Linear Differential Equations in Banach Space (Translations of Mathematical Monographs)

TRANSLATIONS OF MATHEMATICAL MONOGRAPHS

Volume 29

LINEAR DIFFERENTIAL EQUATIONS IN BANACH SPACE

by ..,

S. G. KREIN

AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 02904 1971

JII1HEVIHbIE LII1rI>rI>EPEHlmAJIbHbIE YPABHEHI1H B BAHAXOBOM npOCTP AHCTBE C. r. KPEMH 113AaTeJIbCTBO "HaYKa" rJIaBHaR peAaK~MR M3MKo-MaTeMaTMlIecKol1. JIMTepaTypbI MocKBa 1963

Translated from the Russian by

J.

M. Danskin

AMS 1970 subject class ifications. Primary 34G05, 47D05; Secondary 34K05, 35J15, 35KI0, 35LlO, 35R20, 47A50, 65J05.

Library of Congress Cataloging in Publication Data

..,

Krein, Selim Grigor'evich. Linear differential equations in Banach space. (Translations of mathematical monographs, v. 29) Translation of Lineinye differentSial'nye uravnenila v banakhovom prostranstve. Bibliography: p. 1. Differential equations, Linear. 2. Banach spaces. I. Title. II. Series. 515'.35 71-37141 QA372.K9l7l3 ISBN 0-8218-1579-2

Copyright © 1972 by the American Mathematical Society Printed in the United States of America

PREFACE

This book deals with the theory of linear differential equations in Banach space with unbounded operator coefficients. The origin of this theory dates from the work of Hille and Yosida (1948), in which the first existence theorems were obtainEd for the Cauchy problem for the equation r = A% with A an unbounded operator in Banach space. These were formulated in terms of the theory of semigroups of operators. Yosida, and later Feller, connected these investigations of semigroups with various problems for the diffusion equation. Parallel with this, Hille, and then Phillips, began to construct the theory of the abstract Cauchy problem for equations in Banach space. In 19531954 Lax, Milgram, and Ljance applied semigroup methods to the investigation of various classes of parabolic equations. In 1953 Kato [42] made an essential step forward. He obtained an existence theorem for the solution of the Cauchy problem for the equation %' = A (t) %. with a variable unbounded operator A (t). In their papers Hille, Yosida, Phillips and Kato laid the foundations of the theory of differential equations with unbounded operators, which thereafter became a field of independent interest, attracting the attention of many mathematicians. This book presents a number of fundamental results of the theory of strongly continuous semigroups of operators. However they are mostly formulated in terms of the properties of differential equations. Therefore they are given with complete proofs, and the book may be read independently of the book "Functional Analysis and Semi-groups" by Hille and Phillips [2]. For the understanding of the contents one needs only a knowledge of the general aspects of the theory of operators, which are presented without proof in the Introduction. The book considers only strong solutions of differential equations. Therefore it touches hardly at all on the results of ViAik and Ladyienskaja, and later Lions and others on the theory of weak and generalized solutions of differential equations in Hilbert space. These results are reflected to a considerable extent in the book of Lions [5]. The book investigates the correct statement of the problem for differential equations in Banach space and certain asymptotic and approximate methods for solving them..The limited objective of the iii

iv

PREFACE

book did not make it possible to include in it questions of qualitative investigation of such properties of the solutions as convexity, speed of decrease, stability, stabilization, asymptotic behavior and so forth. Many interesting reslJlts in this field lie outside our field of view. In particular, there is no mention here of such substantial investigations as the papers of Agmon and Nirenberg [116] and Evgrafov [132]. We have only partially treated the work of Ljubic [71]. All the papers of this series known to the author are included in the list of papers directly relating in content to the material expounded in the book. The contents of the book essentially refted the work of the Seminar on Differential Equations at Voronezh University, which the author has directed for more than ten years now. We have paid particular attention in the exposition to the results obtained by the participants in the seminar: M. A. Krasnosel'skii, P. E. Sobolevskii, Ju. L. Daleckii, O. M. Kozlov, B. S. Mitjagin, O. I. Prozorovskaja, Ja. D. Mamedov, L. I. Jakut, G. I. Laptev and N. N. Gudovic. The results presented in the book are not always formulated in the most general form. The author tried to choose some mean level of generality, to which he had to lower some results, and to pull up some others. Compensation for the loss arising in this way will be found in remarks and in the references to the literature at the end of the book. The author would be distressed if the reader, in looking over the book, is not convinced that differential equations in Banach space are an interesting and important object of investigation. The author himself is convinced of this, and has therefore described possible applications of the theory schematically and illustratively. To develop them fully would appear to require another book. The author wishes to express his gratitude to all the participants in the Seminar on Differential Equations at Voronezh University who discussed at various times many parts of the book. Especially fruitful discussions were held with Ju. L. Daleckii, P. E. Sobolevskii, S. D. ~idel'man and S. Ja. Jakubov. The author thanks the Editor of the book, O. I. Prozorovskaja, whose remarks and proposals significantly increased the quality of the exposition.

TABLE OF CONTENTS PREFACE INTRODUCTION • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

i • • •

iii 1

CHAPl'ER I. EQUATIONS OF FIRST ORDER WITH CONSTANT OPERATOR.

SEMIGROUPS • • • • • • • • • • • • • • • • • • • • • • • • •

1. 2. 3. 4. 5. 6. 7. 8.

The Cauchy problem .• . . . . . . . • . . . . . . . • . . • • Uniformly correct Cauchy problem. . . • . . . . • . . . . . Weakened Cauchy problem . . . . . . . . . . . . . . • . . . Equations in Hilbert space. . . . . . . • . . . . . . . . . . . Fractional powers of operators. . . . • . . . . . . . . . . • . Nonhomogeneous equations.................. Equations with perturbed operators. . . . . . . . . . . . . Examples . . . • . . . • • . . • . . . . • . . . . . . . . . . . . . . CHAPl'ER II. EQUATIONS OF THE FIRST ORDER WITH VARIABLE OpERATOR. •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

Unbounded operators depending on a parameter . . • . . Equations with bounded operators. . . . . . • . . • . . . . The uniformly correct Cauchy problem. • . • . . • . • • . Weakened Cauchy problem • . . . . . . . . . . . . . . . • . Abstract parabolic equations with operators having variable domains • • . . . . • . . . . . • . . . . • • . • " . . • . . . CHAPTER III. EQUATIONS OF SECOND ORDER. • • • • • • • • • • • • 1. The hyperbolic case. The Cauchy problem . . . . . . . . . 2. The elliptic case. Boundary problems . • . . . • . . . . . . 3. The Cauchy problem for the complete equation of the second order............................... 1. 2. 3. 4. 5.

CHAPTER IV. ASYMPl'OTIC METHODS. • • • • • • • • • • • • • • • • •

1. Equations with a small parameter on the highest derivative • . . . . . . . . • . . . . . . . . . . . . . . • . . . . . . . . . . 2. Evolution of subspaces in Banach space. . . . . . . . . . . 3. Splitting of an equation into equations in subspaces . . . CHAPTER V. FINITE-DIFFERENCE

METHODS.............

1. Factor-method of solution of operator equations . . . . • 2. Finite-difference factor-method for evolution equations. REMARKS AND REFERENCES TO THE LITERATURE. • • • • • • • • • BIBLIOGRAPHY • • • • • • • • • • • • • • • • • • • • • . • • • • • • • • • • • v

24 24 42 59 82 108 129 143 158 176 176 188 192 209 228 238 238 249 270 283 283 297 309 333 333 340 362 377

lNTRODUcrION

The present book undertakes the study of a number of questions of the theory of linear differential equations in Banach space. In the exposition we suppose that the reader is familiar with the fundamentals of functional analysis as taught in university courses. However, in order to facilitate the understanding of the book, we present in the introduction, mainly without proof, those propositions of the theory of Banach spaces, the theory of functions with values in these spaces, and the theory of operators in Banach and Hilbert spaces, which will be used later on in the basic text. We do not here strive for the maximum generality of formulation. §1 Banach spaees and functionals 1. Banaeh spaee. A linear system E is said to be a oormed linear space, if to each element x E E there is assigned a real number II x II ~ 0, called the OOrTn of the element x, having the following properties: 1) ~x~ = 0 if and only if x = 0; 2) lI>.xll = 1>.lllxll; 3) I x + yll ~ II xII + b~· The number II x - y II is called th~ distance between the elements x and y and has the properties of a metric. In connection with this one says that the sequence x,. E E converges to the element x E E: x,. -+ x, if ~ x,. - x II -+ 0 as n. -+ One introduces in accordance with this the concepts of limit points of sets, and of closed, open, and compact sets of E. The sequence x,. E E is said to be fundamental if II x,. - x". K -+ 0 when m, n. -+ A convergent sequence is fundamental If every fundamental sequence in the space E converges, the space is said to be complete. A complete normed linear space is called a Banach space. Every normed linear space may be completed to a Banach space. A closed linear manifold Y of a Banach space is called a subspace of it. Every subspace is itself a Banach space. One says that the space E is decomposed into the direct sum of the two subspaces Y and .L if any element x E E may be represented uniquely in the form x = y + z, Where yEY and zE.L. Suppose that Y is a subspace of the Banach space E. A residue class relative to the subspace Y is the set X of elements of the form x + u, where x is a fixed element of E and u is any element of Y. The residue (XI.

(XI.

1

2

INTRODUCTION

classes form a linear space ElY under the natural operations of addition and multiplication by a scalar of these classes. The resulting space is called the lactor-space EI Y of the space E relative to the subspace Y. In the factor-space we may introduce a natural norm by means of the formula

I XII =

inf ~x + ull.

"E~

It turns out that in this norm the factor-space ElY is a Banach space. 2. Linear functionals. A functional I(x) defined on the Banach space E is said to be linear if it is additive and homogeneous, and if it satisfies the inequality (1.1)

I I(x) I ~ ell xii

(xEE),

where the constant c does not depend on x. The smallest possible constant in (1.1) is called the norm of the linear functional and is denoted by I III. The following formula holds: (1.2)

I I(x) I I III = sup -11-1-1 = "'EE

X

sup I I(x) I·

1"'1-1

A linear functional is continuous, and, conversely, an additive continuous functional is linear. If on a linear manifold ~ dense in E there is defined a homogeneous functional I(x) having property (1.1) for every x E~, then it is uniformly continuous on ~ and accordingly admits a unique extension to a continuous (linear)· functional defined on the entire space E. In a Banach space E there are sufficiently many linear functionals. More precisely, for each element Xo E E there is a linear functional lo(x) such that I loll = 1 and 10(Xo) = I XoII. Hence there results a formula, dual to (1.2): (1.3)

I x I = max II(x) I = max I~(IXII) I , 1/1-1

H

where in the last term the maximum is taken over all linear functionals defined on E. The following important assertion holds: The Hahn-Banach Theorem. II I(>(x) is a linear functional defined on a subspace .Sf' 01 a Banach space E, then it can be extended to the entire space E, with the norm preserved.

12. FUNCTIONS WITH VALUES IN A BANACH SPACE

3

One defines the operations of addition and multiplication on linear {unctionals in a natural way: (f 4- g) (x) = f(x) + g(x), (>.f) (x) = >..f(x). Thus, all the linear functionals on E form a normed linear space with norm (1.2). This space is complete, i.e. it is a Banach space, and it is called the space E* dual to the space E. The space dual to E * is called the second dual or bidual of E and is denoted by E**. Each element x E E generates a linear functional F" on the space E* by the formula F,,(f) =.f(x). It follows from (1.3) that the norm of the functional F" on E* is equal to the norm of the element x. Thus the space E is isometrically and linearly mapped into the space E**. If in addition the image of E coincides with the entire space E**, then E is said to be reflexive. 3. Principle of uniform boundedness. Suppose that on a Banach space E there is defined a family of continuous functionals ~.. (x) (a E ~ ), having the properties: 1) ~.. (x) ~ 0; 2) ~.. (x

+ y) ~ ~.. (x) + ~.. (y);

3) ~.. (>..x) = >.~.. (x) for >. ~ O. The principle of uniform boundedness asserts the following: if the family ~.. (x) is uniformly bounded at each element x E E, then there exists a oonstant c such that (aE~,xEE).

As an example of a functional satisfying conditions 1)-3), we may take the modulus of a linear functional: ~(x) = I f(x) I. From the principle of uniform boundedness one then obtains the assertion that if a family of linear functionals I{.. (x) I is uniformly bounded relative to a for each fixed x E E, then the family of norms I f .. 1 is uniformly bounded. §2. Functions with values in a Banach space 1. Continuity, difl'erentiability, analyticity. We shall consider functions x(t) , defined on a segment [0, T] of the real axis, whose values for each t are elements of a Banach space E. The function x(t) is said to be continuous at the point to, if

I x(t)

- x(to) 11-+0

as t-+to, and continuous on [0, T] if it is continuous at each point of [0, T]. The norm of a function continuous on [0, T] is a continuous scalar function.

INTRODUCTION

The set of all functions continuous on [0, T] with values in E obviously forms a linear space, which we denote by C(E; [0, T]). It is usual to introduce a norm into this space according to the formula (2.1) In this norm the space C(E; [0, T]) is a Banach space. Convergence in the norm (2.1) means uniform convergence on [0, T]. One says that the function x(t) has a right (left) derivative at the point to if there exists an element y E E such that

II

x(to

+ a:! - x(to)

-

y"

--+ 0

as 6t--+ + 0 (at--+ - 0). One then writes d+x(to) _ dt -Y

(d_X(to) _ ) dt -y.

If the right and left derivatives exist and coincide, then one says that the function x(t) is differentiable at the point to and that its derivative 1Q x' (to)

= dx~to) = y.

A function is differentiable on a segment (interval, semi-interval), if it is differentiable at each point of the segment (interval, semi-interval). In this case the function x' (t) is also a function with values in Banach space. If it is continuous, then one says that the function x(t) is continuously differentiable. If the right derivative of the continuous function x(t) exists at each point of the segment [0, T] and is continuous on it, then x(t) is continuously differentiable on that segment. One introduces in the natural way the concepts of n times differentiable and infinitely differentiable functions. The theorems of analysis on the differentiation of uniformly convergent sequences of functions remain in force. We shall also consider functions x(z), defined on some region G of the complex plane, taking on values in the space E. The element ~ is called the derivative of the function x(z) at the point Zo, if

II

x(Zo+

~-

x(Zo) -

~ II

--+0

12.

FUNCTIONS WITH VALUES IN A BANACH SPACE

5

asat-O. The function x(z) is said to be analytic in the region G if it has a derivative at each point of this region. In the neighborhood of each point Zo E G, an analytic function may be expanded in a series (2.2)

x(z)

=

.

L

a,,(z - Zo)",

were the a" are elements of the space E equal to (l/n!)x(ll)(Zo). Conversely, every power series of the form (2.2) defines an analytic function within a circle of convergence whose radius r can be found from the Cauchy-Hadamard formula

If x(z) is analytic in G, then for any linear functional I the scalar function I(x(z» is analytic in G. The converse assertion is also true. This fact makes it possible to obtain the properties of analytic functions with values in E from the properties of scalar analytic functions. For example, the Liouville theorem holds: if the function x(z) is analytic in the whole plane (i.e. if it is an entire function) and if it is bounded, then it is constant. We note that the norm I x(z) II of a function x (z) analytic in G is a logarithmically subharmonic function in G. Indeed, the modulus If(x(z» I of the analytic function f(x(z» has this property, which means that the function

I x(z) I =

sup If(x(z» I

liN -1

has it as well. 2. Integration. If the function x(t) is continuous on the segment [a, b], then one can define an integral for it as the limit of integral sums:

(2.3)

lim

LN (t ).1t = X

k-l

k

k

ib x(t) dt. II

Here the limit is understood in the sense of convergence relative to the norm of E when the diameter of the decomposition a = to < tl < '., < tN = b tends to zero. The estimate

6

INTRODUCTION

holds, and the mean value theorem:

1

6

x(t)dt

=

ali,

(b -

where i is an element of the closed convex hull of the set of values of the function x(t) on the segment [a, b]. The function

is continuously differentiable and y' (t) = x(t). For any continuous differentiable function the N ewton-Leibniz formula

Iby'(t)dt = y(b) -y(a) holds. Just as in classical analysis, one may introduce the concept of an improper integral. For example, if the function x(t) is continuous on [a, CD), then by its integral on [a, CD) we mean the limit (2.4)

r Jar '" x(t) dt = blim ..... Ja

b

x(t) dt.

If the limit relative to the norm of E exists, then one says that ~he integral (2.4) converges. This integral converges absolutely if

The ordinary convergence of an integral follows from its absolute convergence. One introduces analogously the concepts of improper integral of a discontinuous function and principal value of various improper integrals. One can consider integrals depending on a parameter. The classical theorems on continuous dependence on the parameter, and on integration and differentiation with respect to the parameter, carry over. In the present book we deal with the concepts of integrals of continuous functions and improper integrals. We recall that the most commonly used generalization of the Lebesgue integral is the Bochner integral of a function with values in a Banach space, whose theory is presented, for example, in [2].

12.

FUNCTIONS WITH VALUES IN A BANACH SPACE

7

3. Cauchy integral. In a way analogous to that in which the integral (2.3) was introduced, we can define an integral of a continuous fun~ tion given in a region G of the complex plane along a rectifiable Jordan curve lying in that region. For a function analytic in G the Cauchy

integral theorem remains valid, and also the Cauchy integral formula following from it:

= 21.

x(z)

d

r x(n dr,

Jrt- z

where r is a closed rectifiable Jordan curve bounding the region GI C GI and zE GI • For the derivatives of the function x(z) we have the formulas (II)

_

n!

r

d

x(t)

x (z)-2riJr(t-z)"+1

t.

4. Laplace transform.. Suppose that x(t) is a continuous function on [0, C1D) with values in E satisfying the condition

IIX(t) I

~ Me"'~

Then for ReA> lIJ the integral .t(A) ==

(2.5)

10"

e-Mx(t)dt

converges absolutely. The function .t(A) with values in E and analytic in the halfplane ReA> lIJ is called the Laplace transform of the function x(t). Sometimes we will apply the Laplace transform to functions having a singularity at the origin. In this case the integral (2.5) will be understood as improper relative to the point O. If the Laplace transform of some function x(t) is equal to zero, then x(t) == O. For t > 0 we have the following inversion theorem: (2.6)

x(t)

.

= !!....(~iCl+i .t(A) dt

2r'

CI-i..

eMdA) ,

A

Where a > max(lIJ,O) and the integral is understood in the sense of Principal value. If x(t) is absolutely continuous, or, in particular, continuously differentiable on some segment [tb~] (tl > 0), then formula (2.6) may be simplified:

8

INTRODUCTION

§3. Bounded linear operators 1. Bounded operators. An operator A, defined on a linear manifold of the Banach space E and operating into another Banach space F, is said to be linear if it is additive and homogeneous. The lineM manifold on which the operator is dermed is called its domain 9 (A), and the collection of elements of the form Ax (x E 9(A» is said to be its range !lR (A). If 9(A) = E and for all x E E the inequality (3.1)

IIAxIIF~

cllXllB

is satisfied, then the operator is said to be bounded, and the smallest value of the constant c is called the norm of the operator A and denoted by I A I B_Fo A bounded operator is continuous. Conversely, a continuous linear operator defined on the entire space E is bounded. If 9 (A) is dense in E and (3.1) holds for all elements of 9 (A), then the 'operator A may be extended by continuity to a bounded operator. A linear operator is said to be completely continuous,l) if it is defined on the entire space E and maps every bounded set in E into a compact set in F. Obviously a completely continuous operator is bounded. - If A is a bounded linear operator, then the functional cfI(x) = I Axil F has the properties required of functionals in the principle of uniform boundedness. Therefore, if a family of bounded linear operators is uniformly bounded at each element x of the space E, then the norms of the operators A.. are uniformly bounded. Under the natural definitions of addition and multiplication by a scalar, and with the norm

A:

the family Y (E, F) of bounded linear operators acting from E into F becomes a Banach space. In the space Y (E, F) the concept of strong convergence is very important. One says that the sequence All of bounded linear operators converges strongly to the operator A, if for any x E E I)Editor'B note. "Compact" is also used instead of "completely continuous."

i3. BOUNDED LINEAR OPERATORS

II Ax -

A,.x I F

-

0 for n _

9

co.

The limit operator is obviously linear. Using the principle of uniform boundedness, one shows that it is also bounded. From the convergence of a sequence of operators in the norm of the space .5t' (E. F), it follows of course that they converge strongly. The converse is false. The following assertion is very important. '1HE BANACH-STEINHAUS THEOREM. In order that a sequence 01 bounded linear operators All converge strongly, it is necessary and sufficient that the norms 01 the operators All be unilormly bounded and that the sequences A"x converge lor all x 01 some set dense in E. In what follows we shall frequently use the following easily proved fact. LEMMA 3.1. II a sequence 01 bounded linear operators All converges strongly to an operator A, then it converges unilormly to it on every compact set 01 E.

Suppose that I is a linear functional on the space F and that A E Y (E, F). Consider the functional

g(x) = I(Ax), defined on the space E. It is not hard to verify that it is linear. Thus, to each functional Ie F* there corresponds a functional g E E*. This correspondence is linear. The operator realizing the relation 1- g is called the operator A * dual to the operator A.2) By definition,

(A *f) (x) = I(Ax). The operator A * is a bounded linear operator, acting from F* into E*. Its norm is equal to the norm of the original operator A. 2. Operators depending on a parameter. A function A(t) (0 ~ t ~ T) with values in the space .5t' (E, F) of bounded linear operators will be called an operator depending on a parameter. Since .5t' (E, F) is a Banach space, the concepts of continuity, differentiability and analyticity considered in the preceding subsection extend to an operator depending on a parameter. 2)Editor's note. A * is also called the adjoint or transpose of A in the literature. (See, however, the footnote on p.19).

10

INTRODUCTION

However, in what follows we shall need all these concepts in the sense of strong convergence of operators. A bounded linear operator A(t) is said to be strongly continuous (at a point, on a segmen9) if for each % E E the function A (t) % with values in F is continuous (at the point, on the segment). If the operator A (t) is continuous in norm, then it is strongly continuous. The converse assertion is not true. It follows from the BanachSteinhaus Theorem that an operator A (t) which is strongly continuous on the segment [0, T) is uniformly bounded: I A (t) II E ..... F ~ c. In view of this a strongly continuous operator on [0, T) may be considered as a bounded linear operator mapping the space E into the space of continuous functions C(F; [0, T». The following assertions are verified immediately. LEMMA 3.2. If %(t) is a function continuous on [0, T) and with values in E, and A (t) (0 ~ t ~ T) is a strongly continuous operator from E into F, then the function A(t) %(t) is continuous in F. LEMMA 3.3. II A (t) (0 ~ t ~ T) is a strongly continuous operator from E into F, and B(t) (0 ~ t ~ T) is a strongly continuous operator from F into H, then B(t)A(t) is a strongly continuous operator from E into H.

Suppose that we are given a sequence of strongly' continuous operators An(t). We say that it converges strongly and uniformly for tE [0, T) if for each % E E the sequence of functions An(t) % with values in F converges uniformly on [0, T). In this case the sequence An(t) is uniformly bounded relative to n and t. An application of Lemma 3.1 to the spaces E and C(F; [0, T]) leads to the assertion: LEMMA 3.4. If the sequence of operators An(t) converges strongly and uniformly for tE [0, T), then the sequence of functions An(t)x, where

% runs through a set Q compact in E, converges uniformly for t E [0, T) and for x E Q. In particular, if %(t) is continuous in E, then the sequence An(t)x(t) converges in F uniformly on [0, T). A bounded linear operator A (t) is said to be strongly continuously differentiable in t if for each % E E the function A (t) x is continuously differentiable in F. The formula A'(t)% = (A(t)x)' then defines a strongly continuous operator A' (t), operating from E into F.

11

§3. BOUNDED LINEAR OPERATORS

If the operator A (t) is strongly continuously differentiable, then the sequence of operators (1/ ~t) [A (t + ~t) - A (t)] converges strongly and uniformly relative to t to the operator A' (t). Therefore it is bounded uniformly relative to t and ~t. Thus we have the following. LEMMA 3.5. If the operator A (t) is strongly continuously differentiable, then it is continuous relative to the norm of the space of bounded linear operators, and, moreover, it satisfies a Lipschitz condition:

/I A (t + ~t) - A (t) /I E-.F ~ c I,~t I. In an elementary way one proves the following. LEMMA 3.6. If x(t) is continuously differentiable in E and the operator A (t) is strongly continuously differentiable, then the function A (t) x(t) with values in F is continuously differentiable and

[A (t) x(t)],

=

A' (t) x(t)

+ A (t) x' (t).

Analogously one may formulate a proposition on the derivative of the product of two strongly continuously differentiable operators. In defining an operator A (z) depending analytically on a parameter z running through a region of the complex plane, one confronts two possibilities: to require the existence of a derivative relative to z either in the sense of convergence in the norm of the space Y (E, F), or in the sense of strong convergence. It turns out that both definitions are equivalent. Thus, an operator depends analytically on z if for each x E E the function A (z) x is analytic. Here the operators (1/ ~z) • [A (z + .::1z) - A (z)] tend to the operator A' (z) in the norm of the operators. N ow we shall determine what the smoothness of the operator A (t) says about the smoothness of the inverse operator. Suppose that for t = to the operator A (t) is continuous in norm and that it has a bounded inverse A -l(to). Consider the identity (3.2)

A

(to + ~t) = A (to) [I -

A

-1 (to)

(A (to) - A (to

+ ~t» ].

The operator A -1 (to) (A (to) - A (to + ~t» is a bounded operator acting from E into E, with a norm tending to zero as ~t -+ O. If ~t is such that /I A -l(to)[A (to) - A (to + ~t)]/I E-.E ~ q < 1, then the operator A (to + ~t) has a bounded inverse, given by the decomposition

...

A -l(to

+ ~t) = L 11-0

[A -l(to) (A (to) - A (to

+ ~t»]11A -l(to).

12

INTRODUCTION

From this decomposition one obtains the estimate

Thus, the following lemma holds. 3.7. If the operator A(t) is continuous in norm at the point to and has a bounded inverse operator, then for t from a sufficiently small neighborlwod 01 to there exists a bounded inverse operator A -l(t), continuous at the point Xo. II the operator A (t) is continuous relative to the norm on [0, T] and has lor all t E [0, T] a bounded inverse A -l(t), then this inverse is continuous relative to the norm on [0, T]. LEMMA

N ow suppose that the operator A (t) is strongly continuous and has a bounded inverse. For any y E F the following identity holds: (3.3)

[A -let + at) - A -let) ]y =.A -l(t+ 6t) [A(t) - A(t + 6t)]A -l(t)y.

From the strong continuity of A (t) it follows that on the elements A -let) y E E the operators A (t) - A (t + at) tend to zero as 6t --+ 0, 80 that if the operator A -let + at) is uniformly bounded relative to at, then the whole right side will tend to zero. We have arrived at the following assertion: LEMMA 3.8. II the operator A(t) is strongly continuous on [0, T] and has an inverse operator A -let) uniformly bounded on that interval, then A -let) is strongly continuous on [0, T]. If under these same conditions the operator A (t) is strongly continuously differentiable, then A -let) is also strongly continuously differentiable, and the following lormula ooids:

[A -l(t)]' = - A -let) A' (t) A -let).

As soon as the first assertion of the lemma is proved, the second is obtained from (3.3) by division by a and passing to the limit as M--+O. Analogous facts are valid also for operators depending on a complex parameter z, differentiability being replaced by analyticity. 3. The algebra of operators operating in one space. Resolvent and spectrum. We now consider the space Y (E, E) of all bounded linear operators acting from E to E. In this set it is natural to define the multiplication of operators as follows:

t3. BOUNDED LINEAR OPERATORS

13

(AB)x = A (Bx).

Thus, Y (E, E) is a noncommutative Banach algebra. In this algebra a fundamental role is played by its idempotents, namely the operators which are equal to their own squares, i.e. p2 = P. Such operators are called projection operators or projectors. In the algebra Y (E, E) polynomials in the operator A may be constructed in the natural way. Using various limit transitions, this leads to other functions of the operator A. Thus, if F(z) is an entire function of the complex variable z, then

...

(3.4)

F(z)

=

L

c"z",

,,-0

and then one sets

...

(3.5)

F(A) =

L

,,-0

c"A".

The series (3.4) converges everywhere, 80 that the series (3.5) converges in the norm of the space Y(E, E) for any bounded linear operator A. For us a fundamental role is played by the function etA, defined by the serieS etA

=

'£ -.!,n. (tA)".

,,-0

It is not hard to verify that this function satisfies the basic fundamental relation for the exponential function:

The operator etA is differentiable with respect to the parameter t in the sense of the norms of operators, and

The operator e.zA is an entire analytic function of the parameter z. If we consider the differential equation dxjdt = Ax

14

INTRODUCTION

with a bounded linear operator A, then its solution satisfying the initial condition x (0) = .To is given by the formula x(t) = elAXo. In view of what was said above, all the solutions are entire functions of t. --0 One of the most important functions of an operator is its resolvent. If for a given complex A there exists a bounded inverse operator (A - AI) -I, I being the identity operator, then the number A is said to be a regular point of the operator A, and the operator RA(A) = (A _Al)-1 is called the resolvent of the operator A. The regular points of the operator A form an open set on the complex plane. The closed set forming the complement of this set is called the spectrum of the operator A. The spectrum of a bounded linear operator is never empty. If I AI > I A ~, then the resolvent RA(A) exists. It may be defined using the expansion (3.6) This series will converge absolutely, since ~A-"A"II ~ (IIAII/A)". However, the series (3.6) might converge in a wider region. In fact it converges outside a disk of radius

rA= lim vIIA"II. "....... The number rA ~ ~ A II is called the spectral radius of the operator A. The disk IAI ~ rA is the smallest disk with center at the origin of coordinates which contains the spectrum of the operator A. For the resolvent the following identities are valid:

(3.7) and (3.8)

It follows from the first identity that in the region of regular points the resolvent is an analytic function of ). with values in the space !zf(E, E) of bounded linear operators. Suppose that F(z) is a single-valued analytic function defined in a region which contains the spectrum of the operator A, and r is a rectifiable Jordan contour lying in this region and enclosing the spectrum of the operator A. Consider the operator integral

§4. UNBOUNDED OPERATORS

15

This integral exists as an integral of the continuous function F(A)RA(A) with values in the Banach space ,!£'(E,E), and has the properties of the Cauchy integral. The value of the integral does not depend on the choice of the contour r having the specified properties, and it is a bounded linear operator depending on the choice of the operator A. I t is called a function of the operator A: (3.9) For the case when F(z) is an entire function, the definition (3.9) coincides with definition (3.5). Thus, the Cauchy operator integral makes it possible to extend analytically the class of naturally defined functions of an operator, including the algebra of functions which are holomorphic in neighborhoods of the spectrum of A. It is an essential fact that the correspondence constructed above between the algebra of functions and the set of operators is a homomorphism. Obviously addition and multiplication by a scalar for operators corresponds to addition and multiplication by a scalar for functions. Using (3.7) one verifies that F(A) c)(A) = - 21. d

r F(A) c)(A) RA (A) dA.

Jr

§4. Unbounded operators 1. Closed operators. We shall now consider a linear operator A, defined on some linear manifold ~ (A) of the space E, operating into the space E. We shall call the operator A closed if from the fact that x,,_ x (x" E ~(A» and Ax" - y it follows that x E ~ (A) and y = Ax. A bounded operator is obviously closed. If the operator A is not closed, then for it to have a closed extension (admit closure) it is necessary and sufficient that from x" - 0 (x" E ~ (A» and Ax" - y it follows that y = o. The smallest closed extension of A is called its closure A. If A admits closure, then from XII-X (x"E~(A» and Ax,,-y it follows that y= Ax. This may be written otherwise as

(4.1)

A IIlim x" = lim Ax", ...... CD

, . ...... CD

16

INTRODUCTION

if this limit exists. Applying this relation to such limit operations as differentiation and integration, we obtain

(4.2)

- dx d A -=-(Ax) dt dt '

if both derivatives exist, and (4.3)

if f x(t)dt = fAX(t)dt,

if both integrals exist. For a closed operator the bar over the A in equations (4.1)-(4.3) should be removed. The following fact is essential: a closed operator defined in the entire

space is bounded. If, for a linear operator, Ax = 0 implies x = 0, then it has on its range !if (A) an inverse operator A -t, i.e. an operator mapping !if (A) onto 9 (A) and having the property that A -1 Ax = x (x E 9 (A» and AA-1y=y (YE!if(A». If A is closed and has an inverse A-1, then A -i is closed. We shall say that the operator A has a bounded inverse if A -1 is defined in the entire space (!if (A) = E) and is bounded. If A has a bounded inverse, then it is closed. 2. Resolvent and speetrum. Obviously the operator A - >.1, with domain 9 (A), is closed or not closed along with the operator A. Therefore, if there exists a bounded inverse operator (A - >.1) -1, the operator A is closed. In other words, if the operator has at least one regular point, then it is closed. A closed operator may fail to have regular points (the spectrum may fill out the entire plane). It may have no points in its spectrum (the entire plane consists of regular points)~ For any closed set of the complex plane it is possible to construct a closed operator which has that set as its spectrum. If Ao is a regular point of the operator A and RA (Ao) is its resolvent, then the operator ARA(Ao) is bounded. This follows from the identity ARA(Ao)

= 1+ AoRA(Ao).

For the resolvent, identity (3.7) holds. If the operators A and B have distinct domains, then identity (3.8) loses its meaning. The resolvent is a holomorphic function of >. on the open set of regular points. For the derivatives of the resolvent we have the following formulas: d"R A (") _ 'R"+l() d,," - n. A " .

14.

UNBOUNDED OPERATORS

17

Suppose that R(>.) is a bounded linear operator depending on a parameter A which runs through a set G of the complex plane. In order that the operator R(>.) should be a resolvent of some closed linear operator, it is necessary and sufficient that identity (3.7) be satisfied for all A, pEG and that for some >.0 there exists an inverse operator R-l(>.o), i.e. that R(>.o) % = 0 imply % = O. If the indicated conditions are satisfied, then R(A) = RA(~)' where

A

= >.01 + R -1(>.0).

3. DeeompositioD of an operator. We suppose that the spectrum A of the closed operator A consists of two closed parts Al and A2, where Al is a bounded set. Suppose that r l is a rectifiable Jordan contour encircling the set Al. Consider the operator Pl

= - ~ 211"1

Jrrl RA(>.)d>..

This integral may be considered as an integral of the form (3.9), where the function F(A) is equal to unity in the neighborhood of the set Al and zero in the neighborhood of the set A2• The operator P l indeed has the properties of this function. And in fact, P l is a bounded projection operator: P~ = Pl' The entire space E may be decomposed into the direct sum of two subspaces E = El E 2, where E t = PtE and E2 = (1- Pl)E. The subspace Etlies entirely in the domain !?I CA) of the operator A and is invariant relative to A . The spectrum of the restriction of the operator A to the subspace El coincides with the set Al. The restriction A2 of A onto the set (I - Pl)!?I(A) = !?I(A 2 ) is a closed linear operator, acting in the space E 2 , whose spectrum coincides with the set A2• Thus, the study of the operator A reduces to the study of the bounded operator Al in the space El and of the closed operator A 2• 4. Operators with dense domain of definition. In what follows we shall basically consider closed linear operators ·whose domains are dense sets in E. If for such an operator !?I (A) ~ E, then it is automatically unbounded. We shall suppose that the operator acts into E. If A is unbounded, then its positive powers, generally speaking, have smaller domains than the domain of the operator itself. Thus for example the domain !?I (A 2) consists of those elements % E !?I (A) for Which A% also lies in !?I (A). Then the element A2% = A (A%) is defined. If !?I (A) is dense in E and the operator A has regular points, then the domain of any of its positive powers is dense in E. We shall

-+

18

INTRODUCTION

explain this for the square of the operator. The resolvent RA(A) maps E onto ~ (A), and ~ (A) onto ~ (A2). Further, a bounded opera-

tor maps a set dense in the domain into a set dense in the range, 80 that ~ (A 2) is dense in ~ (A) and hence in E as well. For an operator with a domain dense in E the dual operator is defined uniquely. Suppose that { is a linear functional on E. Consider the functional {(Ax), defined for x E ~ (A). It can happen that this functional is bounded on ~ (A). Then, in view of the denseness of ~ (A), it admits a unique continuous extension to a linear functional g(x) defined on the entire space E. In this case one says that the dual operator A * is defined on {, and we write g = A *{. Thus, the domain ~ (A *) of the dual operator consists of all functionals {E E* for which

I {(Ax) I ~ c I x I , where c does not depend on x. The functional A *{ is then defined on by the formula

~(A)

(xE~

A *F(x) = f(Ax)

(A»,

and then extended to all of E by continuity. The dual operator is always closed. However, the domain of the dual operator may fail to be dense in the space E*. In a reflexive space E, for a closed operator with a dense domain the domain of the dual operator is dense in E*. The spectrum of a dual operator coincides with the spectrum of the original operator. The resolvent of a dual operator is an operator dual to the resolvent of the original operator. §5. Operators in Hilbert space 1. Hilbert space. A Hilbert space H is a Banach space in which the norm is generated by a scalar product according to the formula UxU 2 = (x, x). By a scalar product we mean a functional (x,y) (x,y E H) having the properties: 1) (x, x) > 0 if x ¢ 0; 2) (x,y) = (y, x);- 3) (Xl + ~,y) = (XbY) + (~,y); 4) (),x,y) = A(X,y). The BunjakovskilSchwarz inequality holds:

I (x,y)l

~

I xliii YII·

Two elements x and z of Hilbert space are said to be orthogonal if = o. The element z is said to be orthogonal to the subspace ~ if z is orthogonal to every element of Yo If ! / is a subspace of H, then any element x E H may be represented uniquely in the form x = Y + z, where Y E ! / and z is orthogonal to Yo The element y is called the (x, z)

i5. OPERATORS IN HILBERT SPACE

19

projection of the element x on ~ y = Pyx. The whole space may be decomposed into the direct sum of mutually orthogonal subspaces (orthogonal sum): H =.5fEB 1, where the subspace 1 consists Of all the elements orthogonal to Yo The theorem of Riesz on the general form of a linear functional holds in Hilbert space: every linear functional f(x) on the Hilbert space H may be uniquely represented in the form f(x)

(5.1)

=

(x, u),

where u is an element of H. Conversely, every functional of the form (5.1) is a linear functional in H, and I f~ = I u II. Thus we have established a correspondence between the dual space H'" and the space H itself. However, this correspondence f +-+ U is not an isomorphism, since (>.f) (x) = >.f(x) = >.(x, u) = (x, >.u), i.e. >.f+-+ >.u. In order to get around this difficulty, the operation of multiplication by a scalar in H'" is frequently defined as follows: (>.f) (x) = >.f(x). H'" in this case will be called the adjoint space. 3) It is isometric to the space H itself. 2. Bounded operators. Suppose that A is a bounded linear operator acting in the Hilbert space H. The operator A * adjoint to it will operate in the same space H and will be defined uniquely by the formula (Ax,y)

=

(x, A *y).

