptions is formulated as follows. (C~) The spectrum a(Bo(-r») of Bo(-r) lies in the interior of the left halfplane for all values of -r E [0, T]. In certain important cases the spectrum a(Bo(-r») can fall on the imaginary axis.
Such a situation can be analyzed under certain additional assumptions described by the following condition.
(C v) The phase space 58 is a Hilbert space ~ and there exists a uniformly positive continuously differentiable operator W(-r) (0 ~ -r ~ T) such that the operator Re(W(-r)Bo(-r») is nonpositive. We note that it automatically follows from condition (C v) that the spectrum
°
a(Bo(-r») lies in the closure of the left halfplane. In fact, for any iJ > the operator Bo(-r) - iJI is obviously uniformly W(-r)-dissipative and hence its spectrum lies in the interior of the left halfplane (see Theorem 1.5.1). An example of an operator satisfying condition (C v) is an operator of the form Bo(-r) = S(-r)H(-r), where H(-r) is a strongly continuously differentiable uniformly positive operator and the operator S(-r) is skew Hermitian: S*(-r) = - S(-r). In fact, in this case it is possible to put W(-r) = H(-r). Then Re(W(-r)Bo(-r») = H(-r)S(-r)H(-r) - H(-r)S(-r)H(-r) =
°
and thus condition (C v) is satisfied. This case includes, in particular, the canonical equations (S = if). Our subsequent analysis will be based on two auxiliary propositions. LEMMA 2.1. Suppose the continuous operator function Bo(-r) with values in [58] satisfies condition (C~) while the operator function B1(-r; A) is uniformly bounded in the domain ilAo = Uio,e o = P = peielp ~ Ao;lol< Oo}:
!!B1(-r; A)II ~ c
(A E Ui"e,; -r E [0, T)].
Then the evolution operator U( -r, -ro; A) of the equation (A
~
Ao; -r
E
[0, T])
(2.1)
is uniformly bounded in Ui"e,/or sufficiently small 00 > 0:
IIU(-r,-ro;A)11 ~
Cl
(AEUi"e,;-rE[O, T]).
(2.2)
If the number AO is sufficiently large, the stronger condition II U(-r, -ro; A) I! ~ Ne-IAlvo(~-")
where N, PROOF.
))0
(A E Ui"e,; -r E [0, T]),
(2.3)
> 0, is satisfied.
The totality of values of the continuous operator function eieBo(-r)
2.
343
ESTIMATE OF THE ERROR
(r E [0, T]; 181 < ( 0) is a precompact set of operators in [~] the spectra of which for sufficiently small 80 lie in the interior of the left halfplane and hence to the left of a straight line Re fJ- = - v (v > 0). It follows from Lemma III.6.2 in this case that there exists a constant N > for which
°
II e XBo «)s II = Ilee Bo«)sIXIII i8
We consider a number (} > ('Z', 'Z'o E [0, T])
~ Ne-IXlvs
(A
E
llIo,oo; 'Z' E [0, T]; s ~ 0).
° having the property that when I'Z' - 'Z'ol
IIBo('Z') - Bo('Z'o) II ~ v/2N. Writing the equation for the evolution operator in the form
(2.4) ~
(}
(2.5)
dU('Z', 'Z'o; A) = ABo('Z'o)U('Z', 'Z'o; A) d'Z'
+ A[Bo('Z')
- Bo('Z'o)]U('Z', 'Z'o; A)
+ B1('Z'; A)U('Z', 'Z'o; A),
we obtain the representation U('Z', 'Z'o; A) = eAB,«o)«-
+ A JeAB,«O) «-s) [Bo(s)
- Bo('Z'o)]U(s, 'Z'o; A)ds
<0 <
+ JeABo«o) «-s) Bl(S; A)U(S, 'Z'o; A)ds. <0
When I'Z' - 'Z'ol < (} it follows from this representation and (2.4) and (2.5) that
II U('Z', 'Z'o; A)II
~ Ne-IAIvC<-To)
+ N IAI
v 2N
<
Je-IAlv«-S) II U(s, 'Z'o; A) lids <0
<
+ eN Je-IAIvC<-S) II U(s, 'Z'o; A)
lids.
<0
Using Corollary III.2.2, we conclude that when I'Z' - 'Z'ol < (}
II U('Z', 'Z'o; A) II
~
Ne-(IAlv/2-eN) «-<0).
We obtain the desired estimate for arbitrary 'Z'o, 'Z' E [0, T] by subdividing the interval ['Z'o, 'Z'] by th~ points 'Z'o < 'Z'1 < ... < 'Z'm-l < 'Z'rn = 'Z' so that l'Z'k - 'Z'k-ll < (}. It then follows from the representation m
U('Z', 'Z'o; A) = II U('Z'k' 'Z'k-l; A) k=1
that m
IIU('Z','Z'o;A)11 ~ II IIU('Z'k,'Z'k-l;A)11 k=1 ~ N[«-<0)/o]+le-((1/2) lXiv-eN) «-<0»
(A ED;o.oo)'
(2.6)
VIII.
344
EQUATION WITH A LARGE PARAMETER
We at once obtain estimate (2.2) from this inequality since for it ZOo ~ zo ~ T
E
o~
mo.oo and
Moreover, (2.6) implies
II U(zo, To; it) II
~ ~
Ne-(l!Z)(IAI"-cN-(l/o)ln NJ «-<0) Ne-IAIZ(v-cN/Ao-ln NOAo)«-
It remains to note that the quantity Vo = t(v - cN/ito - In N/ itoo) is positive for sufficiently large ito. The lemma is proved. COROLLARY 2.1. If the coefficients of equation (2.1) are periodic and satisfy the conditions of Lemma 2.1, the spectrum of the monodromy operator lies in the interior of the unit disk for sufficiently large in modulus it E Ufo.o o and sufficiently small
00 >
o.
In fact, if the coefficients of equation (2.1) are periodic with period T, inequality (2.3) can be used to estimate the monodromy operator of this equation: II U(T, O;it) II ~ Ne-IAlvoT. It follows from this estimate that the inequality II U(T, 0; it) II < 1 is valid for sufficiently large lit I· LEMMA 2.2. If the operator Bo(zo) satisfies condition (Cl.')) while the operator B1(zo; it) is uniformly bounded in the strip UAo = U;:. a = {.it I Re it ~ ito; 11m it I ~ a} :
IIB1(zo; it) II ~ c
(zo
E
[0, T]; it
E
UA:. a),
the evolution operator of equation (2.1) is also uniformly bounded in Ulo.a. PROOF.
We consider for fixed x
E
Sj the nonnegative function
1>(zo) = (W(zo)U(zo, zoo; it)x, U(zo, zoo; it)x).
The following estimate holds for its derivative: 1>'(zo)
=d (W'(zo)U(zo,
+
zoo; it)x, U(zo,
TO;
it)x)
(W(zo)[itBo(ZO) + B1(zo; it)]U(zo, ZOo; it)x, U(zo, ZOo; it)x)
+ (W(zo)U(zo, ZOo; it)x, [itBo(zo) + B1(zo; it)]U(zo, ZOo; it)x) = ([W'{zo) + 2Re it Re W(zo)Bo(zo) - 21m it 1m W(zo)Bo(zo)
+ 2Re W(ZO)Bl(ZO; it)]U(zo, ZOo; it)x,
I
~ ([W'{zo)U(zo, zoo; it)x, U(zo, ZOo; it)x)]
+
U(zo, ZOo; it)x)
I
I([W(ZO)Bl(ZO; it) + 2a 1m W(zo)Bo(zo)]U(zo, ZOo; it)x,
U(zo, ZOo; it)x)
I
2. ESTIMATE OF THE ERROR ~
345
{II W-1/2(Z-) W'(-.) W-1I2(-r) I + 2all W-1/2(-r)(Im WB )W-1/2(Z-) I O
+
IIW1I2(-.)B1(z-;A)W-1I2(Z-)II}¢(z-) ~ C2¢(Z-) (0 ~ z- ~ T; A E
m:,a).
Integrating this inequality, we get ¢(z-)
~
¢(z-o)ec,(r-ro)
~
¢(z-o)ecT
and, finally,
I U(z-, -'0; A)x112 ~ I W-111¢(z-) ~ IIW-11IeCT(Wx,x) ~
IIW-IIIIIWlleCTllxI12.
The lemma is proved. 2. Estimate of the solutions of the decomposition equations. The result obtained above will first be used to estimate the operator Ylz-, Z-o; A) (k = 1,.··, n). This operator satisfies the differential equation (l.8), the coefficients of which have been selected so that condition (1.15) is satisfied. It was shown in the proof of Lemma 1.1 that the substitution Yk(z-, Z-o; A) = Q(z-, z-o)Xk(z-, Z-o; A) reduces equation (1.8) to the equation dXk/dz- = [ABJk)(Z-)
+ B{k)(z-; A)]Xk,
(2.7)
where (2.8)
and Bik)(z-; A)
= Pk(Z-O)Q-l(Z-, z-o){Dk(z-; A) - Q'(z-, -'O)Q-l(Z-, z-o)} Q(z-, z-o)Pk(z-o).