Because of the above new definition of multiplication of a functional by a scalar, the relation between the spectra of the operators A and A'" is changed: the spectrum of the adjoint operator is situated symmetrically relative to the real axis in relation to the spectrum of the original operator. The operator A is said to be selfadjoint if A = A "'. The spectrum of a selfadjoint operator is a bounded closed set lying on the real axis. The quadratic form (Ax, x) corresponding to a selfadjoint operator takes on only real values. We have the equation su

I (Ax, x) I. = IA II.

ZE~ (x, x)

8}Translator'snote. The author uses the same term conp._aoellJlOC"PlIIC!ao for the space of (continuous) linear functionalB of Banach space 88 well as the corresponding space of functionalB of a Hilbert space with modified scalar multiplication. The trans1ator prefers to follow YOBida [8] and use "dual space" for the former and "adjoint space" for the latter. This distinction is maintained in the terminology applied to the operator A" defined at the beginning of the next subsection.

20

INTRODUCTION

The greatest lower bound and least upper bound of the values of the quantity (Ax, x) / (x, x) coincide with those of the spectrum o~}he operator A. A selfadjoint operator A is said to be positive if (Ax, x) > 0 when x ¢ 0, and positive definite if (Ax, x) ~ k (x, x) (k > 0). The simplest example of a selfadjoint operator is the operator of orthogonal projection P!e\ which assigns to each element its projection on the given subspace Yo The orthogonal projection operators are characterized by the properties: p 2 = P and P* = P. If YC Y b then P .!fP .!fl = P .!f1 P .!f = P .!f. A linear operator U operating in H is said to be isometric, if II Uxll = II X II (x E H). This is equivalent to having (Ux, Uy) = (x, y). If moreover there exists a bounded inverse operator U- 1, then the operator U is said to be unitary. For a unitary operator U* = U- 1• The spectrum of a unitary operator lies on the unit circumference. Unitary operators form a group relative to multiplication. 3. Unbounded operators. Suppose that A is a linear operator with a domain ~ (A) which is dense in H, operating in H. Then one may define uniquely its adjoint operator A * given by the formula (Ax,y) = (x, A *y) on the elementsy for which I (Ax,y) I ~ cll xii for all x E ~ (A). If the operator A admits closure, then the operator A * also has a domain ~ (A *) which is dense in H. In connection with this one defines uniquely the operator A **, which turns out to coincide with the closure of the operator A: A ** = A. A point ). of the complex. plane is called a point of regular type for the operator A if ~ (A -

>.I)xll

~

all xii

(x E ~ (A), a

> 0).

If ). is a point of regular type and A is closed, then the domain of the operator A -).[ is a subspace. The orthogonal complement to this subspace is called the defect subspace, and its dimension the defect index, of the operator A corresponding to the point ).. If the defect index is equal to zero, then.). is a regular point. It turns out that for each connected set of regular points of the defect index takes on one and the same value. A linear operator A with a domain which is dense in H is said to be symmetric if (Ax,y) = (x,Ay) for any x,YE~(A). A symmetric operator always admits closure. The operator A * adjoint to a symmetric operator is an extension of that operator. All nonreal points of the

§5. OPERATORS IN HILBERT SPACE

21

plane are points of regular type for a symmetric operator. To points lying in the upper halfplane there corresponds one value of the defect index, and to points of the lower halfplane another. These two numbers are called the defect indices of the symmetric operator. If one of the indices is equal to zero, then the symmetric operator is said to be maximal. A maximal symmetric operator does not admit nontrivial symmetric extensions. If the defect mdices of a symmetric operator are equal to zero, then it is selfadjoint:A = A *. More precisely this means that ~ (A) =9' (A *) and (Ax,y) = (x,Ay) for x,yE,q (A). In order that a symmetric operator be selfadjoint it suffices that it have at least one real regular point. In order that a symmetric operator be extendible in the space H to a selfadjoint operator, it is necessary and sufficient that it have equal defect numbers. The spectrum of a selfadjoint unbounded operator is a closed unbounded set of points of the real axis. For the resolvent R A (>.) of a selfadjoint operator one has the. estimate

I RA (>.) II

1 ~ d'

where d is the distance from the point>. to the spectrum of the operator A. 4. Spedral resolution of a selfadjoint operator. A family of orthogonal projection operators E). (< >. < is said to be a spectral resolution of the identity if: 1)' E). is strongly left-continuous in >.; 2) E).E. = E,.EA = E). for>. < ,,; 3) E_ .. = lim). __ .. E). = 0 and E+_ = limA_ .. E). = I, where the limits are understood in the sense of strong convergence. For every bounded continuous scalar function F(A) given on the entire real axis, one can define the Stieltjes operator integral IX)

IX)

i

(5.2)

6

F(A) dE)..

This integral is defined as the limit in norm of integral sums of the form E:_oF(Ai) (EAH1 - E). ), if the segment [a, b] is finite, and as an imor b,= Co. The integral (5.2) is a bounded proper integral if a = operator with IX)

I

i

ll

II

F(A) dEA

II ~ sup I F(A) I· 1I~).~6

If the function F(A) takes on only real values. the operator (5.2) is selfadjoint.

22

INTRODUCTION

If the function F(A) is real and unbounded, then formula (5.2), after assigning an appropriate meaning to the integral, yields a selfadjoint and generally speaking unbounded operator whose domain consists of those and only those elements x for which

f-: I

F(A)l 2d(EAX, x)

<

00.

It turns out that to every selfadjoint operator A there corresponds some spectral resolution EA of the identity with Ax

=

f-:

>..dEAx.

forxE~(A).

The operators EA commute with any operator commuting with A. If A is bounded, and m and M are the greatest lower bound and least upper bound of its spectrum, then EA = 0 for A ~ m and EA = I for A> M, so that Ax

=

f

M+O

AdEAx.

III

If the operator A is positive definite and (Ax, x) Ax

=

f. '"

~ a(x, x),

then

AdEAx.

G

The real regular points of A are characterized by the fact that in their neighborhoods the operator EA is constant. Thus, the points of the spectrum of A coincide with the points of growth of the operator function EA. By using the spectral resolution one may bring into consideration a wide class of functions of an unbounded selfadjoint operator. Thus, for example, for any continuous function F(A) it is natural to put F(A)x

=

f-:

F(A)dEAx,

where E>. is the spectral resolution of the identity corresponding to the operator A. Here, to the operations of addition and multiplication of functions there correspond the operations of addition and multiplication of the corresponding operators. If one considers the function eiA, the corresponding operator

§5. OPERATORS IN HILBERT SPACE

23

eiAx = [~eiAdEAX will be a unitary operator, and conversely, every unitary operator may be represented in this way. If" is a regular point-~of the operator A, then for its resolvent one obtains the representation

RA (,,) =

f

OD

-00

-

1

dEA•

A -"

For a positive operator A one can introduce the concept of its powers with any exponent: A ax

=

!o

OD

A"dEAx.

One need only specify which branch of values of the function Aa one wishes to take.

CHAPTER I EQUATIONS OF FIRsT ORDER WITH CONSTANT OPERATOR.SEMIGROUPS §1. The Cauchy Problem 1. Statement of the Cauchy problem, correctness. Consider in a Banach space E the differential equation dx - =Ax

(1.1)

dt

with a linear operator having a domain 9 (A) everywhere dense in E. DEFINITION 1.1. A solution of the equation on the segment [0, T] is a function x(t) satisfying the conditions: 1) the values of the function x(t) lie in the domain 9 (A) of A for all t E [0, T]; 2) at each point t of [0, T] there exists a strong derivative x' (t) of x(t); 3) the equation x' (t) = Ax(t) is satisfied for all t E [0, T]. Obviously the solution is a continuous function on [0, T]. By the Cauchy problem on [0, T) we mean the problem of finding a solution of the equation on [0, T], satisfying the initial condition (1.2)

x(o)

= XoE9 (A).

DEFINITION 1.2 . The Cauchy problem is correctly posed on [0, T] if:

I) for any Xo E 9 (A) it has a unique solution, and II) this solution depends continuously on the initial data in the sense that if x,,(O) -+0 (x,,(O) E 9 (A», then x,,(t) -+0 for the corresponding solution at every t E [0, T). REMARK 1.1 It follows from the constancy of the operator A that if the Cauchy problem is correct on some segment [0, T] then it is correct on any other segment [0, Td with Tl > 0, i.e. it is correct on the entire semiaxis [0, In order to verify this, it is sufficient to consider the segment [0,2T]. Suppose that Xo E 9 (A) and that x(t) is the solution of the problem (1.1), (1.2) on [0, T). Construct a second solution yet) of equation (1.1) with initial conditions yeO) = x (T) E 9 (A).. Define a function wet) by the equation (X».

24

11. wet)

= { x(t) yeT)

25

THE CAUCHY PROBLEM

for t E [0, T], for t = T+., and TE [0, T].

Evidently wet) is a solution of the problem (1.1), (1.2) on the segment [0,2T]. This solution is unique. Indeed, suppose that WI (t) is another solution of this---problem. Then Wl(t) = x(t) for tE [0, T] ill view of the correctness of the problem on [0, T]. Further, the function Xl(") = wl(T +.,) satisfies equation (1.1) and the initial condition Xl(O) = Wl(71 = x(71. Therefore it follows that Xl(T) == yeT), i.e. Wl(t) = wet) for tE [T,2T] as well. Further, if x,,(O) -+0, then x,,(71 -+0, which means that x,,(t) and y,,(t) tend to zero for all t and '1 E [0, T]. Thus, the problem is correct on the segment [0,2T]. Now consider an operator U(t) which assigns to the element Xo E ~ (A) the value of the solution x(t) of the Cauchy problem (x(O) = Xo) at the moment of time t > o. If the Cauchy problem is correctly posed, then the operator U(t) is defined on ~ (A). In view of the linearity of equation (1.1) and property I), it is additive and homogeneous. In view of property II) it is continuous. Since ~ (A) is dense in E, the operator U(t) may be extended by continuity to a bounded linear operator defined on the entire space E, which will also be denoted by U(t). DEFINITION 1.3. A family of bounded linear operators depending on a parameter t (0 < t < (0) will be said to be a semigroup if (1.3)

(0

< tlJ ~ <

(0).

We shall show that the operators generated .by a correct problem (1.1), (1.2) form a semigroup. Suppose that XoE~(A). Then the function wet) = x(t + '1) = U(t + ")Xo satisfies relative to t equation (1.1) and the initial condition w(O) = X(T) = U(T)Xo. The function w1 (t) = U(t) U(T)Xo also is a solution of (1.1) with the initial value U(")Xo, lying in ~ (A). In view of the uniqueness of the solution we have Wl(t) = wet). Thus the operators U(t + '1) and U(t) U(T) coincides on the entire dense set ~ (A), and since they are bounded they coincide throughout. Now consider the function U(t)Xo for any Xo E E and t> o. Since ~ (A) is dense in E, there exists a sequence of elements xJ,") E ~ (A) such that xJ,") -+ Xo, and accordingly, in view of the boundedness of the operator U(t), %,aCt) = U(t)x/:,)-+ U(t)Xo. Thus the function U(t)Xo is the limit of a sequence of solutions of equation (1.1) on (0, (0) and

26

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

may be called a generalized solution of that equation. However, we do not have any smoothness properties of this function. For the present we can only assert that I U(t)XoII is measurable (as the limit of a sequence of continuous functions). The semigroup property of the operator U(t) makes it possible to strengthen this assertion. LEMMA 1.1. If the Cauchy problem for equation (1.1) is correct, then all the generalized solutions of this equation are continuous on (0, CD). PROOF. We first show that the operators U(t) are uniformly bounded on every segment [a,1/a] (a> 0). In the contrary case there would be a sequence t,. E [a, 1/6] such that t,. --+ l' and I U(t)11 --+ 00 as n --+ CD. In view of the principle of uniform boundedness there then exists an element Xo E E such that I U(tJXoII--+ CD. Without loss of generality we may suppose that I U(t,.)XoII ~ n. The function I U(t)XoII is everywhere finite and measurable on (0, (0), so that one can find on the interval (0,1') a set F with measure m(F) > 1'/2 on which I U(t)XoI is bounded by some number M. Since t,.--+'Y, for sufficiently large n the measure of the intersection of F with the interval (0.. t,.) will also be no less than 1'/2, and then also the measure of the set ~,. = {t,. - T; T E F n (0, t,,) l will be no less than 1'/2. Now we use the semigroup property. We have

n ~ I U(tJXoII = I U(t,. - T) U(T)XoI ~ MI U(t,. - T) II, so that

l U(,,) I

~

n/ M for all "E!JR,..

Writing lim!JR,. =!JR, we arrive at the deduction that I U(,,) I = CD for all "E.5t' and m(!rR) ~ 1'/2. This contradicts the assumption that I U(,,) I is finite for all "E (0, (0). It follows from the uniform boundedness of the operators U(t) on [6,1/a] that the generalized solution U(t)Xo (xo E E) is on [a, 1/6] a uniform limit of pure solutions of equation (1.1), and since the latter are continuous it is continuous as well. The lemma is proved. Lemma 1.1 and the arguments preceding it lead to the following assertion. THEOREM 1.1. If the Cauchy problem for equation (1.1) is correct, then its solution is given by the formula

(1.4)

x(t)

=

U(t)Xo

(XoE 9

(A»,

§1. THE CAUCHY PROBLEM

27

where U(t) is a semigroup of operators strongly continuous for t> O.

We note that the question of the behavior of the semigroup as t - 0 remains open. The limit U(t)Xo for Xo EE 9 (A) as t-O may fail to exist. Further, a generalized solution U(t) may not be differentiable and its values may not lie in the domain 9 (A) of the operator A. On 9 (A) the operator A commutes with the semigroup l!(t). Indeed, for Xo E 9 (A) AU(t)Xo= dU(t)Xo dt

lim U(t+at)Xo- U(t)Xo at

M-+O

= lim U(t) U(at)Xo- Xo = U(t)A.ro. M_O

at

Hence in particular it follows that the derivative of the solution is continuous for t > O. N ow suppose that Xo E 9 (A 2). Then the function

dU~)Xo = AU.(t)Xo =

U(t)AXo

is a solution of the Cauchy problem under the initial condition AXo E 9 (A), so that it is continuous for t ~ 0, and its derivative is continuous for t > O. Thus the condition that Xo lie in the domain of one or another power of the operator A plays the role of the condition that the initial data be smooth; it raises the smoothness of the solution U(t)Xo in t; if Xo E 9(A *) (k ~ 1 an integer), then the solution U(t)Xo for t ~ 0 has a (k - 1) th derivative which is continuous and a kth derivative which is continuous for t > O. We note one further auxiliary proposition: LEMMA 1.2. Suppose that the function x(t) is continuous on [0, T] and continuously differentiable on (0, T], and that its derivative x' (t) has a limit as t-O. If the operator A is closed and the function x(t) satisfies equation (1.1) on (0, T], then it is a solution of that equation. PROOF. We need only verify that the function x(t) satisfies the equation for t = O. It is right differentiable for t = O. Indeed, passing to the limit in the equation

x(t) -X(E) = f'x/(t)dt

as

E-

0, we find that

28

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

x(t) - X(O)

=

1t

X' (t) dt,

from which it follows that x'(O)

= lim x'(t). t_+o

Using the fact that A is closed, we may now pass to the limit as

t- + 0 in the equation x'(t) = Ax(t), and thus arrive at the equation x'(O) = Ax(O). The lemma is proved. 2. Laplace transform. Representation of solution. We now investigate the behavior of the semigroup U(t) as t- CD. To this end we introduce the function /(t) = Inll U(tH (0 < t < CD). The semigroup property (1.3) implies

I U(t1 + ~) I

~ I U(t1)

I ·11 U(~) I ,

and also the subadditivity of the function /(t): /(t 1 +~) ~ /(t 1)

+ /(t2).

It turns out that for every function /(t) which is subadditive the limit

Indeed, suppose that w = inf(f(t) It) is finite. Choose a number o > 0 80 that /(0) < (w + do. Then for (n +. 1)0 ~ t ~ (n + 2)0 we have /(t)

W~t~

n/(o)

+ /(t t

no)

no (

~-t-

)

W+E +

/(t - no) t

•

Since 0 ~ t - no ~ 20, we have I /(t - no) I ~ Mo. Here we are using the fact that a subadditive function is bounded on every segment lying in (0, CD) ([2], Chapter VI, Theorem 6.4.1). For our function Inl U(t) I this is proved in Lemma 1.1. Accordingly the right side of the inequality tends to w+ E as t_ CD. Thus, for sufficiently large t the value of the function /(t) /t differs by arbitrarily little from the number w. Analogously one considers the possible case when W = - CD. Thus,

§1. THE CAUCHY PROBLEM

lim In II

(1.5)

,_...

UW II = Cal < t

29

00.

1.2. If the Cauchy probkm for equation (1.1) is correct, then each generalized solution of it grows to infinity no faster than exponentially, and the exponential types of all solutions are bounded above. THEOREM

The number Cal in (1.5) is called the type of the semigroup U(t) and the type of the Cauchy probkm (1.1), (1.2). It follows from Theorem 1.2 in particular that for the Cauchy problem to be correct it is necessary that the operator A should not have eigenvalues in the halfplane ReA> Cal. Indeed, suppose that z is an eigenvector of the operator A: Az = Az. Then to it there corresponds a solution U(t) = eAlz whose exponential type is equal to ReA, which means that ReA < Cal. If ReA> (iI the operator A - AI has on its range ~ (A - AI) the inverse operator (A - AI) -1. The boundedness of the exponential types of all solutions leads to the applicability of the Laplace transform to their investigation. If XcI E 9'(A) and ReA> Cal, the integral (1.6) is defined. The function U(t) being continuously differentiable on (0,00), we obtain, on integration by parts, 7(A)Xo

= -

~

fN e-AlU(t)Xodt

N_ ..

The limit on the left exists, so that the last integral exists as an improper integral and

30

I. li'IRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

Now suppose that the operator A admits a closure A. Then we may move the operator A across the integral sign and arrive at the equation (1.7)

(A - AI)Jf(>.) %0 = %0

Finally, for a closed operator A we obtain (A - >'I)Jf(>.) %0 = %0

Therefore it follows, first, that the domain 9'(A) lies in the range of the operator A - >.1 for any >. with Re>. > fit, and, second, that on 9J(A) Jf(>.) %0 = (A - >.1) -1%0

Formula (l.6) then takes the form

(1.8) N ow we make the further assumption that the operator A has at least one regular point Ao. Denote by R(Ao) the resolvent of A. If % is any element of E, then R(Ao) %E 9J(A) C ~ (A - >.1) for Re>. > w. Writing z = (A - >.1) -lR(Ao)%, we obtain %= (A - AoI) (A - >.I)z = (A - >.I) (A - AoI)z. Hence it follows that % E !N (A - >.1), i.e. that the range of the operator A coincides with the entire space. The operator (A - >.I) -1 is closed and is defined in the entire space. Hence it is bounded. Thus, the operator A has the resolvent R().) = (A - AI) -1 for all >. in the halfplane Re). > w. If we put %0 = R(Ao)% in (1.8), then with the use of the Hilbert identity we obtain a representation for the resolvent R(>.) on any %EE: (l.9)

R().)% = R(Ao) % - (>. - Ao)

L~

e-MU(t)R(Ao)sdt.

From the fact that A commutes with the semigroup it follows that R(>.) also commutes with U(t). If the element % is such that the function U(t)% is summable on the segment [0, T], then the operator R(Ao) may be taken out across the integral sign, and we arrive at the

formula

§t. THE CAUCHY PROBLEM

31

(1.10)

Our results may be formulated in the form of the following theorem: THEOREM 1.3. Suppose that the Cauchy problem for equation (1.1) is correct and is of type "'. If the operator A has at least one regular point, then it has a resolvent R(A) which may be expressed in terms of the semigroup U(t) according to formulas (1.9) and (1.10). If the generalized solution U(t) is summable on [0, then it has the representation (I) ) ,

(1.11)

U(t) x = - -d ( -1. dt 2rl

f·+ eAJR(A) xdA)i

...

A

.-i'"

(/1 > "',

t> 0).

If on some segment the function U(t) is absolutely continuous, then insUk this segment (1.12)

In particular, the last formula holds for the solution of the Cauchy problem for aU t > O.

Formulas (1.11) and (1.12) follow from the properties of the inverse Laplace transformation described in the Introduction. The last assertion follows from the fact that the solution of the Cauchy problem has a continuous derivative for t > o. Formula (1.9) for the resolvent shows that the norm of the resolvent Indeed, the function U(t)R(Ao)x cannot increase rapidly as Ais continuous for any x E E. Hence from the Banach-Steinhaus Theorem the operators U(t) R(Ao) are uniformly bounded on any finite interval [0, T]. Taking account of (1.5), we may assert that (I).

I U(t)R(Ao) I

~ M.e(o,+d t

for any E > o. Fixing on such an E and writing "'1 = '" + E, we obtain the following estimate for the norm of the resolvent:

I R (A) I ~ M

(1 +

IA - Ao I

Re(A - "'1)

)

(Re A > "'1).

Here M = mull R(Ao) II, M.}. It follows from this estimate that the norm of the resolvent is uniformly bounded on every half-line 1m>. = c, Re>. ~ "'2> "'1. In the entire halfplane ReA ~ "'2 the estimate

32

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

is valid. Thus, the requirement of correctness of the Cauchy problem lays strong restrictions on the resolvent of the operator A. To this point we have been dealing with the norm of the resolvent. Consider the behavior of the resolvent on each element. Suppose that x E ~(A). Then R(A)x

= _~

A

A= tT + iT with T fixed, then in norm, which means that IIR(tT+iT)xll--+O

A

as A--+

CD

as tT--++

CD

If

(1.13)

+ R(A)Ax • the resolvent is bounded (xE~(A».

Further, from (1.10) we find that (1.14)

R(tT

+ iT) X =

-1

m

e-i,te-·tU(t)xdt.

For fixed tT> w and T--+ CD this last integral tends to zero in view of the Riemann-Lebesgue Theorem. Moreover, if tT ~ WI> w, then the family cfJ.(t) = e-·tU(t)x (Wl ~ tT ~ CD) is compact in Y 10 so that the integral (1.14) tends to zero uniformly relative to tT, i.e.

I R(u + iT)xll--+O

uniformly relative to tT (x E

~(A».

Relations (1.13) and (1.14) tell us that

IIR(A)xll--+O as IAI--+ co

(xE~(A».

Since R(tT + iT) is uniformly bounded for fixed T and tends to zero on a dense set ~(A), (1.13) is valid for any x E E. THEOREM 1.4. If for the equation (1.1) the Cauchy problem is correct and the operator A has at least one regular point, then the estimate

(1.15)

I R(A) I

~

M l (1

holds for the resolvent. If x E (1.16) For any xEE

(1.17)

+ IAI> ~(A),

I R (A) x I --+ 0

as

(A = tT + iT. tT ~ W2) then

IAI --+ co

§1. THE CAUCHY PROBLEM

33

3. Construction of solutions of the Cauchy problem. It is natural to attempt to construct a solution of the Cauchy problem using the inverse Laplace transformation according to formula (1.12). The difficulty with this procedure is that the integral in this formula is not as a rule absolutely convergent, since the resolvent cannot decrease more rapidly than 1/ IAI. Knowing, however, the behavior towards infinity of the norm of the resolvent. in many cases we can find elements x = Xo for which the integral in (1.12) and its derivative with respect to t exist in the sense of principal value for all t ~ O. We shall clarify this. Suppose that the resolvent R(A) is defined in the line ReA = a and further at a point Ao with ReAo> a. If the element Xo E 9"(A~, then it is representable in the form Xo = R'(Ao) Zo·

Then R(A)

=

R(A) - R(>.o) R'-1(Ao)Zo

Xo

A->.o R'(A)Zo R(>.o)Zo A - >.0 - ... - (X - Ao)l"

R(A)Zo = (A - Ao)'

If we multiply both sides by eAt and integrate along the line ReX

=

a,

then the integrals of the functions of the form eA1RS(>.o)/(A - >.o)'H-' vanish in view of the Jordan lemma. If in addition the integral of eMR (A) / (A - >.0) I converges absolutely for t ~ 0, then the integral (1.12) will exist in the principal value sense for t ~ O. Write x(t) = -

~ 211"&

(1.18)

i

,,+i .. eMR(A)XodA

,,-i ..

1 i,,+i ..

= - -. 211'"&

,,-i..

M

e

R(X)Zo ,dx (X - Ao)

(Xo = R'(A)Zo).

The function x(t) will be continuous for t ~ 0, if r,,+i .. II R(X) II

J" -i ..

IX _ >.0 I,I dA I <

ex>.

Analogously, it will be continuously differentiable if (1.19)

i

,,+i.. II R(X) II

,,-i ..

--"--~/-li dxl

IX - >.01 -

<

00.

In this case x(t) will be the solution of the equation (1.1). Indeed,

34

I. FIRST"()RDER EQUATIONS WITH CONSTANT OPERATOR

for

t> 0 x'(t)

=

-

(1.20)

~i«+i" >..e Al 2d

«_i..

1

a

= --. 2...,

i

+,..

a-i..

Ae Al

R(A)Zo IdA

(A - Ao)

R(A)Zo 1 IdA+-. (A - Ao) 2...,

i

a

.

+,..

eAlZo

(A - Ao)l

a-i ..

dA=Ax(t).

It follows from Lemma 1.2 that x(t) is a solution of (1.1) for t ~ The solution x(t) satisfies the initial condition (1.21)

x(O)

1

= - -2' ... ,

ia+i.. .

a-'"

(

o.

R(A)Zo ) ,dA . A - Ao

Generally speaking it is not clear how this value of x(O) is connected with Xo. We require in addition that the resolvent be defined in the entire halfplane ReA ~ a, and moreover that the contour of integration may be contracted to the point Ao without changing its value. Then by the theorem of residues 1 x(O) = (l _ I)!

d'-1R(Ao)Zo I dAI 1 R (Ao)Zo = Xo-

From the method just described we obtain the assertion: 1.5. Suppose that the resolvent of the operator A a and satisfies the inequality

THEOREM

for ReA

~

I R(A) I

(1.22)

~ M(1

&S

defined

+ I AI)k

for some k ~ - 1. Then the Cauchy problem is solvable for any initial value Xo E 9(A [k J+3).1) The proof is obtained from the preceding considerations with l = [k] + 3. REMARK 1.2. It is easily verified that for k < 0 condition (1.22) may be weakened somewhat and replaced by the following:

I R(A) ~

~ M(l

+ ITl)k (A = + iT, tT

tT

~ a).

REMARK 1.3. In the proof of the fact that the function x(t) is a solution of equation (1.1), we did not essentially use the fact that in (1.18) the integral is taken on the line ReA = a. All that was important was the fact that the integral of the scalar function eAl/ (A ~ Ao) I should be equal to zero. Therefore we may attempt to construct a solution 1)

[k] is the largest integer not exceeding k.

§1. THE CAUCHY PROBLEM

35

according to the formula (1.23)

x(t)

= - 21. 11"1

reAlR(A)XodA,

Jr

where the contour r is chosen so that on it the function e Al tends to zero as IAI - 00. This makes it possible to obtain a solution of equation (1.1) in the form of the absolutely convergent integrals (1.23) under very weak restrictions on the growth of the resolvent. However as t-O the advantages arising from the presence of the factor eAl steadily disappear, so that as a rule the solution is "spoiled" as t-O. Classes of such solutions will be treated in §3. 4. Generating operator of a strongly continuous semigroup. Among the generalized solutions of equation (1.1) which are not solutions of the Cauchy problem, there may in general be some differentiable functions. Denote by ~ the set of those elements Xo for which the function U(t)Xo, with Xo as its definition at zero, is right differentiable at zero. On the elements of 9 the linear operator (1.24)

lim U(t) Xo - Xo = U' (0) t

'-d.O

Xo

is defined. The operator U' (0) is said to be the generating operator of the semigroup.2)

LEMMA 1.3. If Xo E 9, then the generalized solution U(t)Xo h4s a continuous derivative for t> O. PROOF.

(1.25)

Suppose that I1t>

o.

Then

U(t + I1t)Xo - U(t)Xo = U(t) U(l1t)Xo - Xo. I1t I1t

The right side has the limit U(t) U'(O)Xo, so that the left side has the same limit. If now I1t < 0, then U(t + I1t)Xo - U(t)Xo = U(t + I1t) U( - 6t)Xo - Xo. I1t -l1t The right side again has the limit U(t) U'(O)Xo, i.e. the function U(t)Xo is differentiable and 2)ln (2] (and also in (8]) the pperator U'(O) waa called an infinitesimal operator, and its closure, if one existed, an infinitesimal generating operator.

36

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

dU(t)Xo dt

U(t) U' (0)%0-

The lemma is proved. The right side of (1.25) may be written in the form U(t) U(J1t)Xo - Xo J1t

=

U(J1t) - I U(t) J1t %0-

On passing to the limit as J1t --+ 0, we obtain (1.26)

U(t) U'(O)Xo

=

U'(O) U(t) %0-

The generating operator commutes with the semigroup on its domain. If the Cauchy problem for equation (1.1) is correct. then U(t)Xo is differentiable at zero for XoE9(A). while in view of equation (1.1) U(J1t) Xo - Xo J1t

A --+

%0-

Thus,9(A)C9 and U'(O)Xo=AXo for XoE9(A). An operator A generating a correct Cauchy problem may be extended to a generating operator U'(O) of a strongly continuous semigroup U(t). 5. Equations with a parameter. Analyticity of solutions. If for equation

(1.1) the Cauchy problem is correct, then obviously the problem (1.27)

dx dt

= "Ax,

x(O) = XoE9(A) =9("A),

is correct as well for any ,,> o. Indeed, if x(t) is the solution of the Cauchy problem for (1.1), then x("t) is the solution of the Cauchy problem for equation (1.27) with the same initial condition. Denote by U,.(t) the semigroup corresponding to the Cauchy problem for equation (1.27). Then (1.28)

r

r

Now suppose that is a complex number. The collection of those for which the problem

(1.29)

dx dt = tAx,

x(O)

=

XoE9(A)

is correct is called the correctness set of the Cauchy problem for the operator A, and is denoted by K A • From what was said above it is clear that this set consists of a collection of rays issuing from the point

r = o.

37

§1. THE CAUCHY PROBLEM

Therefore in order to describe it it suffices to indicate its intersection with the unit circumference I l"1 = 1. Denoting by U,(t) the semigroup generated by the problem (1.29). we put (1.30)

U(t)

=

U,(1).

From (1.28)

CJ' > 0).

(1.31)

LEMMA 1.4. If the operator A has at least one regular point "0. then the operators UW form a semigroup in the sense that if l"b l"2. and l"1 l"2 lie in KA> then

+

(1.32) PROOF.

U«l"l

Suppose that XoE9'(A). Then in view of (1.31) the function U'1+l2(t)Xo = x(t) is a solution of the Cauchy problem

+ l"2)t)Xo =

dx dt = (l"1 + t2)Ax.

(1.33)

Now consider the function yet) its derivative for t > O. We have fl.y fl.t

=U

(t

=

Xo-

= U(l"lt) U(l"2t)Xo.

+ fl.t) U r2(t + fl.t) Xo -

We shall calculate

U r2(t) Xo

fl.t

l'J.

+

x(O)

Url(t

+ fl.t) fl.t

Url(t) U () 1'2 t Xo·

Since U r2 (t)XoE9'(A). then the second term tends to l"IAUrl (t)U r2(t)Xo as fl.t-+O. In view of the continuity of the semigroup Url(t) for t> 0 the first term tends to U r1 (t) l"2AUr2(t)Xo

as fl.t -+ O. Thus for t

=

l"2AUrl(t) U r2 (t)Xo

>0

Let us investigate the behavior of yet) as zero. For this we suppose that .xoE9'(A 2). and represent it in the form Xo = R("o).zo. where ZoE9'(A). Then

+ [Ur1 (t)Xo - Xo] Ur1(t)R(Ao) [Ur2 (t)Zo - Zo] + [Ur1(t)Xo -

yet) - Xo = Url(t) [Ur2 (t)Xo - Xo] =

Xo].

38

I. FIRST-QRDER EQUATIONS WITH CONSTANT OPERATOR

The operators U r1(t)R(X) are uniformly bounded in view of the Banach-Steinhaus Theorem. As t - 0 they converge to the operator R(>.o). Therefore both terms tend to zero as t-O, and y(t)-.xo. As we have already calculated, y' (t) = U.. + l"J U r1 (t) Ur2 (t)A.xo. If finally one assumes that %oE~(A2), then y'(t)-(l"1+ l"2)Axoas t-O. By Lemma 1.2,y(t) is a solution of the Cauchy problem for equation (1.33) with the initial value .xo. In view of the assumed uniqueness of such a Solution y(t) == x(t); that is, for %0 E ~(A 3) we have (1.34)

Since ~(A3) is dense in E and all the operators in (1.34) are bounded, this relation is valid for all %0 E E. Putting t = 1, we arrive at (1.32). The lemma is proved. It follows from the lemma in particular that the operators U(r) commute with one another. THEOREM 1.6. Urukr the conditions of Lemma 1.4 the operator function UW is analytic at each interior point of the set K A •

The collection of all interior points of the set KA is the sum of no more than a countable number of open sectors. Suppose that 1'0 is an interior point of KA and that S.1.2{ l": #/11 < arg l" < "'2} is the open sector of KA containing it. Write argl"o. Suppose,that %oE~(A). The fun~tion U(e i9ot) %0 = U ~o(t)%o is a solution of equation (1.29) for l" = e~, so that PROOF.

"'0=

limU(fo +re i9lO )xo - U(fo)%o r--+O

re~

= lim U. i9l0,( I tol r--+O

+ r)%o -

Uei9lO ( I fol )%0

re"o

= AUei\6o(1 toD%o = AU(l"o).xo. Thus the function U{f) has at the point to a derivative in the direction corresponding to the angle "'0 with the real axis. N ow suppose that the angle '" is such that < + '" < Then as r-+O

"'1 "'0

(1.35)

"'2'

39

§1. THE CAUCHY PROBLEM

in view of the fact that U(fo)XoE9(A). The arguments are rather complicated if r-+ - O. Fix on the ray fo + reiC·oH ) a point fl with negative r = - a (a > 0), lying in the sector 8.1• 2• Then fo = fl + aei (.oH), and the difference quotient in (1.35) may be represented in the form U,i.(#'HO) (a

+ ~)- U,i.(H~) (a) re'(#'Hr}

U(":) )'1

Xo-

Since U(f1) Xo E ~ (A) and the semigroup U.i(H~ (t) is differentiable on elements of 9 (A), this ratio tends to AU.i(H~)(a) U(f1)Xo

= AU(fo)Xo.

Thus the derivatives of the U(r)Xo exist and coincide along the two noncollinear directions. Hence, as usual, it follows that the U(f) Xo are differentiable functions of the complex variable f. Indeed, suppose that f tends to fo in any direction: r = ro + re il E K A • Then U(r)Xo _ U(fo)Xo

t- to

U(r)Xo

~ U( ro+ rSin~~ 1/10) ei(tHol) Xo

=

re"

The first term may be written in the form U (fo + r

Sin~in~ 1/10) ei(H~)

U(71)'; - Xo

where 71

=

r

sin(l/Io + 1/1 - 0) ;... . e~u sml/l

varies along a ray lying in 8~.2. Hence it is clear that this term tends to U(,. ) A ...- sin(l/Io + 1/1 - 0) i(.o-,) )0 -u sin 1/1 e •

The second term, from (1.35), tends to

40

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

from which it follows that dU(ro) Xo

(1.36)

dr

= A UUi ) 0

Xo-

Thus, for Xo E g(A) the analyticity of the function U(r)Xo is proved. N ow suppose that Xo is any element 01 E. We construct a sequence xIIEg(A) such that Xn-+Xo. We construct sectors "0 - E ~ arg r ~ rPo + E and r = rl + 'I, rPo + '" - E~ arg'l ~ rPo + '" + E, where rl is the same point as above. We choose the number E > 0 so small that this sector does not fall outside the limits of S~1.2 and such that the point rl lies outside the FIGURE 1. first sector. The intersection of these sectors yields the quadrilateral ABeD (Figure 1). From the proof of Lemma 1.1 it follows that the semigroup UW is uniformly bounded on the segments AB and 'CD. In view of the semigroup property (1.32) they are also uniformly bounded on the sides AD and Be. Therefore the sequence of functions UWxlI will converge uniformly on the boundary of the region ABeD to the function U(r)xo, from which uniform convergence on' the interior follows, and thus the analyticity of the limit function. The theorem is proved. We note that from the preceding arguments it follows that the derivatives AUWxlI converge to the derivative dU(r)Xo/dr inside ABeD. Since the operator A is closed we find that . (1.37)

dUd?Xo

= AUWXo.