(2.9)
The operators BJk)(Z-) and B?)(z-; A), as can be seen from expressions (2.8) and (2.9), act in the subspace 5
346
VIII.
EQUATION WITH A LARGE PARAMETER
is nonpositive. Thus in this case Lemma 2.2 is applicable to equation (1.8). We have arrived at the following proposition. LEMMA 2.3. Suppose that the operator Bo(7:) satisfies condition (C~) or condition (C.fj). Suppose further, that the operator Bk(7:) has a continuous derivative of order p + I - k (k = O,l,···,p). Then the solution y k(P)(7:, 7:0; A) of the equation
(A
E
ill,;
7:, 7:0 E
[0, TD,
(2.10)
where the ()ks(7:) (k = 1,.··, n; s = 1,.··, p) are the operators (1.30) and (1.35), that satisfies the initial condition Y k(7:o, 7:0; A) = Pk(7:0) is bounded: (7:0,7: E
[0, T]; AE ill,; k = I,.··,n).
COROLLARY 2.2. If in addition to the conditions of the lemma the operator B(P+l)(7:; A) is boundedfor A E ill, and A E [0, T], so is the operator (1.26): (7: E
[0, T]; A E ill,).
(2.11)
3. Asymptotic estimates of the approximations. We consider in conjunction with the operator (1.7) the operator (2.12) which differs from it by a factor C(7:0; A) that is independent of 7:. If this factor is bounded for A E ill" the operator Vp{7:, 7:0; A) satisfies an equation of the same type as (1.13). We choose C(7:0; A) so as to provide a suitable initial value for the operator Vp(7:, 7:0; A), thanks to which it will differ little from the evolution operator U(7:,7:o; A) of equation (1.1). Inasmuch as the operator Vp(7:, 7:0; A) must by construction serve as a pth approximation to U(7:, 7:0; A) it is natural to require that Vp(7:o,
7:0;
A)
= 1+ O(A-(P+1).
(2.13)
This can be done as follows. From (2.12) we get Vp{7:o,
7:0;
A)
= {I +
El El
A-SVks(7:0)}Cp(7:0; A).
For sufficiently large Athe operator in braces has an inverse which can be calculated by summing an infinite geometric series. We take as Cp{7:o; A) the expression (2.14)
2. ESTIMATE OF THE ERROR
347
which is obtained by replacing this series by its pth partial sum. As a result, condition (2.13) will be satisfied. LEMMA
2.4. Suppose the operator function O,(-c, 'Z"o; A) satisfies the differential
equation dO('Z"d;O; A) = AB(-c'; A)O('Z", 'Z"o; A) ('Z", 'Z"o
E
+ A-Pl/fp(C; A)
(2.15)
[0, T]; A E DJ.,),
where IIWp('Z"; A) II ~CI ('Z" E [0, T]; A E D}-,). If the evolution operator U('Z", 'Z"o; A) of equation (l.l) is bounded: ('Z", 'Z"o E [0, T]; A E DAo)'
(2.16)
the condition
II 0('Z"0, 'Z"o; A) - III ~ C3A- P
('Z"o
[0, T]; A E DA,)
E
(2.17)
implies the estimate
II O('Z", 'Z"o; A) - U('Z", 'Z"o; A) II ~ C4A- P
('Z", 'Z"o E [0, T]; A E DAo)'
(2.18)
And if the operator U('Z", 'Z"o; A) is subject to the stronger condition
II U('Z", 'Z"o; A) II
~ Ne-Av(T-To)
('Z", 'Z"o
E
[0, T]; A E DAo )'
(2.19)
('Z"o
E
[0, T]; A E DA,)
(2.20)
the inequality
II 0('Z"0, 'Z"o; A) - I II ~ C3A-(P+1) implies the estimate
II O('Z", 'Z"o; A) - U('Z", 'Z"o; A) II ~ C4 A- CP +1) PROOF.
('Z",'Z"0
E
[0, T]; A E DAo)'
(2.21)
A solution of equation (2.15) can be written in the form T
O('Z", 'Z"o; A)
=
U('Z", 'Z"o; A)O('Z"o, 'Z"o; A)
+ A-P JU('Z", s; A)WP(S; A)ds, To
which implies the inequality
II U('Z", 'Z"o; A) - O('Z", 'Z"o; A) II T
~ IIU('Z"~.'Z"o;A)IIIIO('Z"o,'Z"o;A) -
111+ CIA-P ToJ IIU('Z",s;A)llds.
If conditions (2.16) and (2.17) are satisfied, inequality (2.22) implies that
II U('Z", 'Z"o; A) - O('Z", 'Z"o; A)II ~ A-P(C2C3 + CIC2 T ). And if conditions (2.19) and (2.20) hold,
II U('Z", 'Z"o; A) - O('Z", 'Z"o; A) II
~
NC3A-CP+I)
+ CIA-PN Je-AvCT-s)ds ~ To
A-CP+1)N(C3
+ C~N).
(2.22)
348
VIII. EQUATION WITH A LARGE PARAMETER
The lemma is proved. The final result now follows from Lemmas 2.1- 2.4. THEOREM 2.1. Suppose the operator Bke) (0 ~ 7: ~ T; k = 0, 1, 2,.··,p) has a continuous derivative of order p + 1 - k while the operator B(P+1l(7:; .it) is continuous and bounded for 7: E [0, T] and .it E ill,. Suppose the operators Vks (7:), Qk.(7:) (k = 1,.··,n; s = 1,···,p) have been calculated by formulas (1.30), (1.35), (1.32) and (1.37) while the operators Yk(7:, 7:0; .it) (k = 1,2,.··,n) satisfy equations (2.10) and the initial condition (k = 1,2,.· ·,n). ()o
Then, if condition (Gil) is satisfied, for sufficiently large .ito and sufficiently small the operator
(2.23)
is subject in the domain ill,
= ill:,o, to the estimate (7:,7:0 E [0, T];.it E ill;,o.).
And if condition (C,fi) is satisfied,for sufficiently large ;1.0 and a > 0;(7:,7:0;;1.) =
the operator
k~JI + S~1.it-SVkS(7:)]Y ?l(7:, 7:0;;1.) x {I
is subject in the domain ill,
°
(2.24)
+
E:(- 1)t~
)J}
(2.25)
st/- SVk.(7:0
= il{;,a to the estimate (7:,7:0 E [0, T]; .it E llf;.a)'
(2.26)
REMARK 2.1. Inequality (2.26) obviously continues to hold if one simplifies expression (2.25) by discarding all terms in it of order not less than O(;1.-P). REMARK 2.2. In the simplest case p = 1, which is the most important for the applications, formula (2.23) takes the form
0 1(7:,7:0; .it) =
Ell + l
Vk1(7:)]ypl (7:, 7:0; ;1.(1 -
l k~1
Vk1(7:0)] (2.27)
2.
349
ESTIMATE OF THE ERROR
while formula (2.25) takes the form D{(r, ro; A)
=
E1
(I + 1Vk1(r))yp)(r, ro; A).
After making the simplification indicated in the preceding remark, we obtain the formula (2.28)
Under condition (Gll) the operator (2.27) satisfies the estimate U(r, ro; A)
=
D1(r, ro; A)
+ 0(1/).2),
while under condition (C v) the operator (2.28) satisfies the estimate U(r, ro; A) = D{(r, ro; A)
+ 0(1/1.).
In the latter case to obtain a more exact estimate with error of order 0(1.-2) it is necessary to consider a second approximation, which after the appropriate simplifications takes the form
+ t~l
{Vj1(r)yk(2)(r, ro; A) -
yp)(r, ro; A)Vj1(ro)}
+
o( 12).
4. Canonical case. We assume that Bo(r) = i,lYf(r), where ,I is an invertible Hermitian operator satisfying the condition ,12 = I and Yf(r) (0 ~ r ~ T) is a uniformly positive operator in a Hilbert space:p. We have already noted above that the operator Bo(r) satisfies condition (C v) in this case. Here the evolution operator U( r, ro; A) of the equation
dCP/dt
= ABo(r)cp
is ,I-unitary for real A. We will now show that under the above conditions the operator I:Z=l Yk 1)(r, ro; A), viz. the first approximation to the evolution operator, is also ,I-unitary for real A ~ 1.0 • We first note that the operator Q(r, ro), which tracks the invariant subspaces )!h(r) of the operator iBo(r), is ,I-unitary (see § IV.l), i.e. satisfies the condition (2.29)
Let y k(l)(r, ro; A) = Q(r, ro)Xk(r, ro; A).