THEOREM 1.7. If the operator A has at least one regular point, then for each Xo E g(A) there exists a solution of equation (1.37), defined and analytic in the interior of the correctness set KA of A. The exponential types of all solutions along a ray belonging to the closed sector of the interior of the set KA is uniformly bounded.

In the proof we need the two following assertions. Suppose that the sector ~ arg r ~ lies in the interior of K A • We choose the subset Q = {r: 1 ~ I rl ~ 2, "1 ~ arg r ~ "2} of it. Since in this portion the semigroup U(r) is analytic, the operators U(r) are uniformly bounded:

"1

"2

§1. THE CAUCHY PROBLEM

I UWII

~M

41

(rE Q).

N ow suppose that .so = rei. is any point of the sector lying beyond Q, and suppose that n + 1 ~ r ~ n + 2. Then

I U(re~) I = I U(ne i 4» U«r - n)e i4» ~ ~ I U(e~) 11"11 U«r - n)e~) II. Since (r - n)e~ E Q and e~ E Q, we have

Without loss of generality we may suppose that M> 1. Writing lnM = "" we have The theorem is proved. We note that there may be operators in the correctness set which consist only of zero. Indeed, if, for example, the spectrum of the operator A lies on the real and imaginary axes and goes out to infinity in the directions ± 00 and ± i,ll!", then for any r;;
4 2 1 . FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR If the spectrum of the operator A is bounded, then, as we proved in the Introduction, the entire space decomposes into a direct sum E = El -+- Ez of invariant subspaces El and E2 such that the operator A is bounded in El and its spectrum in Ez is empty. Suppose that %0 E Ez and that f is a bounded linear functional on E. The function f(R(A) %0) is analytic in the whole FIGURE 2. plane. Suppose now that the correctness set contains three rays not lying in one halfplane. Then out~ side the triangle formed by certain perpendiculars to the rays symmetric to the given ones relative to the real axis (Figure 2), estimate (1.15) will hold for the resolvent. It therefore follows that the function f(R(A)%o) grows no faster than a linear function, which means that it is in fact linear. Since, moreover, from (1.17) it tends to zero in the halfplane, it is identically zero. It follows from this that R(A) %0 = 0 and %0 = o. Hence, E2 consists of zero, so that the operator A is bounded, which contradicts the hypothesis. The theorem is proved.

§2. Uniformly correct Cauchy problem 1. Uniform correctness. Semigroups of class Co. A correctly posed Cauchy problem is said to be uniformly correct if x,,(O) --+0 implies that x(t) --+0 uniformly in t on each finite interval [0, T]. THEOREM

(2.1)

2.1. For a uniformly correct Cauchy problem, for any x E E, lim U(t) x '_+0

= x.

PROOF. We prove first that the operators U(t) are uniformly bounded in t on each finite interval [0, T]. Indeed, if we suppose the contrary, then there exists a sequence of points tIl E [0, T] and elements w" E 9(A) with norm II w,,11 = 1 such that ~ U(t,,)w,,11 ~ n. Put I)" = w,,/n. Then 111),,11--+0, and II U(t")u,, II ~ 1, which is impossible in view of the uniform correctness of the problem. It follows from the definition of correctness that the operators U(t) converge weakly to I as t--+O on an everywhere dense set 9(A). By

12.

UNIFORMLY CORRECT CAUCHY PROBLEM

43

the Banach-Steinhaus Theorem, they converge strongly to I on any %EE. The theorem is proved. DEFINITION 2.1. The semigroup U(t) belongs to the class Co if it is strongly continuous for t> 0 and satisfies the condition !im,""-Hl U(t) %= % for any %EE. Theorem 2.1 may now be reformulated 88 follows: THEoREM 2.1'. The semigroup U(t) generated by a uniformly correct Cauchy problem belongs to the class Co-

We shall study semigroups of class Co independently of differential equations. The generating operator U' (0) is again defined by formula (1.24) on those elements where the limit exists. THEoREM 2.2. If a semigroup belongs to the class Co. then the following estimate holds for it:

II U(t) II

(2.2)

~

Me ..t•

Vet)

PROOF. We shall denote sUPo=,t:ii111 II by M. This supremum is finite in view of the Banach-Steinhaus Theorem. We may write any t > 0 in the form t = n + T, where 0 ~ T < 1. Then U(t) = IF'(1) U(T) and

I U(t) II

~

II U(1) II "M = Me"in I U(1l I.

If Inll U(1) H~ 0, then ~ U(t) II however Inll U(t) I > 0, then ~

U(t) II

~

M, i.e. (2.2) is valid for w =

~ Me(II+~llnlU(1ll

o. If

= Me tin I U(ll I,

i.e. (2.2) is valid for w = Inll U(I) H. The theorem is proved. 3) If w ~ 0, then the semigroup is bounded, II U(t) II ~ M, and the Cauchy problem is uniformly correct on [0, CD). In this case it is possible to introduce a Euclidean norm into the space E, for example I %111 = SUPo=,I< .. II U(t)%I, in which the operators U(t) have norms not exceeding 1. Indeed,

I U(t)xIl1 = supU U(T) U(t)xll = supll U(t + T)xll ~

~

= supll U(s)%11 ~ II %111. • il:I

a)

This theorem also follows from Theorem 1.2, but is more easily proved directly.

44

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

If II U(t) I semigroup.

~

1 (0 ~ t

<

00), the semigroup is said

to be a contraction

THEOREM 2.3 If a semigroup belongs to class Co. then the domain ~ of the generating operator U' (0) is everywhere dense in the space E . ..Voreover, the set of elements for which all powers of the operator U' (0) are defined is everywhere dense in E. PROOF. Suppose that R is a class of numerical functions k(.,.) , finite on (0, 00). We consider the set E(R) of elements of the form

y =

1

m

k(.,.) U(.,.) wd.,.,

wEE.

For these y, as h---+O we have

1

1

rm

Ii [U(h)y - y] =:' Ii Jo k(.,.) [U(.,. + h) - U(.,.)] wd.,. =

rm 1

J" Ii [k(.,. -

rIa k(.,.) U(.,.)wd.,.

1

Jo

h) - k(.,.) ]U(.,.)wd.,. - Ii ---+ -

Analogously we find that [U' (0)

1"

ty exists for

[u'(o)]ny = (_l) n

l

m

k'(.,.) U(.,.)wd.,. = U'(O)y.

any n, and

k(n)(.,.)U(.,.)wd.,..

Now we will show that E(R) is dense in E. If this were false, there would exist a bounded linear functional W* ~ e such that W*{ E(R)} = o. Then

Hence

W*[U(.,.)w] == 0, .,.> o. Since U(T)W---+W as .,.---+0 and W* is continuous, this identity yields W*[w] = 0 in the limit for any w, i.e. W* is the null functional. The resulting contradiction indicates that E(R) is dense in E. Since E(R) cnn~{[U'(o)]n}, the therorem is proved.

45

§2. UNIFORMLY CORRECT CAUCHY PROBLEM

THEOREM 2.4. II a semigroup belongs to the class Co. then its generating operator is not closed.

PROOF. Suppose that

w.. is a sequence of points of 9

such that W.. -Wo and U'(O)w.. -Zo. We shall show that U'(O)wo is defined and equals Zo. For w.. E 9 we have the equation

1 flo, 1 f"dU(T)W.. 1 Ii Jo U(T) U (O)W..dT = Ii Jo dT dT = Ii [U(h) Passing to the limit as n -

cx>,

I]w...

we get

Ii Joflo U(T)ZodT = Ii1 [U(h) 1

I]wo-

N ow suppose that h - o. Then the left side tends to Zo, i.e. Zo = U' (0) woo The theorem is proved. THEOREM 2.5. II a semigroup belongs to the class Co, then, lor all A with sufficiently large real part, the operator U' (0) has a resolvent.

PROOF. Introduce the operator

J

(A) according to formula (1.6).

We shall prove that J (A) = [U'(O) - AI]-l. In fact, suppose that woE9. Then J

(A) U' (0) Wo

= -

L~

e-AtU(t) U' (0) wodt

= _ f ~ e_AtdU(t)wo dt = _ e-AtU(t)wo

Jo

dt

10

- ALa> e-AtU(t)wodt = wo+ AJ (A)wo. Hence we get (2.3)

J(A) [U'(O) -AI]wo=wo for woE9.

On the other hand, for any wEE U'(O)J(A)w

= _ lim U(&) - I f~ e-AtU(t)wdt

J == - lim l [ f" e -AtU(t + tlt) wdt - f - e -AtU(t) wdtJ. 41--+0tlt Jo Jo

41--+0

tlt

0

46

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

In the first integral we make the substitution t + ~t =

T.

Then

U'(O)Y(A)w =

-lim.!..[ ra> e-A(~-M)U(T)wdT- ra> e~AtU(t)Wdt] M..... O~t

= -

lim-1 ia>

M ..... O~t

+ lim.!.. M .....

=-

JM

>.

O~t

1-

Jo

e- A• ]U(T)WdT

[e-A(.-At) -

M

rMe-AtU(t)wdt

Jo

e-AtU(t)wdt+ w

= AY(A)w+ w.

We thus find that [U'(O) - >.1] ~(A)W = w for any wEE. Hence it follows from (2.3) that Y(A) is the resolvent of the operator U'(O). The theorem is proved. Thus, the spectrum of the generating operator of a semigroup of Co always lies in some halfplane. ReA ~ Col. Now we tum to the uniformly correct Cauchy problem for equation (1.1). As we found in §1.4, the operator U'(O) is an extension of the operator A. THEOREM 2.6. If the Cauchy problem for equation (1.1) is uniformly correct, then the closure of the operator A coincides with the operator 0.' (0). PROOF. The operator U'(O) is closed and A C U'(O), so that A admits closure and A C U'(O). As we showed in (1.7), in addition

(A - A1)Y(>.) x = x

(xE~(A».

In view of the boundedness of Y(A) this equation is valid for any x E E. But the resolvent Y(A) of the operator U' (0) maps the entire space E into the domain ~(U'(O». Therefore ~(U'(O» C ~(A), so that A = U'(O). The theorem is proved. The question arises as to whether the problem (2.4)

dw dt

(2.5)

w(O)

= U'(O)w, = woE~,

is correct, where U' (0) is any operator of a semigroup of class Co. The following theorem gives a positive answer to this question.

62. THEOREM

UNIFORMLY CORRECT CAUCHY PROBLEM

47

2.7. The problem (2.4), (2.5) is uniformly correct.

PROOF. For any Wo E ~ the function wet) = U(t)wo is a solution of equation (2.4) and, in view of the estimate (2.2), depends continuously on Wo uniformly in t on every finite interval. For the proof of the theorem it will suffice to establish the uniqueness of the solution wet). Suppose that WI(t) is another solution of (2.4) on the interval [0, T], satisfying the same initial condition WI(O) = w(O). Construct the function vet) = U(t)wo - WI(t). This is a solution on [0, T] of equation (2.4) and satisfies the null initial condition v(O) = O. Obviously vet) E~. Define the function

VI (t )

={

0 ~ t ~ T, v(t) , U(t- T)v(T), T~t< co.

This function satisfies equation (2.4) for all t transform of VI (t):

x (>.) =

1'"

~

O. Consider the Laplace

e -MV1 (t) dt,

which is defined for Re>. > '" in view of the estimate (2.2). Reasoning just as in the derivation of (2.3), we get (U'(O) - >.I)X(>.)

= VI(O) = O.

Since the operator U'(O) for Re>. > '" has a resolvent, X(>.) == 0 (Re>. > ",). In view of the uniqueness theorem for the Laplace transform VI (t) == 0, so that vet)

=

U(t)wo - WI(t)

== 0

(0 ~ t ~ T).

The uniqueness of the solution U(t) Wo is established. The theorem is proved. Thus we arrive at the following fundamental result: THEOREM 2.8. In order that the problem (1.1), (1.2), where A is a closed operator, should be uniformly correct, it is necessary and sufficient that A should be a generating operator for a semigroup of class Co-

Thus, if we restrict ourselves to equations with closed operators, then the class of equations for which the Cauchy problem is uniformly correct coincides with the class of equations for which the operator is a generating operator for a semigroup of class Co. 2. Construction of solutions by approximation. If the operator A in equation (1.1) is bounded, then the Cauchy problem obviously is

48

I. FIRS'l'-oRDER EQUATIONS WITH CONSTANT OPERATOR

uniformly correct and the semigroup U(t) may be represented in the form U(t) = etA. It turns out that every uniformly correct problem is, in a certain sense, limiting for problems with bounded operators. THEOREM 2.9. In order that the problem (1.1), (1.2), where A is a closed operator, should be uniformly correct, it is necessary and sufficient that the operator A have a resolvent for sufficiently large ). > 0 and be a strong limit of a sequence of bounded operators AM commuting among themselves, such that (2.6)

where M and

(r,)

do not depend on n.

PROOF. SUFFICIENCY. Consider the identity

etA" - e'A", =

L'

e (1-.lA,. (A" -

A",)eM~,

gotten by integrating the expression

d ds

(e(I-.lAneMm) = e(I-.lA"(A,,, -

AJe,A",.

Since the operator A" commutes with A"" it commutes as well with esAm• Taking account of this and (2.6) for any v we find that:

IletAnv - etAmvll (2.7)

~ L'lle(HlA"llllesAmllll (A" ~

M2te""11 (A

II -

A",)vllds

Am)vll.

Suppose that voE9(A). Then A"vo-Avo, so that

II (A" -

Am)voll-O.

It then follows from (2.7) that the functions etA"Vo converge uniformly in t in some finite segment. Since 9(A) is dense in E, the operators etA" are uniformly bounded on some segment [0, T], the functions etA"v converge uniformly in t on this segment for any v. The limit of the sequence etA"V will be denoted by U(t) v. Obviously U(t) is a bounded linear operator strongly continuous in t for t ~ o. Passage to the limit in the identity

49

§2. UNIFORMLY CORRECT CAUCHY PROBLEM

establishes that the operators U(t) form a semigroup. Since the operators etA" are equal to I for t = 0, we have U(O) = I, and therefore the semigroup U(t) belongs to the class Co. If we pass to the limit in the identity

=

i'

eaAnA"vods

U(t)vo - Vo =

.£'

U(S) Avods.

etA"vo - Vo

for Vo E

~(A),

then we get

Hence it follows that U'(O)vo = Avo.

This shows that U' (0) :> A. In view of Theorem 2.5 the operator U' (0) has a resolvent for ReA::;> w. The operator A has a resolvent R(A, A) by hypothesis. Then [U'(O) - AI]R(A,A) = [A - AI]R(A,A) = I

on

~(A).

Applying the operator U' (0) to both sides of the resolvent, we get R(A,A) = R(A, U'(O».

Hence U'(O) = A. From Theorem 2.7, the Cauchy problem is uniformly correct. NECESSITY. Suppose that the problem is unformly correct. Since the operator A is closed, by Theorem 2.8 it is a generating operator for some semigroup U(t): A = U'(O). The resolvent of the operator A is given by formula (1.8), so that, in view of (2.2), the inequality

IIR(A) I ~ReA-w M

(2.8) is valid. Consider the operators

A" =

-

A"I - A!R(A,,),

where the An are real and A" - + 00. The An are bounded. We shall show that they converge strongly to the operator A. Suppose that VoE~(A). Then (2.9)

A"vo =

-

A"R (A,,) Avo.

50

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

The operators )."R().,,) converge strongly to the operator - 1. Indeed, in view of (2.8), for woE9(A) (2.10)

I )."R().,,)wo + woll = I R(A,,) Awo I M

~

IIAwoll-O

for n-

00.

A" - '"

Further, the norms of the operators A"R(A,,) are uniformly bounded, since (2.11)

I A"R (A,,) I ~ A~~"'" ~ 2M

for )." > 2w.

By the Banach-Steinhaus Theorem, it follows from (2.10) and (2.11) that )."R (A,,) w _ - w

for any",. Putting w = Avo, we find from (2.9) that A"vo-AvOo

N ow we will show that estimate (2.6) holds for the operators AN We

have etA" = e-A"t-AiR(A,.>t = e-A,.t

i

(>.!t)kRk(A,,) (- 1)'.

k-O

k!

Differentiating formula (1.8) k - 1 times with respect to A, we get (2.12)

di -

r-

1

dA k- 1 R(A)W = (- 1)4 Jo tk-1e-UU(t)wdt.

The legitimacy of differentiation under the integral sign is guaranteed by the estimate (2.2). On the other hand, dk - 1 d).le-l R().)w

(2.13)

=

(k - 1) !Rk().)w.

From (2.12), (2.13), and the estimate (2.2) it follows that (2.14)

I Rk(A) I s

M

- (k - I)!

fo0

t'-le- (Re>..- ..ltdt

=

M 4. (ReA - "')

N ow we shall estimate the norm of the operator etA,,:

§2. UNIFORMLY CORRECT CAUCHY PROBLEM

IletA"lI (2.15)

...

"A~"M

~ e~A"'L , " • 4-0 k. ("A" - w)

= MetA".,/(A,.-.,)

~

51

= Me-A,.teA~t/(A,,- ..)

M e 2t.>l for "A" > 2t.I.

The operators A" obviously commute with one another and accordingly satisfy all the conditions of the theorem. The theorem is proved. REMARK 3.1. In the proof of the second portion of the theorem, the only property of the operator A we used was estimate (2.14) for "A = >'k. Therefore one proves analogously that for the construction of a sequence of operators A" satisfying the conditions of Theorem 2.9, it suffices that there exist numbers wand M, and a sequence of real numbers >." -+ co, such that

Here the operators A" are again constructed according to the formula (2.16)

All =

- "A"I - >.!R(>.,,).

On the other hand, if the problem (1.1), (1.2) is correct and the operator A is closed, then its resolvent satisfies inequality (2.14) for ReX> w. Thus, we may formulate the theorem: THEOREM 2.10. In order that the Cauchy problem for equation (1.1) with a closed operator A should be uniformly correct, it is necessary and sufficient that the resolvent R(X) of A should satisfy the condition

(2.17)

I R"(>') II ~

M (Re>. -w)"

(Re"A > w)

for some wand M. Here, for the corresponding semigroup, we have the inequality

I U(t) I

~

Me"".

This last assertion is obtained by a passage to the limit as n -+

co

in one of the intermediate inequalities in (2.15): II etAnx II ~ Me tA"..1(A,,- ..) "x" . Condition (2.17), generally speaking, is difficult to verify. It is ob"ious that for it to hold it is sufficient that the inequality

52

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

(2.18)

be satisfied. If such an estimate holds, then the semigroup satisfies the inequality

I U(t) I ~ e..t. In particular, if w = 0, then II U(t) I ~ 1 and the semigroup is a contraction semigroup. One should be careful to note that the satisfaction of the condition (A> w)

alone with M > 1 does not guarantee the uniform correctness of the Cauchy problem. This is shown by example 2 of §1 of Chapter XII of the book [2]. 3. Uniqueness and correctness. In certain cases the correctness of the Cauchy problem follows from the existence and uniqueness of its solutions. Suppose that the following condition is satisfied: ~. Suppose that the operator A in equation (1.1) is closed. Then for any Xo E ~(A) there exists a unique solution of the Cauchy problem which is continuously differentiable on [0, T]. We note that for a uniformly correct Cauchy problem the derivative of the solution is continuous on [0, <Xl): dU(t)Xo dt

=

U(t)AXo

.

We introduce the norm I x II A = II x I + I Ax II into ~ (A), thus converting it into a Banach space EA. We denote by C(EA) the space of all functions x(t) continuous on [0, T] with values in E A , with the usual norm

To each Xo E EA we assign a solution x(t) of the Cauchy problem, which, in view of condition ~, lies in the space C(EA ). Thus we define a linear operator from EA into C(EA): FXo = x(t). We shall show that this operator is closed. Suppose that x~") --+ Xo in EA and Fx~") = X,.(t) -+ vet) in C(EA ). Then in particular dx,./dt = Ax,.(t) -+ Av(t) in E uniformly in t, and therefore one can pass to the limit in the identity

§2. UNIFORMLY CORRECT CAUCHY PROBLEM

Xn

()

(n)

t = Xo

53

+ Jortdxnd Cit t.

We obtain v(t)

= Xo +

Sot Av(t) dt.

Hence it follows that v(t) is a solution of the Cauchy problem with initial value Xo, continuously differentiable on [0, T]. In view of the uniqueness of such a solution, v(t) = FXo, i.e. the operators F is closed. The fact that the operator F is closed implies that it is bounded, since it is defined on the entire space EA. Thus, for the solutions of the Cauchy problem we have the inequality (O~t~T).

(2.19)

Repeating the arguments of §1, we may represent the solution of the Cauchy problem in terms of a certain semigroup U(t) of the operators bounded in E A : x(t) = U(t)Xo (the boundedness of the operators U(t) follows from inequality (2.19». From condition 5:f it follows that this semigroup- satisfies the Co-condition. It is natural to ask how one obtains the generating operator of the semigroup U(t). In order to answer this question we first show that on !?J (A 2) the operators U (t) and A commute. Suppose that Xo E !?J (A 2). Then the functions U(t)AXo and dU(t) Axo/dt = AU(t)AXo are continuous on [0, T], so that (2.20)

U(t) AXo = AXo

+ Sot A U(t) AXodt =

A [ Xo

+ Sot U(t) AXodtJ.

The function in square brackets, z(t)

= Xo +

Sot U(t) AXodt,

is continuously differentiable on [0, T], and for its derivative dz dt

= U(t) Axo = Az(t)

in view of (2.20). From the uniqueness of the solution of the Cauchy problem it follows that z(t) = U(t)xo. Then (2.20) may be rewritten in the form U(t)Axo

= Au(t)xo

54

.

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

It follows from this relation that for Xo E .99 (A 2) and at - 0 U(at) - I A at Xo- Xo

(in E)

and A U(at) - I x = U(at) - I A _ A 2 at 0 at Xo Xo

(inE),

i.e. the function U(t) Xo is differentiable at zero in the norm of the space EA' Its derivative at zero is equal to AXo. Thus, the operator U' (0) is defined on .99 (A 2) and coincides on it with A. Conversely, if XoE.99(U'(O», then U(at) - I A at Xo- Xo

(in E)

and the expression A U(at) - I at

Xo

also has a limit in E. Since A is closed, then Axo E .99(A) , i.e. Xo E .99(A 2). We have arrived at the assertion: THEOREM 2.11. If for equation (1.1) condition !r£' is satisfied, then for that equation and the initial condition x(O) = Xo E .99 (A ~ the Cauchy problem will be uniformly correct in the space EA'

Under an additional condition, 5:f implies the uniform correctness of the Cauchy problem in the original space. 2.12. If condition !r£' is satisfied and the operator A has at least one regular point, then the Cauchy problem is uniformly correct in the space E. THEOREM

PROOF. Suppose that >.0 is a regular point of the operator A, Xo E .99(A) and Yo = R(>.o)Xo. Obviously, Yo E .99 (A 2), so that

U(t)Xo

=

U(t)Ayo - >.oU(t)yo

=

A U(t) Yo - >.oU(t)yo'

Hence

II U(t)XoII In view of (2.19)

~ I A U(t)yo I

+ 1>.0111 U(t)yo II ~ Moll U(t)yo I

EA'

§2. UNIFORMLY CORRECT CAUCHY PROBLEM

Since

55

I U(t)XoII ; ;r; MoC(boll + I AYol1> = MoC(IIYoll +IIXo+Ao.Yoll> ;;;r;;M1(IIXoII + boll}. boll = I R("o) Xo I ; ;r; I R("o) IIII Xoll, we have IIU(t)XoII ;;;r;;MIIXoII (O;;;r;;t;;;r;;T),

from which the uniform correctness of the Cauchy problem follows. The theorem is proved. Suppose that condition SR is replaced by the weaker condition: SRo• The operator A is closed, and for any Xo E 9(A) there exists a unique solution of the Cauchy problem, whose derivative is continuous on (0, T] and summable on [0, T]. Then for an operator A with a regular point it is possible to prove the uniform correctness of the Cauchy problem. This is done in a way which is analogous to the preceding, with the difference that the operator F which puts into correspondence with Xo a solution x(t) of the Cauchy problem is now considered as an operator from the space EA into the space S(EA ) of functions continuous on (0, T] and summable on [0, T]. This space is complete in the metric

d [ ] ~ 2 -II I x - Y II II x, Y = f:'o 1 + I x - y I II ' where

Inequality (2.19) is obtained with a constant depending on t. The correctness. of the Cauchy problem results from this. It has not yet been possible to obtain the correctness of the Cauchy problem from only the existence and uniqueness of the solution of the problem, since there appears to be no topology in the space of functions which are only continuous in the norm of EA on (0, T] in Which it would be complete and the operator F closed. In a way analogous to the proof of Theorem 2.12, one proves the following assertion: 2.13. If the operator A has regular points and if for each O(A ") (n fixed) there exists a unique n times differentiable solution of the Cauchy problem for the equation (1.1), then the Cauchy problem for that equation is uniformly correct. THEOREM

~E

• 56

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

In the proof ~(A ") is converted into a Banach space with the norm

Ilxll,,= L" IIAkxll· 10-0

4. Sets of uniform correctness. Analyticity. In analogy to what was done in §1.5, we define the set of uniform correctness of the operator A to be the set of points t of the complex plane for which the Cauchy problem dx dt

= tAx,

x(O)

=

xoE~(A),

is uniformly correct. We denote this set by K':... This set, as did K A , consists of rays issuing from the origin of coordinates. Suppose that the point to = e i9 lies in the set of uniform correctness. In view of Theorem 2.10, it is necessary and sufficient for this that the inequalities

(n

be satisfied for some M

>0

= 1,2, •.. )

and real w, or

I R"(ue- i • + Te i (r/2-.» I ~

M"

(u - w)

We denote a variable point on the ray arg t = -,p by z: z = ue -i9, one on the line arg t = ± 7r /2 -,p perpendicular to it by.", and the point we -"by Zo. Then the last inequality may be rewritten in the form (2.21)

IIR"(z +.,,>11

~ Iz-MZo I"

(n

= 1,2", .).

Thus we have the following theorem. THEOREM 2.10'. In order that the ray arg t =,p should belong to the set of uniform correctness of the operator A, it is necessary that condition (2.21) should be satisfied, where M> 0, Zo is a point on the ray argz = -,p, z any point on that ray with Izl > IZoI and ." any point on the line perpendicular to that ray. A sufficient condition is that (2.21) should be satisfied for." = O.

The set KA is a part of the set K A , and, in view of Theorem 1.8, lies in some halfplane.

§2. UNIFORMLY CORRECT CAUCHY PROBLEM

57

THEOREM 2.14. The set of uniform correctness of the operator A is a closed or open sector with an angle of opening 1J;: 0 ~ 1J; ~ '£, or a line. PROOF. For the proof it suffices to show that along with the two rays arg r = q,l and arg r = q,2 (0 < q,2 - q,l < '11""), any ray arg r = q, with q,1 ~ q, ~ q,2 must also lie in the set of uniform correctness. Thus, suppose that e;"1 E KA and ei~2 E K A• This means that the estimates

(2.22) hold.

FIGURE

(2.23)

N ow choose 711 and 712 specially, as follows. We erect at the points ZI and Z2 perpendiculars to the rays argz = - q,1 and argz = - q,2 respectively (Figure 3). We denote the intersection of these perpendiculars by ~ and put 711 = ~ - Zl and 712 = ~ - Z2. We introduce the operator Al = A - ~I. An easy calculation shows that the resolvent of this operator satisfies the inequalities

3.

I R1

(z)

I

~ I~~

and

I R"(Z) I

~ I~:'

where z varies on the rays argz = - q,1 and argz = - q,2 respectively

(O
I ZRR A1 (z) I

~ Icos( (q,2~ q,l) /2) I";

and in view of (2.23) its values on the sides of the angle are bounded: ~Z"RAI(Z)

I

~

M = max(MlI M 2 ).

From the Phragmlm-LindelOf principle it follows that this last estimate is valid inside the angle as well, i.e. (2.24)

. 58

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

By Theorem 2.10' the Cauchy problem is uniformly correct for the operator rA 1 - rA - HI for all r m. the sector 4>1 ~ arg r ~ 4>2. The same holds also for the operator A, i.e. rE K~l' The theorem is proved. From the theorem just proved, and Theorem 1.6, we find the following. If the Cauchy problem is uniformly correct for the two rays arg r = 4>1 and arg r = 4>2 (0 < 4>2 - 4>1 < r), then its solutions may be continued to the sector 4>1 < arg r < 4>2, within which they are analytic functions. The corresponding semigroup U(t) wiU be analytic within that sector, and it satisfies the estimate CoROLLARY.

(2.25) REMARK. 2.2. Using a theorem of Nevanlinna,~ it is possible to obtain an expression for the constant M in (2.24) which is somewhat less rough than max(Ml>M2):

M =

M1~-·II(~-.1)M~·-·1)/(~-.1).

REMARK 2.3. If Zl = 0 and ~ = 0, then ~ = 0, and we obtain the estimate I U(r) II for the semigroup. In particular, if on the two rays the semigroup is contractive (M = 1), then it is contractive inside the sector as well. We note that in the case when the Cauchy problem is correctly solvable on the ray arg r = 4>, the resolvent is defined in the halfplane - (r/2 + 4» ~ argr ~ r/2 -;. If the Cauchy problem is uniformly correct in the sector 4>1 ~ arg f < 4>2 and estimate (2.25) holds for the corresponding semigroup, then the resolvent is defined in the sector - r/2 - 4>2 < argz < r/2 - 4>10 whose opening angle is equal to r + 4J2 - 4>1' This sector contains inside itself the rays argz = 4>1 and argz = 2. On each ray argz = 4> (4)1 < 4> < 4>2) the resolvent satisfies the inequality

IIR"(z) I ~ Iz-we M 0.1" This inequality extends by continuity to the boundary rays argz1 = - 1iJ1 and argz = - 2. Thus the Cauchy problem will be uniformly correct in the closed sector 1iJ1 ~ arg t ~ 4>2' We have arrived at the following assertion: f) R. Nevanlinna, Eindeutige analytische Funktionen, Springer, Berlin, 1936; Russian transl., ONTI, Moscow, 1941, p. 48.

§3. WEAKENED CAUCHY PROBLEM

59

THEOREM 2.15. For the semigroup UH) corresponding to the Cauchy problem to be analytic in the sector ~1 < argz < ~2 and to satisfy condition (2.25) there, it is necessary and sufficient that the Cauchy problem should be uniformly correct on the rays arg t = ~1 and arg t = ~2

In the following section we shall once again tum to the question of the analyticity of the semigroup UH) in a sector and give a more effective criterion for analyticity (Theorem 3.8). §3. Weakened Cauchy problem 1. Statement of the problem. For many applications it is necessary to extend the concept of solution of the Cauchy problem.

3.1. A weakened solution of the equation x' = Ax on the segment [0, T] is a function x(t) which is continuous on [0, T], strongly continuously differentiable on (0, T] and satisfies the equation there. By a weakened Cauchy problem on [0, TJ we mean the problem of finding a weakened solution satisfying the initial condition x(o) = Xo. Here the element Xo may already not lie in the domain of the operator A. Thus, we have relaxed the requirements on the behavior of the solution at zero. On the other hand, we have required the continuity of the derivative of the solution for t> 0. However, as we showed on page 27, for a correct Cauchy problem this requirement is automatically· satisfied. Suppose that .L is some linear set of E. We will say that for the set.L the weakened Cauchy problem is solvable on the segment [0, TJ if the solution of this problem exists for each Xo E .L. The weakened Cauchy problem is uniquely solvable if it is solvable and the solution is unique on [0, T]. It is correct or uniformly correct if the solution in addition depends continuously on the initial data of .L for each t E [0, T] or uniformly there. We note that in distinction from the ordinary Cauchy problem, the indication of the segment [0, TJ is essential. The arguments of §l connected with the extension of the solution beyond [0, T] cannot be carried over, since it is not known in advance whether the values of the solution x(T) lie in the set for which the weakened Cauchy problem is solvable even on any segment. Even if x(T) lies in this set, it is unclear whether it is possible to fit the solutions for t ~ T and t ~ T together smoothly. If the operator A has at least one regular point, there is a simple DEFINITION

.. 60

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

connection between the solutions of the ordinary and weakened Cauch~ problems. Suppose that the function x(t) is a weakened solution of equation (1.1). Consider the function Xl(t) = R(Ao)x(t), and calculate its derivative for t > 0: dXl dx (3.1) dt = R(Ao) dt = R(Ao)Ax(t) = x(t)

+ AoR(Ao)x(t) =

AXl(t).

From the next-to-Iast expression for the derivative dXl/dt it is clear that it is continuous on [0, T] and that it has the limit x(O) + AoR(Ao)Xo as t-+O. It follows from Lemma 1.2 that Xl(t) is continuously differentiable on [0, T] and that it is a solution of the Cauchy problem there with initial conditions Xl(O) = R(Ao)x(O). We note further that, as is clear from formula (3.1), the function xl(t) has a second derivative which is continuous on [0, T]. Conversely, if Xl (t) is a solution of the Cauchy problem, continuously differentiable on [0, T] and twice continuously differentiable on (0, T], then the function x(t) = (A - Aol)x1 (t) will be a weakened solution of the Cauchy problem with the initial conditions x(O) = (A - Aol)Xl(O). Indeed, x(t) is continuous on [0, Tl, since x(t) = dxtldt - Ao%l(t), and has a continuous derivative dx/dt = d2Xl/dt2 - ~xddt on (0, T]. Finally, since the operator A is closed d2Xl = dAxl = A dXl dt 2

for t

> 0,

dt

=

A 2Xl

dt

which means that dx dt

=

2

A Xl - AoAXl

= Ax

(tE (0, T]).

Thus, the resolvent R(Ao) realizes a one-to-one mapping of all the weakened solutions of equation (1.1) onto the set of all those pure solutions which are continuously differentiable on [0, T] and twice continuously ditIerentiable on (0, T]. The study of the weakened solutions reduces in essence to the study of solutions of the Cauchy problem with spectral properties. However in practice it is frequently more convenient to study the weakened solutions directly rather than their smooth images. If the Cauchy problem is correct for equation (1.1), and the operator A has a regular point, then every weakened solution x(t) is a generalized solution, i.e. x(t) = U(t)Xo, where U(t) is the semigroup generated by the Cauchy problem. Indeed, the function Xl(t) = R(Ao)x(t)

61

§3. WEAKENED CAUCHY PROBLEM

is a solution of the Cauchy problem with the initial condition = R(>.o)x(O), so that

Xl (0)

Xl(t) = U(t)R(>.o)x(O) or Xl(t) = R(>.o) U(t)x(O).

Equating R(>.o)x(t) = R(>.o) U(t)x(O), we obtain x(t) = U(t)Xo. Since by hypothesis x(t) is continuously differentiable for t > 0, the weakened solution may be represented in terms of the resolvent according to formula (1.12):

(3.2)

x(t)

1 = - -2. '11"'

i·+i
2. Weakened Cauchy problem, correct on 9 (A). If the weakened Cauchy problem is correct on the set ..L~ 9(A) for all t E (0, 00), then we shall say for short that it is correct on 9(A). Repeating the argument carried out for the correct Cauchy problem, we arrive at the conclusion that a solution of the weakened Cauchy problem which is correct on 9(A) is given by the formula

=

x(t)

U(t)Xo,

where U(t) is a strongly continuous semigroup of bounded operators. If in addition, the operator A has a regular point >.0, then the operators U(t) R(Xo) will be uniformly bounded on every segment [0, T] and will converge weakly to R(>.o) as t-O. Further, the function R(>.o)U(t)Xo for XoE9(A), as was shown above, is a solution of the Cauchy problem with the initial value R(>.o)Xo. In view of the uniqueness of the weakened solution R(>.o) U(t)Xo

=

(Xo E 9"(A».

U(t)R(>.o)Xo

The operators on both sides are bounded, so that the equation is valid for any X E E, i.e. the semigroup U(t) commutes with the resolvent. Therefore it follows that on 9(A) the operators U(t) commute with the operator A. Indeed, AU(t)Xo = A U(t)R(>.o)(A - >.o1)Xo

= AR(>.o) U(t) (A - >.o1)Xo = U(t) (A - >.o1)Xo

+ >.oR (>.0) U(t) (A -

>.o1)Xo = U(t)AXo,

Xo E 9(A).

If the Cauchy problem is not uniformly correct, and the weakened Cauchy problem is correct on 9"(A), then the relation lim U(t)Xo

,~-

.

=

Xo

62

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

holds for the semigroup U(t). if U(t) are unbounded as t -+ o.

XoE~(A).

However. the operators

The assertion on the connection between uniqueness and correctness presented in §2.3 is valid for the weakened Cauchy problem. In fact if the operator A has a regular point, and for each Xo E ~(A) there exists a unique weakened solution of the Cauchy problem on (0, having a derivative which is continuous on (0, and summable on each fmite segment, then the weakened Cauchy problem is correct on ~(A). 3. Uniqueness of the weakened solution. Suppose that x(t) is 'a weakened solution of equation (1.1). In view of the continuous differentiability of x(t) for t> 0 one can integrate by parts: (X».

(X»

Passing to the limit as

E -+

0, we get

t 1 !o e-ATx(T)dT = - - e-ATx(t) o >.

x(O) +-+ -1!ot e-ATAx(T)dT. >. >. 0

If A has a resolvent at the point >., then it is closed, and it can be taken out of the last integral sign, so that

(3.3)

!ote-ATX(T)dT = R(>.) (e-Atx(t) - x(O».

Now suppose that x(t) is a weakened solution for which x(O) = From (3.3) we have for t = T that (3.4)

R(>.)x(T)

=

!oT eA(T-T)x(T)dT

=

o.

!oT eJ"x(T - s)ds.

This equation tells us that the function R(X)x(T) is a function of exponential type. If it is known in advance that the type of this function is less than T, then it will follow from the identity (3.4) that the function x(T - s) is equal to zero close to s = T. i.e. the function X(T) is equal to zero near T = o. More precisely, we refer to the following well-known fact: 5) LEMMA. Suppose that cJ>(s) is a summable function, satisfying the condition 5) E. C. Titchmarsh, Introduction to the Theory of the Fourier Integral, Russian transl., pp. 412-413.

§a. WEAKENED CAUCHY PROBLEM

63

Then ,,(s) = 0 almost everywhere on [h, T].

Applying this assertion to identity (3.4), we arrive at the following theorem. THEOREM 3.1. Suppose that the operator A has a resolvent for sufficiently large positive A and

(3.5) If the weakened solution x(t) of equation (1.1) on the segment [0, T] is equal to zero for t = 0 and hA < T, then x(t) == 0 for 0 ~ t ~ T - hA. In particular, if hA = 0, then x(t) == 0 on [0, T], i.e. the solution of the weakened Cauchy problem on the segment [0, T] is unique.

Indeed, in view of the lemma it follows from the identity (3.4) and from (3.5) with hA < T that x(T - s) is equal to zero for almost all s on [hA' T]. In view of the continuity of x(t) we have x(t) == 0 on [0, T - hAl. The theorem is proved. We note that hA = 0 for a resolvent whose norm grows no faster than some power of A, so that in this case the theorem on the uniqueness of the solutions of the weakened Cauchy problem holds. This theorem holds also for hA < CD, if the weakened solution is considered on the entire semiaxis (0, CD) (T = CD). We shall present an example showing that Theorem 3.1 reflects the essence of the uniqueness question quite faithfully. Suppose that E = ~[O, h]. Let the operator A be the closure of the differentiation operator djds, given on continuously differentiable functions u(s) satisfying the boundary condition u(O) = O. If u(s) (s ~ 0) is a smooth function and uo(O) = 0, then the function x(t) = uo(t + s) with values in 5f2 [0,h] will be a solution of the equation dx dt

= Ax

(= OX) . as

Now suppose that Uo(s) is identically zero on [0, T] and positive for t> T. Then the solution x(t) == 0 for 0 ~ t ~ T - h, and x(t) ~ 0 for T - h < t ~ T. We note that this solution is not extendible be-

64

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

yond the segment [0, T], since for t> T the function u(t + s) does not satisfy the boundary condition and accordingly x(t) EE ~(A). We shall estimate the resolvent of A. From the equation du/ds - Au = v and the condition u(O) = 0 we find that

= fo' eMv(s - a)da.

R(A)v(s)

Hence

I R(A)V~ 2 = LII

[L'

eMv(s - a)da

J

~ LIIe2A'dsl vl

ds

2

=

e~~ 111 vii 2,

ie.

On the other hand, put vo(s)

Then

I voll =

I, R(A) Vo = (e A,

~R(A) ~ ~ IIR(A)voll

1) /A Vh and

-

= =

1

= y'h'

A~ ~LII(eA' -1)2ds' 1

Ay'Wi

.Je2AII _ 4e AII + 2Ah + 3 •

From the upper and lower bounds for the norm of the resolvent just obtained it follows that lim In I R(A) I = h. A

Hence Theorem 3.1 is sharp. 4. Construction of weakened solutions. Their properties. In order to construct weakened solutions of the Cauchy problem we again use the technique of the Laplace transform. THEOREM 3.2. Suppose that the operator A has a resolvent for ReA and that the estimate

(3.6)

I R(A) ~

~ M(1

+ IAI>

(Re A ~ a)

~ a

t3. WEAKENED CAUCHY PROBLEM

65

holds. Then every weakened solution of equation (1.1) ,s representable in the form (3.2). This assertion is a strengthening of what was said in subsection 1, since (3.6) is satisfied if the Cauchy problem is correct. We use identity (3.3) for the proof. Its right side can be considered as the Laplace transform of a function equal to xes) for 0 ~ s ~ t and to zero for 8 > t. Since this function is continuously differentiable on the interval (0, t), then for 0 < 8 < t the following inversion formula is valid: (3.7) It follows from estimate (3.6) that the operators R(X) f>. are uniformly bounded for large I>.1. If x E 9 (A), then (3.8)

R(>.) x

= _~

+ R(>'~Ax ,

80 that R(>.)x/>.-O as 1>.1- 00. In view of the Banach-Steinhaus Theorem, R(>.)x/x-O for any x E E. It then follows from the identity (3.8) that, for xE9(A), R(>.)x-O as 1>.1_00. In (3.7) the element x(t) E9(A); therefore R(>.)x(t)-O as 1>.1_ 00. Hence in view of the Jordan lemma

i~::" e-1(t-.) R(>.)x(t)dx = O. The theorem is proved. N ow suppose that the resolvent satisfies a weaker condition for Rex ~ a: (3.9)

(1) fJ> 0, 1m>. = T).

Then, according to Remark 1.2 to Theorem 1.5, there exist solutions of the Cauchy problem for all x E 9 (A 2). As was shown in the proof of the theorem, these solutions are continuously differentiable on [0, (0). They tum out to be infinitely differentiable on (0, (0). In fact, by Theorem 3.2 they are representable in the form (3.2). Integrating by parts n - 1 times and using (3.9), we arrive at the formula x(t) =

(-l)"~n - I)! 2 ... ,t,,-1

i,,+i .

eAtR"(>.)x(O)d>..

,,-i ..

For the kth derivative we obtain the expression

66

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

(3.10) which has meaning for n it follows that

> (k + 1) / fl. Since

dkx _=AkX(t) dt k

A is closed (see page 60)

(t> 0).

I t follows from the arguments of subsection 1 that the weakened Cauchy problem will have a solution on (0,00) for any Xo E 9(A). These solutions are also infinitely differentiable on (0, co). Again we consider the operator U(t) (t> 0) which assigns to the initial value Xo the value x(t) of the solution of the weakened Cauchy problem with the initial condition Xo: x(t) = U(t)Xo. The operator U(t) is bounded. 'Indeed, from (3.10) with k = 0 we find that

Therefore the operator U(t) may be extended to the entire space-- by continuity. Condition (3.5) with hA = 0 follows from condition (3.9), i.e. the uniqueness theorem holds for the weakened solutions. Thus, the weakened Cauchy problem is correct on g(A) for tE (0,00), which means that the operators U(t) form a strongly continuous semigroup. Estimates of type (3.11) can be obtained for the derivatives of U(t) Xo as well, so that the semigroup U(t) will be infinitely strongly differentiable for t > O. We note that the infinite differentiability of the semigroup for t > 0 already follows from the strong differentiability of the semigroup (differentiability of U(t)Xo for any Xo and t> 0). Indeed, if 0 < a < t d 2U(t)Xo d d dt2 = dt (AU(t)xo) = dt (AU(t - a) U(a)Xo)

d

= dt

(U(t - a)AU(a)Xo)

=

AU(t - a)AU(a)Xo.

Putting a = t/2, we have d 2U(t)Xo dt2

=

[AU(!) ]

2

.

Xo

Analogously we show the existence of all derivatives, and obtain the formula

13.

67

WEAKENED CAUCHY PROBLEM

d"U(t)Xo dt"

= [AU(!.)J" . n

Xo

Assembling all of these assertions, we arrive at the theorem: THEOREM

3.3. Suppose that for ReA ~ a the operator A has a resol-

vent for which condition (3.9) is satisfied. Then the weakened Cauchy problem is correct on the set 9'(A). All of its solutions are infinitely differentiable for t> O. They are given by the formula x(t) = U(t)Xo, where U(t) is a semigroup of bounded operators, strongly differentiable for t> 0 and having the property that lim,-ooU(t)Xo = Xo for Xo E 9'(A). The last property follows from the definition of a weakened solution. We have not included estimate (3.11) in the formulation of the theorem, since it may be made more exact. It follows from inequality (3.9) that the resolvent is defined in a wider region than the halfplane Re A ~ a. In fact, decomposing into a series R( 0'

+ IT') =

~(O' - a)" d"R(a + iT) n! dx"

L.J

,,-0

~ ( 0' -

= L.J

,,-0

a

)"R"+l(a

+') IT,

we see that this series converges if

where q E (0,1). Here (3.12)

I R(O' + iT) I ;;;;

E10'(1 -+ ITI)I"M"+l m

a

(,,+1ll1 ;;;;

For fixed q we find that the resolvent exists and is bounded (because of (3.12» in a region situated to the right of the curve r, having the equation 0' = a - (qj M) (1 + ITI )11, as in Figure 4. The integral

f

M

(1 + ITI)II(1 _ q)

\-----1 A

eAtR(A) Xod>.,

taken along the contour of thE! curvilinear trapezoid ABCD, is equal to zero. The integrals along the segments AB and CD, because of estimate (3.12), do not exceed

#-----fj)

FIGURE

4.

68

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

MIIXoII SO

evtdu =

fa

(l-q)(l+ITI>~

MIIXoII

eat

(1-q)(1+ITI>~t'

-..

that they tend to zero as T-+ ex>. Thus for Xo E g(A) the solution is

(3.13)

This last integral converges already for any Xo E E and accordingly yields a formula for all the generalized solutions of the Cauchy problem. We shall estimate that integral. We have

II U(t) Xo II s ~ r e[a-(q/M)(1+I./)~]t-,--_~M,-----:-~ -2rJrq (1-q)(1+ITI>~

x I d (a -

(3.14)

s

M - 11"(1 - q)

J1

it

(1 + I

(3.15) ;£

III XoII

+ q2 eat r~-(q/M)(l+.lt dT Ilxoll. M2 Jo (1 + T)~ "

We make the substitution (q/M)(l II U(t) II ;£

TI>~ + iT) + T)~t = s.

Then

M 1 eat.!L ()l-l/~L" t e-'s--:2+1/llds 11(1 - q) M qI/M

Moeattl-l/~r (~

- 1) .

Analogously one estimates the derivatives of the semigroup U(t). Indeed, differentiation of (3.13) produces a factor >.. under the integral sign, whose modulus satisfies 1>"1 =

~[ a -

!

(1 +

T)~J +

T2

;£ e(l +

T),

where

e=max(l,lal)+icJ/M='Y+q/M. Under the substitution (1 + T) ~t = s we also get a factor (qt/ M) -l/~ outside the integral sign, and the power of s under the integral sign goes up by 1/11. We have arrived at the following assertion: (q / M)

THEOREM 3.4. Under the condition (3.9), for the semigroup U(t) and its derivatives, the estimates

69

i3. WEAKENED CAUCHY PROBLEM

(3.16)

II d;t~ II ~ Moekea't1-(k+l)/"r (k ; 1 -

/wid for t

1 ),

k= 0,1, ... ,

> 0.

REMARK 3.1. The boundedness of the semigroup U(t) , given by the formula

(3.17)

U(t)

1

= - -2' 11"'

ia+i. m eAtR (A) dA, a_1m

can be proved under a weaker condition on the resolvent than (3.9). It suffices that the resolvent should decrease faster than 1/In 11" I. Then for any E > 0, for sufficiently large 1" we will have the inequality

"R(a + iT)" ~ In(b ~ H)

(with some b > 1) •

. Repeating the arguments used in the proof of the theorem, we arrive at the integral

r

Jo

m

e-(Qt/.)In(b+ r )

which, under the substitution In(b

r'" Jo

dT In(b + T)'

+ T) = S,

e-(Qt/')8e 8

r~duces

to

ds • S

This integral will converge if f is taken so small that qtlE> 1. A more detailed analysis shows that in the case at hand the weakened Cauchy problem is also solvable for any Xo E ~(A) (see [2], Russian pp. 394-397). REMARK 3.2. Suppose that condition (3.9) is satisfied with fJ = 1, i.e. "R(A)" ~ M(l

(3.18)

+ 1TI)

-1.

Then for the corresponding semigroup the estimate (3.19)

" U(t) "

~ M' eat ( 1 + max { 0, In ~ } )

holds, and for its derivatives the estimate (3.16) with fJ = 1. In estimating the derivatives we did not use the fact that fJ < 1. For the estimate of the semigroup itself we arrive at the estimate (3.15) in the form

70

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

Integration by parts yields

II U(t) I ~ M1e,rt [e-(q/M}tlnt + (3.20)

k f'" r(q/M)'lnSds]

~ M#"tln ( e + ~ ) .

3.3. In order to obtain the integral (3.2) we used only the behavior of the resolvent on the line A= a + iT for a fixed a. The information on the behavior of the resolvent in the halfplane ReA ~ a was necessary for the proof of existence of solutions of the Cauchy problem in Theorem 1.4. The question naturally arises as to when the solutions of (1.1) are analytic functions of t. We note that if the solution U(t)Xo is representable in the form of an absolutely convergent inverse Laplace transform, then REMARK

(3.21) The first term generates an analytic function in the upper halfplane: (3.22)

1• U1(z)Xo = - -2 ... J

1"+;'" eAzR(A)XodA

(z

0

=

t

+ is, s> 0),

and the second correspondingly in the lower halfplane. If each of these terms admits analytic continuation into a region containing the nonnegative real axis, then the solution U(t)Xo will be an analytic function of t. Such a situation arises when condition (3.18) is satisfied for the resolvent of the operator A. As we indicated above, the first integral may be replaced by an integral along the ray a = a - (qIM) (1 + T). It then becomes clear that it is an analytic function of z = t + is under the hypothesis that

! > sup [ _!L . 1 + T 1 = _!L.

t • M T J M Analogously, replacing the path of integration in the second term, we find that it is analytic under the condition sit < ql M. Thus, the solution U(t)Xo will be analytic in the sector Isltl < ql M, and, since q is any number on (0,1), also in the sector

71

§3. WEAKENED CAUCHY PROBLEM

Is/tl < l/M.

(3.23)

Estimating the integral (3.22) in the same way as in the proof of Theorem 3.4, we find that

IU

(z)

1

I ~ ~ ~1 + -

1- q

q2 eale'

M2

Jor'" e-(l+r)(q/M)I+.)~. 1+ T

-r.

If s > 0, one may drop the factor e under the integral sign, and we arrive at the same estimate as in (3.19). If s < 0, then we may drop the factor e' in front of the integral, and then

I U1 (z) I ~ ~' eat

(1 + max {O, In t + (S~/q) }) .

Performing the same computations for the second term in (3.21), we arrive at the inequality

~U(z)11 ~M'eat(l+~max {o,In}} +~max {o,In t _

(3.24)

IIM/ }) s

q

~ M'eatIn (e + t -IS~ M/q)· We have proved the following. THEOREM 3.5. If condition (3.18) is satisfied, then a generalized solution of (1.1) is an analytic function of z = t + is in the sector (3.23), and it satisfies the estimate

Ilx(t + is) I

~ M'e"'In (e + t _ (~/q) lSI) Ilx(O) I

for any q E (0, 1) in the sector Is I ~ qt/ M.

FIGURE

5.

FIGURE

6.

72

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

REMARK 3.4. It is clear from the proof of the theorem that \~e sector of analyticity of the semigroup U(t) may be found as follows: if inequality (3.18) holds for the resolvent on the rays Iarg(a /" "') I = 1r /2 + 41, then U(z) is analytic in the sector Iarg z I < 41 (Figure 5). REMARK 3.5. In inequalities (3.19) and (3.24) it was sometimes important to have the best way of estimating the behavior of U(t) towards infinity. Denote by '" the least upper bound of the real parts of the points of the spectrum of the operator A lying to the left of the line ReX = a, at whlch inequality (3.18) is satisfied. Suppose al > "'. Then the same inequality with a new constant M will be satisfied on the line ReX = al as well. Indeed, as was shown above, inequality (3.18) with the constant M / (1 - q) is satisfied in the region IReX - al ~ (q/M) ImX, which contains the entire line ReX = al with the possible exclusion of the finite segment AB depicted in Figure 6. This segment lies in the region of regularity of the operator, and inequality (3.18) may be guaranteed on it at the cost of the choice of a large constant M. Thus, in inequalities (3.19) and (3.24), by changing appropriately the constants M' and M the number a may be replaced by any al > SUPAES(A) ReX. 5. Abstract parabolic equations. Equation (1.1) is called an abstract parabolic equation if the weakened Cauchy problem for it is uniquely solvable for any Xo E E. On the basis of what was said in subsection I, the unique solvability of the weakened Cauchy problem implies the unique solvability of the Cauchy problem on R(Ao)E = ~(A), and all the solutions are It then follows from: Theorem continuously differentiable on [0, 2.12 that: (X».

THEOREM 3.6. If in an abstract parabolic equation the operator A has at least one regular point, then the Cauchy problem is uniformly correct on every finite segment.

Conversely, if the Cauchy problem for (1.1) is uniformly correct and all of its solutions are twice continuously differentiable for t > 0, then the equation is abstract parabolic. The existence and uniqueness of a weakened solution for any Xo E E follows from the considerations of subsection 1 (see also Theorem 3.1). The semigroup generated by an abstract parabolic equation will be strongly continuously differentiable for t > 0, from which it follows as indicated above that all weakened solutions are infinitely differentiable functions for t > O. Here we see reflected a characteristic

13.

WEAKENED CAUCHY PROBLEM

73

property of partial differential equations of parabolic type: however "bad" the initial conditions may be the solution for any t > 0 will be infinitely smooth. In our case this means that x(t) E~(A") for any n. This fact explains the name which we have given to the equation. Under the conditions of decrease of the resolvent along a line parallel to the imaginary axis considered in the preceding subsection, the differentiability of all the generalized solutions was proved. Therefore if the Cauchy problem is uniformly correct for the corresponding equations, then they are abstract parabolic. Using Remark 3.1, we obtain the following: THEOREM 3.7. For equation (1.1) to be abstract parabolic, it suffices that condition (2.17) and the condition

(3.25)

lim ITI-"

Inl Till R(a + iT) I =

0

~

/wld for some a.

The following is very important in the applications. THEOREM

3.S. If the operator A has a resolvent for ReA> (0) and

(3.26)

(ReA

~

(0),

thl!n equation (1.1) is abstract parabolic. The corresponding semigroup is analytic in some sector of the complex plane containing the semwx,s t> O. In this sector it satisfies the estimate (3.27)

II U(t+ is) II ~ Qe"l' «(0)1> (0). choose a > fII and positive. Repeating

PROOF. We the argument of the proof of Theorem 3.4, we first coarsen inequality (3.6) to the inequality

~R(a + iT) II ~~M, a T

a'

=

a - (0).

We then arrive at the estimate (3.15) in the form

II U(t) II ~ Mgea"

r"

)(q/M)a"

e-· ds • s

This estimate is true for all sufficiently large a'. For sufficiently small = l/t. Then we obtain

t we may take a'

II U(t) II ~

Moe!"

q/M

e-· ds = C, S

74

I. FIRST-ORDER EQUATIONS WITH ;CONSTANT OPERATOR

i.e. the semigroup U(t) is bounded close to zero and the CauchY<J>roblem is uniformly correct on every finite interval. The analyticity of the semigroup was proved in Theorem 3.5, so that the equation is abstract parabolic. In estimates (3.19) and (3.24) the term with the logarithm may be dropped for any a> "'1. Putting a = "', we arrive at estimate (3.27). The theorem is proved. ~ It turns out that in order for a semigroup with the estimate (3.27) to correspond to a uniformly correct Cauchy problem it is necessary that the resolvent satisfy inequality (3.26) in some halfplane. We shall prove this. Let X = a + iT, where 8 a > '" and T > o. Consider FIGURE 7. the zero-valued integral, taken along the contour OABO (Figure 7),

o

Ie->.zU(Z)xodZ

=

0,

where OB lies inside the sector in which the semigroup is analytic and satisfies (3.27), and AB is an arc of a circle of radius R. The integral along this arc tends to zero as R _ ClCl. Indeed,

\\LB

e-AZU(z)XQdz \\

~ R I_o. e<-a",,",+rsinl)RQe"lRd~ ~

QRe<-acoeH

" l )R.

(we suppose that", > 0; otherwise the second factor is replaced by Q). If the angle tP is sufficiently small, then acostP> "'it and the last expression tends to zero. Thus R(>.) =

1-

1"'

e- Al u(t)dt=e- i40

Then

~.SIDtP [

Q

a - -"'1costP

+T ]

e-A,..-i+U(re-i+)dr.

75

f3. WEAKENED CAUCHY PROBLEM

where Ql = Q/sinq, and W2 = wdcosq,. Hence (3.26) holds as an estimate for the resolvent in the halfplane ReA ~

WI.

As was shown in Remark 3.3, estimate (3.26) for the resolvent implies an inequality for the derivative:

II dd~ II ~ ~l e

a

(3.28)

'.

This inequality in turn implies the analyticity of the semigroup U(t). Indeed, it follows from (3.28) that

"d;t~ I = II[ AU(£)

r I ~ Mlnn

n

eat,

so that the Taylor series U(t + ,,)

= U(t) +

E:; d:~(t)

has a nonzero radius of convergence: r

=

--IiF;==;=~=:='~:;r ~

1

.!!.....:..._~--'-:........!!....- lim ~M~nn tnn!

t

- Mle' -

Here we are using the fact that in view of Stirling's formula n! We summarize:

> (n/e)n.

THEoREM 3.9. For a uniformly correct Cauchy problem the following assertions are equivalent: 1. The semigroup generated by the problem is analytic in some sector and satisfies (3.27). 2. The resolvent has the property (3.26) in some halfplane. 3. The semigroup is strongly differentiable, and (3.28) holds for the derivative.

6. Solutions with retarded smoothness. Analyticity and quasi-analyticity. In the preceding subsections we started from estimates of the resolvent on lines parallel to the imaginary axis. These estimates made it possible to transform the integral (3.2) into an integral along contours on which the function e"t tends to zero as I>.1-+ CD. One can consider the case when it is directly known that the resolvent is defined on an appropriate contour, as we shall do in this subsection. Consider on the complex plane a curve r given by the equations

76

I. FlRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

where ~(T) is a smooth nondecreasing concave function defined for T> 0 and such that ~(T) - 00 as 1 ' _ 00. Suppose that this curve consists of regular points of the operator A and that its resolvent satisfies

I R(A) I

~ ce- bo

(A

=

0'

+ iT E r).

Introduce the integral x(t) = - 21 ,

(3.29)

r£

r

Jr eAtR(A)dA.

The integrand may be estimated in modulus in terms of the quantity From the concavity of ~(T) it follows that the derivative ~'(T) does not increase, so that IdAI = 1- ~'(T) + ildT ~ MldT. If we choose t so that

ceo(t-b).

«(j > 0), then the integrand can be estimated for large T by the quantity ~'(T)e-6t1(.)/2, so that the integral (3.29) converges absolutely. If we consider the integral obtained from (3.29) by differentiating n times under the integral sign:

~ r A"eAtR(A)dA' 2r£ Jr

(3.30)

then it suffices for its convergence that

lnlTI +-ll Iln~'(T)1 t>b+ n -1· un ~(T) m ~(T)

(T_OO).

Consider some special forms of the curves r. 1) Suppose that ~(T) = a lnl Tj. Then the integral (3.29) will converge absolutely and uniformly in t for t ~ to> b + l/a, and the integral (3.30) for t> tIl > b + (n - 1)/a. For t ~ tl x' (t)

=

-.1.... r XeA1R(A)dA =.1.... reA1dA 2r£ Jr 2r£ Jr

21, 11"£

r

Jr eAtAR(A)dA.

In the first term the contour of integration may be contracted to a point, so that the integral is equal to zero and x' (t)

=

Ax(t)

(t ~ t l ),

§3. WEAKENED CAUCHY PROBLEM

77

i.e. x(t) satisfies (1.1). Making the substitution yet) = x(t + t 1), we obtain a solution yet) of (1.1) on [0, 00). As is clear from what went before, this solution will have the property that its smoothness steadily increases with increasing t. 2) Now suppose that If(T) =aITI P (O
If(~

so that x(t) is defined, satisfies (1.1), and is infinitely differentiable for t> b. If we make the substitution yet) = x(t + t 1), where t1 > b, then yet) is an infinitely differentiable solution of (1.1) on [0, 00). If P = 0, i.e. the resolvent is bounded on r, then x(t) will satisfy (1.1) for t > O. It is not hard to find for it the estimate II x(t) II

~ t~P

(t> 0).

3) Put 1I;(T) = alTI - a. In this case everything said above remains valid, and moreover x(t) is analytic for t> b. Indeed, for its derivatives we get II x(")(t) II

1

~ C1

e(.. -al
= 2c1ea(t-b)[a(t - b) ]nH = 2c1ea(t-b) [aCt -

L"

e-zz"dz

b) ]"Hn!,

so that x(t) is analytic in a disk of radius aCt - b). 4). It is natural to expect that somewhere in the passage from infinite differentiability (case 2) to analyticity (case 3) there must appear the case when the solution belongs to quasi-analytic classes. Indeed, SUppose that the curve r satisfies the following additional condition: 10 • For certain a ~ 1 and TO > 0 such that a TO > 1 the function lnlf(T)/InaT increases monotonically for T ~ TO and tends to 1 as T--+ 00. Fix a number T1 ~ TO and write qo = In 11' (TO) /In aTo and q1 = lnlf(T1)/aTl < 1. Then qo ~ q1 and for T> T1we have 1 >In 11' (T)/InaT > q1 or aT> lfh) > a q1 Tq1 > a'lOT V1 =P oT q1. Further. since If(T\) = a Q1 TQ1, Q and since If(T) > a Q1T 1 for T> Th we have If'(T1) ~ q1aQ1Tf1-1 ~ qo If(T1)/Tl' In view of the arbitrariness of the choice of T1 ~ TO this last inequality is valid for any T > T1' We suppose that TO is so large that on the curve r we have 0' = - If(T) for ITI ~ TO' Then the inequality obtained above may be rewritten in the form

78

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

(1'

~

1'1)

~'(T) ~ qo

and

1;

r

(1'

~

Passing in the integral to the variable of integration s = and using inequalities (3.31). we get

TO).

(J

=

~(T)

II x(n)(t) II ~ cM1 eat (l + a) nT~ 11"

cM1 f." e -at ( s+ ( -S ) +-BJ. Po

q l/ 1 ) "

11"

~ C1eat(1

+ a)"T~

+ C~l { max ( 1,

:J +

(s)

-1 qoS Po

11q1

1 r+ 1) /q1 t-(,,+l)/9tr

ds

(n ~ 1).

For a fixed interval 0 ~ t ~ T of the variable t, using the fact that if t ~ 1 and t-(nH)/9 ~ t-("H) if t ~ 1, we may )/ql ~ t-(n+l)/qo, write the resulting inequality in the form t-(n H

II x(")(t) II

~ pnT~ + Q"r

(n ~ 1)

(O~t~T),

where P and Q are certain constants. Now we take 1'1 = n. Then ql = In~(n)/alnn. The Stirling formula leads to the inequality max V'llx(n)(t) II O~t~

~ Rnln"/In#-(n).

T

In view of the Denjoy criterion x(t) will lie in a quasi-analytic class if the following condition is satisfied: 2°. .,

1

2: n Inn/In~(n) =

co.

n=1

We have arrived at the following theorem: THEOREM

3.10. If the resolvent of the operator A is bounded on a curve

r for which conditions 1° and 2° are satisfied, then the function x(t) is a solution of equation (1.1) lying in a quasi-analyticity class.

§3. WEAKENED CAUCHY PROBLEM

79

It is not hard to verify that an example of a function satisfying conditions 10 and 20 is

7. The Cauchy problem inverse to the correet problem. For equation (1.1),

dx -=Ax dt

we pose the problem of finding on [0, T] a solution, given its value on the right endpoint of the segment: (3.32)

x(T)

=

Xl

E g-(A).

If we introduce a new variable T = T - t and write x(t) = x(T - T) = y( T), then for the function y( T) we arrive at the ordinary Cauchy problem: dy

(3.33)

dT

= -Ay,

yeO)

= Xl E g-(A)

with the operator-A. The problem (1.1)-(3.32) or, what is the same thing, problem (3.33), will be called inverse to the problem (1.1)-(1.2). Analogously one introduces the concept of a problem inverse to the weakened Cauchy problem. We suppose that both the direct and the inverse weakened Cauchy problems are correct on ~(A), and we denote their semigroups by U(t) and Ul(t) respectively. Suppose that Xo E ~(A). Then both the functions Ul(T) U(t)Xo and U(t) Ul(T)Xo are solutions of (1.1), the operator A commutes both with Ul(T) and with U(t). They satisfy the same initial condition x(l) = Ul(T)Xo and therefore coincide. Thus the operators U1(T) and U(t) commute on ~(A), and, since they are bounded, everywhere. Further, let us consider the function z(t) = U(t) Ul(t)Xo. Its derivative is dz dt

= A U(t) Ul (t) Xo -

U(t) A Ul (t) Xo

= O.

This means that U(t) Ul (t) Xo is a constant. Now suppose that A has a regular point Xo and that Xo E ~(A 2). Then

80

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

U(t) U 1 (t) Xo - Xo

=

U(t) R(~) (U1(t) - I) (A - >.01) Xo

+ U(t) Xo -

Xo-+~ r

In view of the uniform boundedness- of the operators U(t) R(~) and the fact that Xo E 9(A) and (A - >.o1)Xo E 9(A). Thus for x E 9(A2)

U 1 (t) U(t) Xo = U(t) U 1(t) Xo = Xo. In view of the boundedness of the operators U(t) U1 (t) this relation holds for all Xo E E. Thus the operators U(t) and U 1(t) are inverse to one another. For t < 0 write U(t)

=

U1(-t).

Then the operators U(t) will form a group U(t) U(T)

=

U(t

+ T)

(- CD

< t, T < CD).

From the group property it follows that the norms of the operators U(t) are uniformly bounded on each finite interval of t-variation (see the proof of Lemma 1.1). Hence in particular it follows that U(t)Xo-+Xo for any Xo E E and t --+ 0, i.e. that the direct and inverse Cauchy problems are uniformly correct. This means that the operators A and - A must satisfy conditions (2.17). We have arrived at the following assertion: THEoREM 3.11. If the operator A has at least one regular point. then in order that the direct and inverse Cauchy problems should be correct it is necessary and suffieient that they should be uniformly correct, i.e. that the following condition be satisfied:

M

~ R"(A) II ~ (IReAI _ (aI)" for IReAI > III.

(3.34)

Conditions (3.34) are sufficiently restrictive, so that in many examples the direct Cauchy problem is correct and the inverse problem incorrect. We introduce a new definition: DEFINITION 3.2. The inverse Cauchy problem (1.1)-(3.32) is said to be correct in the class of bounded solutions on the segment [0, T], if for every M, E> 0 and to E (0, T] there exists a Cl (M, E, to) > 0 such that for every solution x(t) of equation (1.1) satisfying the conditions (3.35)

II x(t) II

the inequality

~

M (0 ~ t ~ T) and II x(T) II ~ Cl(M, E,to),

81

§3. WEAKENED CAUCHY PROBLEM

I x (to) I

(3.36)

~

E

(0

< to < 1').

is satisfied. From this definition of correctness it follows that the inverse Cauchy problem has a unique solution. It differs from the usual definition of correctness in that it does not assume the existence of a solution and in that continuous dependence on the initial data is required only on each class of solutions bounded by one constant. However this definition has a practical meaning, since in many problems both the solution itself (whose existence is clear from physical considerations) and possible "perturbations" of the solution can be estimated a priori. THEOREM

3.12. 1/ the direct Cauchy problem is correct in the sector is correct in

Iarg rl < q,o (q,o> 0), then the inverse Cauchy problem the class 0/ bounded solutions on any segment [0, T]. 8~:

PROOF. It follows from Theorem 1.7 that the solution of the Cauchy problem is analytic in the sector 8~. Suppose that 0 < q,1 < q,o and a > O. Consider the sector 8~1: Iarg rl ~ q,1 and the sector 8:1 gotten from it by a shift in the direction of the positive semiaxis by an amount a. In view of (2.25), the following estimate holds for the semigroup generated by the Cauchy problem in the sector 1:

8:

I U(r) I

~ M 1e"'ll1

(rE 8:1)

or (3.37)

where WI = W/COSq,l and M =M1e"'l tJ

tJ•

We denote by G the region obtained from the sector 8:1 by making a cut along the ray a + T ~ t < CD (Figure 8). Now suppose that x(t) = U(t) Xo is some solution of the Cauchy problem for equation (1.1). We shall estimate x(r) on the boundary of G. On the sides of the sector 8~, we hav'e from (3.37) that

8

FIGURE

8.

(3.38)

I xU') I

~

M e"'lRe (t- lI) I xoll· tJ

On the cut we have x(t) = U(t)xo = U(t - 1') U(1')xo, and since t - TE 8:1, then

82 (3.39)

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

~ x(t)

I

=

I U(t - T) U(T)xoll

I

~ M,j!"'1(t-T-a) x(T)

II.

Now we apply the Nevanlinna theorem to the function y(r) = e-1(t-a)x(r), which is analytic in the region G. On one part of the boundary (the sides of the sector) we have

I y(~) I

(3.40)

~

Mall xoll

from (3.38). On the other portion (the cut) we have

I y(r) I

~

M,j!-"1 T I x(T) I

from (3.39). Thus, at any point of the region G (~E

(3.41)

G),

where Co) (~) is the harmonic measure of the ray a + T ~ t < region G. 6) Hence I x(r) I ~ M,j!"'1IRet-a-T..cm I xoI1 1-..(I) I x(T) I ",(ll. (3.42)

ex>

in the

Suppose that to E (0, T). Take a = 1to. Then for t = to inequality (3.42) will hold with a = t tOo If for a given E > 0 we write ~(M,E,to)

) IMtol , 2 = Me"'1 T (MaE M e-("1/ )to

then, in view of (3.42), if r =

to and a = t to.

(3.35) implies (3.36).

§4. Equations in Hilbert space 1. Equations with negative definite selfadjoint operators. Consider equation (1.1) in Hilbert space H. We suppose that A = - B, where B is a positive definite selfadjoint operator in H. Then equation (1.1) may be written in the form

(4.1)

dx dt

+ Bx = O.

For definiteness we will suppose that 6) We should not be disturbed by the sidered the portion of it with Re r ~ N. have the estimate (3.40), which means measure of the segment a + T ~ t ~ N letting N tend to infinity.

fact that G is infinite. We could have conOn the boundary Re r = N we would also (3.41) as well, where ... (r) is the harmonic in the region GN. We arrive at (3.41) by

§4. EQUATIONS IN HILBERT SPACE

inf(Bx, x)

(4.2)

=1

(x, x)

83

(xE~(B».

For the resolvent R(A) = (A - AI) -1 = - (B + AI) -1 of the operator A the estimate II R(A) II ~ l/d will hold, where d is the distance from the point Ato the spectrum of the operator A,sothat in the halfplane ReA ~ -1 (4.3) It follows from this estimate that equation (4.1) is an abstract parabolic equation with an analytic semigroup. Further, comparison of (4.3) with inequality (2.18) shows that the corresponding semigroup is contractive. Moreover, (4.4) THEOREM 4.1. Equation (4.1) under condition (4.2) is abstract parabolic, and the semigroup corresponding to it is given by the formula

U(t)

(4.5)

= e- Bt =

i"

e-AtdE).,

where E). is the spectral resolution of the identity generated by the operator B.

In order to prove the last assertion, we specify that for any Xo E ~ (B) the function x(t)

= e-BtXo =

i"

is a solution of the Cauchy problem. For " x(t

+ i1~ -

x(t)

+ Bx(t)

e-).tdEAXo i1t

> 0 we have

112

(4.6)

Thefunction (e1)/u + 1 tends to zero as u-+O and to 1 as u-+ 80 that it is bounded for u > O. For Xo E ~(B) the integral U -

00,

84

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

i" ).

2yJ, (EAXo, Xo)

converges, so that the second term in (4.6) may be made arbitrarily small by choosing a sufficiently large N, and then the first term arbitrarily small by choosing a sufficiently small Il.t. Thus the right derivative of the function x(t) exists for t ~ 0 and is equal to - Bx(t). Analogously one verifies that for t> 0 the left derivative has the same value. In view of the uniqueness of the solution of the Cauchy problem for :to E 9(B) we have U(t)Xo = e-BtXo.

Both of these operators are bounded, so that they coincide for all Xo E H. The theorem is proved. The explicit expression of the semigroup in terms of the spectral resolution makes it possible to deduce all of its properties directly from formula (4.5). We can find, for example, an estimate for the derivatives of the semigroup, or, more generally, an estimate for the operator B"U(t) for any a ~ o. An easy calculation yields

I B"U(t) II

=

(4.7) =

II i" )."e->.tdE>.11 ~ ~~8f )."r At {(alet) .. for t ~ e- t for t ~ a.