The operator Xk(r, ro; A) is a solving operator in the subspace (2.7):
~k(rO)
of equation
350
VIII. dXk('d:O; A)
EQUATION WITH A LARGE PARAMETER
=
Q-l(-C', 'C'o)Pk(z,)Bo('C')Pk('C')Q(-C', 'C'o)Xk('C', 'C'o; A).
(2.30)
The operator Bo('C') is by assumption /-skew-Hermitian, i.e . .IBo('C')
=
Bt('C')/.
-
Multiplying this equality from the left and from the right by the projections N('C') and Pi'C') respectively, we reduce it to the form PZ('C')/Pi'C')Bo('C')
Bt('C')N('C')/Pk('C').
= -
With the use of this relation and the relations Q('C', 'C'o)Pk('C'O)Q-l('C', 'C'o) = Pk('C');
we deduce that P:('C'o)/Pk('C'O)Q-l('C', 'C'o)Pk('C')Bo('C')Pb)Q('C', 'C'o)
= Pt('C'o)Q*('C', 'C'o)/Q('C', 'C'o)PbO)Q-l('C', 'C'o)Pk('C')Bo('C')Pk('C')Q('C', 'C'o) = Pt('C'o)Q*('C', 'C'o)/Pk('C')Bo('C')Pk('C')Q('C', 'C'o) =
Q*('C', 'C'o)N('C')/Pk('C')Bo('C')Pk('C')Q('C', 'C'o)
=
-
Q*('C', 'C'o)Pt('C')Bt('C')Pk*('C')/Pi'C')Q('C', 'C'o)
= -
Q*('C', 'C'o)Pt('C')Bo*('C')Pt('C')Q-l*('C', 'C'o)Pt('C'o)Q*('C', 'C'o)/Q('C', 'C'o)Pi'C'o)
=
[Q-l('C', 'C'o)Pk('C')Bo('C')Pk('C')Q('C', 'C'o)]* Pt('C'o)/Pi'C'o).
-
This equality means that the coefficient of equation (2.30) is a Pt('C'o)/Pk('C'o)Hermitian operator. Therefore the evolution operator Xk('C', 'C'o; A) of this equation is Pt('C'o) /Pk('C'o)-unitary for real A, i.e. xt('C', 'C'o; A)Pt('C'o)/Pi'C'o)Xk('C', 'C'o; A)
=
Pt('C'o)/Pk('C'o)
(A ~ Ao).
(2.31)
We note that the /-Hermitianness of the operator iBo('C'o) implies the /-Hermitianness of its spectral projections. . Therefore
(j ¥= k). Taking into account these relations, we deduce from (2.31) the equality
.E j
X!('C', 'C'o; A)/
.E
Xk('C', 'C'o; A)
=
.I,
k
denoting the /-unitariness of the operator .EkXk('C', 'C'o; A). Since a product of /unitary operators is .I-unitary, the operator
.E k
yk(l)('C', 'C'o; A) = Q('C', 'C'o)
.E k
Xk('C', 'C'o; A)
2. ESTIMATE OF THE ERROR
351
is ,I-unitary. 5. Example. We consider as an example the equation d¢/dt
(2.32)
iAf£'o(t)¢
=
in a Hilbert space .p, assuming that £'o(t) is a twice continuously differentiable in t uniformly positive periodic operator function: :YPo(t + T) = £'o(t). The spectrum of the ,I-positive operator f£'o(t) lies the real axis. We assume that this spectrum consists of a finite number of points:
satisfying for each
'C E
[0, T] the condition
(2.33)
(j # k).
/li'C) # /lk('C)
In this elementary case we will find under the assumption of large A an approximate expression for the monodromy operator U(T; A). In the invariant subspace .pl'C) corresponding to a point /lk('C) the spectrum of the operator ,I£'o('C) consists of just this point. Since this operator is ,I-positive,
= /lk('C)Pk('C),
f£,o('C)Pk('C)
Therefore n
,I£'o('C)
=
.I; /lk('C)Pk('C), k=l
Equation (2.30) now takes the form
~k =
iA/ll'C)Q-l('C, 'Co)Pk('C)Q('C, 'Co)Xk = iA/lk('C)Pk('Co)Xk,
Consequently,
and, finally, Ui.('C, 'Co;A) = Q('C, 'Co) k"f1e -
n
fAS'
I'.(s) ds
<0
Pk('Co)
+ 0 ( A1 ) .
From this result 'we obtain the following representation for the monodromy operator: U(T; A) = Q(T, 0) k~l eiAM.Pk(O)
where Mk
=
g /lk(s)ds.
+ o( 1)
(ReA
~ An; IImAI ~ a),
(2.34)
352
VIII.
EQUATION WITH A LARGE PARAMETER
When the space .p is finite dimensional, this representation permits one to investigate the spectrum of the monodromy operator for large Aand to establish the presence of stability bands of equation (2.32) for arbitrarily large real A. The fact that the representation (2.34) is also valid for complex Ain a neighborhood of the ray A :?; Ao plays an important role in such an investigation. Another important problem that arises in connection with equation (2.32) is the boundary problem with a condition of the form E ¢(O) + F ¢(T) = O. Such a problem is regarded (for reasons which we do not cite here; see I. C. Gohberg and M. G. Krein [2]) as selfadjoint if E*,1E
=
(2.35)
F*,1F.
When the operator F is invertible it is possible without loss of generality to take F = I, in which case (2.35) becomes the ,1-unitariness condition E*,1 E = ,I for the operator E. We thus arrive at a boundary problem for equation (2.32) with the condition E¢(O)
+ ¢(T) = 0,
(2.36)
where E is a ,I-unitary operator. Substituting the expression ¢(T) = U(T; A) ¢(O) into (2.36) and using (2.34), we obtain the relation [E
+ Q(T, 0) ~leiJ.M. Pk(O) ]
¢(O)
=
0(1/ A).
(2.37)
This expression can be used to investigate the spectrum of the boundary problem in question. For example, in the finite-dimensional case it is known that the leading term in an asymptotic expansion of the spectrum does not depend on the choice of the ,I-unitary operator E. This permits one to put E = Q(T, 0) and reduce equation (2.37) to the particularly simple form n
¢(O)
+ I: eiJ.M·Pk(O) ¢(O)
=
O(1/A)
k=l
or, after multiplying by Pj{O), (1
+ eiJ.M')Pj{O) ¢(O) =
0(1/ A).
In order for this relation to be satisfied when ¢(O) =1= 0 it is necessary (though not sufficient) that the following equality hold for somej(j = 1, ... , n): 1
+ e iAM, =
0(1/ A).
Thus our boundary problem has n possible series of eigenvalues, which are defined by the asymptotic formula
2k+1
Ajk= gflj{S) ds
1r
(I)
+ 0 --;r .
(2.38)
3.
353
RAPIDLY OSCILLATING COEFFICIENTS
An additional investigation would be required to prove the fact that formula (2.38) actually represents eigenvalues for large k. Such an investigation, which essentially makes use of the assumption (2.33) and the asymptotic expression (2.34), has been carried out in the book of 1. C. Gohberg and M. G. Krein [2].
§ 3. The equation with a rapidly oscillating coefficient 1. Description of the method. In this section we again consider the equation
dcp / d-r = AB(-r; A)CP,
(3.1)
but under somewhat different assumptions on the coefficient. We assume that (3.2) It is assumed for the sake of simplicity that the operator Bo is a constant. As before, we will suppose that it satisfies one of the conditions (C~) or (C p) and, in addition, has a spectrum with a decomposition O'(Bo) = UZ=l O'k into spectral sets. We further assume that the following representation is valid for sufficiently large real A ~ Ao:
Bk(-r; A) = 1::: Bt)(-r)eimAWT
(k = 1, ... ,p),
(3.3)
m
the coefficients of which are continuously differentiable a sufficient number of times. In order to avoid convergence questions we will suppose that only a finite number v of terms in the expansion (3.3) are different from zero. Finally, we assume that the operator function B CP+ 1) (-r; A) is continuous in -r and bounded:
(-r
E
[0, T]; A ~ Ao).
(3.4)
As in §l, we will seek an approximate expression for the evolution operator U(-r, -ro; A) of equation (3.1) in the form
Ui-r, -ro; A) where
YC~)(-r,
=
El
ttA-SVk.(-r; A)}
Y~)(-r, -ro; A),
(3.5)
-ro; A) satisfies in the subspace 18k = Pkl8 the equation
dYcP) td-=
{P
A PkBo + S~l A-sOks(-r; A)
}
YCl)
(3.6)
and the initial condition (3.7)
But in accordance with expansion (3.3) we assume that the following analogous expansions hold:
354
VIII.