We note that e-t < (a/et) .. for t> a, so that the estimate

(4.8)

I B"U(t) I

~

(a/et)"

is valid for all t> O. In particular,

Ild;t~ II ~ (:tf· It also follows directly from formula (4.5) that the semigroup U(t) is analytic inside the right halfplane and that throughout this halfplane the inequality

II U(t + is) I is satisfied.

~ e- t

85

§4. EQUATIONS IN HILBERT SPACE

2. Equations with eontradion semigroups. Dissipative operators. Equation (4.1) had the largest collection of "nice" properties among those which we have considered for the Cauchy problem. The question arises of characterizing those operators for which equation (1.1) has only a portion of these properties. THEOREM 4.2. In order for the Cauchy problem for equation (1.1) with a closed operator A in Hilbert space to have a contraction semigroup, it is sufficient that the following conditions be satisfied:

Re(Ax,x) ~ 0,

(4.9)

(4.10)

Re(A*y,y) ~

xE9(A),

0, y E 9(A*).

PROOF. Suppose that condition (4.9) is satisfied. For x E 9(A) and A with Re A> 0 we write y = Ax - Ax. Taking the scalar product by - x, we get

- (y, x) = - (Ax, x)

+ A(x, x).

In view of (4.9) - Re(y,x) ~ ReA(x,x).

Since - Re(y, x) ~ IIYII I xII, we have (4.11)

IIxil

1 eA

1 eA

~-R IIYII =-R IIAx-Axll·

Thus A is a point of regular type for the operator A. Since A is closed the range 9R (A - Al) is closed. We shall show that it coincides with the entire space. In the contrary case there would be an element Z ~ 0 orthogonal to ~ (A - Al): (Ax - AX,Z)

=0

It follows from this equation that

(xE9(A». ZE

9(A*) and A *z =

Re(A*z,z) = ReA(z,z)

iz. But then

> 0,

which contradicts (4.10). Hence for ReA> 0 the operator A has a resolvent. and, from (4.11), (4.12)

I R(A)xll

1 ~ ReA" xII·

The assertion of the theorem now follows from Theorem 2.10.

86

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

The conditions of Theorem 4.2 may be verified on many examples. However there is sometimes difficulty in verifying condition (4.10). For it a complete description of the domain of the operator A* adjoint to A is needed, and this is not always an easy problem. As is clear from the proof, condition (4.10) is needed in establishing the solvability of the equation Ax - >.x = y for all y E H, which sometimes can be done directly. DEFINITION 4.1. A linear operator A with a domain dense in a Hilbert space is said to be dissipative if Re(Ax,x)~O

for xE~(A),

and maximal dissipative, if it is dissipative and does not have a nontrivial dissipative extension. A dissipative operator always admits closure. Indeed, suppose that x,.-+O (x" E ~(A» and Ax" -+ y. For any x E ~(A) and complex a we have

+ aX,,), X + ax,,) ~ 0, Re(AX,x) + Rea(y,x) ~ o.

Re(A (x

SO that in the limit we get This inequality can be satisfied for any complex ex only in the case when (y, x) = O. Because of the denseness of ~(A) in H, y == o. Obviously the closure of a dissipative operator will also be dissipative. As was shown in the proof of Theorem 4.2, all the points>. with Re>. > 0 will, for a dissipative operator, be points of regular type. Hence in particular it follows that a dissipative operator is closed if and only if the range ~ (A - >.1) of the operator A - >.1 is closed for Re>. > o. If for some >. with Re>. > 0 the range ~ (A - >.1) coincides with the entire space, i.e. the operator has a resolvent, then it will be a maximal dissipative operator. Indeed, in the contrary case let A be a nontrivial extension. Then there would be an element Xo ~ 0 such that (A - >.1) Xu = 0, which contradicts the dissipativeness of A. We shall show that if the dissipative operator A is closed and the range ~ (A - >.1) for Re>. > 0 does not coincide with the whole space, then it has a nontrivial dissipative extension. For a fixed >. we denote by N the orthogonal complement to ~(A - >.1) in H. On the set ~(A) + N we define an operator A by the equation

A(x+ u)

=

Ax - ~u

(xE~(A), uEN).

For the operator A to be uniquely defined, it is necessary and sufficient

87

§4. EQUATIONS IN HILBERT SPACE

that x + u = 0 should imply that x = 0, U = o. The defect subspace N consists of the eigenvectors of the operator A *: (4.13)

A *u = Au

so that x

(xEN),

= - u would imply (Au, u)

=

(u, A *u) = A(u, u),

and since Re X> 0, then u = 0, so that x = o. We shall show that the linear operator A is dissipative. We have

(A(x

+ u), x + u) =

(Ax,x)- ~(u,u)

+ (Ax, u) -

X(u,x).

Further, in view of (4.13), .

(Ax,u) = (x,A*u) = (x,~u) = x(x,u). Then

Re(A(x + u),x + u) = Re(Ax,x) - ReX(u, u) ~ 0, i.e. A is dissipative. The operator just constructed is maximal dissipative. Indeed, if Y E H, then y = z + w, where z E ~ (A - XI) and wEN. Then there existsanxE9(A) such that AX - Xx=z=y- w. Writingu = -w/2ReX, we obtain A(x + u) - A(x + u) = Ax - Xx - 2ReXu = y, i.e. y E ~(A - xI). We have proved the following theorem: THEOREM 4.3. Every dissipative operator admits extension to a maximal dissipative operator. A dissipative operator A is maximal dissipative if and only if for any X with Re X > 0 the range ~ (A - xl) coincides with the whole space.

We shall establish one further criterion for maximal dissipativeness. THEOREM 4.4. For a dissipative operator to be maximal dissipative, it is necessary and sufficient that it be closed and that condition (4.10) be satisfied.

The sufficiency of the conditions was proved in Theorem 4.2. NECESSITY. Inequality (4.9) in the definition of a dissipative operator may be rewritten in an equivalent form, as follows: (4.14)

II Ax + x 112 ~ I Ax - x 112

Now using the identity

(xE9(A».

88

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

«A - I) x, (A *

+ I) y) =

«A

+ I) x, (A * -

1) y),

valid for any x E .9J(A) and y E .9J(A *), we find using (4.14) that I«A-I)x, (A*+I)y)1 ~ II (A

+ 1) x ~ I (A * -

I) y I ~ I (A - 1) x III (A * - 1) y II·

If the. operator A is maximal dissipative, then (A - 1) x runs through all of H, so that

I (A * + 1) yll

~ I (A * - I) yll,

which is equivalent to (4.10). The theorem is proved. Theorem 4.2 may be strengthened. THEoREM 4.5. In order for the Cauchy problem for equation (1.1) with a closed operator A in Hilbert space to have a contraction semigroup, it is necessary and suffu:ient that A should be a maximal dissipative operator, i.e. that conditions (4.9) and (4.10) should be satisfied.

For the proof of necessity we note that for a contraction semigroup the function I U(t)XoIl 2 is nonincreasing, since ~ U(t

+ r)Xo11 2 =

I U(r) U(t)XoIl 2 ~ I U(t)Xo 112

(r> 0).

Therefore the derivative of this function is nonpositive. If Xo E .9J (A), then 2Re(AXo,Xo)

=

(AXo,Xo)

+ (Xo,AXo) = dtd (U(t)Xo, U(t)Xo) 1,-0 ~ o.

Thus A is dissipative. The fact that it is maximal dissipative follows from the fact that it has a resolvent for Re). > o. The theorem is proved. 3. Equations with isometric subgroups. CODSenative operators. A linear operator A with a dense domain is said to be conservative, if Re(Ax,x) = 0

(x E .9J(A».

It is clear that the operator - A will also be conservative. Every conservative operator is dissipative. If we write A = iB, then obviously Im(Bx, x) = 0, i.e. B is a symmetric operator. The converse is also valid: if B is symmetric, then A = iB is conservative. Every symmetric extension 11 of the operator B will yield a conservative and therefore dissipative extension of the operator A: A = ill Therefore, if A is a conservative and maximal dissipative, then B is a maximal symmetric

89

§4. EQUATIONS IN HILBERT SPACE

operator. Conversely, if B is a maximal symmetric operator, then either

A or - A is a maximal dissipative operator. Indeed, the defect index of B is equal to zero either at the point i or at the point - i. Further, (a - I) = i(B + il) and (- A - I) = - i(B - iI), which means that for one of these operators the range coincides with the whole space. Finally, if A is closed and A and A * are both conservative, then A = iB, where B is a selfadjoint operator. This follows from the fact that B and B* will simultaneously be symmetric operators. THEOREM 4.6. For equation (1.1) to have a semigroup of isometric operators, it is necessary and sufficient that A should be a maximal dissipative and conservative operator. PROOF. The necessity follows from Theorem 4.5 and the fact that for xEg(A)

d

d

2 Re(Ax, x) = dt (U(t) x, U(t) x) 1,-0 = dt (x, x) =

o.

For the proof of sufficiency we need only verify that the operators U(t) are isometric. For x E g(A) we have

d dt (U(t)x. U(t)x)

=

2Re(AU(t)x, U(t)x)

=

0,

so that (U(t)x, U(t) x) = (U(O)x, U(O)x) = (x, x). Since g(A) is dense in H, this last relation is valid for all x E H. The theorem is proved. THEOREM 4.7. For equation (1.1) to have a semigroup of unitary operamrs, it is necessary and sufficient that A = iB. where B is a selfadjoint operator.

Indeed, in view of Theorem 4.6, the conditions of the theorem are equivalent to having both semigroups U(t) and U*(t) isometric and thus unitary. In the case at hand the semigroup U(t) may be extended to a group of unitary operators, if one puts

U( - t) = U*(t) = U-1(t), For this group the spectral representation

U(t) x =

f- ~ eiA1dEAx

t>

o.

90

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

holds, where E). is the spectral resolution corresponding to the operator B. The verification of this fact is the same as in subsection 1. 4. Equations reducing to equations with dissipative operators. In a number of cases equation (1.1) reduces to an equation with a maximal dissipative operator. 1) The operator A is said to be bounded to the right, if (4.15)

~

Re(Ax, x)

(x E ~(A».

w(x, x)

N ow suppose that A is closed and that A and A * are bounded to the right, i.e. (4.15) is satisfied and (4.16)

~ w(y,y)

Re(A *y,y)

(y E D(A

*».

Then the operator A - wI is a maximal dissipative operator. The semigroup Vet) generated by the equation dv dt

(4.17)

-

=

(A - wI)v

'

is contractive. But the semigroups of equations (1.1) and (4.7~ are connected by the relation U(t) = ewtV(t). Thus, in the case at hand the Cauchy problem for equation (1.1) is uniformly correct, and

I U(t) I

(4.18)

~ ewt.

Conversely, if equation (1.1) has a closed operator A and generates a semigroup satisfying (4.18), then V(t) = e-"'U(t) will be a contraction semigroup, and the operator A - wI will be maximal dissipative. Thus the operator A satisfies conditions (4.15) and (4.16). Suppose that the operators A and A * have a common domain and that the operator (A + A *)/2 is bounded. Then (x,Ax) R e (A x,x) -_ (Ax,x) + 2

=

2 x, x ) ~ ( A+A*

IIA+A* I Ilxll 2

x

~w(x,x)

for x E

~(A), .

i.e. (4.15) is satisfied. Inequality (4.16) is also satisfied, since Re(A *y,y) = Re(Ay,y). Thus the Cauchy problem is uniformly correct, and (4.18) is satisfied. 3) Suppose that the operator A is represented in the form (4.19)

A=QB,

91

§4. EQUATIONS IN HILBERT SPACE

where Q is a selfadjoint positive definite bounded operator, and B is a maximal dissipative operator. We introduce into the space H a new scalar product, using the formula

[x,y] = (Q-lx,y).

(4.20)

Since by hypothesis

~ (x, x) ~ (Q-1x,x) ~!

(x,x)

(m> 0, M

<

(0)

(m and Mbeing the lower and upper bounds ofQ), the norm I xiiI = [X,X]2 is equivalent to the original norm of the space: (4.21)

1

v'Mllxll

1 ~llxlll~Vmllxll.

We shall show that with this new scalar product the operator A is dissipative: (4.22)

Re[Ax,x] = Re(Q-1Ax,x) = Re(Bx,x)

~

o.

Further, it will be closed. Indeed, if ,x,.-+x (xnE~(A» and Axn-y, then Q- 1Axn = BXn-+ Q-ly• Since B is closed it follows that x E ~(B) = 9'(A) and Bx = Q-ly, i.e. y = QBx = Ax. Finally, ~ (A - AI) coincides for A > 0 with the entire space H. In the contrary case the defect space N),. would be nonempty, i.e. there would be an element u such that

A*u = B*Qu = Au. But then

Re(B*Qu, Qu) = A(u, Qu)

> 0,

which contradicts the maximal dissipativeness of the operator B. Thus, if the operator A is represented in the form (4.19). then the Cauchy problem is uniformly correct, the corresponding semigroup is contractive in the new norm. and thus. in view of (4.21).

II U(t) I ~ v'MTiii". where M and m are the upper and lower bounds of Q. 4) Suppose that A has the form

A=BQ, where B is a dissipative operator and Q is a selfadjoint positive definite

92

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

operator. Along with equation (1.1) we consider the equation

~; = Q1!2 BQl/2Z.

(4.23)

The operator Q1!2 BQl/2 is dissipative: for x E Re( Ql/2BQl/2X, x) = Re(BQl/2x, Q1!2x)

~(Ql/2 BQl/2) ~

we have

o.

If one requires that this operator have a bounded inverse, which would be the case for example if B or Qlf2B or BQ1!2 had a bounded inverse, then it would have regular points close to zero with Re ~ > 0, and thus it would be maximal dissipative. We denote by Z(t) the contractive subgroup generated by equation (4.23):

I Z(t) II

(4.24)

~ 1.

If z(t) is a solution of the Cauchy problem with z(O) = Zo E for equation (4.23), then Az Jl.t

~

Ql/2 BQl/2Z(t) (t

~

~(Ql/2 BQl/2)

0) and z(t) ~ Zo for t ~ o.

Considet the function x(t) = Ql/2Z (t). In view of the boundedness of the operator Ql/2, Jl.x ~ BQl/2Z(t) Jl.t

=

BQx(t)

Ax(t)

=

(t ~ 0) and x(t) ~ Ql/2z0'

i.e. x(t) is a solution of the Cauchy problem for equation (1.1) with the initial condition Xo = Q-l/2z0.1f Zo runs through ~(Ql/2BQl/2), then Xo runs through ~(Ql/2BQ) = ~(Ql/2A). This solution is given by the formula x(t)

=

Q-l/2 Z(t) Ql/2XO.

The operators (4.25) form a semigroup of unbounded operators with a common domain If we convert this set into a Hilbert space Hl/2 by introducing into it the scalar product [X,yh/2 = (Ql/2X, Ql/2y) , and then instead of the operator A consider its minimal restriction Alt operating in H 1/ 2: ~(Al) = ~(Ql/2A), then for the equation ~(Q1!2).

dx - =A1x dt

93

§4. EQUATIONS IN HILBERT SPACE

the Cauchy problem in the space H1/2 will be uniformly correct, and in view of (4.24) and (4.25) the corresponding semigroup U(t) will be contractive. The operator A1 is maximal dissipative in Hm. We note that the operators of the so-called Hamiltonian equations have the structure considered in the last case. In them the operator B = J, where J2 = - I and iJ is a bounded selfadjoint operator. 5. Uniformly correct Cauchy problem. Now we consider any uniformly correct Cauchy problem in Hilbert space H with a semigroup U(t). We shall endeavor to explain the structure of the generating operator A corresponding to it. Suppose that", is the type of the Cauchy problem and "'1 > "'. We introduce into H a new scalar product by the formula [x,y] =

L'"

(U(s)x, U(s) y) e- 2"'18ds.

Since

I U(s) I

~

Me"',

we have

i.e. the new norm is not stronger than the original norm in H. _ We denote the completion of H relative to the new norm by H. Let us estimate the new norm of U(t). We have

I U(t) xii ~ = =

So '" I U(s) U(t)xI1 2e-2..1·ds So" I U(s + t)xI12e-2011Bds

=

e2..1'

i '" I

U(T)xI12e-2w1TdT

~ e2 1t li xii f, 0/

i.e. (4.27)

I U(t) 111 ~ e0/

1t•

A semigroup U(t) on H is strongly continuous for t ~ 0 in the norm of H, which means that it is strongly continuous in the norm of H as well. Then in view of (4.27) it may be extended by closure to a semigroup U(t), which will be strongly continuous for t ~ 0 in H and for

94

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

which the same estimate holds. As was shown in considering ~ase 1) of subsection 4, it follows from condition (4.27) that for the semigroup U(t) the operator A - "'II is dissipative, A being the generating operator of this semigroup. If (U(t)xo - xo)/t--+Axo in the norm of H, then this will also hold, because of (4.26), in the norm of 'ii, i.e. A is an extension of A. The range of the operator A - AI for Re ~ > "'I coincides with the space H and is therefore dense in 'ii. Since A - ~I has a bounded inverse for these~, it is the closure of the operator A - ~I. We shall show that the domain of A lies in H. For this we use the representation of the resolvent of A

R(~)x =

-

So '" e-A1U(t)xdt,

valid for Re ~ > '" and thus for Re ~ > "'I. From this we get

IIR(~)xI12 ~

[So'" e-(ReA-"'llllle-"'IIU(t)XlldtT

or (4.28)

IIYII ~ v'

1

2(Re~

- "'I)

I (A -

AI) yh .

Hence it is clear that when in closing the operator A there is a sequence y" --+ Yo in 'ii such that Ay" --+ Ayo in 'ii , then (A - ~I) y" --+ (A - ~I) Yo, and, because of (4.28), y" converges in H. Hence Yo E H. THEOREM 4.8. If A is a generating operator for a uniformly correct Cauchy problem of type '" in Hilbert space, then the operator A - "'I I ("'I > "') may be extended by closure to a maximal dissipative operator A - "'II, operating in a wider Hilbert space. The domain of the operator A lies in the original space H.

As is well known, the new scalar product [x, y] in H is always generated by some bounded selfadjoint positive operator in H. We shall denote this operator by Q-l: (4.29)

[x,y]

=

(Q-1x,y)

=

(Q- 1/2 X, Q-1 /2y ).

If the sequence x" E H is fundamental in the new norm, then

I x" - xmll ~ = I Q-1/2X" -

Q- 1/2x m I1 2--+ 0

(n, m --+ CD),

95

§4. EQUATIONS IN HILBERT SPACE

i.e. the sequence Q-l/2X,. converges in H. If x,.--+x of H in the norm of 'it, then Q-l/2X,. --+ Q-l/2X; if x,. converges to an "ideal" element x E 'it, then Q-l/2 X" converges to some element Y E H, which we shall denote by Y = (1-1/2X• Thus one defines an extension (1-1/2 of the operator Q-1/2, mapping the space 'it into the space H. If x and z are arbitrary elements of 'it, then, constructing sequences x,. --+ x and z" --+ z in the norm of 'it such that x", z" E H, we obtain (4.30)

[x,z] = lim [x,.,z,,]

= lim (Q-l/2X",Q-l/2Z,.) = «(1-1/2X, (11/2Z ).

Now we shall show that the operator (1-1/2 maps the space the entire space H. Suppose that Y E H. Then for x E H

'it

onto

I (Q-l/2x ,y) I ~ I Q-l/2xllllYll = Ilxllllly~· Hence it follows that «(1-1 /2x ,y) is a bounded linear functional in 'it, defined on the set H dense in 'it. By the Riesz theorem it is uniquely representable in the form (Q- 1/ 2x,y) = [x,v]. From (4.30) (Q-1/2 x ,y) = «(1-1/2 X, (1-1/2 V) = (Q- 1/2 X, (1-1/2V). It follows from the density of the set Q- 1/2 X (x E H) in H that y = (1-1/ 2V•

As a by-product it is clear that the operator (1-1/2 has an inverse, which we denote by (11/2• Obviously it is an extension to all of H of the operator Ql/2. N ow we introduce an operator B by the equation B = Q- 1/2(1-1/2(A - w2I)

(W2> WI)'

This operator is defined on ~(A) and operates into H. It is dissipative: if x E ~(A'> C H, then Re(Bx, x) ( 4.31)

=

Re( Q-1/ 2(1-1/2(A - (21) x, x) 'X

,.."

,.."

= Re(\:i-l/2(A - w2I) x, Q~1/2X) = Re[(A - W2I) x, x]

~

O.

In view of (4.28) and (4.29),

I (A -

w2I) xii 1 =

I (1-1/2(A -

w2 I )xll

= I Q1/2Bxll ~ V2(W2 - WI) I xii,

i.e. the operator Ql/2B has a bounded inverse. The domain ~(A) is Inapped by the operator A - w21 onto all of li, and the space li by

96

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

the operator 11-1/2 onto all of H. Therefore the operator Q1!2J3 maps ~(A'> onto H, and accordingly (Ql/2Bj -1 is defined on the enti~e space. If now we consider the operator B only on 9'(A). then Bx = Q-1/2(1-1/2(A - w2I)x = Q-1(A - w2I)x

(x E ~(A».

We have arrived at the following theorem. 4.9. If the Cauchy problem is uniformly correct for equation (1.1) with A closed, and is of type w, then the operator A may be represented in the form THEOREM

(4.32) where W2 > wand Q is a selfadjoint positive definite operator, and B is a dissipative operator such that the operator Ql/2 B has a bounded inverse defined on the entire space H.

It would be desirable to be able to prove the inverse assertion: if a closed operator A is representable in the form (4.32), then the Cauchy problem is uniformly correct for the equation (1.1). However, this we are not able to do. In this direction we have obtained only the following: in the space H it is possible to introduce a norm according to formula (4.20) and to complete it to the space H, and to extend the operator A to the operator A given by

A=

w2I

+ (11!2Q1/2 B,

defined on ~(Ql/2B). Since B is dissipative in H, it follows from (4.32) that the operator 11: - w2I is dissipative in H. Further,

I (A -

w2 I ) x 111 = I Q1/2 Bx I

~ I (Ql/21B) -111 11x II,

and the range of the operator 11: - W2I = (11/2 Ql/2 B coincides with the entire space H. Thus, zero is a regular point of the operator A - w2I, and thus that operator is maximal dissipative. THEOREM 4.10. If the operator A in equation (1.1) is represented in the form (4.32), th£n the space H may be densely imbedded in another Hilbert space H, and the operator A extended by closure to an operator A such that for the equation

dx "-=Ax dt

14. EQUATIONS IN HILBERT SPACE

97

the Cauchy problem will be uniformly correct, and for the corresponding semigroup U(t) the estimate

I U(t) 1111 ~ e"'Jf will hold. 6. Analyticity of solutions. Suppose that the ray arg t = t/J lies in the uniform correctness set of the closed operator A in Hilbert space. We suppose that the corresponding semigroup is contractive. According to Theorem 4.5 a necessary condition for this is that Re(e i16 Ax, x) ~ 0

(xE g'(A» ,

or (4.33)

cos I/> Re(Ax, x)

~

sin I/> 1m (Ax, x)

(x Eg'{A».

If moreover the operator has a resolvent at at least one point of the

ray arg t = 41, then these conditions are sufficient as well. Now suppose that A is maximal dissipative. Let us ask when the corresponding semigroup admits an analytic extension to a contraction semigroup in sector - 1/>0 < arg t < 1/>0. According to Theorem 2.14 it is necessary and sufficient for this that condition (4.33) should be satisfied for I/> = ± 1/>0, i.e. that (4.34)

tan 1/>01 Im{Ax, x) I ~ IRe(Ax, x) I (x E g'(A».

A maximal dissipative operator for which there exists a constant c > 0 satisfying the condition (4.35)

cl Im(Ax, x) I ~ IRe(Ax, x) I,

is said to be regularly dissipative. Condition (4.35) means that the numerical range of the operator A, that is, the set of values of the form (Ax, x), lies inside a sector containing the negative real axis and symmetric relative to it, with an opening angle of 1/1 = 2 arcot c. THEOREM 4.11. For the closed operator A to be a generating operator of a contraction semigroup U(t) which is analytic in some sector containing the real positive semiaxis, it is necessary and sufficient that it should be regularly dissipative. If (4.35) is satisfied, then the semigroup is analytic in the sector

Iarg tl

~

arctan c.

In a special case Theorem 4.11 implies the following.

98

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

THEOREM 4.12. For a closed operator A to be a generating operator contractive semigroup, analytic in the right halfplane, it is necessary and sufficient that it be a negative selfadjoint operator.

01 a

PROOF. NECESSITY. If the contractive semigroup U(r) is analytic in the right halfplane, then inequality (4.34) must be satisfied for all ,p; 0 ~ 1/1 ~ 1f/2, from which it follows that Im(Ax,x) = O. This means that the operator A is symmetric. It has a resolvent for Re" > 0 and therefore is selfadjoint. It is dissipative and therefore negative. The sufficiency of the conditions was shown in subsection 1. The theorem is proved. The results of subsection 4 make it possible to consider the question of the analyticity of the solutions of a uniformly correct Cauchy problem in Hilbert space. The preceding considerations and Theorems 4.9 and 4.10 make it possible to formulate the following assertion: THEOREM 4.13. In order that all the generalized solutions of a uniformly correct Cauchy problem should admit analytic extension into the sector - 1/10 ~ arg ~,po with the estimate

r

(4.36)

~ x(r)

I

~ Mew1tlll Xoll

(I arg)1

<,po ~ 1f/2),

it is necessary that the operator A should admit the representation (4.37)

A

= wI + QB,

where Q is a positive definite selfadjoint operator and B is a dissipative operator for which

tan,pol 1m(Bx,x) I ~ IRe(Bx, x) I. COROLLARY. If under the preceding hypotheses the solutions are analytically continuable into the right halfplane I/Jo = 1f/2, then B is a negative symmetric operator. THEOREM 4.14. If the closed operator A has the representation (4.37), then the space H be densely imbedded into a Hilbert space 'it in which all the generalized solutions of equation (1.1) are analytic in the sector Iarg rl < ,po, and the estimate (4.36) holds in the norm of II.

may

7. Sesquilinear forms and dissipative operators. A function a(u, v) (u, v E V), defined on a linear set V of Hilbert space H, is said to be

a sesquilinear form if it is linear in u and antilinear in v, i.e. a(alul + a2UZ, v)

= ala(uu v) + a~(uz, v)

14. EQUATIONS IN HILBERT SPACE

99

and a(u, alVI

+ a2V2) = ala(U, VI) + a~(u, V2)'

An example of a sesquilinear form defined on the entire space H is the scalar product (u,v). The sesquilinear form a(u, v) is said to be bounded above if it is defined in all of H and (4.38)

la(u,v)

I ~cllull IIvll;

It is said to be bounded below if (4.39)

The form a*(u, v) = a (v, u)

is said to be conjugate to the form a(u,v). If a*(u,v) = a(u,v), the form is said to be symmetric. A form which is bounded above and below may be represented in terms of the original scalar product in the form (4.40)

a(u,v)

=

(Qu,v)

=

(u,Q*v),

where Q is a bounded linear operator with a bounded linear inverse Q-l. Indeed, for a fixed v the quantity a(u, v) is a linear functional on H relative to u and therefore is representable in the form (4.41)

a(u, v) = (u, w)

(wEH)

with I wll ~ ell vII, so that the element w is determined uniquely. Putting w = Q*v, we arrive at (4.40). Taking u = v in (4.41), we find from (4.39) that kllvll2~la(v,v)1

= I (Qv,v)l ~IIQvll

IIvll·

Thus

and, analogously, kllvll ~ I Q*vll ~

ell vII •

It follows from the two left inequalities that the operator Q-l is bounded and defined in the entire space. We shall say that the Hilbert space HI is densely imbedded in the

100

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

Hilbert space Ho if HI identifies with a linear subset of Ho dense in Hoand

Ilxll o~ cOlli xii I· Now suppose that a form a(u, v) (u, v E HI) is defined and bounded above and below on HI. In view of the preceding considerations a(u,v)

=

(QU,VI)I.

We denote by ~ the collection of all those elements u E HI for which the form a(u, v) is an antilinear functional in v, continuous in the norm of the space HOo For such u the form admits the representation (4.42)

a(u,v)

=

(uE~,

(Au,v)o

AUEHo, vEHI).

This representation is unique in view of the density of HI in Ho. Thus there is defined a linear and generally speaking unbounded operator A with domain ~(A) = ~CHI and values in Ho. By continuity, relation (4.42) may be extended to all elements v E Ho (we will not change the notation for the form): (4.43)

a(u,v)

=

(Au,v)o

(uE~(A),

vEHo).

LEMMA 4.1. The domain ~(A) of the operator A ~ dense in HI and therefore in Ho. The operator A has an inverse operator which is defined in the entire space Ho and bounded as an operator from Ho into HI.

PROOF.

Suppose that z E Ho. For any v E HI we have'

(4.44) It therefore follows that the quantity (z, v)o is an antilinear functional, bounded in HI. Then

(4.45)

(z,v)o

=

(w,vh

= a(Q-Iw,v).

Writing Q-Iw = u and comparing (4.45) with (4.43), we find that z = Au. Further, (4.44) implies the inequality

II will ~ Codlz 110, which leads us to the estimate

I ulll ~ I q-llbll will ~ I Q-1111COlII Aullo. Thus the operator A -1 is defined on all of Ho and is bounded as an operator from Ho into HI.

§4. EQUATIONS IN HILBERT SPACE

101

We shall show that 9(A) is dense in HI. In the contrary case there would be a nonzero element YI E HI such that (u, YI) = 0 for any u E 9(A). Write Xl = (Q*) -IYI E HI. Then

0= (U,YI)I = (u, Q*XI)I = a(u, Xl) = (Au, XI)O. Since Au runs through all of H o, then Xl = 0, and so YI = o. The lemma is proved. Analogously, relative to the form a*(u,v), one constructs an operator A * mapping 9(A *) C HI onto Ho and having a bounded inverse. For it a*(v,u)

=

(vE9(A*), uEHI)

(A*v,u)o

or, what is the same thing, (4.46)

a (u, v) = (u, A *v)o

(v E 9(A *), u E HI).

If u E 9(A) and v E 9(A *), then (4.47)

(Au, v)o = (u, A *v)o.

From the denseness of 9(A) in Ho it follows that the operator A has a uniquely defined adjoint operator, which we shall temporarily designate by A'. Relation (4.47) shows that A * CA'. Suppose that vE9(A'). Then (Au, v)o = (u, A'V>O (u E 9(A».

Denote by w the solution of the equation A *w = A'v. Then (Au, v)o = (u, A *w)o = (Au, w)o. so that v = w and A *v = A'V. Hence A * is the operator adjoint to A in H o•

We gather all these conclusions together. 4.15. If the space HI is densely imbedded in the space Ho and a sesquilinear form a(u, v) bounded above and below is given on Hh then (4.42) defines on a set 9(A) dense in HI an operator A having an inverse operator which is bounded as an operator from Ho into HI' Equation (4.46) defines on a set 9(A *) dense in HI an operator A * which is the operator adjoint to A in Ho. REMARK. If the form a(u, v) is symmetric, then the operator A is selfadjoint. The situation often arises when the sesquilinear form a(u, v) is THEOREM

102

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR ~

defined on some linear set Ho which is dense in HOt and the space Hl is already constructed relative to the same form. We shall explain this first on a simple case. Suppose that the form a(u, v) is symmetric and nonnegative, i.e. a(u, u) ~ O. Such a form is called closed if whenever Un --+ U (Un E HOI u E Ho) and a (un - u m, U" - Um) --+ 0 it follows that uEHa and a(u,.,un)--+a(u,u). We introduce onto the space Ha a new scalar product according to the formula (u, V)l = (u, v)o

+ a(u, v).

Since the form a(u, v) is closed it follows that the space Ha in the corresponding norm will be a complete space. We denote that by H 1• Now we may apply the Remark to the form (u, v) 10 Then (u, v>O

+ a(u, v) = (u, vh = (Bu, v)o

(u

E g-(B» ,

where B is a selfadjoint positive definite operator in Hoo Writing S = B-1, we arrive at the formula a(u, v)

=

(Su, v)o

(u

E g-(A), v E Ho),

and S is a selfadjoint nonnegative operator. It follows from Theorem 4.15 that the domain of the operator B, and thus of the operator S, is dense in Hlo i.e. for any element u E H1 = Ha there exists a sequence of elements Un of g-(8) such that

II u -

unll~

+ a(u -

u'" u - u,,) --+0.

Hence it follows that Un --+ U in Ho and (4.48)

a (un - u m, Un - uml = (S(u n - u m ), Un - u m)

= nS 1I2un - S 1I2 um Il 2 --+O.

Since the operator S1I2 is closed, the element u E g-(S1I2). Conversely, S1I2 is the closure of its restriction to g-(S) , and so for u E g-(Sl/2) there is a sequence Un E g-(S) such that Un --+u in Ho and (4.48) is satisfied. Since the form a(u, v) is closed, u E Ha. Thus the domain Ha of the norm a(u, v) coincides with the domain ' g-(S1I2) of the operator S1I2, while a(u, v)

=

(Sl/2U, Sl/2V) 0

(u, v E Ha).

N ow we tum to a more complicated situation. Suppose that the form a(u, v) is nonsymmetric. Introduce the symmetric forms

14. EQUATIONS IN HILBERT SPACE

103

1

= 2" [a(u,v) +a*(u,v)]

aB(u,v)

and aI(u, v)

=

:i [a(u, v) - a*(u, v)],

which are called the real and imaginary parts of the form a(u, v). A sesquilinear form a(u, v) with a domain H/J dense in Ho is said to be regularly dissipative if its real part is closed and nonpositive, and the imaginary part satisfies the inequality (4.49)

(uEH/J)'

The smallest possible value of the constant b is called the regularity index of the form. THEOREM

4.16. A regularly dissipative form a(u, v) admits the repre-

sentation (uE~(A),

a (u, v) = (Au, v)o

vEHo),

where A is a regularly dissipative operator with domain ~(A) C Htb and is uniquely defined by the form. There is an operator A * corresponding analogously to the form a * (u, v), adjoint to the operator A in the space HoPROOF. We construct the space HI relative to the symmetric closed negative form - aR(u, v), according to the schema described above. We have

I ull ~ = II ull~ -

aR(u, u)

(u E Ha).

From (4.49) one easily obtains the inequality (4.50)

IaI(u, v) I ~ bV -

aR(u, u)

V-

aR(v, v) ~ b I u Ib I vlb,

from which it follows that the form a(u, v) is bounded above on the space HI:

Consider the form al (u, v)

= - a(u, v)

+ (u, v)

0-

It is already bounded below: lal(u, u) I ~ (u, u)o - aR(u, u)

=

I u IIf.

104

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

By Theorem 4.15 the form al(u,v) is representable in the form al(u, v)

=

(u-E 9J(B) , v E H o),

(Bu, v)o

where the operator B has a bounded inverse defined on the entire space Ho. Put A = I - B. Then (4.51)

a(u,v)

=

(Au,v)o

(uE9J(A), vEHo).

The operator A is dissipative, since

= aR(u, u) ~ o. From the fact that the operator A - I = - B has a bounded inverse Re(Au, u)o

it follows that A is a maximal dissipative operator. Inequality (4.49) tells us that it is regularly dissipative. Finally, the last assertion of the theorem follows from Theorem 4.15. The theorem is proved. Further we consider a selfadjoint nonnegative operator defined by the form - aR(u, v): (4.52)

(u, vEH..).

We shall call it the real part· of the operator A. Inequality (4.50) may then be written in the form (4.53)

lal(u,v)

I

~

bIIS l /2ulloIS l /2vllo.

from which it follows that the form a l (u, v) may be considered as a function of S1/2U and S1!2V: al(u, v) = a (Sl/2U, Sl/2.v), where a(u', v') is a sesquilinear form defined on the range 9i'(Sl/2) of the operator S1/2 and bounded above. Repeating the arguments of page 99, we arrive at the representation (4.54) where Q is a bounded selfadjoint operator, which may not be unique (if 9i'(Sl/2) is not dense in Ho). It follows from (4.53) that II QII ~ b. Formulas (4.53) and (4.54) lead to the equation a(u, v)

=

«- 1+ iQ) Sl/2U, Sl/2V)

0-

If we assume that u E 9J(A) and compare the equation just obtained

with (4.51), then we are led to the conclusion that Au

= 8 1/ 2 ( -

Thus the operator 81/2 ( - I

+ iQ) 8

I

+ iQ) Sl/2U.

1/ 2

is an extension of A, and since

105

14. EQUATIONS IN HILBERT SPACE it is obviously dissipative, and A maximal dissipative,

A = 8 1/ 2 ( - I

+ iQ) 8 1/ 2•

8. Abstract hyperbolic equation. It is natural to try to select from among the uniformly correct Cauchy problems those which have the properties of the Cauchy problem for hyperbolic partial differential equations. As is well known, for such equations there is no preferential choice of the direction of time. The solutions have the same properties for both t > 0 and t < O. Therefore one must restrict attention to those problems for which the operators U(t) form a group. The simplest case here is that of the group of unitary operators, when the operator A = iB, B being a selfadjoint operator (Theorem 4.7). If we add a bounded operator to such an operator, then the direct and inverse Cauchy problems remain uniformly correct (see Theorem 7.5), so that the operators U(t) will also form a group (see §3.) However the group property is not sufficient for distinguishing the class of hyperbolic problems. Thus, for example, the solutions of the Schrodinger equation have the same property. What is specific for hyperbolic equations is the distribution of perturbations along the characteristics with a finite velocity. For the construction of an abstract model of this phenomenon we suppose that a selfadjoint operator X is given in the space H and has a spectrum which is simple and fills out the entire real axis. We shall. roughly speaking, require that the initial disturbances, concentrated on a finite portion of the spectrum, spread out along the spectrum of X in a definite < A < IX» be the spectral function way. More precisely, let E), (of X. We shall require that for each interval .:1 of the real axis the relation IX)

(4.55)

U(t)EIJ.