EQUATION WITH A LARGE PARAMETER
Vks(r; A)
=
V~";)(r)eim}.wT
1:;
(3.8)
m
We will see that these expansions consist of a finite number of terms. The coefficients Vr;:;) and Qks must be chosen so that the operator D(-r, -ro; A) satisfies an equation of the form (3.9)
Proceeding in the same way as in § 1 and, in addition, isolating the coefficients of the functions eim}.WT, we obtain the system of relations (3.10) (k (Bo - imOJI)V~~)(-r)Pk =
= 1,2, ... , n; m = 0, ± 1, ... );
V~~)(-r)PkBo
V~~)(-r)Qk1(-r)Pk
+
V'~~)(-r)Pk -
1:; B(~-r)(-r)V~~)(-r)Pk r
(k = 1,2, ... , n; m = 0,
=
V~~)(-r)Qks(-r)Pk
+
V'~~~1(-r)Pk
+
(3.11 )
± 1, ... );
s-1
1:; V~~~j(-r)Qki-r)Pk
}=1
(3.12)
s-1
- 1:; 1:; B;~7) (-r) V~}(-r)Pk }=o
r
(s = 2, ···,p;k = 1,2, ···,n;m = 0,
± 1, ... ).
The following expression is obtained for the function WP( -r; A) in this connection:
2P
P
zp
p
s=P+1
j=1
+ 1:;
A-(s-P-1) 1:; Vks-i-r; A)Qki-r) s=P+1 j=1
1:; A-(s-P-1) 1:; Bs-i-r; A)Vki-r; A) - B(P+1)(-r; A) st/-SVk.(-r; A)}
(3.13)
Y~l(-r, -ro; A).
If we manage to determine all of the coefficients Vr;:;) and QkS from equations (3.10)-(3.12), a subsequent investigation and estimate of the deviation of the operator D(-r, -ro; A) from the solving operator of equation (3.1) can be carried out
3.
355
RAPIDLY OSCILLATING COEFFICIENTS
in exactly the same way as in § 2. We will therefore confine our attention to equations (3.10) - (3.12). Our conduct of this investigation depends on whether or not the spectrum O"(Bo) contains points that are congruent modulo iw (the resonance case). 2. Nonresonance case. Equation (3.10) is satisfied for m = if we put v~g\..-) == P k (k = 1,.··,n). But if m =f. 0, the fact that the spectra of the operators Bo and Bo - imw/ do not intersect implies by virtue of the results of § I.3 that this equation has only the trivial solution, and we put
°
From relation (3.11) for m = projection Pk " the equation
°
(m
=f.
0; k
=
1,.··,n).
we obtain, after multiplying it from the left by a
BOPk,V~~)("-)Pk - Pk,v~~)("-)PkBo = Pk,Qkl,,-)Pk - Pk,B~O)(,,-)Pk'
If k = k', this equation is satisfied for Qkl("-)
=
P kQkl(,,-)Pk = PkB~O)("-)Pk
and
PkVk~) Pk
= 0.
When k =f. k' it has a unique solution, which can be found by using the formulas of § 1.3 and then putting
°
At present the right side of equation (3.11) has been determined. When m =f. equation (3.11) has a unique solution (since there is no resonance), which can be found by using the same formulas of § I.3. The treatment of equations (3.12) is completely analogous. 3. Resonance case. We consider this case under additional simplifying assumptions. We assume that if the sets O"k(Bo) + imw and O"k,(Bo) intersect for some k, m and k' = ¢(k, m), each of the spectral sets O"k(Bo) and O"k,(Bo) consists of a single point Ak and Ak' respectively, so that (3.14) In addition, we assume that the invariant subspaces of Bo corresponding to these spectral sets are eigenspaces: (3.15)
Let us multiply equation (3.10) from the left by the projection P k " In this way we obtain the equation (Bo - imwl)Pk,Vi~) P k - Pk,vi~) PkBo = 0,
which is treated in exactly the same way as in the resonance case if k' Thus we put
(3.16) =f.
¢(k, m).
356
VIII. EQUATION WITH A LARGE PARAMETER
=
1,.··,n)
(3.17)
(m =F 0; k' =F ¢(k, m).
(3.18)
(k
and
But when k'
=
¢(k, m) equation (3.16) becomes an identity, and the operators
Z k'f
=
Pk , Vk'O) Pk
remain indeterminate for the time being. Multiplying equation (3.11) from the left by P k" we now obtain the equation
(Bo - imwI)Pk,VkT') (r)Pk - Pk, VkT') (-1:)PkBo (3.19)
- P k,Bim)('1:)Pk + Zk'f°klPk. When m
=
0 and k' = k we put, as before, PkVk~) P k = 0
(3.20)
and obtain the relation
Okl = PkOk1Pk = ~ (~PkBi-r)Pk"Zk0;,1) + PkB iO)('1:)Pk. r*O k" When m = 0 and k' =F k equation (3.19) reduces to the equation BOPk,Vk~)('1:)Pk - Pk,Vk~)('1:)PkBo = -
~ Pk,Bi-r)('1:)Vk~)('1:)Pk'
(3.21)
(3.22)
r
which, if the right side is known, has a unique solution determined, as above, by the formulas of § 1.3. An equation of the same type is also obtained when m =F 0 if k' =F ¢(k, m): (Bo - im))I)Pk,VkT') P k - P k' VkT') PkBo = - ~ Pk,Bim - r) V~) Pk. (3.23) r
It remains to consider the case when k' = ¢(k, m). In this case, as in the case of equation (3.16), the left side of equation (3.19) vanishes identically. We therefore obtain the relation
(Zk'1)' -
~
r~,O
(:£:
Pk,Bt- r)('1:)Pk',zt/k) - Pk,Bim )(-1:)Pk + Zk'1oklPk = 0,
k"=l
which together with (3.21) leads to a system of differential equations for the unknown operators Z k'f'} :
+ zk~1pkBiO)('1:)Pk + Zk'1 ~ - Pk,Bim )('1:)Pk = 0
[:£:
PkBi-r)('1:)Pk"Zk~}kJ
r*O k"=l (m = 0, ± 1,.··; k'
=
¢(k, m).
(3.24)
3.
RAPIDLY OSCILLATING COEFFICIENTS
357
System (3.24) is a system of Riccati operator equations for fixed k with m = 0, = p(k, m). In order for the principal asymptotic part of D('ro, 'ro; }.) to be the identity operator:
± 1,.·· and k'
lim D('ro, 'ro; }.)
=
I,
A~OO
it suffices, as is easily verified, to take Z k':'k('ro) = O. Thus system (3.24) should be considered under zero initial conditions. We remark that it then generally has a nonzero solution provided (k' = p(k,
m».
We note that if for a given k there exists only one index k' that is representable in the form k' = p(k, ml), system (3.24) reduces to the single equation (Zk~)'
+ Zk~) PkBt m ,) Pk,Zk~) + Zk':',,') PkBiO) Pk - Pk,BiO) Pk,Zk~) - Pk,Bim ,) Pk = O.
(3.25)
In general, system (3.24) need not have solutions that are extendable onto the whole interval [0, T]; but if a solution exists, then, after finding all of the operators Z k':i1 , we can determine the operators (3.26) Furthermore, the right sides of equations (3.22) and (3.23) become known and it thus becomes possible to determine all of the operators P k ' Vk'{') P k except those which correspond to k' = p(k, m). The calculations at subsequent steps (s = 2,.··,p) are carried out in exactly the same way. At each step the operators Pk,vk~~lPk left undefined at the preceding step are determined. We note, incidentally, that the differential equations for these operators turn out, as is easily verified, to be linear and not of second degree, as in the case s = 1. The last step (s =p) can be carried out by choosing the remaining indeterminate operators arbitrarily, for example, by putting (k'
=
p(k,
m».
For these operators are not subject to any other relations, as was the case for s <po The final construction of an operator approximating the evolution operator U('r, 'ro; }.) is carried out in the same way as in § 2. We will therefore construct only the first approximation. Consider the operator (3.27) From (3.5) we get
VIII.
358
UI(ZOO, ZOO; A)
EQUATION WITH A LARGE PARAMETER
+ -5.-
kfJVkO(zoo, ZOO; A)
=
Vkl(ZOO, ZOO; A)} PkC(ZOo)
i; {m*O I: V~)(zoo, ZOO; A)eimAVT
=
k=1
+
Vk8) (zoo, ZOO; A)
l
+
Vkl(ZOO, ZOO; A)} PkC(ZOo)
=(1 + l ~ Vkl(ZOO, ZOO; A)Pk) C(ZOo)· It therefore suffices to take C(ZOo) = 1 -
T1
~
Vkl(ZOO, ZOo; A)Pk.
Substituting this expression into (3.27) and neglecting terms of order O(A-I), we obtain the formula (3.28) We note that this expression contains (in contrast to the nonresonance case) rapidly oscillating terms with amplitudes that do not tend to zero as A -4 00. 4. Example. Let us illustrate the method presented above with a simple example. We consider the scalar equation of second order
y"(t)
+ [1
- sp(st) cos 2t]y(t) = 0,
(3.29)
which contains a small, slowly varying coefficient sp(st). When this coefficient is a constant, equation (3.29) is a special case of Mathieu's equation. We reduce (3.29) to a system ofform (3.1). To this end we introduce a new variable zo = st and put A = l/s. Then
~~ + AZ[ 1 -
l
p(zo)cos 2AZOJ y =
o.
(3.30)
Equation (3.30) is equivalent to the system
J~~
= AYh
11;
= -
A[1 -
-5.- p(zo)cos 2AZOJ y,
which can be written in the form of a single equation with matrix coefficients for the vector w
= (~J:
~~
=
{A(_
~ ~) +
p(zo) (~
In this equation we make the substitution
~) cos 2AZO}W.
(3.31)
3.
359
RAPIDLY OSCILLATING COEFFICIENTS
in order to reduce its principal part to diagonal form. We write the resultant equation in the form
0) + . cjJ("C) (- 1 4 1
d¢={Ai(1 d"C
- 0
1
(3.32)
I
This equation is of form (3.1) for p = 1, with
(1 _ 0)l ' B(1) _ BH) _
_ 2 B - . » - , °- I 0
1
BiO) = 0,
-
1
. cjJ("C) (- 1
I
-
4
1
- 11)')
(3.33)
B(Z)("C; A) = O.
The operator Bo obviously satisfies condition (C~;). Its spectrum consists of the two points Al = i and Az = - i, to which correspond the spectral projections (3.34)
Since Al - Az = 2i = wi, we have only the resonance case to deal with. We will find the first approximation for the solving operator of equation (3.32). Using formulas (3.17) and (3.18), we get
l
=PI =(~ ~), V~) =Pz =(~ ~), P1Vg)P1 =0, PzVi6)P1 =0, P2V~Ol)P2 =0, I Vi8)
(3.35)
PI V~Ol) Pz = 0
and thus V~Ol) =
vg) = 0,
O.
(3.36)
In addition, from (3.18) it follows that PI viz 1) PI = 0,
P zV~6) Pz = O.
We put p zvi( 1)p1 = Zl and P1V~6)Pz = Z2' Then viol) = Z10
V~6) = Z2'
(3.37)
Equation (3.25} for the operator Zl has the form Z{
+ Z l P1BP) P2Z 1 -
pzBi-l) PI = O.
(3.38)
The operator Zl maps the subspace ~h into )8z. Its matrix must therefore be proportional to the matrix C? Setting
g).
Zl("C, "Co)
=
iz("C,
"Co)(~ ~),
(3.39)
we obtain from (3.38) and (3.39), after some simple calculations, the equation
VIII.
360
z'(zo,
EQUATION WITH A LARGE PARAMETER
ZOO)
+ ¢(ZO)[Z2(ZO, ZOO)
1]
-
=
o.
Its solution satisfying the condition z(zoo, zoo) = 0 has the form T
z(zo, zoo) = - tanh S ¢(s )ds. TO It can be shown in an analogous way that
= iz(zo, zoo)(~
Z2(ZO, zoo)
(3.40)
~),
-
(3.41)
where z(zo, zoo) is the same function as (3.40). Using formula (3.21), we get
Dn =
¢(zo)z(zo)(~ ~),
D21
¢(zo)z(zo)(~ ~).
=
(3.42)
Thus equation (3.6) takes the form dY?)ldzo = (0.
which implies that Y (1)( 1 ZO,
) _ ZOo
-
+ ¢(zo)z(zo, zoo»)P1YiIl,
;,l(T-To) f~o¢(S)'(S.To)ds
e
e
e;,l(T-To) P h S rI.()d cos ~o'f' S S l'
P _ 1 -
(3.43)
.
Analogously, we get y~Il(zo, zoo) =
e-;,l(T-TO)
cosh
S
~o¢(s )ds
P 2•
(3.44)
Substituting these expressions into (3.28), we obtain the formula ei,l(T-To) U(zo, zoo; A) = (PI + ZIC 2iAT ) cosh S~o¢(s)ds .
, + (P2 + Z2 e2 ,,lT)
e-iA(T-To) cosh S~o¢(s )ds
( 1)
+ a A'
which, after returning to the original notation, permits us to approximately calculate the fundamental matrix Wet, to; c) for the solutions of the original equation (3.29):
Wet, to; c)
(_? 6)sin(t -to) tanh c I¢(cs)ds [(? 6) cos(t + to) + (6 _?) sin(t + to)]}
={(6 ?)cos(t - to) x
+
1 cosh c S:o¢(cs)ds
+
O(c)
(0
~
to; t
~
Tic).
361
EXERCISES
We note that if we had considered in place of (3.29) the equation
+ [1
y"
- e¢(et) cos 2wt]y(t) = 0
with w "# 1, we would have encountered the nonresonance case. Some simple calculations show that in this case W(t, to; e)
=
(6 ?) cos(t -
to)
+ (_?
6) sin(t -
to)
+ O(e).
EXERCISES 1. Consider the inhomogeneous equation d
+ He; A)eUOCr>,
(0.1)
where B('C; A) is the same operator as (1.9) and j('C; A)
= £; ji")A- k + 0
(.1.- 0 + 1»).
k~O
Find an asymptotic representation for the solutions of this equation. The construction of asymptotic approximations for such an equation essentially depends on whether the values of the function k('C) = 8'('C) for certain 'C can appear in the spectrum of the operator BO<'C) (resonance) or not (absence of resonance) (see S. F. Fescenko, N. I. Skil' and L. D. Nikolenko [1], Ju. L. Daleckii and S. G. Krein [1] and Ju. L. Daleckii [1, 21). Hint. If the points of the curve .1.= k('C) (0:0;;:":0;;: T) appear in the spectral sets O'j('C) (j = 1.. ··, s) but not in any other spectral sets of Bo('C), a solution should be sought in the form q(,,; A)
= eaOCr ) [ j t Vi'C; A)7}j("; A) + g(,,; A)],
where the 7}j ('C; A) are vector functions satisfying the equations (j= 1,.··,s),
the operator functions Vj' OJ and vector functions hj , g being sought in the form of expansions of type (1.11), (1.12). 2. Obtain under condition (C,!!) or (Cp) the following estimate for the pth approximation to a solution of the equation of the preceding exercise: (0:0;;:
'C:O;;:
T).
Obtain the following improved estimate for the solution with zero initial value in the nonresonance case:
3. Show that the expression
which arises when equation (0.1) is solved by making use of the general formula (111.1.2) and the expansion (1. 7) for the evolution operator is of order 0(.1. - ) if the values of 8' ('C) do not appear in
362
VIII. EQUATION WITH A LARGE PARAMETER
the spectral set I1k(r) and the functionf(r; A) is differentiable (Ju. L. Daleckii and S. G. Krein [1]). 4. Construct asymptotic approximations for the equation of second order A(r; s) d 2if;/dr2
+ sB(r; s) dif;/dr + C(r; s) if; =
0,
assuming that the coefficients have expansions of type (1.9) for s = ,1-1, the operator Ao(r) is invertible and the spectrum of the operator Aol(r)Bo(r) decomposes into nonintersecting spectral sets. Obtain an estimate of the approximations, assuming that the operators Ao(r) and Bo(r) are positive (Ju. L. Daleckii and S. G. Krein [1]). 5. Consider the boundary problem fdif;/dt = iV£o(t)if;, \Ecf;(O) + if;(T) = 0,
(0.2)
where £o(r) is a twice continuously differentiable uniformly positive periodic operator function. A point A for which this problem has only the trivial solution is called a regular point of the problem. The complement of the set of regular points is called, as usual, the spectrum of the problem. a) Show that if the spectrum of the operator / £ oCr) is of form (2.33), the spectrum of problem (0.2) under the special choice E = Q(T, 0) is contained in the system of intervals with endpoints
1) 2k+l Ajd O (k = flPi(S)ds
7r
± 0 (1) k
(j
= 1,2"",n; k = 1,2,.··).
b) Extend the preceding result to the case when the operator /£o(r) has a countable number of noncoincident eigenvalues pir) (k = 1,2,.··) for rEO [0, T] and the corresponding system of eigenspaces is complete. Hint. Make use of the arguments of § 2.5. In the case of a countable collection of eigenvalues consider the decomposition of the spectrum 11(/£o(r» = (PI(r») U"'U (Pn-I(r») Ul1 n(r), where I1n(r) = (pir),.··, Pn+p(r),.··). 6. Consider in a Hilbert space H the boundary problem {
fr (A(r)
~~) + B(r)u + A2C(r)u = u(O)
0,
= u(T) = O.