=

E.(/,IJ.)U(t)

hold, where s(t, X) is some function realizing for each t a one-to-one continuous mapping of the entire real A-axis onto itself, and s(t,.:1) is the image of the interval .:1 under this mapping. If the property (4.55) is present the group U(t) will be said to be hyperbolic, and the function s(t, A) the characteristic of this group. It follows from (4.55) that for any t and '( E.(t+T,!J.) U(t

+ T) =

U(t

+ T)EA

= U(T)E.(/,A)U(t) = E.(T,B(t,!J.)U(t

Hence we find that (4.56)

s(t + T,.:1)

=

S(T, s(t,.:1».

+ T).

106

I. FlRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

: l

Since by definition s(O,~) =~, the inverse of the function set, X) is the function s ( - t, >')'. Suppose that 1/>(>') is a bounded continuous function on the axis QO <). < 00. Then the operator I/> (X) is bounded and defined by the formula

The identity (4.57)

U(t) I/> (X)

= 1/>(8(- t,X»

U(t)

holds. In fact, it follows from (4.55) that U(t)

(4.58)

L"

"-1

I/>(>.,,)E~ =

L" 1/>(>.,,)E.(t.6.Ii)U(t)

"-1

= L"

"-1

I/>(s( - t,,,,,»E.(,.4~iU(t), \

where the system {~} is a finite subdivision of the axis, )." E ~ and "" = s(t,).,,). Relation (4.57) is obtained from (4.58) by passing to the limit. The identity (4.57) may be used for an abstract" treatment of the method of characteristics. Suppose that a generalized solution x(t) of equation (1.1) has been found, taking on the initial value x(O) = %0Then one may define a generalized solution x.(t) with the initial condition x.(O) = I/> (X) %0 using the formula (4.59)

x.(t) = U(t) I/> (X) %0 = I/>(s(- t,X» U(t) %0

= I/>(s(- t,X»x(t).

It is not hard to extend this last equation to a wider class of functions I/> (>.) . A particularly interesting case is that in which %0 is a generating element uo, eigenelement or not, of the operator X. Then the knowledge of the solution of the Cauchy problem with this initial condition Uo makes it possible with formula (4.59) to find the solutions for the everywhere dense set of initial conditions I/> (X) uo. Now we shall suppose that the characteristic s(t,>.) has a derivative with respect to t, which for each>. is continuous and uniformly bounded in >. for t = O. If %0 E ~(A),
64. EQUATIONS IN HILBERT SPACE U(t) Aq, (X) Xo

= -

q,'(s( - t,X»s:( - t,X) U(t)Xo

+ q,(s( -

X,

we get

Putting t = 0 and recalling that s(O, X) = (4.60)

107 t,X» U(t)AXo.

Aq,(X)xo - q, (X) Axo = - s;(O,X)q,'(X)xo (Xo E ~(A), q,(X)

XO

E

~(A».

If one regards the operator X and the function q, (X) as fixed, and considers the class of operators A satisfying the last relation, it is at once clear that the operators of this class remain in the class when an arbitrary function of the operator X is added to them. Let us find a "particular solution" to equation (4.60). To do this, denote again by Uo a generating element of the operator X. Consider a set F of elements Xo of the form Xo = I(x) uo, where I(x) is a continuously differentiable finite function, the set F being dense in H. On this set an operator Ao is defined as follows:

(4.61)

AoXo = Ao/(X) Uo = s;(O, X) f' (X) uo.

It is not difficult to verify that this definition is single-valued. The function q,(X) I(x) will also lie in the indicated class of functions, so that Aoq,(X) I(X)Uo = s:(O,X) [q,'(X) I(X)

+ q,(X) I' (X) ]uo

= q, (X) Ao/(X) Uo + s:(O, X) q,' (X) I(X) Uo or in other words Aor/I(X)Xo - q,(X)AoXo = s;(O,X)q,'(X)xo-

Thus Ao satisfies (4.60) for all Xo E F. In what follows our arguments are of a qualitative character. Under certain restrictions on the operators A in question we may conclude that the general solution of (4.60) is of the following form: (4.62)

A

=

Ao+b(X),

where b(X) is any function of the operator X. Every operator X with a simple spectrum generates an isomorphism of Hilbert space with the space 5/2" (see [5]). Under this isomorphism the operator X corresponds to the operator of multiplication by A, and functions of that operator correspond to multiplication by the corresponding functions of A. The generating element in this representation may be taken to be the function uo(A) == 1. Then on the continuously differentiable functions {(X) the operator A will be given by the formula

108 (4.63)

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

At =

a(X) :(

+ b(A),

with a(A) =S:(O,A), as follows from (4.61) and (4.62). We write equation (1.1) in the form (4.64)

ax(t, A) at

( ) ax(t, A) aA

+

a A

=

b() A.

Thus, we in fact obtain a partial differential equation of hyperbolic type. We arrived at equation (4.64) from the identity (4.60). It is not difficult to see that, by approximating the function tPo(A) == A by functions tPn(A) for which (4.60) is satisfied, we can, using a limit transition, obtain the relation (4.65)

AX - XA

=

a (X)

(a(A)

= s:(O, A»,

generalizing the well-known indeterminacy relation of quantum mechanics. One can show that under natural conditions (4.~5) implies (4.60), so that the operator A may be represented in the form (4.63). Knowing the relation (4.65) makes it possible to recover the characteristic S(t,A) of the group U(t). Indeed, S:(O,A) = a(A). Differentiating relation (4.56) with respect to T and putting 'T = 0, we obtain the differential equation of the characteristic: as(t,A) _

at

«

»

- a s t, A

•

Weare interested in the solution of this equation satisfying the inital condition s(O, A) = A.

If the characteristic is found, then the group itself can be constructed using the identity (4.57).

§5. Fractional powers of operators 1. Non-single-valued fractions of operators. In §3 of the Introduction we considered analytic functions of bounded operators. The semigroups studied in the preceding sections were in fact functions etA of unbounded operators. In this section we study certain non-single-valued analytic functions of operators. The simplest possible construction of such functions may be carried out along the following lines. Suppose that I(A) is analytic in }. in the plane, with a finite number of cuts. Suppose

15. FRAClIONAL POWERS OF OPERATORS

109

that all the cut points are regular points of the operator A. Then the resolvent R (~) of A is defined and is analytic in some neighborhood of the cuts. Select a contour r enclosing the cuts and lying in that neighborhood, and construct the integral 1. - -2

(5.1)

r f(~) R(~) d~.

'11"£ Jr

If the function f(~) is such that this integral is absolutely convergent, then we obtain a bounded operator which it is natural to call a function f(A) of that operator. For a fixed cut the functions defined in the way just described ordinarily form some algebra ~ of operators. Obviously, if f(A),g(A)E~, then af(A)+bg(A)E~. We shall show that under certain natural conditions f(A) g(A) E ~ as well. To this end we choose a contour r' encircling the cut and lying closer to it than r. In view of the analyticity of f(~) and R(~) the integral (5.1) may be transformed into an integral along the contour r'. If the contours r and r' are unbounded, then in order for the integral not to change, we need certain conditions on the decrease of f(~)R(~) towards infinity, which usually have the same character as the condition of absolute convergence of the integral (5.1). If such a transformation is legitimate, then f(A)g(A) = =

(5.2)

~)2 r r f(~)g(p.) R(~) R(p.) d~dp. (2'11"£ Jr,Jr

r r

_1_. f(~)g(p.) R(~) - R(p.) d~dp. (2r£)2Jr'Jr ~-p.

1 = -2' r£

f

r'

f(~) R(~)

i

( f ( f 1 -2' '11"£

1 - -2' g(p.) R(p.) '11"' r

g(p.) ) d~ --dp. r ~ - p.

1 -2' '11"'

-..R~ -d~ ) dp.. r'1\ - P.

Ordinarily the first integral in brackets is equal to zero, since the point ~ lies outside the contour bounding the region of analyticity of the function g(p.), and the second is equal to - f(p.) , since the point p. lies inside the contour bounding the region of analyticity of the function f(~). Here we are taking account of the direction of integration. Therefore (5.3)

f(A)g(A)

~2~

(f(p.)g(p.) R(p.) dp. '11"' J;

= fg(A) E ~.

110

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR ~

We shall show by example how these heuristic considerations~may be effected rigorously. Suppose given a closed operator A with a dense domain such that the entire negative real semiaxis consists of its regular points, while at these points M

II R(A) I ~ 1+ I AI' or in other words (5.4)

II R( - 8) II = ~ (A + sl) -111 ~ 1~ s

(s> 0).

Decomposing the resolvent R ( - s + 1') into Taylor series at the point - 8 and using estimate (5.4), we find that this series converges in a disk where

11'1

1

1+8 < M· In a disk with center at we have the estimate

8

and radius r = (1 + 8)qjM with q < 1 M

1

M

IIR(A) I =IIR(-s+I')11 ~1+8 l-II'IMj(1+8)~1+8

1 l-q

M _1_1+8+11'1 -1+1-8+l'll-q 1-8 S

M

1

(

q)

M

~1+IAI1-q 1+ M =1+IAI· AB s varies from 0 to co these disks fill out a region bounded by the lines .,. = ± (8 -1) tan I/> (arc sin I/> = qjM) and the circumference 8 2 + .,.2 = q2 j M (Figure 9). In this region S the estimate (5.5)

IIR(A) II

holds. We denote by contour consisting arg(A - a) = ± (11" N ow consider a in the plane with a

M

~ 1 + IAI

ra (0 ~ a ~ qj M) the of the two halflines 1/».

function {(A), analytic cut along the negative

FIGbRE 9

15. FRACTIONAL POWERS OF OPERATORS

111

real axis, and having the property that for some k > 0, depending on the function, >.. "1(>..) -+ 0 as Ixl -+ 00, uniformly in the region - 11" + E ~ arg X ~ 7r - E, for any E with 0 < E < 11". In view of (5.5), for each such function the integral I(A)

(5.6)

=

21.

r

1(>..) R(X) dx

7rlJra

converges absolutely. The family F of functions I(x) is obviously an algebra, and for the functions of F all the operations performed in (5.2) are legitimate. Therefore relation (5.3) holds, so that operators of the type (5.6) form an algebra !l. 2. Fractional powers of operators. In the algebra F defined above there entered functions I(x) = x-a (0 < a < 00), considered as singlevalued in the plane with a cut along the negative axis, with the condition 1(1) = 1. Therefore one can define negative fractional powers of the operator A by the formula (5.7)

A

-a

= 21. 11"&

r X-aR(x)dx

(0

Jra

< a < 00 I a> 0).

The operators A -a are bounded. When a is an integer, a = n, then in the integral

the contour of integration may be contracted to zero, so that by the theorem of residues 1 -2· 11"&

f

-n

_

1

>. R(>..)d>.. - (n _ 1)' ra .

In view of (5.3) the operators A

d n - 1R(>..) d n-l

-II

A

I

10-0

= R n (0) = A -n•

form a semigroup

A -IIA -/I = A

-(,,+/I).

The integral (5.7) converges uniformly for a E [~, 1/~] (~> 0), SO that the semigroup of operators will be continuous in norm for a > O. In the investigation of its behavior near zero it is expedient to obtain a new representation for the operators A -". Suppose that 0 < a < 1. We shall contract the contour ret into the negative real axis. In view of the estimate (5.5), the integral (5.7) will not change. Then we obtain

112

I. PlllST-oaDBa BQUATIONS WITH CONSTANT OPERATOR

where the integrals are taken along the lower and upper sides of the cut respectively: A = se- ri and A = serio Hence eari. A -a = -

fa> s-aR( -

211'l

0

e- ari s)ds - . 2'l1'l

1

a>

s-aR( - s)ds

0

or

(5.8)

(O
Now we shall estimate the norm of A

IIA -all ~ sina1l' 11'

-a.

We have

r'" s-a~ ds = M. l+s

Jo

Thus the semigroup A -a is uniformly bounded on [0,1]. N ow we shall show that the semigroup A -a satisfies the Co-condition. Suppose that XoE9'(A). Then sina1l' A-a Xo - Xo =_11'

J:"' s-aR(-s)xods--Sina1l'f"" --dsxo s-a ol+s 0

11'

Sina1l'l"' s-a ' --R(-s) (l-A)Xods. 11' 0 l+s

=--

Hence

The integral on the right tends to unity as a --+ 0, so that

(5.9) In view of the uniform boundedness of the operators A -a (0 < a ~ I) and the denseness of 9'(A), this last relation is valid for any Xo E E, i.e. the Co-condition is satisfied. The function A -la+i,8) (0 < a < (0) also belongs to the algebra F, so that it is possible to define a complex power of the operator A using the analogous formula (5.10)

A -(,,+ill) = ---.;. 211'&

f

A -(a+i,a) R(A)dA rca

(0
<

(0).

§S. FRACTIONAL POWERS OF OPERATORS

113

Thus the semigroup A -a admits analytic extension to a semigroup A -% analytic in the right halfplane. There is a very essential question as to which conditions guarantee that the operators A -(II+i/l) strongly converge as a - t 0 to a bounded operator, which it is natural to call A -i/l. This question has not been cleared up. In other words, no class of operators has been singled out for which the pure imaginary powers are bounded operators. We formulate the result of our considerations. THEOREM 5.1. If conditions (5.4) are satisfied, the semigroup of powers of the operator A, defined by formulas (5.7) and (5.10), satisfies the Co-condition and is analytic in the open right halfplane.

It is also possible to define positive fractional powers A a (a > 0) of A as the operators inverse to the negative powers. These inverse operators A'" exist for any a. Indeed, if A -II Xo = 0 for any Xo, then A -JlXo = A -(.8-11) A -IIXO = 0 for 'Y > a and, since the semigroup is analytic, A -zxo == 0 (Rez> 0), In particular A -lXo = 0, which is impossible. The operators A II are closed, being inverses of bounded operators. The domain g(A ") is dense in E. Indeed, suppose that n ~ a < n + 1. If X E g(A "+1), then x = A -(,,+l)y (y E E). Define z = A,,-(n+1ly . Then A -"z

=

= A -("+1ly = X,

A -"A II-(n+lly

i.e. xEg(A"). Since g(A"+1) is dense in E, so is g(A") .. The positive powers of an operator form a semigroup of unbounded operators in the sense that for x E g(A II+Il) (5.11)

Indeed, if x E g(A"+Il) , then (5.12)

x

= A -
A

-/IA

-"y.

It follows from this equation that x E g(AIl) and AIlX

=

A-IIy ,

and therefore that AIlX E g(A") and y=AaAll x .

Comparing with (5.12), we obtain (5.11). If x E g(A ") and a> p >0, then A -aAllx

= A -(a-lilA -IiAIiX = A -(a-lil x .

If on the other hand 0 < a < p, then

114

I. PIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

A -"AIIX = A -"A"AII-"X

= AII-"X.

Thus for any a and {J

= AIIAax =

AaAll x

Aa+ ll x ,

X E ~(A'Y), where 'Y = max[a,{J, a + {J]. In particular, if x E ~(A) we obtain a formula for the positive fractional powers (0 < a < 1) of the operator A:

if

A"x

r ).a-1R(>.)Axd>. sinra 1:'" sa-lR( - s)Axds.

= A,,-lAx = ~ 211'l

(5.13)

J rca

= -r 0

This formula could have been applied to obtain the positive fractional powers of A on ~(A). It remains only to observe that the entire operator A a is obtained by closure from its restriction to ~(A). Indeed, if x E ~(A "), then for the sequence x" = nRC - n) x E ~(A), in view of the estimate (5.4), X,,---+X and A"x,,=nA,"R(-n)x = nR(- n)A"x-+A"x. That A" and R(- n) commute on ~(A) follows from the commutativity of A -,. and R( - n) on all of E. 3. Estimates of fractional powers. Suppose that n < a < n + 1. Consider the operator A"R"+l(- 8). We have ~ A CI R"+l( _

s) ~

(5.14)

= II A ,,-(10+1) A 11+1 RlI+l( - s) I ;i ~ A .-(10+1) ~ I AR( - s) ~ 11+1 ;i MI I ;i

M[ 1 +

18: ]"+1 S

;i

- sR( - s) ~ "+1

M(1 + M)lI+l.

Thus the operators AClR"+1( - s) are bounded. However it is natural to think. that, since a < n + 1, the norms of these operators must decrease to zero as s ---+ co. This is in fact the case. In order to obtain the appropriate estimate, we put a - (n + 1) = - (J and calculate A-IIAR(-s)

r >.-IIR(>.)AR(-s)d>. = ~ r ). -JlA R(>.) - R( - 8) d>.

=~ 2rl

Jra

211'l

Jr"

1: -).+-

= -1 . 211'l

-II

r,,>'

8

). + s

AR(>.)d>.

1:

1 -2' rl

-

>. -II

rca).

+ S d>.AR(- s).

115

§S. FRACTIONAL POWERS OF OPERATORS

The second integral is equal to zero, and we may replace the contour of integration in the first by ro- Taking account of the equation of fo: A = re=i(.. -.), we may estimate this integral as follows: IA-fJAR(-s) I

1 [

~211'

r'"

r-fJdr

Jo lre-i("-')+sl

r" .r-fJdr

+ Jo lre'("-') + 81 ~

1 + Ml 11'

r"

J0

.

)

~AR(re-'("-')11

~ AR(re i (.. -.» II]

1" -fJdT -fJ IT + e'!" .1 I 8 •

Thus Hence

I A "R"+1( -

8) I ~ IIA -fJAR( -8) ~ IIAR( - s) II"

-~~(1+M)"-~ 8fJ - s,,+I-a'

In order for this last estimate not to .arouse suspicions about the existence of a singularity at zero, we may combine it with (5.14): (5.15)

IIA aR"+l( _

8) I ~ - (1

e + 8)"+1-"

(n ~ a

< n + 1).

The following result is important in the applications. THEOREM 5.2 (MOMENT INEQUALITY). For any a the following inequality holds:

(5.16)

I A"xll

~

< P < 'Y,

if X E

eta, p, 'Y) I A "Yxll (fJ-,,)/!7-a) 1/ A"xl h-fJ)/h- a ).

Consider first the special case when 'Y = 0, {:J = - PI (0 < PI < al) and n ~ al < n + 1. The expression PROOF.

~(A"Y)

a

= -

ah

tnay be integrated n times by parts. Here, in view of the estimate (5.5), the terms outside the integral will vanish. Then we arrive at the equivalent formula

116

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

Contracting the contour onto the negative semiaxis, we obtain

(5.16') A-Il =sintJI'll" r

n!(-l)ra

r"sn-IlIRn+l(_s)ds.

(1 - tJI) ••• (n - tJI)

J0

N ow we calculate A -IlIX = _Sin_tJI_'II" n! (- 1)n 'II" 7-(l~--tJ~l)-·-'-·(~n---tJI~)

x [!oN Sra-IlIA "IRn+l( - s)dsA - a1 x +

L"

sn- Il IR n+l( - s)dsx ].

Applying inequality (5:15) to the first integral and (5.4) to the second, we estimate

Minimizing the expression in square brackets relative to N, we arrive at the inequality II A -/l1x II ~ k(ab tJI) ~ x II (al-~I)/alll A alx lilil/"!, N ow we tum to the general case. Suppose that a < tJ < 'Y and ~(A T). We apply the last inequality to the element A TX with al = 'Y - a and tJI = 'Y - tJ. Then

xE

IIAlixl1

=

IIA -lilATxll ~ k(y - a, 'Y - tJ) IIATxll,
The theorem is proved. 4. Resolvent of a fractional power. The algebra F of functions which we introduced above contains the function

~u(X) =

1

U

+ Xa

for u > 0 and 0 < a < 1. It is natural to expect that the corresponding function of an operator ~u(A)

1 = -2' '11"&

i

1 - R(A.) dA. rau+A. a

will be the resolvent of the operator A a (0 < a < 1) at the point - u. The operator ~u(A) is obviously bounded. Further, suppose that x~ ~(AL Then

117

§S. FRACTIONAL POWERS OF OPERATORS

Using the identity ).,,-1

1

u

u+)."

).

A(u+).a)

and taking account of (5.3), we find that ~,,(A)A"x

=

=x-

(A -1_ u~,,(A)A -l)Ax

u~,,(A)x

or ~,,(A) [Aa+ uI]x

(5.17)

= x.

If now x E ~ (A "), then the element A "x may be approximated by a sequence Y"E~(Al-a). We write x"=A-"y,,. Then x"E~(A), x" --+ x and A "x" --+ A ax. Passing to the limit in the equation ~,,(A) [A"

+ uI]x" =

x"

as n--+ 00, we find that (5.17) is valid for Further, for x E ~ (A a) ~.. (A)x

=

~,,(A)A -"A"x

xE~(A").

= A -"~.. (A)Aax =A -"[x -

In view of the boundedness of the operators ~,,(A)x

=

~.. (A)

u~,,(A)x].

and A -a,

A -"[x - u~.. (A)x]

for all x E E, so that [A"

+ uI]~.. (A)x = x.

Thus we have proved the following theorem. THEOREM 5.3. The operator A" has a resolvent at the points of the negative semiaxis, and

L+

1 RA ",( - u) = -2. -1- R().)d).. 1r£ r,,)." u

The formula for the resolvent may also be given a real form. To this end we contract the contour r. onto the negative real axis and obtain

(5.18)

R ,,(-u) =~fO R(~s)ds _~ r"R~-s)ds :.t. 21r£ ~ s"e-'"'' + u 21r£ Jo s"e'"'' + u =

sinn 1r

r~

Jo

. s"R( -

s). ds. (s"e-· .. + u) (s"e'"'' + u) a

118

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

Finally R

(5.19)

(_ u)

= sinra ('"

Jo

r

A"

s"R(- s)

S2"

+ 2s"u COSra + u 2

ds.

Formulas (5.18) and (5.19) define an analytic function of - u in the sector ar < Iarg( - u) I ~ r, and accordingly the resolvent of the operator A" exists in this sector. Moreover, the following holds. THEOREM 5.4. TM resolvent R A,,(IL) of the operator A" is defined in the sector a(lI' - q,) < IargILI ~ r, and in every sector interior to it tM

estimate (5.20) holds. PROOF.

We write the resolvent RA\aI(IL) in the form (5.18): R ,,(IL) =sin1l"a (..

(5.21)

r Jo

A

. s"R(-s). ds. (s"e-''''' - 1') (s"e'"'' - 1')

As we already stated, this integral is an analytic function of IL in the

sector all' < Iargl'l ~ r. For definiteness we suppose that argl' > o. Again we deform the contour of integration and contract it onto a ray A = re- i• with a small angle of inclination /c. Then the integral (5.22)

R

sin ra

,,~) = - - '

A

11'

1'" 0

r"e -i." R ( - re -i<) . . (r"e-'("+<)Cl< - IL) (r"e,( .. -<)a

-

IL)

dr

.

will still be an analytic function of IL in a sector turned through an angle /C (Figure 10). Thus we obtain an analytic continuation of the resolvent into a new sector, where (r - /C) a < argl' < ra. We may continue this process of deformation of the contour so long as /C ~ q" since for those I( the estimate I R( - re-k)ll -.c ~ Md (1 + r) holds, so that the integral does not change. Repeating PHC. 10. the same arguments for argl' < 0, we arrive at the conclusion that the resolvent of the operator A a is defined in the sector a(r - r/» < Iarg IL I ~ 11". We shall estimate the resolvent using formula (5.22). Suppose that" /.& = IILleioJ-. Then

-

15. FRACTIONAL POWERS OF OPERATORS R

II A"(Il) II (5.23)

sin 'Ira

r"

sinra

r"

~ - r - Jo

= rallli Jo

sinra Ml

119

r" II R ( - re -iK) II dr Ir" _ llei("+K)"llr" _ Ile-i(.. -K)al IIrR(-re-iK)IIdT IT_ei(r+
r'"

dT

~ --;;- r;r Jo IT - ei("+
e-i(r-
The last integral is uniformly bounded in the entire sector interior to the sector indicated by dotted lines in Figure 10. Therefore in such a sector (5.24)

II RA"(Il) II

M'

~~.

It is clear from formula (5.19) that the resolvent of the operator A" is defined at zero. (5.20) follows easily from this and from the

estimate (5.24). The theorem is proved. REMARK. For a real II. = - u one finds from (5.19) the estimate (5.25) An important consequence of Theorem 5.4 is the fact that if 0 ~ a ~ t the resolvent of A" is defined in the left halfplane and the estimate (5.20)is valid for it there. From this and Theorem 3.B it immediately follows that: THEOREM 5.5 If the operator A satisfies condition (5.4), then the operators - A" for a ~ t are generating operators of analytic semigroups satisfying the Co-condition.

5. Semigroups constructed relative to fractional powers. For semigroups with the generating operator - A" we may write the integral representations in terms of the resolvent of A. The point is that if ct ~ 1, t> 0 and - r+ E ~ argA ~ r - E, then the quantity

tends as IAI --+ ex> to zero uniformly relative to arg A faster than any Power of IAI. Therefore the function e -). belongs to the algebra F. Then it is natural to assume that

120

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

(t> 0, 0 ~ a

~

i>

will also be a semigroup generated by the operator - A The semigroup property of the operators U(t) follows from (5.3). Further 0.

dU dt

= _~ 211'&

r

Jr"

X"e->.atR(X)dX.

The function X"e->'o, E F, so that in view of (5.3) A _"dU dt

= _~ 21/'&

r e->.atR(x)dx = -

J r"

U(t),

i.e. dU

(it = - AaU(t).

(5.26)

Suppose that Xo E U(t)Xo - Xo

=

~ (A a).

Then Xo = A -ay(y E E). We calculate

U(t) A -ay - A -"y

= 21.

rX

..., Jr"

-"(e->.al

-

1) R(X)ydX •

The integral converges absolutely and uniformly in t, so that it may be replaced up to arbitrary accuracy E > 0 by an integr81. along a bounded contour, and this last made less than E by taking a sufficiently small t. Thus ~ U(t)Xo - XoII-+O

as t-+O. This means that the function U(t)Xo for XoE~(Aa) is a solution of the weakened Cauchy problem for the equation dx dt

= -

A"x(t).

In view of the uniqueness theorem this function is a solution of the Cauchy problem, so that U(t) is a semigroup generated by the opera-

tor - A-a. We could have proved the uniform boundedness of the semigroup U(t) directly, by passing to its real representation

(5.27)

U(t)

= ~ f" e ....at........ sin(satsin ...a)R( -

... Jo

s)ds .

The passage to such a representation by contracting the contour

§S. FRACTIONAL POWERS OF OPERATORS

121

onto the negative semiaxis is legitimate when 0 < a < t . The case = t is particularly important in the applications. The integral

a

(5.28)

1.£-

U(t) = r

sinvstR(-s)ds

0

in this case does not converge absolutely, but it simply converges, which one ca@)lscertain in the usual way, by integration by parts. The fact that it is equal to U(t)

r e-VAtR(A)d>..,

= 21.

r' Jra

may also be proved by integrating by parts in the last integral, and then contracting the contour of integration onto the negative semiaxis. 6. Raising powers to powers. In view of the estimate (5.20) the operator A", for 0 < a < I, satisfies also the condition (5.4), so that one may define fractional powers for it. For fJ > 0 we have (A")-6= 21.

r >..-6RAa (>..)dA.

r' Jr;.,

We present the resolvent of A a by the integral (5.13), taken along a contour rb with b > a. Then

= -1(2ri)2

-

1

-. 2d

i i R(A)

IO

i"

A-6

r;., - A

""-

+ ,," dAd"

-abR()d - A-a6

IO

.

7)

Further, (Aa)~ = [(Aa)-6]-1 = [A -"6]-1 = Aa6• Thus (Aa)6 = Aa6 for any fJ and 0 < a < 1. 7. Fractional powers of generating operators. If B is a generating operator of the semigroup UB(t), satisfying the Co-condition, then

I RB
~ (A ~"')"

for A> "'.

7) By r~ we mean a contour obtained by deformation from r .. and pulled in 80 close to the negative semiaxis that it lies between that semiaxis and the image of the contour rb after the transformation

"a.

122

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

Then for the operator A = - B IIRA(-s)1

.

M

= IIRB(s) II

~--

S-w

and, if w < 0, it follows that estimate (5.4) holds for the operator A. Accordingly, one may define fractional powers of the operator A. Suppose that 0 < a < 1. Then

Using the representation (1.10) of the resolvent in terms of the semigroup, we obtain

Since 1 r(1 - a) r(a)

sin ...a ...

we have (5.29)

Since the left and right sides are analytic for Rea> 0, formula (5.29) is valid for all such a. In the entire right halfplane the estimate

I RB(P) II ~ R~M_ep - w

(Rep

holds, so that in the left halfplane Re). IIR A().)

~

0)

<0 M

M II ~ -Re).-w = -1).lcoso-w'

where 0 = arg).. In the sector 101 = Iarg).1 ~ .../2 estimate

+

E

we obtain the

15. FRACTIONAL POWERS OF OPERATORS

123

Thus, for the quantity q, for which estimate (5.5) is valid, one may take 1('/2 - E for any E E (0, 1('/2). It then follows from Theorem 5.4 that, for the operator A" (0 < a < 1), estimate (5.20) will be valid for largl'l > a (1('/2 + E). For a fixed a we may choose E so small that inequality (5.20) will be satisfied in the entire left halfplane. We arrive at the following assertion:

5.6. If B is a generating operaror of a semigroup of type satisfying the Co-condition, then for 0 < a < 1 the operator - (- B)" is a generating operaror of an analytic semigroup satisfying the Co-condition. THEOREM

w < 0,

8. Fractional powers of operators with unbounded inverses. To this point we have been constructing the theory of fractional powers under the assumption that the operator A has a bounded inverse operator. In doing this we have naturally used the fact that its negative powers form a uniformly bounded semigroup. This significantly eased all the arguments. Now we tum to the consideration of the wider class of operators for which the estimate (5.30)

IIR(-s)11

M s

~-

(s> 0)

holds. Arguing as in subsection 1, we find that the inequality (5.31)

I R(,,) I

~

Ml fir

is valid in some sector containing the negative real semlaxlS. We shall say for short that the operator A has type (w, M) if (5.30) is satisfied for it with s > 0 and (5.31) for all " of a sector interior to the

sector 1(' - w ~ arg" ~ 1('. Now we introduce the operators A, = A from (5.30) that (5.32)

IIRA(-s)11 =IIR(-s-E)II •

+ d(1 >

M

~-

E+S

M

E

> 0). It follows 1

<-. -. E l+s

We note that in the sector in which (5.31) is satisfied (5.33)

124

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

I

i.e. for each A the resolvents R<\(A) are bounded uniformly in f. It follows from (5.32) that all the powers a fr.om - 00 to 00 are defined for the operators A" We shall be interested in a E (0,1). We introduce the integral 1(1')

= sin'll"a 'II"

r'"

Jo

s"R(- s) ds. s2.. _ 2s"p.cos'll"a + 1'2

In view of estimate (5.30) this integral converges absolutely and is an analytic function of I' in the sector a'll" < Iarg 1'1 ~ '11". It is analogous to the integral (5.19) for the resolvent of a fractional power. We have

UY(-p) -RA:(-p)

I ~sin'll"a II,S"[R<-S) -R(-s-d]ds

Jo

'II"

s2"+ 2s"p.cos'll"a +1'2

II

r'

~ sin'll"a { s"[11 R( - s).11 + II R( - s -:- f) II] - 'II" Jo Is" + p.e"'" I Is" + pe-'"'' I +

r" s"ll R( -

f

s) I! II R( - s -.f) II Is'" + p.e''''' I Is" + p.e'''''1

J.

ds

ds}.

Since ~ R A ( - s - E) II ~ MIs (0 ~ E), the first integral converges and may be made arbitrarily -small by taking 6 sufficiently small. For the same reason the second integral converges, and for fixed 6 the second term is arbitrarily small for small E, so that IY(-p.) ~RAI'(-p.)II--+O as E--+O.

Hence in particular it follows that the operator Y (A) satisfies the resolvent identity (5.34) on the negative real axis, and, because of analyticity, in the sector a"lr Iarg I' I ~ 'II" as well. Further, IIpY(-p.)x-xll "

=

II"sin'll"a 'II"

~ II'I sin"lra

-

'II"

r'" S2"s,,-l[sR(-s)x-x] ds II +2s"p COS'II"a + p2

.1

rN

.1

+ 1"lsin'll"a "Ir

s,,-l(M + l)dsllxll Is "+2s"p.cos'II"a+p. 2 1 2

i"" I N

s,,-lllsR(- s)x - xii ds 2 • S2" + 2s"p. COS "Ira +" I

~

15. FRACTIONAL POWERS OF OPERATORS

125

Since sR ( - s) x -+ x as s -+ 00, for sufficiently large N we will have IsR( - s)x - xii < E, so that the whole second term will be less than E. For fixed N and sufficiently large Ip.1 the first term will also become less than E. Thus (5.35)

Therefore it follows that for given x E E and for sufficiently large the element J ( - p.) x ~ O. From the resolvent identity (5.34) it follows that J(>.) x ~ 0 for all >. with a'll" < I arg>.1 ~ '11". The operators Y'(>.) have closed inverses. Introduce the operator

Ip.1

A" = [J(>.) ]-1 + ).[.

Because of (5.34) the operator A a does not depend on >.. It has a domain ,q(A a) dense in E. Indeed, it is obvious that elements of the form p,Y'(-p.)xE9'(Aa) and p.Y'(-p.)x-+x for any xEE. Thus we have defined the powers A a of the operator A for 0 < a < 1. The operator J(>.), by its construction, is a resolvent of the operator Aa. Repeating the arguments given in the proof of Theorem 5.4 and the remark to it, we can show that the operator A« has the type (aw, M). If aw < '11"/2. then the operator - A" will be a generating operator of an . analytic semigroup with the Co-condition, Let us consider the connection between the operators A: and A", For xE9'(A), using formula (5.8), we obtain

IIA:X - A:xll =sin'll"'II"a

ilL"

s,,-l[R(-S-E)A.-R(-s-,,)A.]xds

~ sin'll"'II"a {L' s,,-l[11 A,R( -

+ IE- ,,1 ~sin'll"a 'II"

{2(1+M)

a

s - E) I

i '" s" I

R(-

+ I A,R( - s - ,,) I ]ds s-

E) I

I R ( - s- ,,) I ds } I x I

~a+ M2 IE_"I~"-l} ~xll. 1-a

Minimizing the expression in brackets relative to ~, we get

I A:X - A:xll Where c depends only on M and a.

~

II

cl E- ,,1 "II xii,

126

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

Thus, the operators A~ converge on ,q(A) to some limit, which we shall denote by B. We have

IIA:X - Bxll

~

etall xii

(x E ,q(A».

In other words, we may write that Bx=A~x+Qx

(xE,q(A»,

where Q is an operator bounded on ~(A) and therefore on the entire space E. As was shown in subsection 2, the operator A~ is obtained by closure with ,q(A.) = ,q(A), so that 11 = A~ + Q and ,q (B) = ~(A~). Hence in particular it follows that the domain ,q(A~) does not depend on E. For the operator B the following inequality holds as well: (x E

(5.36)

~(B)

= ,q(A:».

Consider the resolvents Rii( - 1) and R.« - 1). Since the latter are uniformly bounded, it easily follows from (5.36) that Rl; l- 1) converges uniformly to Rii( - 1). On the other hand, as was shown, RA;'( - 1) -+-"(-1) = R Aa (-I). Hence it results that R A ,,(-I) = Rii(-I), so that B = A a. Thus ,q(Aj = ,q(A ") and

IIA:X - Aaxll

(5.37)

~

etall xii

(x E ,q(Aa».

We have incidentally proved that the operator A is obtained by closure of its restriction to ,q (A) . Relation (5.37) has many important consequences. 10 • For a < fJ the domain ,q (A j contains the domain ,q (A~), so that II

< fJ). (0
2°. The operatorsAa and Indeed, suppose that xE,q(A:+~

Then

A~x

E ,q(A:)

=

A~

= ,q(A"H)

=

limA=A~x .,,-+0

A~

E

=

limA~A=x . , ..... 0

is closed, then Aax E ,q(AII) and AaA~x

Now we let

nD(AII).

,q(A a). We have

AaA~x

Since the operator

C~(Aa)

= A~Aax.

tend to zero. Then since Aa is closed we get A"'AIIX = A"A"x.

127

§S. FRACTIONAL POWERS OF OPERATORS

3°. For a, {3 > 0 and a + {3

< 1 the semigroup identity

(5.38)

holds. Indeed, the left side of the equation is defined as before. We have (5.39)

Further, ~A:A:% - A"AtI%11

~ II (A: - A") A:% II + IIA"(A: - AtI)%11 ~ CEal1 A:%II +11 (A: - AtI)A a%11 ~ CEal1 A:%II +CEtlll A a%~ ~o.

Passing to the limit on both sides in (5.39), we get (5.38). 4°. For 0 < a < (3 < 'Y ~ 1 and %E9(A'Y) II AtI%11 ~c(a,{3,'Y)IIA'Y%II~-a)/h-a)IIA"% II h-tl)/h- a).

(5.40)

The validity of this inequality follows from the fact that the constant in (5.16), as is clear from its proof, depends only on the constant Ml in inequality (5.31) and on the sector in which this inequality is satisfied. Since for RA.(A), in view of (5.33), the sector and constant may be chosen independently of E, (5.40) is gotten by passing to the limit from the same inequality A,. We can show that formulas (5.27) and (5.28) for the semigroups generated by the powers A a of A for 0 < a ~ i remain valid in the case at hand. Finally, if %E 9(A), then, writing formula (5.13) for the operators A, and passing to the limit as E ~ 0, we obtain a formula for A a: Aa%

sinra

= -11"

!om sa-1R( -

s)A%ds

(% E 9(A), 0

< a < 1).

0

9. Fractional powers of operators in Hilbert space. The simplest example of an operator in Hilbert space for which condition (5.4) is satisfied is a selfadjoint positive definite operator. The fractional POwers of such an operator may be defined in terms of the spectral resolution:

where a is the lower bound of the operator A. I t is not hard to verify that this definition coincides with the other definitions given above.

128

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

For the semigroup of negative powers, the generating operator will be the operator InA, which is also defined in terms of the spectral resolution:

InA =

i"'InAdE~.

A number of the inequalities obtained for the general case may be sharpened for selfadjoint operators. For example, the moment inequality takes the following form: II A/lx~

(5.41) for any a

< fj < l'

II Allxl12 =

i

m

~

II A 'Yxll (,i-cr)/h-a) II A axil h-/l)/h-a)

and x E

g- (A 'Y). Indeed,

A2I1d(E.,.x,x)

=im

X2Y18-..)/h-a)A2a!7-/l)/h-cr)d(E"x, x).

Applying the Holder inequality with the exponents p = h - a)/(fj - a) and q = h - a)/('Y - fj) respectively, we obtain (5.41). One can also sharpen the constants in other inequalities. Another example is the dissipative operator. If B is a maximal dissipative operator, then, as was shown in §4, for Re" > 0 there exists a resolvent R B (,,) , and the inequality 1

II R B (,,) II ~-R ell

is satisfied. Accordingly, for A = - B, IIRA(-s) II ~ l/s

(s> 0),

i.e. condition (5.30) is satisfied. Thus for every maximal dissipative operator one may introduce the concept of its fractional powers. Moreover, in every sector 7l'/2 E ~ I argXI ~ 7l'wehave

+

1

HRA(A) II

1

~IReAI ~sinE

1

'1Af'

Therefore the operator A will be of type (... /2,1). As was shown above, the operator A" will now be of type (a... /2, 1), SO that the operator - Aa will be the generating operator of a contraction semigroup, admitting analytic extension into some sector containing the negative semiaxis. There it follows in tum that the operator - A a = - (- B)" is also a maximal dissipative operator.

129

§G. NONHOMOGENEOUS EQUATIONS

i6. Nonhomogeneous equations 1. Solution of nonhomogeneous equations. Now we tum to the consideration of the nonhomogeneous equation dx dt

(6.1)

= Ax + I(t) ,

where I(t) is a given continuous function with values in E. The concepts of solution, weakened solution, Cauchy problem and weakened Cauchy problem, remain for this equation the same as in equation (1.1). THEOREM 6.1. Suppose that the weakened Cauchy problem for the homogeneous equation (1.1) i8 correct on ~(A) and the operator A has at least one regular point. If x(t) is a weakened solution of equation (6.1),

then x(t)

(6.2) PROOF.

= U(t)x(O)

+

1t

U(t - 8) f(8)ds.

We apply to both sides of the equation x' (s) = AX(8)

+ f(s) ,

satisfied by x(s) for 8> 0, the operator U(t - s), supposing that 0<8
U(t-s)x'(s)

= U(t-s)Ax(s)

+ U(t-s)f(s).

On the other hand,

!!... (U(t ds

8)X(S»

= lim [U(t _

._0

+

8 _

0') x(s

+ 0') 0'

x(s)

U(t - s - 0')X(8) - U(t - 8)X(S)] 0'

.

At the point t - s > 0 the operator-function U(t) is strongly continuous, and the function x(s) is by hypothesis differentiable for 8> O. Therefore the limit of the first term is equal to U(t - 8) x' (s). The quantity x(s) E ~(A), in view of which the limit of the second term exists and is equal to - AU(t - S)X(8) = - U(t - s)Ax(s) (see the discussion following Theorem 1.1). Thus (6.3) goes into the equation

d

ds -(U(t - s)x(s»

= U(t - s) 1(8).

130

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

,/

The right side is continuous ins everywhere on [O,t] with the possible exclusion of the point t itself. Therefore, on integrating with respect to s from 0 to t - h, we obtain U(h)x(t - h) - U(t)x(O)

=

J:-

Ia

U(t - s) I(s)ds.

In order to clear up the behavior of the first term for h --+ 0, we write the last equation in the form U(h)R(>.o)(Ax(t - h) - >.ox(t - h»

=

U(t)Xo

+

J:-

h

U(t - s)f(8)ds.

As we already said above (see 13.1), the operators U(h) R(>.o) converge strongly to the operator R(>.o). The weakened solution x(t) has a continuous derivative for t> 0, so that the function Ax(t) is continuous for t > o. Hence it follows that the expression on the left has a limit as h--+O. Passing to the limit as h--+O, we obtain (6.2). REMARK 6.1. It is clear from the proof that the function I(s) is not necessarily continuous. For example it may have a summable singularity at the point s = O. Then the function U(t - 8) {(8) also has a summable singularity, and the proof remains valid. Thus the problem of finding the weakened solutions of the nonhomogeneous equation (6.1) has a ready answer: if' a solution exists, then it is given by formula (6.2). It remains only to check whether the function (6.2) for a given I(t) is a solution of (6.1). If Xo E 9J(A), then the first term in the right in (6.2) is a solution of the homogeneous equation (1.1), so that in checking we need only the second term,

(6.4)

y(t)

=

Q{(t)

=

J:I-O U(t -

8) I(s)ds.

The integrand may have singularity at the point 8 = t on account of the factor U(t - 8), so that even the existence of the integral cannot be proved for an arbitrary continuous function {(8). We need to impose various additional restrictions on {(8). For simplicity of exposition in what follows we suppose that there is a regular point >.0 = 0 of A at zero. If this is not so, we make the substitution x(t)

in equation (6.1).

= eAotz(t)

131

§S. NONHOMOGENEOUS EQUATIONS

The function z(t) will satisfy the equation dz dt

=

(A - Ao1)z + e-AoI/(t).

For the operator A - Ao1 zero is a regular point, and those properties of the functions x(t) and z(t), /(t) and e -Aot/(t) which will interest us, are correspondingly the same. Thus we suppose that the operator A -1 exists and is bounded. If the function /(t) is such that /(t) E .9J(A), and A/(t) is Bochner integrable, then the integral (6.4) converges absolutely and

II y(t) I ~ L'II U(t -

~

s)A -11111 A/(s) lids

sup I U(T)A- 111 ('IIA/(s) lids <

Jo

O~T~'

m.

o.

Thus the function y(t) is defined for all t ~ If in addition the function A/(t) is continuous, then y(t) is also continuous. Indeed, consider the function U(t) /(s). We shall show that it is continuous in the square

o ~ t, U(t

s

~

T. We have

+ .:1t) /(s + .:1s) -

U(t) /(s)

=

+ .:1t)A -1[A/(s + .:1s) - A/(s)] + U(t + .:1t) /(s) - U(t) /(s).

U(t

The first term tends to zero because of the continuity of A/(s) and the uniform boundedness of the operators U(t + .:1t) A -1, The second term tends to zero because U(t) /(s) is a weakened solution of equation (1.1). The continuity of the function U(t) /(s) implies the continuity of the function U(t - s) /(s) as a function of the two variables t, s (0 ~ s ~ t ~ n, which means also that the integral (6.4) is continuous. In order that the function y(t) should be a weakened solution it is necessary that y(t) E .9J(A) for t> 0, and for this, since A is closed, it is sufficient that the integral (6.5)

Ay(t)

=

i '-

o A U(t - s) /(s)ds

=

L'-o

U(t - s)A/(s)ds

exist. If A/(s) E .9J(A) and the function A 2/(S) is continuous, then, by the foregoing, the integral converges absolutely and yields a continuous function Ay(t). Under this hypothesis the function U(t)A/(s) will be a continuous function of two variables. Further, since U(t)A/(s)

132

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

is a weakened solution of (1.1), it follows (see §3.1) that ,the function U(t) f(s) = A -1 U(t)Af(s) will be a continuously differentiable solution of (1.1) on [0, <XI), so that

dUet - s)f(s) = AU(t _ s)f(s) = U(t - s)Af(s) ds

(0 ~s ~ 1').

Thus the integrand in (6.4) is continuous in (t,s) and has a derivative in t, which is also continuous in (t, 8). Therefore the legitimacy of the differentiation of the integral (6.4) with respect to the parameter by the usual rules is established. We have

:

= f(t)

+ !o' AU(t -

8) f(s)ds

= f(t)

+ Ay(t)

(O~t<<XI).

We have arrived at the following assertion: THEOREM 6.2. Under the conditions of Theorem 6.1 the function (6.4) is a solution of equation (6.1) for all t ~ 0, if the function f(t) is such that f(t) E 9'(A 2) and A2f(t) i8 continuous for t ~ o.

One ought not to suppose, however, that it is in any way necessary for the values of f(t) to lie in 9'(A 2) or even in 9'(A) for the function (6.4) to be a weakened solution of (6.1). This situation is illustrated by the following fact: LEMMA 6.1. If the free term in equation (6.1) is constant: f(t) = fo E E, then formula (6.4) yields a weakened solution of equation (6.1).

PROOF.

We have

yet) = !o'-o U(t - s) fads =

(6.6)

1'-0

rt-OdU(t - s)A -Vo dt ds

= Jo

U(t - 8)AA -lfods

= -

rt-OdU(t - s)A -1/0 ds ds

Jo

= U(t) A -lfo - A -lfoThe first term yields a weakened solution of the homogeneous equation, and the second is obviously a solution of the nonhomogeneous equation. The lemma is proved. The identity (6.6) makes it possible to transform the integral (6.4) into the form

16. NONHOMOGENEOUS EQUATIONS

133

yet) = fu'-o U(t - s) (f(s) -/(t)]ds + fu'-o U(t - s) I(t)ds

(6.7)

= fu'-o U(t - s) (f(s) -/(t)]ds

+ U(t) A -l/(t) -

A -l/(t).

This form often happens to be useful, since the expression in brackets inside the integral vanishes for s =. t and may cancel the singularity of the factor U(t - s). In connection with Lemma 6.1 it is natural to presuppose that the rapidity of variation of the function I(t) will be related to the question of existence of a solution to equation (6.1). Suppose that I(t) has a continuous derivative I' (t). Then I(t)

= 1(0) + fu'I'(t)dt.

We shall transform the integral (6.4). Using the identity (6.6), we obtain formally yet) =

(6.8)

fu'

U(t - s)f(O)ds +

l'

U(t - s)

1t

1'1'

= U(t) A -1/(0)

- A -1/(0)

+

= U(t) A -1/(0)

- A -1/(0)

+ fut U(t -

-l' =

(T)dTds

i t U(t - s) I' (T)dsdT T)A -11' (T)dT

A -11' (T)dT

U(t) A -1/(0)

+

l'

U(t - T) A -1/, (T) dT - A -l/(t).

The first term is a weakened solution of the homogeneous equation (1.1), so that for the study of the properties of yet) it suffices to consider the function

(6.9) The integral in this formula has the same form as the integral (6.4), but the function I(t) in it has been replaced by the function A -I/, (T).

134

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

The investigation of such integrals carried out earlier leads us directly to a theorem: THEOREM 6.3. II under the conditions 01 Theorem 6.1 the lunction I(t) has a continuous derivative f' (t) such that I' (t) E ~(A) and A'/(t) is continuous lor t ~ 0, then lormula (6.4) yields a solution 01 equation (6.1).

The proof of the differentiability of z(t) and the rule for calculating the derivative follow from the preceding discussion. Therefore

!; =

A -If' (t)

+A

it

U(t - T) A-II' (T) dT - A-II' (t)

= Az(t) + I(t).

Further, z(O) = - A -1/(0). We note further that it follows from the inclusion 1(0) E ~(A) that U(t) A -1/(0) is a solution of the homogeneous equation (see §3.1), so that the function yet)

= z(t)

+ U(t) A -1/(0)

is a solution of equation (6.1) with the initial condition yeO) = O. By Theorem 6.1 it is representable in the form (6.4), which justifies the transformations carried out in (6.8). The proof is complete. We may continue our transformations, assuming that I(t) has a continuous second derivative. Then, analogously to (6.8), (6.10)

yet)

= U(t) (A -2f' (0) + A -1/(0»

Here already, aside from the continuity of the second derivative it is not necessary to impose other restrictions, and

t=

U(t) (A -If' (0)

+A =

I" (t),

+ 1(0»

l'

U(t - T)A -21" (T)dT - A-II' (t)

A[ U(t) (A -2f'(0)

+ A -1/(0» +

l'

U(t - T)A -21"(T)dT - A -2f' (t) - A -1/(t) ]

+ I(t) = Ay + I(t).

135

16. NONHOMOGENEOUS EQUATIONS

THEOREM 6.4. II under the conditions 01 Theorem 6.1 the lunction I(t) has a continuous second derivative, then formula (6.4) yields a weakened solution of (6.1). If moreover the consistency condition f(O) E !?I(A) ~ satisfied, then the function (6.4) will be a solution of equation (6.1).

The second part of the theorem is connected with the presence in (6.10) of the term U(t) A -1/(0), which for arbitrary 1(0) will be differentiable only for t > o. Theorems 6.1-6.4 deal with the well-known equivalence of smoothness conditions relative to t of I(t) and conditions that its values should lie in the domains of the powers of the operator A. This fact will also appear in what follows. 2. Uniform correetness of the Cauchy problem. In the case when the Cauchy problem is uniformly correct for the homogeneous equation (1.1) corresponding to equation (6.1), the investigation of formula (6.4) is greatly eased. The integral (6.4) will exist for any continuous function I(t) and will, as is easily seen, yield a continuous function y(t). Further, if I(t) is such that f(t) E !?I(A) and the function Af(t) is continuous, then

dU(~t- s) =

U(t - s)Af(s) ,

and this function is continuous in t and s jointly (0 nection with this

Ie = Jor U(t -

d

t

~

s

~

t ~ T). In con-

s)Af(s)ds + f(t) = Ay + f(t).

If the function f(t) is continuously differentiable, then formula (6.8) also yields a solution of equation (6.1). Thus we have the following theorem. THEOREM 6.5. If the Cauchy problem is uniformly correct for equation (1.1), then lormula (6.2) yields a solution of the Cauchy problem for equation (6.1) for Xo E !?I(A) and a function f(t) satisfying one of the two conditions which follow: 10 • The value I(t) E !?I(A) and the function Af(t) is continuous. 20 • The function I(t) is continuously differentiable.

For any Xo E E and continuous f(t), formula (6.2) yields a continuous function, which it is natural to call a generalized solution of the Cauchy problem for equation (6.1).

136

I. FIRST-QRDER EQUATIONS WITH CONSTANT OPERATOR

The following is frequently useful in investigations of the question of the solvability of equation (6.1): LEMMA 6.2. If the Cauchy problem for equotion (1.1) is uniformly correct and if for some t the value of the /unction (6.4) lies in 9(A). then at that point it has a right derivative satisfying equation (6.1). PROOF.

yet

For l1t > 0 we calculate

+ l1t) -

yet)

l1t

(6.11)

=.!.. l1t

(tHI

Jt

+ U(l1~t -

U(t

+ l1t _ s) [f(s) _ I(t) ]ds

I A -l/(t)

+ U(l1~t -

I

Lt

U(t - s) I(s)ds.

The first term does not exceed

max II U(t) II max

t:ii.;:;;t+~

O;:;;t:iiT

II f(s)

- f(t)

II

in norm, and tends to zero as l1t -+ 0 because of the continuity of f(t). In the remaining terms the operator [U(l1t) - 1]1 l1t applies to elements of ~(A), so that (6.12)

d

;; = f(t)

(t

+ A Jo

U(t - s)f(s)ds = I(t)

+ Ay(t).

The lemma is proved. It is not hard to verify that a continuous function which has a right derivative continuous on a segment is continuously differentiable on that segment. Therefore, in order to prove that the function (6.4) is a solution of equation (6.1) under the condition of uniform correctness of the Cauchy problem for equation (1.1), it suffices to verify that

l'

(6.13)

AU(t - s)f(s)ds

exists and is a continuous function of t. 6. Abstract parabolic equations. If equation (1.1) is abstract parabolic, then the operators A U(t) are bounded and strongly continuous in t for t > O. Therefore the integral

(6.14)

L'h

AU(t - s) f(s) ds

=

it

AU(T)f(t - T)dT

(h

~ t)

exists for each continuous function I(t) and is a continuous function on [h, T], in view of the uniform boundedness on this segment of the

137

~.NONHOMOGENEOUSEQUATIONS

operators AU(t). The difference between the integrals (6.13) and (6.14) may be represented in the form

LA AU (T) I(t -

T)dT

=

!oil

A U(T) [/(t - T) -/(t) ]dT

+ U(h) I(t) -I(t).

(6.15) If the Cauchy problem for equation (1.1) is uniformly correct, then U(h)/(t) -/(t), and, in view of the uniform boundedness of U(t) (0 ~ t ~ 1') and the continuity of I(t), this convergence is uniform for t E [0, T]. Thus we arrive at the conclusion that the function yet) of (6.4) satisfies equation (6.12), if (6.16)

LII~AU(T)[/(t-T)-/(t)]lldT-O ~

h-O

for the given t, and equation (6.1), if this last relation holds uniformly in some neighborhood of the given value of t. The integral (6.16) may for example be estimated in terms of (6.17)

loA ~ A1-"U(T) ~ II Aer/(t -

T) - Aer/(t) II dT

(0

~ a ~ 1).

In writing down condition (6.17) it is supposed that the semigroup U(t) is of negative type, so that fractional powers of the operator A are defined. The following considerations are then applicable for estimating the first factor under the integral sign. We write (6.18) Ifthe operator A is unbounded, then we will have U(T) -0 as T-O. We now use the moment inequality (Theorem 5.2). We have (6.19)

~ A l-"U(T) II ~ c~ U(T) Ilerll A U(T) 11 1 -er ~ ul:~(T) •

THEOREM 6.6. II equation (1.1) is abstract parabolic and il the operator A has a regular point, then in order that lormula (6.4) should yield a weakened solution to equation (6.1) it suffices that lor some a and h _ 0 the integral (6.17) tend to zero unilormly on each segment [~, T] (~> 0). REMARK 6.2. If the convergence of (6.17) to zero is uniform in the neighborhood of a given t, then the function (6.4) satisfies equation (6.1) in this neighborhood.

138

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

N OW we consider that special form of abstract parabolic equations in which the operator A generates an analytic semigroup. For this, from §3.4, we have the following estimate:

I A U(t) I ~ ~1 e'"

(6.20)

(t> 0).

The condition of Theorem 6.6 will be satisfied if the function f(t) satisfies the H older condition

I f(t)

(6.21) with an arbitrary

'Y

> o.

- f(s) I

~

cit - s 17

Indeed, then for ex

l"IIAU(T) IIII f(t - T) -

=0

f(t) IldT ~ C1h",

where C1

= -1 cM Ie"T• 'Y

If the semigroup U(t) is of negative type, then (6.19) implies that

(6.22)

I A 1-.. U(t) I

~ tl~".

The preceding considerations and estimate (6.20) make it possible

to formulate the following assertion. THEOREM

6.7. If the operator A is a generating operator of an analytic

semigroup satisfying the Co-condition and of negative type, then formula (6.4) yields a weakened solution of equation (6.1) if the function f(t) satisfj£s one of the following requirements: 10 • The Hjjlder condition (6.20) is satisfj£d. 2°. For some ex> 0 the quantity f(t) E!?J(A"'), and the function A"'f(t) is bowJded on [0, T]. REMARK 6.3. If the Holder condition is satisfIed it is not necessary to require that the type of the semigroup be negative, and the Holder condition itself may be replaced by the condition that

(til f(t -

Jo

T) - f(t)

I dT

T

should converge uniformly in t. 4. The weakened Cauchy problem. Subgroups with weak singularities. It is not necessary that the semigroup U(t) be bounded for the integral

139

16. NONHOMOGENEOUS EQUATIONS

(6.4) to exist for each continuous function f(t). One may require for example only that II U(t) II be summable:

50 Til U(t) II dt <

(6.23)

<x>.

The function y(t) is then continuous: Iy(t+~t) -y(t) II ~

(tHt

Jt

II U(T) I

Uf(t+~t-T)lldT

+ ./:tll U(T) I I f(t + ~t - T) ~ f(t - T) I dT ~ ~~TII f(tn

(t+4t

J,

I U(T) IldT

+ O~I~T max I f(t + ~t - T) -

f(t -

T) I Jo(III U(T) I dT-+O.

We suppose that for the homogeneous equation (1.1) the weakened Cauchy problem is correct on 9(A). We introduce the functions

(,-h

(6.24)

Yh(t) =

Jo

U(t - s) f(s)ds.

We shall suppose them defined on a segment [6, T], where 0> h. If the function Af(s) is defined and continuous, then, because of the correctness of the weakened Cauchy problem on g-(A), the function U(t - s) f(s) will be differentiable in t for t> s, and its derivative AU(t - s) f(s) = U(t - s)Af(s) will be a continuous function of two variables in the region t - s ~ h. Hence the function Yh(t) is differentiable and (6.25)

yHt)

= U(h) f(t - h)

+ ./:'-h A U(t -

s) f(s)ds.

The first term U(h) f(t - h) = U(h) A -lAf(t - h) tends to f(t) uniformly on [6, T] because of the corrrectness of the weakened Cauchy problem on g-(A). The second term also tends uniformly to a limit in view of the summability of II U(t) II and the continuity of Af(s). Thus, the function y(t) of (6.4) becomes continuously differentiable and y'(t)

= f(t)

+ So' U(t -

s)Af(s)ds

= f(t)

+ Ay(t) ,

140

I. PIRST-oRDER BQUATIONS WITH CONSTANT OPERATOR

i.e. yet) is a solution of equation (6.1). If the function I(t) has a continuous derivative on [0. T]. then. defining yet) by the last equation in (6.8) and repeating the preceding arguments. we arrive at the conclusion that in view of the term U(t) A -1/(0). this function is a solution of equation (6.1). Thus the following theorem holds. THEOREM

6.8. If the weakened Cauchy problem for eqUGtion (1.1)

is correct on 9(A) and the semigroup corresponding to it satisji£s condition (6.23), then formula (6.4) yields a solution 01 equation (6.1) if condition 1 0 01 Theorem 6.5 is satisji£d. and a weakened solution if condition. 20

01 that

theorem is satisfied.

Consider a class of operators whose resolvents satisfy the condition (A =

(6.26)

0'+ iT. ReA ~ 0).

If fJ > i, then. in view of Theorem 3.3 and estimate (3.15), the operator A satisfies the conditions of Theorem rS.8. Further. the semigroup generated by the weakened Cauchy problem in this case is infinitely differentiable, and the inequalities (6.27)

I U(t) II ~ t 1l:- 1

and

~AU(t) r ~ t2/~-~

hold. The presence of these estimates makes it possible to relax somewhat the condition on the function f(t) in Theorem 6.8. Using (6.6). we may represent the derivative of the function (6.24) in the form yW)

= U(h) [t(t - h) - f(t) ] + U(t) 1(0)

+ !o'-A A U(t -

s) [f(s) - I(t) ]ds.

Now we suppose that f(t) satisfies the Holder condition (6.21) with exponent 'Y> 1/{J - 1. Then. from (6.27). the first term tends to zero as h-O uniformly on [a. T). However the condition 'Y> l/fJ- 1 does not guarantee the convergence of the last term to a limit. Impose the requirement: 'Y > 2 (1/(J - 1). Since 'Y cannot exceed unity. such a choice of'Y is possible if and only if fJ < 2/3. Then in view of (6.27) the integral

§S. NONHOMOGENEOUS EQUATIONS

L'

141

A U(t - s) [t(s) - f(t) ]ds

converges absolutely and uniformly in t, and therefore the derivative of yet) exists: y' (t)

= U(t) f(O)

+A

L'

U(t - s) [t(s) - f(t) ]ds.

The integral is again represented in the form of a difference of integrals with f(s) and f(t). In view of (6.6) the second integral will lie in ~(A), so that the first also lies in ~(A). Then yet) = f(t)

+A

1t

U(t - s) f(s)ds = f(t)

+ Ay(t).

THEOREM 6.9. If condition (6.26) is satisfied with /3 > 2/3, then formula (6.4) yields a weakened solution of equation (6.1) for a function f(t) satisfying the Holder condition (6.21) with 'Y> 2(1//3 - 1).

As is clear from the foregoing, for problems which are not uniformly correct, besides the behavior of the integral term in (6.25) it is necessary also to investigate the term U(h) f(t - h). Suppose that the following condition, somewhat stronger than (6.26), is satisfied: (>. =

(6.28)

g+ iT, ReA ~ 0).

Then for the operator - A one may introduce fractional powers and study the more detailed behavior of the semigroup U(t) at zero. 6.3. Suppose that condition (6.28) is satisfied. Then for /3 the operators (- A) -a U(t) are uniformly bounded on [0, 00).

LEMMA

a

>1-

PROOF. It suffices to prove the lemma for tation (3.13):

U(t)

a

< 1. We use the represen-

i . 1f'

1 = - -2'

eAtR(A)d>.,

rq

where the contour rq has the equation g = - (q/M) (1+ITlil) (0 < q < 1), and the representation (5.7) for the fractional powers (- A)a, from which it follows that (_ A)a =

~ 21fJ

r (_ p) -aR(p)d

J r'_o

p,

142

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

where r'-a has the equation O'=-a+kIT/ (OO). MUltiplying these two representations and using the resolvent identity and the properties of the Cauchy integral, we obtain (- A) -aU(t)

= 21. ( (- A) -aeIJR(A)dA. d

Jrq

Hence

('" dT & c1 Jo T a (1 + Til) =

C2

<

CD

since a + P > 1, a < l. The lemma is proved. It follows from Lemma 6.3 that the operator U(t) A -a is strongly continuous on [0, Tl when a> 1 - (3. Indeed, if x E 9'(A), in view of the correctness of the weakened Cauchy problem U(t) A -ax -+ A -ax. From the uniform boundedness of the operators U(t) A -" it follows that this relation is valid also for any x E E. THEOREM 6.10. Suppose that condition (6.28) is satis/i.€d. If tM function Aaf(t) is defined and continuous for some a> 2(1 - p), tMn tMre exists a solution of the weakened Cauchy problem for equation (6.1) with the initial condition x(O) = Xo E 9'(A'), wMre ,,> 1 - p. PRoOF. It is natural to seek the solution in the form (6.2), The first term U(t)Xo = U(t) A -'YA'Yxo, in view of what was said above, is a weakened solution of the homogeneous equation (1.1). We shall investigate the second term with the aid, once again, of (6.25). The first term U(h) f(t - h) = U(h) A -a Aaf(t - h) will converge uniformly to f(t), since a> 2(P - 1) > P - 1. For the proof of convergence of the integral (6.5) we estimate

1t H

A U(t - s) f(s) H ds

~ J:~II A 1-a U(t -

s) I

I A af(s) I ds.

From the moment inequality (Theorem 5.2) and (6.27) we get ~A 1-aU(t - s) H ~ cH U(t - s) H"IIAU(t - s) n1-" ~ (t _

c s)(2-a)//I-1'

By hypothesis (2 - a)/p - 1 < I, so that the integral (6.5) converges absolutely and uniformly. The remainder of the argument folloWS the standard schema. The theorem is proved.

17. EQUATIONS WITH PERTURBED OPERATORS

17.

143

Equations with perturbed operators

1. Comparison of operators. In this subsection, unless specified to the contrary, A will denote a closed unbounded operator with a domain ,q(A) dense in E. Suppose that the operator B admits closure and has a domain ,q(B) containing ,q(A): ,q(B):::> ,q(A). Then the inequality (7.1)

I Bxll

~

c(llxll + I Axil>

(xE,q(A»

holds. Indeed, we turn ,q(A) into a Banach space EA by introducing the norm

IlxiiA = I xii + IIAxll· The operator B is defined on the entire space E A , and is closed. Indeed, if x,. -+ x in EA and Bx,. -+ y in E, then x,. -+ x in E. Hence y = Bx. But the closure B coincides with B on ,q(A), so that y = Bx. A closed operator B defined on all of EA is bounded, from which (7.1) follows. DEFINITION 7.1. The operator B is said to be subordinate to the operator A if ,q(B):::> ,q(A) and

I Bxll

(7.2)

~cllAxll

(xE,q(A».

The following lemma results from what has been said earlier. LEMMA 7.1. If the operator A has a bounded inverse, and the operator B admits closure and ,q(B) :::> ,q(A), then B is subordinate to A. PROOF.

It follows from (7.1) that

I Bxll

~c(llxll

+ I Axil> ~c(IIA-lll + 1)IIAxll·

The lemma is proved. DEFINITION 7.2. The operator B is subordinate to the operator A with order a (0 ~ a ~ 1), if ,q(B):::> ,q(A) and (7.3)

I Bx I

~ ca(B) I X I I-a I Ax 11«·

It follows from the definition that operators subordinate to A with order 0 are bounded. Suppose that the operator A satisfies the condition (5.4):

(7.4)

(s> 0).

If the operator B is subordinate to a fractional power A a of A, then it is subordinate to A with order a. This follows from the moment inequality (Theorem 5.2). Indeed,

144

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

I BXII

~

C11Aax11

~ clllxP- aIIA x ll a•

LEMMA 7.2. If the operator B admits closure and is subordinate to th£ operator A with order a, then its closure B is subordinate to the fractional power A~ for fj > a with order a/ fj.

PROOF.

We use formula (5.16') with n

I Bxll =IIBA~A-~xll ~11"~~~~)

~11"~~~~) c,,(B)

1"

= 1.

Then for xEP}(A)

s1-!' IIBRA(-s)AIIRA(-s)xllds

1'" sl-~IIRA(-s)APRA(-s)xlll-" xIIARA(- s)APRA(- s)xll"ds.

Using estimates (7.4) and (5.15), we get

I

Bx

I

Si-llds [ (N ~ c Jo (1 + s) 1-"(1 + s) 1-/i I xii

+

L" (1 + S)~;I~:~l +

s)..11

~ c[~Nall xii + _l_N"-PIl APxll a fj - a

APx~ ]

J.

Minimizing the expression in brackets over N, we arrive' at the inequality

I Bxll

~

c' I xii I-alII I APXII alP.

Since the operator A~ is gotten by closure from its restriction to P}(A), the analogous inequality for the operator B and any x E P}(A~) is obtained by passing to the limit. The lemma is proved. Suppose that the operator B also satisfies condition (7.4) :

I RB ( LEMMA

s)

I

Ml

~1

+s

(s > 0).

7.3. If the operators A and B satisfy condition (7.4) and B

is subordinate to A. then the fractional power B" is subordinate to the fractional power PROOF.

A~

for any fJ > a.

From the moment inequality

I B"xll

~

cll xlll-"~ BXII a ~ cI11 xl11-allAxll ".

The assertion then follows from the preceding lemma:

§7. EQUATIONS WITH PERTURBED OPERATORS ~ B'xll ~

145

C.,pl APxll.

In Hilbert space there are deeper results. THEoREM 7.1. (HEINZ' INEQUALITY). Suppose that A and B are two

selfadjoint positive definite operators operating in Hilbert spaces H and HI respectively. Let T be a bounded linear operator with norm M, operating {rom H into Hh such that T~(A) C ~(B) and //BTxl ~ Md/Axll

(7.5)

Then

T~(A") C~(B')

(xE~(A».

and

(7.6)

PROOF. Take x E ~(A") and y E ~(B). The function TAzx will be analytic for Re z < a and continuous for Re z ~ a. The function B" -'y is analytic for Re z > a - 1 and continuous for Re z ~ a - 1. We introduce the scalar function

4>(z) = (TAZx, B"-Zy). It will be analytic in the strip a - 1 < Rez < a and continuous in the closed strip a-I ~ Rez ~ a. We estimate:

/4>(a+iT)/ =/(TA,,+iTX, BiTy)/ ~IITA"+iTXIl I/B iTY Il ~MIIA"x/1 I/yll and

/4>(a - 1 + iT) I = / (TA,,-l+iTx, Bl+iTy) I = I (BTA"-1+ iT X, BiTy) I ~IIBTA"-l+iTxl/

I/BiTYl/

~MdIA"+iTxl/

I/yll

=

M1l/A"xl/ I/YII.

Here we are using the fact that for x E 9J(A a) the element A ,,-l+iTX E~(A), which means that TA,,-l+iTX E ~(B). By the theorem on three lines we then have

or

I(Tx,B"y)/

~Ml-aMl1/A"xl/

·IIYII.

Since the operator B" is obtained by closure from its restriction to ~(B), it follows from the last inequality that Tx E ~(B") and (7.6) is valid. The theorem is proved. REMARK 7.1. In the proof we used only nonnegative powers of B. Therefore it remains valid also for a positive operator B. If A is also

146

I. FIRST-ORDERED EQUATIONS WITH CONSTANT OPERATOR

positive, then we may pass to the positive definite operator A. = A Then

+ d.

and so (7.7) AB was proved in §5, the operators A: converge strongly to the operator Aa, and (7.6) is gotten by passing to the limit as E-+O from (7.7). For H = Hl and T = I we arrive at the following assertion. 7.1. If A and B are positive selfadjoint operators, and B is subordinate to A, then the fractional power B" of B is subordinate to the fractional power AU of A. In addition it follows from (7.2) that CoROLLARY

I :B"xll

~

c"IIA"xll·

DEFINITION 7.3. We will say that the operator B is fully subordinate to the operator A if it is subordinate to it and if for every sufficiently small 7J

(7.8) where .(x) is a continuous convex functional of x E E. We note that if B is subordinate to A with order a, (0 ~ a < 1), then it is fully subordinate to it. Indeed, (7.3) and the Young inequality imply the inequality

I Bxll

(c 1/(1-"111- a/(1-"la,,/(l-"lllxII>1-,, «l1la) I Axil>" ~ (1- a)cl/(l-all1-alU-alaa/(l-alllxll + 7JIIAxll, ~

which is a special case of (7.8). It is convenient for us to apply this inequality using for 11 the quantity 11 = (1 - a)1-"ca"E 1 - a. Then (7.9)

I Bxll

(1- a)l-aca"(E-"llxll + E1-"IIAxll> == c,,(B) (E-"llxll + E1-"IIAxll>. ~

If inequality (7.9) is satisfied for all E, then, minimizing the right side with respect to E, we arrive at inequality (7.3). Sometimes it is possible to establish (7.9) only for small E ~ EO' If here the operator A has a bounded inverse, then (7.3) still follows from (7.9). Indeed, if Emin=allxV(1-a)IIAxll ~EO' then on substituting this into (7.9) we get (7.3). If on the other hand a II X II I (l - a) II Ax II ~ EO, then I Axil ~ (al(l - aho) I xii and

17. EQUATIONS WITH PERTURBED OPERATORS

147

Thus (7.3) is satisfied with a constant equal to

lC~~)max{ (

a: 1):

foGIIA-lIIG}.

LEMMA 7.4. Suppose the operator A is such that the operator A * adjoint

to it has a domain dense in E*. If on 9(A) (7.10)

Bx = TAx,

where T is an operator admitting approximation by linear finitedimensional operators to any accuracy, then B is fully subordinate to A. PROOF. For a given 11 we construct a finite-dimensional operator QIX = Ll-d.(x)XII such that I T - Qlll < 11/2. Since 9'(A *) is dense in E"', the functionals f. may be approximated by functionals g. E 9(A *). One can then construct an operator Qx = Ll-18.(x) XII so that I Q - Qlll < 11/2. Then II

I Bxll ~ I QAxl1 + I (T - Q)Axll

~

L IA *g.(x) I I xii + 1I11Axll· II-I

The lemma is proved. REMARK 7.2. Suppose that the operato... A has a dense domain and admits closure and that the space E has a basis and is reflexive. Then the assertion of the lemma follows from relation (7.10), if the operator T is completely continuous. Indeed, it follows from the reflexivity of E that 9(A *) is dense in E* (see [1]). Since E has a basis it follows that T is the limit of linear finite-dimensional operators. An assertion which is in a certain sense converse holds in a reflexive space: LEMMA 7.5. If for sufficiently small 11 > 0 inequality (7.8) holds with a weakly continuous functional cI>,(x), and the operator A-I is boUnded, then the operator B admits the representation (7.10) with a completely continuous operator T.

PROOF. Consider the linear operator T = BA -I, defined in the entire space. We shall show that it is completely continuous. Suppose that 8 is the unit ball of the space E. The set A -18 is bounded, so that, in

148

I. F1RST-ORDER EQUATIONS WITH CONSTANT OPERATOR

view of the reflexivity of the space, one can select from each sequence A -lYII (II YIIII ~ 1) a weakly convergent subsequence A -lUll ~ z. Then

I TYII - TYml1

+ '111 YII- Ymll ~ cI>,(A -lYII - z) + cI>,(z - A -lYm) + 2'1. ~ cI>,(A -lYII - A -lYm)

Taking '1 = E/6 and taking into account that cI>.(A -ly" - z) -+ cI>.(0) = 0, so that for sufficiently large m and n the sum of the first two terms on the right is less than 2E/3, we arrive at the inequality I TYII - TYml1 < E. The lemma is proved. 2. Resolvents of perturbed operators. In this section we consider the question as to how the addition to an operator A of an operator B subordinate to it (perturbation of the operator) is reflected in the properties of A. It is clear that the addition to A of an operator simply subordinate to A may sharply change its properties. Thus, the operator itA is obviously subordinate to the operator A, for any complex factor p.. Adding the operator itA to A in the differential equation (1.1) can make a correct problem incorrect, etc. However, if the perturbing operator is subordinated to A with a small coefficient, then in a number of cases the basic properties of the resolvent of A do not change. LEMMA 7.6. Suppose for the operator A that the quantity II>.RA(X) I is bounded on some set of the complex plane. If the operator B is sub-

ordinate to A. then for sufficiently small E the quantity I XRA+.B(X) II, corresponding to the operator A + EB, is also bounded on the same set. PROOF.