Suppose that the operators A(r) and C(r) are uniformly positive and twice continuously differentiable, and that the operator B(r) is bounded and continuous. Suppose, further, that the operator A-I(r)C(r) has a spectrum consisting of a sequence of positive numbers p}(r), with p}(r) *- p~(r) (j *- k, 0 ~ r ~ T), and that the sequence of corresponding eigenspaces ~j(r) is complete in.p. Obtain the asymptotic formula Ajk
± o(t)
=
fl~(r)dr ± 0
(U
for the endpoints of a system of intervals containing the spectrum of the problem. Hint. Reduce the equation to a canonical equation of first order in the space .p(2) = .p .p. 7. By developing the arguments in § 2.5, prove the e.xistence of stability bands of the finitedimensional equation (2.32) for arbitrarily large A under the assumption that the spectrum of the operator /£o(r) is of form (2.33). Hint. Use formula (2.34) to obtain an asymptotic formula for the spectrum of the monodromy operator U(T; A). To prove the existence of eigenvalues near the values obtained from the asymptotic formula, use Rouche's theorem.
+
NOTES
363
To find the stability bands, determine those values of), for which the characteristic multipliers are all of the same kind (see M. G. Krein [6]; for the definition of characteristic multipliers see Chapter V). NOTES Asymptotic methods for solving differential equations containing a parameter were first used by Liouville to investigate expansions in the eigenfunctions of a selfadjoint boundary problem. These methods were developed for arbitrary equations of nth order and systems of equations of first order in the fundamental papers of G. Birkhoff [I] and J. Tamarkin [I] and later in the works of W. Trjitzinsky [I], V. S. Pugacev [I] and others. In all of these works it was either assumed that the eigenvalues of the principal part of the equation are simple or assumed that their multiplicities do not vary over the entire interval [0, T]. A number of results on the asymptotic representation of solutions of ordinary differential equations were obtained in connection with problems in the spectral theory of differential operators by I. M. Rapoport [1]. The case when the spectrum of the matrix of the principal part of an equation decomposes into nonintersecting (for all t) spectral sets in the interiors of which the multiplicities of the eigenvalues can vary arbitrarily was first considered in the finite-dimensional case by S. F. Fescenko [I]. S. F. Fescenko indicated a procedure for asymptotically decomposing a finite-dimensional system of form (1.4) into subsystems corresponding to nonintersecting (for all t) spectral sets lying in the left halfplane. These and other results obtained by combining the methods applied in the above mentioned papers with the methods developed by N. M. Krylovand N. N. BogoIjubov [I] (see also N. N. Bogoljubov [1]) can be found in the subsequently published book o[S. F. Fescenko, N. I. Skil' and L. D. Nikolenko [I] (in particular, this book contains a detailed account of the history of the question and a bibliography). We note that in the original papers of S. F. Fescenko the validity of the method employed was not completely established, since he did not prove the differentiability (needed for carrying out the next step) of certain expressions arising at each step of the process. S. G. Krein and Ju. L. Daleckii extended these results to the case of equations in Hilbert space giving a complete substantiation of the method (Ju. L. Daleckii and S. G. Krein [I] and Ju. L. Daleckii [1, 2]). The results of these papers are presented in § §1 and 2 for the case of bounded operators. We note that the assumptions underlying the error estimate in §2 have been generalized. The results presented in § 3 are due to Ju. L. Daleckii [5]. The example given in § 2.5 was taken from the book of I. C. Gohberg and M. G. Krein [2] while the example in § 3.4 is from the article of M. K. Belkin and Ju. L. Daleckii [I].
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S.
M. LOZINSKIi
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A. D.
MAizEL'
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z.
NOTATION INDEX A+A* 2 A-A* A3 = ImA = 2i lB Banach space (real or complex) [lBb lBzl Banach space of bounded linear operators acting from lBl into lB2 Am
= ReA =
[lB]= [lB, lB]
C"-n dimensional complex space with a positive definite Hermitian scalar product .p-Hilbert space (real or complex) H,.9',G-Hermitian operators belonging to [.p] H» O-uniformly positive operator: Am(H) 0 H";p O-nonnegative operator: Am(H) ~ 0 1= I!!J-identity operator in lB ,I-Hermitian operator in.p having the property ,12 = I Jr-skew-Hermitian operator (Jr = -J;) in a real space.p having the property J} = - I
>
AM(H) = ~~ '(H)
"m
(Hx, x)
IfXlf2
. f (Hx, x) = ~~\i IfXlf2
P-projection belonging to [lB]: p2 = P R"-n dimensional Euclidean space u(A)-spectrum of an operator A u(A) = u +(A) U u _(A)-decomposition of a spectrum into spectral sets lying respectively in the interiors of the right and left halfplanes u(A) = u.{A) U u .(A)-decomposition of a spectrum into spectral sets lying respectively in the interior and exterior of the unit disk b
Ja -sign for a multiplicative integral with the indicated order of elementary factors .
dx
x=Tt=x
,
377
AUTHOR INDEX Abraham, R., 365 Adamjan, V.M., 143, 365 Agmon, S., 4, 365 Andronov, AA., 332, 365 Anosov, D.V., 195-197, 365 Antosiewicz, H.A, 333, 365
Frolov, P.A., 248, 373 Fuchs, L., 275, 368
Barbasin, E.A., 3, 143, 144, 365 Belkin, M.K., 363, 365 Bellman, R.E., 144, 148, 365 Bernstein, S.N., 63 Birjuk, G.I., 332, 365 Birkhoff, G.D., 334, 363, 365 Bogoljubov, N.N., 3, 332, 333, 363, 365, 372 Bohl, P., 3, 93, 123, 146-148, 196, 197, 329, 332,366 Brodskii, M.L., 68, 148, 366 Brouwer, L.E.J., 146 Bylov, B.F., 61, 146, 331, 366 Cesari, L., 366 Coddington, E.A., 141, 239, 275, 332, 366 Coppel, W.A., 3, 197,248,366 Courant, R., 89, 228, 366 Dahlquist, G.G., 62, 366 Daleckii, Ju. L., 2, 68, 141, 191, 192, 197,275, 332, 333, 361, 362, 363, 365, 366, 367 Demidovic, B.P., 3, 330, 332, 367 Derguzov, V.I., 4, 68, 249, 250, 367 Dieudonne, J., 3, 367 Diliberto, S.P., 248, 367 Domslak, Ju. I., 4, 367 Duffin, R.J., 92, 368 Eidel'man, S.D., 63, 89 Erugin, N.P., 197, 250, 368 Favard, J., 192, 368 Fescenko, S.F., 361, 363, 368 Floquet, G., 201, 218, 239, 240, 249, 368 Fomin, V.N., 4, 249, 368 Frobenius, G., 275, 368
Gantmaher, F.R., 275, 368 Gavrilov, N.I., 148, 368 Gel'fand, I.M., 61, 68, 248, 249,368,369 Germaidze, V.E., 328, 330, 369 Gihman, 1.1., 333, 369 Ginzburg, Ju. P., 64, 67, 68, 139, 369 Gohberg, I.e., 45, 192,246, 352, 353, 363, 369 Grobman, D.M., 61, 146, 331, 366 Hadamard, J., 196, 369 Halilov, Z.I., 4, 369 Halmos, P.R., 240, 370 Hartman, P., 3, 197, 370 Hilbert, D., 89, 228, 366 Hille, E., 274, 275, 370 Hirsch, A, 63 Howe, A., 248, 366 lohvidov, I.S., 64, 68, 369, 370 Ivanov, L.A, 191, 370 Jakubovic, V.A., 2, 142, 146, 197, 242, 245, 248-250, 370, 372 Jonas, P., 68, 370 Kantorovic, L.V., 65, 370 Kapica, P.L., 245, 370 Karaseva, T.M., 244, 370 Kato, T., 197, 371 Korobkova, I.K., 275, 367 Kovtun, 1.1., 68, 250, 371 Krasnosel'skii, M.A., 44, 277, 327, 333, 371 Krasovskii, N.N., 3, 330, 369, 371 Krein, M.G., 1-3, 35, 44, 45, 64, 67, 68, 92, 93, 143, 147, 148, 192, 196, 225, 242-247, 249, 250, 332, 352, 353, 363, 367, 370--372 Krein, S.G., 4, 63, 89, 192, 197, 277, 327, 333, 361, 362, 363, 367, 371, 372 Krylov, N.M., 333, 363, 372 Kucer, D.L., 2, 144, 148, 330, 372
378
AUTHOR INDEX
KUhne, R., 92, 372 Kurzweil, J., 3, 333, 373 Landau, L.D., 245, 373 Langer, H., 68, 92, 372, 373 Levin, B. Ja., 63, 373 Levinson, N., 141, 239, 275, 332, 366 Levitan, B.M., 373 Lidskii, V.B., 68, 242, 248, 249, 369, 373, 375 Lifsic, E.M., 245, 373 Liouville, J., 363 Ljapunov, A.M., 2, 32, 68, 93, 146, 197, 198, 201, 241-243, 247, 249, 279, 285, 287, 332, 373 Ljascenko, N.Ja., 147, 197, 373 Ljubarskii, G.Ja., 225,372 Ljubic, Ju.I., 92, 373 Lozinskii, S.M., 62, 146, 373 Lumer, G., 240, 370 Lykova, O.B., 3, 333, 374 Maizel', A.D., 194, 196, 373 Malkin, I.G., 145, 147, 196, 330, 373 Mandel'stamm, L.I., 333, 374 Marcus, M., 63, 374 Markov, A.A., 35, 374 Markus, L., 197, 374 Massera, J.L., 3, 6, 148, 163, 196, 197, 239, 240, 249, 374 Melik-Adamjan, F.E., 143, 372 Millionscikov, V.M., 148, 374 Mil'man, D.P., 44 Mine, H., 63, 374 Mirocnik, L. Ja., 241, 242, 374 Mitropol'skii, Ju.A., 3, 332, 333, 366, 374 Myskis, A.D., 146, 374 Naimark, M.A., 334, 375 Neigauz, M.G., 242, 375 Neimark, Ju.l., 332, 375 Nemyckii, V., 61, 146, 331, 366 Nikolenko, L.D., 361, 363, 368 Nirenberg, L., 4, 365 Ostrowski, A.M., 68, ·375
379
Papaleksi, N.D., 333, 374 Perov, A.I., 92 Perron, 0., 123, 145, 147, 148, 196, 329, 332, 375 Persidskii, K.P., 147, 148, 375 Phillips, R.S., 67, 68, 375 Pliss, V.A., 375 Poincare, H., 249,331,332 van der Pol, B., 333 Potapov, V.P., 64, 139, 375 Povzner, A.Ja., 328, 375 Pugaeev, V.S., 363, 375 Rabinovic, I.M., 146, 374 Rapoport, I.M., 193, 363, 375 Robbin, J., 365 Rosenblum, M., 68, 375 Rutman, M.A., 2, 35, 147, 332, 372, 376 Scerbakov, B.A., 329, 332, 376 Schaffer, J., 3, 6, 148, 163, 196, 197, 239, 240, 249,374,376 Schneider, H., 68, 375 Sibuya, Y., 192 Silov, G.E., 61, 369 Sinai, Ja.G., 195-197,365 Sircenko, Z.S., 333, 376 Skil', N.I., 361, 363, 368 SmuI' jan, Ju.L., 68, 372 Starzinskii, V.M., 249, 376 Stokalo, I.Z., 250, 376 Svarcman, P.A., 275, 376 Sz.-Nagy, B. von, 34, 68, 376 Tamarkin, J.D., 334, 363, 376 Taussky, 0., 68, 376 Trjitzinsky, W., 363, 376 Vinograd, R.E., 61, 146, 331, 366 Vitt, A.A., 332, 365 Vorel, Z., 333, 373 Wasow, W., 192, 376 Weyl, H., 4, 197, 376 Wintner, A., 62, 146, 148, 197,376
SUBJECT INDEX A-operator, 143 Adjoint, 31 Adjoint equation, 100 Algebraic multiplicity, 59 Almost periodic function, 87 Fourier coefficient of, 92 mean value of, 92 module of, 192 spectrum of, 92 Alternating multiplicative integral, 140 Amplitude, angular, 65 Angle, 64 Angular amplitude, 65 deviation, 65 distance, 156 operator, 67 Approximate decomposition, 337 Asymptotic equivalence operator, 143 phase, 305 Asymptotically comparable equations, 125 equivalent equations, 115 stable solution, 279 Attraction correspondence, 115 Autonomous equation, 281 Averaging principle for a finite interval, 319 for the real line, 327 Balanced operator, 66 Banach space, 10 Banach spaces direct sum of, 13 isometric isomorphism between, 10 isomorphic, 10 isomorphism between, 10 Banach's theorem, 11 Bernstein's theorem, 63 Bistable equation, 113 Bochner integrable function, 95
integral, 95 Bochner's theorem, 87 Bohl exponent lower, 118, 119, 127 upper, 118, 119, 127 strict, 122, 127 at zero, 282 interval, 118, 119 spectrum, 174 Boundary problem regular point of, 224, 227, 362 selfadjoint, 352 spectrum of, 224, 362 Bounded operator, 10 Canonical equation, 142, 218, 225 of positive type, 222, 226 stable, 219, 225 strongly stable, 220, 225 Cauchy operator, 101 sequence, 10 Cayley transform, 45 Central stability band, 221 Characteristic multiplier, 224 of the first kind, 225 of the second kind, 225 Coalescing shift, 265 Compact set, 134 Comparable spectral sets, 273 Complementary projections, 12 Complemented subspace, 12 Complex hull, 14 Condition (C j ), (C 2), (C 3), 323 (C'll), (Cj;), 342 S"L, 136 Cone, 56 Cone inequality theorem, 57 Conjugation equation, 152 Continuously differentiable function, 54 Contraction, 52
380
SUBJECT INDEX
uniform W-,38 with coefficient q, 66 Contraction principle, 52 Convergence property, 330 Countably valued function, 94 Decomposable operator, normally W-, 48 Decomposition of an equation, approximate, 337 of the identity, 13 of a spectrum, normal, 265 stably normal, 265 Definite subspace, W-,47 Deviation, angular, 65 Dichotomic equation, e-, 73, 162, 195 operator, e-, 20 u-,41 Dichotomy, regular exponential, 163 Differentiable function, 54, 96 continuously, 54 Direct complement, 12 sum of Banach spaces, 13 of Hilbert spaces, 14 of subspaces, 12 Disjoint projections, 13 subspaces, 155, 156 Dissipative operator, 33, 34 uniformly, 34 uniformly, W-, 34 Domain, stability, 248 e-dichotomic, equation, 73, 162, 163, 195 operator, 20 Eigenvalue algebraic multiplicity of, 59 index of, 59 normal, 59 mot manifold of, 59 root space of, 59 Element negative, W-, 37 neutral, W-, 37 nonnegative, W-, 37 nonpositive, W-, 37 positive, W-, 37 Elementary shift, 265 Entire function of exponential type, 63 of minimal type, 255
Equation adjoint, 100 autonomous, 281 bistable, 113 canonical, 142,218,226 conjugation, 152 e-dichotomic, 73, 162, 163, 195 Hamiltonian, 142,218 of positive type, 222, 226 quasilinear, 280 reducible, 159 with a regular singularity, 252 resolvent, 15 Riccati operator, 194 with slowly varying coefficients, 335 stable left, 113 right, 112, 219, 225 strongly, 220, 225 uniformly left, 113 uniformly right, 112 unlimitedly, 146 variational, 281 Equation operator, 67, 208 minus, 208 plus, 208 Equations asymptotically comparable, 125 asymptotically equivalent, 115 integrally comparable, 114 kinematically similar, 157 L-diagonal systems of, 193 Evolution operator, 100 Expansion, uniform W-, 206 Exponential dichotomy, regular, 163 operator function, 25 splitting, 174 Exponentially dichotomic operator, 20 Favard's theorem, 192 Floquet representation, 201 of second order, 239 Force gyroscopic, 90 Rayleigh, 90 resistance, 90 Fourier coefficient, 92 Fn!chet derivative, 54 of second and higher orders, 54, 55 Function almost periodic, 87 Bochner integrable, 95 of class K A ,17
KA.B,22
381
382 L~,
SUBJECT INDEX
307
(M,q), (M,q)+, (M,q,p), (M,q,p)+, 297
countably valued, 94 differentiable, 54, 96 entire of exponential type, 63 of minimal type, 255 Green, 80, 167 incomplete, 87, 176 periodic, T-, 85 piecewise analytic, 17 principal, 81, 168 strongly integrable, 95 strongly measurable, 95 Fundamental operator, 101 solution, 252
-
Gaps, 174 Generalized Ljapunov theorem, 32 Green function, 80, 167 incomplete, 87, 176 periodic, T-, 85 principal, 81, 168 Gyroscopic force, 90 Hamiltonian, 218 equation, 142 Hamilton's equation, 218 Hausdorff's theorem, 63 Hermitian, 218 of positive type, 222 Hermitian operator, 31 indefinite, 37 skew-,36 spectral length of, 215 W-,45 Hilbert spaces, orthogonal sum of, 14 Homoclinic motion, 332 Hyperboloids minus, 74 plus, 74 Identification operator, 66 Imaginary part, 32 Incomplete Green function, 87, 176 Indefinite Hermitian operator, 37 scalar product, 37 Index, 59 Integral Bochner, 95 continuity, 320 manifold, 307 multiplicative, 102
alternating, 140 Stieltjes, 103, 140 Riesz,20 shift, 265 Integrally bounded operator function, 116 comparable equations, 114 . Intertwining property, 152 Invariant subspace, 20 Inverse, 11 Inverse function theorem, 55 Isometric isomorphism, 10 Isomorphic Banach spaces, 10 Isomorphism, 10 Kinematically similar equations, 157 Kreln-Krasnosel'skil-Mil'man theorem, 44, 45 L-diagonal systems, 193 Law of inertia, 66 Left 9'-set, 170 stable equation, 113 uniformly, 113 Linear operator, 10 Ljapunov exponent lower, 118, 127 strict, 26 strict upper, 122, 127 upper, 26, 90,117,127 spectrum, upper, 117 theorem, generalized, 32 Logarithmic norm, 62 Lower Bohl exponent, 118, 119, 127 Ljapunovexponent, 118, 127 Manifold integral, 307 (p,7j )-, 307 (p,7j,+)-,(p, 7j,-)-, 315
root, 59 . Markov's theorem, 35 Maximal "-nonnegative subspace, 67 norm, 13 Mean value, 92 Minus equation operator, 208 hyperboloids, 74 Module, 192 Monodromyoperator, 199,207 Multiplicative integral, 102 alternating, 140
SUBJECT INDEX
Stieltjes, 103, 140 Multiplier, characteristic, 224 of the first kind, 225 of the second kind, 225 Negative element, W-, 37 operator, 32 uniformly, 32 subspace, uniformly W-, 38 W-,38 Neutral element, W-, 37 Nonenlargement operator, W-, 64 Nonnegative element, W-, 37 operator, 31 W-,63 subspace maximal fl' -, 67 W-,37 Nonpositive element, W-, 37 operator, 32 subspace, W-, 38 Norm of an element, 10 maximal,13 Norm of an operator, 10 logarithmic, 62 Normal decomposition, 265 stably, 265 eigenvalue, 59 operator, 58 set, 265 Normally W-decomposable operator, 48 Normed space, 10 Norms, topologically equivalent, 10 w-limit operator, 135 Operator, A-, 143 adjoint of, 31 of algebraic type, 59 angular, 67 amplitude of, 65 deviation of, 65-asymptotic equivalence, 143 balanced, 66 bounded, 10 Cauchy, 101 contraction, 52 uniform W-, 38 with coefficient q, 66 decomposable, normally W-, 48 dichotomic e-,20
u-,41 equation, 67, 208 minus, 208 plus, 208 evolution, 100 expansion, uniform W-, 206 exponentially dichotomic, 20 fundamental,101 Hermitian, 31 indefinite, 37 skew-,36 W-,45 identification, 66 imaginary part of, 32 inverse of, 11 linear, 10 logarithmic norm of, 62 monodromy, 199,207 negative, 32 uniformly, 32 nonenlargement, W-, 64 nonnegative, 31 W-,63 non positive, 32 norm of, 10 logarithmic, 62 normal,58 w-limit, 135 positive, 31 uniformly, 32 quasinilpotent, 63 real part of, 32 resolvent of, 15 right spectral bound of, 139 signature, 248 of simplest type, 57 solving, 100 spectral radius of, 199 spectrum of, 15 stable, 34 unitary, 34 strongly stable W-, 49 W-,45 Volterra, 63 Operator function exponential, 25 integrally bounded, 116 of positive type, 226 precompactly valued, 135 stationary at infinity, 136 strongly continuous, 96 Operator halfplane, right, 67 Operators, product of, 11 Orbital asymptotic stability, 305
383
384
SUBJECT INDEX
Orthogonal elements, W-,37 projection, 14 sets, W-,37 sum, 14 vectors, W-, 37 Phase space, 69 Piecewise analytic function, 17 Plus equation operator, 208 hyperboloids, 74 Point of regular type, 42 Positive element, W-,37 operator, 31 uniformly, 32 subspace, W- 37 uniformly W-, 37 Potential, 142 Precompact set, 135 Precompactiy valued operator function 135 Principle ' averaging, for a finite interval, 319 contraction, 52 for the real line, 327 uniform boundedness, 11 Product of operators, 11 Projection, 12 orthogonal, 14 spatial, 307 spectral, 19 uniformly conjugated, 157 Projections mutually complementary, 12 pairwise disjoint; 13 Property ~(J), N), 122 ~(J), N, p), 282 convergence, 330 intertwining, 152 DiN, if», 329 Quasilinear equation, 280 Quasinilpotent op;rator, 63 Rayleigh force, 90 Real part, 32 Reducible equation, 159 Regular exponential dichotomy, 163 point of a boundary problem, 224, 227, 362 of an operator, 15 Resistance force, 90 Resolvent, 15
Resolvent equation, 15 (p, 7])-manifold, 307 (p,7], + )-manifold, 315 (p, 7], - )-manifold, 315 Riccati operator equation 194 ' Riesz integral, 20 Right operator halfplane, 67 9'-set, 169 spectral bound, 139 stable equation, 112,219,225 uniformly, 112 Root manifold, 59 space, 59 9'-set left, 170 right, 169 Scalar product, indefinite, 37 products, topologically equivalent 32 Selfadj~int boundary problem, 352 ' SeparatIOn, a-, 305 Set compact, 134 normal,265 precompact, 135 9'-,169 Sets, orthogonal, W-, 37 Shift coalescing, 265 elementary, 265 integral, 265 Signature operator, 248 Skew-Hermitian operator, 36 Slow time, 326, 335 Solution asymptotic phase of, 305 fundamental, 252 stable, 279 asymptotically, 279 uniformly, 279 Solving operator, 100 Space Banach,10 complex hull of, 14 normed,1O phase, 69 root, 59 Spaces B~nach, direct sum of, 13 Hilbert, orthogonal sum of, 14 isometric isomorphism between 10 isomorphic, 10 '
SUBJECT INDEX
isomorphism between, 10 Spatial projection, 307 Spectral bound, right, 139 length, 215 mapping theorem, 19 projection, 20 radius, 199 set, 19 sets, comparable, 273 Spectrum a-separation of, 305 Bohl,174 A, 90
Ljapunov, upper, 117 normal decomposition of, 265 stably, 265 of a boundary problem, 224, 362 of an almost periodic function, 92 of an operator, 15 Stability band, central, 221 domain, 248 point, 221, 226 strong, 221, 226 stability, orbital asymptotic, 305 Z-tests for, 249 Stable canonical equation, 219, 225 strongly, 220, 225 equation left, 113 uniformly, 113 right, 112 uniformly, 112 unlimitedly, 146 operator, 34 solution. 279 asymptotically, 279 uniformly, 279 W-unitary operator, strongly, 49 Stably normal decomposition, 265 Stationary at infinity operator function, 136 Stieltjes multiplicative integral, 103, 140 Stone's theorem, 37 Strict Ljapunov exponent, 26 Strict upper Bohl exponent, 122 Ljapunovexponent,122 Strongly continuous operator function, 96 integrable function, 95 measurable function, 95 stable canonical equation, 220, 225 stable W-unitary operator, 49
385
Subspace, 12 complemented, 12 definite W-,47 direct complement of, 12 invariant, 20 negative, uniformly W-, 38 W-,38 nonnegative, maximal f, 67 W-,37 nonpositive, W-, 38 positive, uniformly W-, 37 W-,37 Subspaces direct sum of, 12 disjoint, angular distance between, 156 Sz.-Nagy's theorem, 34 Theorem Banach, 11 Bernstein, 63 Bochner, 87 on a boundary point of the spectrum, 28 cone inequality, 57 Favard,192 generalized Ljapunov, 32 Hausdorff, 63 inverse function, 55 Krein-Krasnosel'skii-Mil'man, 44, 45 Markov, 35 spectral mapping, 19 Stone, 37 Sz.-Nagy, 34 Topologically equivalent norms, 10 scalar product, 32 Transform, Cayley, 45 Transformer, 99 u-dichotomic operator, 41 Uniform boundedness principle, 11 W-contraction, 38 with coefficient q, 66 W-expansion, 206 Uniformly conjugated projection, 157 dissipative operator, 34 left stable equation, 113 negative operator, 32 positive operator, 32 right stable equation, 112 stable solution, 279 W-dissipative operator, 34 W-negative subspace, 38 W-positive subspace, 37
386 Unitary operator, 34 strongly stable W-, 49 W-,45 Unlimitedly stable equation, 146 Upper Bohl exponent, 118, 119, 127 strict, 122, 127 at zero, 282 Ljapunovexponent, 26, 90,117,127 strict, 122, 127
SUBJECT INDEX
Ljapunov spectrum, 117 Variational equation, 281 Vectors angle between, 64 orthogonal, W-, 37 Volterra operator, 63 Z-tests, 249