We have A

+ EB -

xl = (l

+ EBRA(X» (A -

>.1).

Further, from (7.2),

e(1 + sup I XRA(X) II)· E < 1/2c(1 + sup I xRA(X) II) the resolvent

I BRA (X) I This means that for exists and

~

ell ARA(X) I

I RA+.B(X) I

~

~

RA+.B(X)

2~ RA (X) II·

The lemma is proved. If the perturbing operator is not small, then more rigid restrictions on the way in which it is subordinated to the operator A have to be imposed in order to preserve the properties of the resolvent.

149

§7. EQUATIONS WITH PERTURBED OPERATORS

LEMMA 7.7. If the operator A is closed and the operator B is fully subordinate to A and admits closure, then the operator A B, considered on 9(A), is closed.

+

PROOF.

Suppose that x" -+ x and (A + B) x" -+ y (x" E ~(A». We have

II Ax" - Axmll ~ II (A + B) x" - (A + B)xmll + II B(x" - xm)1I ~ II (A + B) x"

+ ~,(x" Take

'II ~

1.

- (A + b)xmll

xm)

+ 'II II Ax" -

Ax",l.

Then

t II Ax,. -

Ax",11 ~ II (A + B) x" - (A + B)xm~

+ ~,(x" -

x)

+ ~,(x -

xm)

-+

o.

Because A is closed we have x E 9 (A) and Ax" -+ As- It then follows from the relation (A + B)x,,-+y that Bx,,-+Bx = Bx (x E 9(A». Hence y = (a + B)x. The lemma is proved. THEOREM 7.2. II for an operator A having regular points the equation (1.1) is abstract parabolic and the corresponding semigroup is analytic in a sector containing the positive semiaxis, then the equation

dx dt

(7.11)

=

(A

+ B)x,

where the operator B is lully subordinate to A, wilt also be abstract parabolic. The corresponding Cauchy problem is uniformly correct. The corresponding semigroup is analytic in a sector containing the positive semiaxis. PROOF.

From Theorem 3.9, inequality (3.26) holds for the resolvent

of A: M ~RA(A)II ~IA-wl (ReA>w). We again use the identity (7.12)

A

+B -

AI = (I

+ BRA (A» (A -

AI).

We shall estimate the norm of the operator BRA(A). For ReA> w we have IIBRA(A)xll ~ ~,(RA(A)x)

+ .,,11 ARA (A) x"U

~{II~,IIIA ~wl

+."

(1 +I~~A~I)} IIxll·

150

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR (

Choosing " so small that the second term in brackets is less than "/2 for all A ~ w' > w, and then IAI ~ large that the first term will be the same, we get (7.13)

II BRA (A) II

<" < 1 for ReA ~ w' > w and IAI

~ R(,,).

The existence of the resolvent RA+B(A) then follows from (7.12) and (7.13), along with the estimate

Ml

II RA+B(A) II ;:;; IA- wd ' where Ml = M/(1-,,) and W1 = maxtw',R(,,) t. The assertion of the theorem follows from Theorem 3.8. The theorem is proved. REMARK 7.3. If B is simply subordinated to A, then, by Lemma 7.6, the assertion of Theorem 7.2 remains valid for the equation dx dt

- =

(7.14)

(A

+EB)x

for sufficiently small E. Now we tum to the consideration of the case when the weakened Cauchy problem is correct on the set ~ (A). LEMMA 7.8. Suppose that the resolvent RA(A) of A' satisfies for some A the condition (7.15) Suppose that the operator B is subordinate to A with order a. Then the following inequality holds:

(7.16)

II BR (X) II A

S; -

(2 + M)c,,(B) • (1+lxl)lI-"

PROOF. In view of (7.9) we have ~BRA(X)X~ ;:;;c,,(B) (E-a~RA(x)xll +E1-"~ARA(X)xll>

~C"(B{(l+~XI>IIE-Cl+ (1+ (l~II:ll>p) e -"Jilxll. 1

Putting E= 1/ (1 + IXI), we arrive at inequality (7.16). The lemma is proved.

151

§7. EQUATIONS WITH PERTURBED OPERATORS

THEOREM 7.3. If the operator A satisfies condition (7.15) for ReA ~ W and 0 < fl < 1, and the operator B is subordinate to A with order a < fl, then the weakened Cauchy problem is correct on ~(A) for equation (7.11). All of its solutions are infinitely differentiable for t> O.

It is clear from (7.16) that for each 6> 0 there exists an Wl ~ W such that I BRA (X) I ~ 6 for Re X ~ Wl. Taking 6< 1 and again making use of identity (7.12), we arrive at the inequality PROOF.

II RA+B(X) I

~

M

1 _ 6 (1 + IXI) -{J (Re X ~ Wl).

The assertion in question follows from Theorem 3.3. The theorem is proved. In analogy with Remark 7.3 we may consider the case of an equation with a small perturbation. THEOREM

7.4. If under the conditions of Theorem 7.3 the operator B

is subordinate to the operator A with order fl, then the assertion of the theorem remains valid for equation (7.14) for sufficiently small

E.

If we tum to the consideration of equations for which it is known only that the Cauchy problem is uniformly correct, the situation becomes more complicated. If the operator satisfies condition (2.18), then, under the hypotheses of Lemma 7.8, it may, on being perturbed, no longer belong to that class of operators. But if we consider conditions (2.17), these are connected with estimates of powers of the resolvent, which are not expressed so simply as in (7.12) in terms of the corresponding powers of the resolvent of the unperturbed operator. In this direction only the following assertion has been obtained: THEOREM 7.5. If the Cauchy problem is uniformly correct for the closed operator A, and the operator B is bounded (subordinate to A with order 0), then the Cauchy problem is also uniformly correct for equation (7.11).

For the resolvent of A inequalities (2.17) are satisfied, so that the norm of the operator BRA (X) for large X is less than 6 < 1 (when MIIBII(x-",)-l<6). Then from (7.12) we find that PROOF.

RA+B(X) =

{t RA (X) [BRA (x)]i }

n.

J-O

We regroup the terms of this series according to powers of the operator B. The terms of this series have the form of products, where

152

I. li'IRST-ORDER EQUATIONS WITH CONSTANT OPBBATOR (

factors B alternate with the first or second power of R A1A). If the factor B enters k times, then R A (,,) enters n + k times. It follows from (2.17) that such a term does not exceed in norm the quantity

Mk+11IBll k (A _

w) -(nH ).

The collection of terms of the series containing the factor B k times will be equal to the coefficient ak of zit in the series

Therefore IIRA+B(A) I ~

...

LakMkHIIBII"(A-Cd)-(nH) k-O

= M(A- Cd) -n(1_ Mil BI ("- w) -l)-n = M(A - w - Mil BII) -n = M(A - Cd1) -n.

(7.17)

Thus the operator A + B satisfies the conditions of (2.17). The theorem is proved. REMARK 7.4. It follows from inequalities (7.17) that for the corresponding semigroup UA+B(t) the inequality • UA+B(t) I ~ Me(.o+MlBUI

is satisfied. 3. Comparison of semigroups. For simplicity we suppose that the operator A has a bounded inverse. If A is a generating operator of a semigroup with the Co-condition, and the operator B is subordinate to A, then for Xo E 2J(A) the operator BU(t)Xo = BA -lU(t)AXo

is defined. If the semigroup generated by A is strongly differentiable, i.e. the operator AU(t) is defined and bounded, then the operator BU(t) will be bounded as well. Thus, if A satisfies (7.15), then from (3.16) we have IIBU(t) I ~IBA-IIIAU(t)1 ~MIHBA-llle..ttl-I/6.

It is natural to presuppose that in the case when B is subordinate to A with definite order a < 1, the behavior of BU(t) at zero should be better. Indeed, applying (7.9), we get

I BU(t)XoII

~ c,,(B) ~

(E-"II U(t) I + E1 - a IIAU(t) II> IIXoII

Mc.(B)eo>'(E -at 1-

1/ 6

+ El-atl-2/1i) UXon·

17. EQUATIONS WITH PERTURBED OPERATORS This inequality is valid for any

E

> O.

Put

E

153

= tl/fJ. Then

(7.18) Consider the two differential equations dx dt

(7.19)

= Ax and

dx dt

=

(A

+ B)x = A1x.

Suppose that for each of them the weakened Cauchy problem is correct, on 9(A) and 9(A 1) respectively. We denote the semigroups generated by them by UA(t) and UA1 (t) respectively. Our problem is to o~tain an estimate for the difference of the solutions UA (t) Xo and UA1 (t)Xo.

Suppose that XoE9(A 2 ). Then for x(t) = UA(t)Xo we have dx/dt

= Ax =

(A

+ B)x -

Bx.

If Al = A + B is defined on 9(A) and x(t) E 9(A) C 9(A 1), then the transformation just written down has meaning. The resulting equation for x(t) may be considered as a nonhomogeneous equation of the form (7.20)

dx/dt

=

A1x

+ f(t).

where the function f(t) = - BA -lUA(t)AXo is continuous because AXo E 9(A) and the operator BA -1 is bounded. Hence x(t) is a weakened (and even pure) solution of equation (7.20), and, in view of Theorem 6.1, x(t)

=

UA(t)Xo = U A1 (t)Xo - !at U A1 (t - s)BUA(s)ds.

Finally, (7.21)

U A1 (t)Xo- UA(t)Xo= LtUA1(t-S)BUA(S)Xods (XoE9(A 2

».

This identity is fundamental when the problem is one of obtaining various estimates for the difference of semigroups. We note that when the Cauchy problem is uniformly correct for equation (1.1), then identity (7.21) is valid for Xo E 9(A), since the function BA -1 UA(t)AXo is in this case continuous. If moreover the Cauchy problem is correct for the second equation in (7.19), then (7.22)

IIUA+B(t)XO- UA(t)XoII ~c(t)IIBA-IIIIIAxoll,

where c(t) is easily expressed in terms of the functions estimating the

154

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

growth of the semigroups UA+B(t) and UA(t) relative to t. Estimate (7.22) has the defect that on the right side it has the factor ~ Axo~ and not I xo~. Because of this it does not enable us to estimate the difference of the semigroups in terms of the norm. This may sometimes be accomplished when the semigroup is strongly differentiable for t > O. We shall consider again the case when the operators A and B satisfy the conditions of Theorem 7.3 with (3 > 1/2. It is clear from the proof of Theorem 7.3 that estimates (3.16) hold for the semigroup UA+B(t). In particular

I UA+B(t) I

~

Moewlltl-l/lI,

where No and Wl depend continuously on the constants c,,(B). Using that inequality and (7.21), we have

I UA+B(t)Xo -

UA(t)xoll

~ M'MoC .. (B)

.£1

eOll(t-s) (t

- s) l-l//leOl'sH!+a)//lds I xoll

~ M' MoCa(B)e(Wl+")I.£1 (t -

s) l-l//lsH!+a)//ldsll XoII.

If a < 2{3 - 1 < fJ, then the integral converges and

(7.23)

I UA+B(t)

- UA(t)

I

~ Qca(B)e( 1+.,)lt3 -(2+"j//l. 0I

It follows from the condition a < 2{3 - 1 that 3 - (2 + a)/fJ > 1 - 1/fJ, i.e. the difference of the semigroups behaves itself as t -+ 0 "better" than each of the semigroups separately. 4. Comparison of fractional powers of operators. Suppose that the operator A satisfies condition (7.4), and that the operator B is subordinate to A with order a. Just as in Theorem 7.3, we find using Lemma 7.8 that the operator A + B satisfies the inequality

I R A+B( -

M 1 s) I ~ 1 _ ~ 1 + s for s ~ a ~ O.

This means that for the operator A + B + al == A + Bl inequality (7.4) is already satisfied, so that its fractional powers may be defined. The operator Bl = B + al is also subordinate to A with order a. THEOREM 7.6. The fractional powers (A + Blrr and A'Y (0 ~ 'Y < 1) of the operators A + Bl and A differ by an operator subordinate to A with any order ,,> a + 'Y - 1.

155

§7. EQUATIONS WITH PERTURBED OPERATORS PROOF.

C = [(A

Consider on 9(A) the operator

+B

l )'" -

A r]A - I = sin'Y1I" 'I/"

roo s.,-lRA+BJ.( Jo

s)sBlRA ( - s)A -«ds.

The operator B'lA -. will be subordinate to A with order

CIt

CIt -

/(10

where

+ 'Y -1 < /(1 < /(. Indeed, in view of Lemma 7.2 B'l is subordinate to the

power Aa+I-ll of A. Hence it follows that B'lA -. is subordinate to the power Aa- l of A. Then it follows from Lemma 7.8 that I

I B'I A -IRA( -

s) I ~ c/(1

+ SP-"+Il,

so that in view of (7.4) 1 fooo S..,-l__ M c ds< I CII s-.. 0 1 - 6 (1 + s) 1-11+"1

CD

•

since 1 + (/(1 - CIt - 'Y + 1) > 1. The assertion of the theorem follows from the boundedness of the operator C. The theorem is proved. 5. Operators with difJ'erent domains. If neither of the domains of the operators A and Al is contained in the other, it is inappropriate to pose the question of their comparison. Even if the intersection of 9(A) and 9(A 1 ) is dense in the entire space, but the operators do not admit closures from their restrictions to that intersection, comparison of their values on the intersection is not of much use. But frequently certain functions of A and Al have in fact a common domain, and then it is possible to consider comparing them. In this subsection we consider the case when the fractional powers of the operators A and Al have a common domain. Thus suppose that the operators A and Al satisfy condition (7.4). for simplicity with the same constant M. We suppose that the fractional powers A' and Ai (0 < II < 1) of A and Al have a common domain and differ on it by an operator B. subordinate to A with order CIt < II. LEMMA 7.9. For the difference of the resolvents R A1 ( - s) and R A ( - s) of the operators Al and A, the estimate

(7.24)

I R A1 ( -

s) - R A ( - s) I ~ c./(1

+ s) 1+(.-,,)-,

holds for any f > O. PROOF.

identity

Suppose that I = kll + 'Y, where 0

~

'Y

< II. On 9(A l ) the

156

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

R A( - S) - R A1 ( - S)

(7.25)

= A 1R A1 (- s)RA( -

S) - R A1 ( - s)RA( - s)A

Ie

=

LA~-i'RA1( - s)B,RA( - s)A U-l), i-I

+ R A1(AI -

A'I')RAA1-'I'

holds, where B, = Ai - A'. In view of Lemma 7.2 the operators B, are subordinated to the operator A"+' for any E> O. Therefore, using inequality (5.15) with n. = 0, we obtain

II Al-i'R A1 ( -

s)B,A -(a+,) A.,+,+U-l)'RA( - s) II ~ cjl(l

+ s) 1+(,-.,),.

Further, from what was said in §5.6, the operators A'I' and A«+< may be considered as the operator A' raised to the powers "(Ill and (a + dill respectively. Applying Theorem 7.6 to the operators A' and Ai = A' + B., we arrive at the conclusion that the operator A'I' - Al is subordinate to the operator (A') (<<+'1'+'1)/,-1 = A«+'I'+'1-'. Then \I R A1( - s) (AI - A'I')A -("+'1'+'1-') A 1+(0-'+'1) R A ( - s) I

~

c/(l + s)1+(,-«I-'I.

Combining the estimates of all the terms of thersum (7.25), we arrive"at the inequality (7.24). The lemma is proved. REMARK 7.5. If the operator B, is subordinate to A a and "( = 0, i.e. Il = 11k, where k is an integer, then from the proof of the lemma it is clear that the inequality

II R A1 ( -

s) - R A( - s)

I

~ cl(l

+ S)I+o-«

holds. Now we tum to the comparison of the semigroups U A1 (t) and UA(t) under the assumption that the operators Al and A satisfy in addition to condition (7.4) also condition (7.15) with "( > t for all A with Re A ~ o. Before doing this we note that formula (7.21) in this case may fail to have meaning, since the operator B = Al - A may fail to be defined on the solutions UA(s)Xo. We need to replace it by another formula, taking account of the fact that the operators Ai and A' have a common domain. Suppose again that Xo E ~(A 2). Consider the function y(t) = A 1 (1-0)UA(t)xo. This function will satisfy the equation dyldt

= A 1 (l-')AUA(t)xo = AiUA(t)Xo + (A 1 (1-')A - Ai> UA(t)xo = A 1y + (A 1 (l-,) A - Ai> UA (t) Xo.

17. EQUATIONS WITH PERTURBED OPERATORS

157

Here we take into account the fact that UA(t)XoEPJ(A) cPJ(AP) = PJ(Ai}. The functions AI(l-"AUA(t)Xo = AI(l-"UA(t)AXo and Ai UA(t) Xo = AiA -. UA(t) A'Xo are continuous on [0, T], so that, from Theorem 6.1, yet) = UAl(t)yO -

.£'

UAl(t - s) (Ai - A I (1-P'A) UA(s)xods

or AI(l-"UAl (t)xo - A I (1-·' UA(t)xo

=

L'

U Al (t- s)(Ai -

AI(l-., A)UA(s)xods·

Suppose again that 1 = k" + 'Y. U sing identity transformations, we obtain A I (1-P'UA1 (t)Xo - A I (1-"UA(t)Xo

"'L-lL'

(7.26)

i-I

0

+

U A1 (t - s)AliP(Ai - AP)Ai·UA(s)xods

L'

UAl(t- s)A I (1-"(Ar - A'Y)A l -7UA(S)Xods.

From the point of view of a possible singularity at the point s = 0 the worst term in the preceding sum is the last one, containing A l-'YUA(s) Xo. If the operator A I-p is applied to both sides of the equation, then in the sense of behavior at the point s = t the worst term is the first, containing At-'UAl (t - s). We can write down an identity in which the exponents 1- 'Y and 1 - " exchange roles. Analogously to (7.26), we get A -(1-7)UA1 (t)Xo - AI(l-'Y)UA(t)Xo =

(7.27)

+

t Jor'

.£'

UAI(t - s) (Ar - A7) UA(s)Xods

UA1(t - s)Ali-(Ai - AP)AV- I)'+7UA(S)Xods.

i-l

Estimates (7.26) and (7.27) make it possible to estimate the norms of operators of the type AI"(UAI(t) - UA(t» A -".

If in addition the difference of the operators Ai - AP is an operator subordinate to A with order a < II, then on using (7.26) the number

158

I. FIRST-QRDER EQUATIONS WITH CONSTANT OPERATOR

IC may be chosen 80 that the function At-'--UA1(t - s) has , summable singularity at s = t, and the number " in such a way that the function AP UA(s) Xo, for some p > 1- "y +" - 1-', has a summable singularity for s = o. The computation of the corresponding exponents for operators satisfying (7.15) will not be carried out here.

§8. Examples 1. The Cauchy problem for partial differential equations with constant coefficients. Consider a differential equation of the form (8.1)

av/at = A(D)v,

where v is a vector-function v = (Vb· •• , vm) of t and x,

A(D) =

L

ACl Da ,

lal :or

= (0£10 ••• ,a,,) a multi-index, I" I ~ "1 + "2 + .• . + a", D" = Dil ••• D:", D" = i a/ax" (k = 1,2, •.. , n), x = (Xb ••• , x,,) a point of n-dimensional space R", and the coefficients ACI are given constant square matrices of order m xm. The number r is called the order of the system. By the Cauchy problem for equation (8.1) we mean the problem of (t, x), satisfying the condition finding its solution

a

,=,

(8.2)

,(O,x)

= q,(x) ,

where the vector-function q, (x) is given throughout the space R". If we apply a Fourier transform relative to the space variable x to both sides of the system, and denote the image of the function ,(t,x) by v(t,p), we arrive at the system of ordinary equations (8.3)

d'TJ/dt = A (P)'TJ,

where A (P) is a matrix with elements which are polynomials in = (Pb ••• ,p,,). For the system (8.3) one poses the dual Cauchy problem, of finding a solution with the condition

p

(8.4)

'TJ(O,p)

= 4, (P) ,

where $(p) is the Fourier transform of t/>. The application of the Fourier transform is particularly convenient in considering solutions of (8.1), lying for each t in the space ~(R,,) (Yx-theory), since the Plancher~ Theorem guarantees the equality of the norms of, and its image u in the spaces 5f2 (R,,). Thus, every estimate in the ~(R,,) norm for

159

§S. EXAMPLES

a solution of equation (8.3) yields an estimate for the corresponding solution of equation (8.1). If equation (8.3) is considered in the space E = ~(RJ of vectorfunctions of p, then it will be a linear differential equation with an unbounded operator A, the operation of multiplication by the unbounded matrix A (P). The domain of A is naturally taken as the collection of all those u(p) E ~(R,,) for which A (p) u(P) E Y2(R,,).In general, A is not closed, but admits closure. The last fact follows from the fact that the adjoint operator A * is defined on a set of finite functions dense in ~(R,,) as the operator of multiplication by the adjoint matrix A *(p). We note that for the operator B of multiplication by a matrix B (p) with bounded continuous elements, the norm in ~(R,,) is calculated according to the formula ~ BII

= sup I B(P>l1 m, pER,.

where I B(P) I m is the norm of the matrix B(P) as an operator in mdimensional Euclidean space. The solution of the system (8.3) under the initial condition (8.4) is given by the formula ~(t,p)

= e tA (P)4>(P).

For the operator U(t) of multiplication by the matrix etA(p) to be bounded for fixed t > 0, it is necessary and sufficient that for this value of t sup I etA(p) I m

(8.5)

<

ex>.

pER,.

Denote by "'l(P)," • ,,,,,,(P) the eigenvalues of the matrix A(P). Then the numbers etl'j(P) will be eigenvalues for the matrix etA(P). The norm of the matrix is not less than the modulus of its eigenvalues, so that condition (8.5) implies that

Ietl'j(P) I =

etRel'j(P)

~

M(t)

(P E R", i

= 1, .. "m).

Therefore it follows in tum that (8.6)

Re"'j(p) ~ c.

A system (8.1) for which condition (8.6) is satisfied is said to be correct in the sense of Petrovskil.

160

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

We have arrived at the following assertion: LEMMA 8.1. For the operator U(t) of multiplication by the matrix etA(p) to be bounded in 5f2(R,,) for any t > 0, it is necessary that the system (8.1) be correct in the sense of Petrovskil.

If the operators U(t) are bounded for all t> 0, they obviously form a semigroup of bounded operators. Unfortunately, condition (8.6) is not sufficient for the boundedness of the operators U(t). Indeed, consider the following example. Suppose given the system (8.7) gotten from the one-dimensional wave equation a2ujat 2 = a2ujax2 by the substitution u = Vb auj at = v2. After a Fourier transformation we get (8.7') Here

(8.8)

1'1(P) = ip and 1'2(P) = - ip, i.e. the system is correct in the sense ofPetrovskii. The solution of the problem (8.3)-(8.4) for this system is obtained by a simple calculation: (8.9) The operator of multiplication by p sinpt is unbounded, so that the semigroup operator corresponding to (8.9) is also unbounded. In [232], Chapter I, §3.3, the following estimate for the norm of the matrix etA(p) is obtained: (8.10)

IletA(P)llm~ (1+2tIIA(P)llm+ ... + (2t)m-lIIA(p)II:::-l)etmaxRe~i(P).

The norm of A (P) increases no faster than (1 + Ipi)', so that for a given t the right side in (8.10) will be a bounded function of p if

maxRel'i(P)

~

«m -l)rjt) In(1 + Ip I>

+ a.

It follows from this inequality that Rel'l(P)-+-co as Since the numbers l'i(P) are roots of the characteristic IA(P) -I'll = 0, whose left side is a polynomial in p and 1', from the last relation that there exist constants h > 0 and b

Ipl-+oo. equation it follows such that

161

§S. EXAMPLES

(8.11)

(see [235], pp. 110-111 = 112). The systems (8.1) which satisfy (8.11) are said to be parabolic in the sense of Silo v, and the constant h is called the exponent of parabolicity of the system. If condition (8.11) is satisfied, the operators U(t) are bounded for all t> 0, and, moreover, the operators of the form A k U(t) will be bounded as well for all k = 0, 1, .. " and t > O. Thus the semigroup U(t) for t > 0 is infinitely differentiable. Thus we have established the following result. LEMMA 8.2. If the system (8.1) is parabolic in the sense of Silov, the semigroup U(t) of operators of multiplication by etA(P), for t> 0, is an infinitely differentiable semigroup of operators bounded in 5!;(Rn).

Condition (8.11) guarantees the "nice" behavior of the semigroup U(t) for t > O. However it does not guarantee the "nice" behavior of the semigroup U(t) as t~O. In particular it does not guarantee the uniform boundedness of the semigroup close to zero. In order to verify this, consider as an example the system (8.12)

{

dvddt = _ p2~, dV21dt

= pkVI

-

P 2V2

(k> 0),

which for integer positive k obviously corresponds to a parabolic system in the sense of Silov of the type (8.1). The solution of the Cauchy problem for this system has the form (8.13)

{ VI

=

e-p2t~h

V2

=

p k te-p2t'J;1

+ e- \P2' P

The maximum of the function p kte- p2t is a quantity or order t 1 - k/ 2• It therefore follows that

I U(t) II = c(t) Itl1, where {J = max{kl2 -I,D}, and c(t) is a bounded continuous function on [0,(0) with c(O) > O. Thus it is only for k ~ 2 that the semigroup U(t) is uniformly bounded, and the Cauchy problem for (8.12) is uniformly correct in 5!;(Rn). In the general case the semigroup U(t) may have any degree of growth as lit --+ 00. It is interesting to investigate the resolvent of the operator A of multiplication by the Illatrix A(P)

=

( _p2II P

0) -p

2'

162

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

The resolvent of A (P) has the form _lj(P2+ >.)

RA(P)(>',p)

= ( _ pkj(P2+ >.)2

Therefore it is clear that for k > 4 the operator A has no regular points. For k ~ 4 the operator A is closed and the resolvent RA (>.) is defined, for example, in the right halfplane, and the estimate liRA (>.) I

~

cjl AImin(l,2-k/2)

holds for it. According for 2 < k < 4 equation (8.3) lies in the class which we studied earlier in §3.3. For these equations the weakened Cauchy problem is correct on 9(A). For k ~ 2 we encounter the equations considered in Theorem 3.8, for which the Cauchy problem is uniformly correct and the corresponding semigroup is analytic in a sector containing the nonnegative semiaxis. It is curious that both for k > 4 and for k ~ 4 the weakened Cauchy problem has a solution (8.13) for arbitrary initial data on 9(A), this solution being analytic in t in the right halfplane. This tells us that investigation of the weakened Cauchy problem employing only the properties of the resolvent of the operator A, apparently, cannot be exhaustive. We tum to the conditions for uniform correctness in Y2(R,,) of the Cauchy problem (8.3)-(8.4) or, what is the same thing, the problem (8.1)-(8.2). Estimate (8.10) also yields the following. THEOREM 8.1. If the system (8.1) is parabolic in the sense of Silov and its exponent of parabolicity coincides with its order, then the Cauchy problem (8.1)-(8.2) is correct in Y 2 (R,,). All of its solutions are infinitely differentiable functions of t on t> O.

Indeed, under the hypotheses of the theorem, for 0

II U(t) I ~

sup I etA(P) I pER,.

~t ~

1.

~ Ml sup (1 + ~ (tlpl,)k) e-ctlplr pER,.

= Ml sup O
k=O

(1 + "-1) L eTk

cr

~ M.

k-O

The second part of the theorem follows from Lemma 8.2. The conditions of Theorem 8.1 are satisfied by systems which are parabolic in the sense of Petrovskii (see [232], Russian pp. 131-132).

§S. EXAMPLES

163

The use of estimate (8.10) leads to comparatively rough sufficient conditions for the uniform correctness of the Cauchy problem in .5!2(Rn). A complete solution of this problem was given in the paper [243]. 8.2. For tM Cauchy problem (8.1)-(8.2) to be uniformly correct in ~(Rn)' it is necessary and sufficient that tMre should exist a rwnsingular matrix S(P) and a constant K such that THEOREM

1) II S(P) II ~ K, II S-l(P) II ~ K; P.l(P)

cdp)··· Clm(P)

2) S(P)A (P)S-l(P) =

o 3) K

~

Rep.l(P)

~

•••

~

Rep.m(p);

4) ICij(P) I ~K(1+IRep.j(p)I>.

In the paper [234] systems of the type (8.1) were investigated, arising from a single differential equation of the type (8.14) after the substitution (8.15) The following theorem was proved. 8.2. For the operators U(t) of multiplication by the matrix etA(p) to form an infinitely differentiable (for t > 0) semigroup of operators bounded in .5!2(R,,) for the system (8.1) gotten from equation (8.14) by the substitution (8.15), it is necessary and sufficient that the system should be correct in the sense of Petrovskil and such that for each a > 0 there should exist a b such that tM inequality THEOREM

Rep.j(p) ~ - a Inl Imp.j(p) I + b

should hold for all pER" and all eigenvalues of the matrix A (P). For a system (8.1) of arbitrary form, conditions 8.2 are not sufficient for the boundedness of the operators U(t). For example, for the system

diMdt = - P2Uh d~/dt

= P"VI - V;

164

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

the conditions of Theorem 8.1 are satisfied, and the solution of the problem (8.3)-(8.4) for them has the form

VI = e-p2tih V2=

,

p

1-p

2

[e- p2 , - e-']il + e-ti2,

and obviously defines an unbounded operator U(t) for k > 2. 2. The Cauchy problem in spaces with a ditl'erentiable ~-norm. We return to the example (8.7). It is well known that the Cauchy problem for the wave equation u; = u;' is an example of a well-posed problem of mathematical physics. The fact that after passing to the system (8.7) the Cauchy problem becomes incorrect in the space ~(RI) suggests that the space ought to be replaced by a more restrictive one, in which the problem may become uniformly correct. This subsection deals with this question. Suppose that B(P) = (bil/(P», for each pER", is a positive definite matrix of order mxm with polynomial coefficients. Suppose that (8.16)

where the constant c does not depend on p. The norm defined by the formula (8.17)

lI'un =

(Bu(P), u(P»",dp

will be called the differential Y2-norm. The collection of all it E Y 2 (R,,) for which the norm (8.17) is finite is a Hilbert space .Sf'B relative to the scalar product (u,vh =

f

(Bu,v) ",dp.

The preimage of .Sf'B under the Fourier transform will be called the space Y B• It may be obtained in the following way: to the matrix B(P) there corresponds some elliptic differential operator B(D). Into the collection ~(B) of all functions u of Y 2 (R,,) for which B(D) u E Y 2 (R,,) one introduces the norm

Ilullt= f(B(D)u,U)",dX. The completion of ~(B) relative to this norm is the. space Y B • Now suppose that the system (8.1) is correct in the sense of Petrovskii. Without loss of generality we may take c < 0 in (8.6), i.e. suppose that (8.18)

Re#ti(P) ~ - ~

< o.

165

§S. EXAMPLES

Then, as was shown by Ljapunov (see [258]), there exists a positive definite matrix C(P) such that

+ A *(p) C(P) =

C(P)A (P)

-

I.

If this last equation is considered as a system of equations relative

to the elements of the matrix C(P), then the matrix of this system has as eigenvalues all possible sums I'i(P) + I'j(P) (i, j = 1,2", .,m). Hence its determinant .1(p) will be positive, in view of (8.18). Multiplying C(P) by 1.1(P>1 2, we obtain a matrix C1 (P) = 1.1(P) 12 C(P) with polynomial coefficients, for which C1(P)A(P) +A*(p)C1(p) = -1.1(P)1 2 I.

In [3] the formula C(P) =

L"

e,A'(P)elA(P)ds

was given for the determination of the matrix C(P). In addition, the estimate (8.19}

--

(C(P)u, u) .. =

Jor'" (e

1A(p)-

ds 00 - -) u)", ~ I A I .. (u, U III

1A(p)-

u, e

was given, where 00 is an absolute constant. For the matrix C1(P) the following inequality will then be satisfied: (8.20)

(C1(P)u,u)", ~ (00!.1(P)1 2/IIAII",) (u,u)",.

The quantity I A",II grows like (1 + Ip 1)'. The quantity 1.1(P) 12 is bounded below by the constant (20) 2m. Hence the coefficient in the right side of (8.20) decreases as Ip I -+ 00 no faster than some negative POwer of Ip I. Multiplying the matrix C1 (P) by 1 + IP 121 , we can arrange that for the polynomial matrix B(P) = (1 + Ipl21)C1 both (8.16) and (8.21)

B(P)A(P)

+ A * (P) B(P) =

-

b(p)I

are satisfied, where b(P) is a positive scalar function. It follows from (8.21) that (d/dt) (B(p)etA(P)~, etA(P)~)", ~ 0,

and so

166

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

Assuming that R", we obtain

i

E

YB

and integrating' in the last inequality over

I U(t) ill B ~ I ill B· Thus the semigroup U(t) is contracting in the space YB. We have arrived at the following assertion. THEOREM 8.3. For there to exist a differentiable ~-norm II··· I B with respect to which the problem (8.1)-(8.2) is uniformly correct in the space Ya. it is necessary and sufficient that the system (8.1) should be uniformly correct in the sense of Petrovskit.

The necessity is proved in the same way as in Lemma 8.1. It was proved in the paper [243] that a necessary and sufficient condition for the uniform correctness of the Cauchy problem in the original space Y 2 (R,,) is that there should exist an operator B(P) satisfying condition (8.21) and such that the norm (8.17) is equivalent to the norm ~(R,,). The sufficiency of this condition is verified directly. Its necessity is established by a direct construction of the matrix B (P) using Theorem 8.2. We recall that for the general case of a uniformly correct Cauchy problem in Hilbert space we have also introduced a new scalar product in which the operator A becomes dissipative (Theorem 4.8). However it has not been possible to prove that this scalar product is bounded below. Theorem 2.14 makes it possible to consider the question of analyticity of the solutions of the Cauchy problem. THEOREM 8.4. For there to exist a differentiable ~-norm I ••. I under which the semigroup U(r) of operators of mUltiplication by e tACp) wiU be analytic in a sector containing the nonnegative semiaxis and satisfying there the inequality

(8.22) it is necessary and sufficient that the eigenvalues #Lj(P) of the matrix A(P) should lie in some sector containing the negative semiaxis, with an opening angle of 1r - 24>0, or in other words that the inequality

(8.23) should hold, where b is a constant. PROOF. It follows from Theorem 2.14 that for U(z) to be analytic and for (8.22) to hold it is necessary and sufficient that the Cauchy

167

§S. EXAMPLES

problem should be uniformly correct for the system with operators e±i,;oA(p). The eigenvalues of the matrix e±i';oA(p) are equal to e±i';Op.k(P) (k = 1••. . ,m). The condition of correctness in the sense of Petrovskii of the system with operators e±<.IoA (D) leads to the

necessity of inequality (8.23). Conversely, if condition (8.23) is satisfied, then systems with the operators e±itloA(p) are correct in the sense of Petrovskii, so that for each of them there exists in accordance with Theorem 8.3 a differential Yrnorm in which the Cauchy problem is uniformly correct. However, generally speaking, these norms will be distinct, and we cannot apply Theorem 2.14. In connection with this one establishes the following. LEMMA 8.3. Consider two systems of linear differential equations with constant coefficients,

du/dt

= Alu and du/dt = A 2 u,

of the first order. Suppose that the matrices Al and A2 commute and have eigenvalues satisfying (8.18). Then there exists a common Ljapunov matrix for these systems.

Indeed,

is such a matrix. One verifies directly that CAl + AtC = -

.£'" eaA~e·A2dO' < 0

CA 2 + AtC = -

L'"

and

eSAie'Alds

< O.

II A211>

u).

(8.19) implies the estimate (CU, Il) ~ (a~/ II Adl

(Il,

In order to complete the proof of Theorem 8.4 we need to construct a common Ljapunov matrix C(P) for the matrices ei4>oA (P) and e-i
Of course one has first to make a substitution which shifts the spectrum to the left.

168

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

( that the resulting matrix B (p) will be a polynomial matrix and satisfy condition (8.1G). In the polynomial norlJl constructed relative to B(p), the Cauchy problems relative to the systems dvjdt = eioA(p)v and dvjdt = e-i"oA(p)V'

will be uniformly correct in the space Y B• The assertion of the theorem then follows from Theorem 2.14. The theorem is proved. REMARK 8.1. We could have dropped the requirement of uniformity in p of the positive definiteness of the matrix B (p). Then it would not have been necessary to adjust the matrix C1 (p) by adding the factor (1 + 1p 121). In this case one can prove the uniform correctness of the corresponding Cauchy problem in a space !/B which may already contain functions not lying in !z!;.(Rn) , and even generalized functions. From a simple calculation we find that for the system (8.12) the norm of Theorem 8.3 may be determined by the formula

2+21 P 2k- 4 1-1 2 1 P k-2(--::; 1 -112 Vl v B= f~{1-12 _ .. Vl + 1-1 V2 VI +"2 VIV2+ V2-::)}d p. For the system (8.7) the eigenvalues lie on the imaginary axis. Therefore in order to obtain a positive definite Ljapunov matrix one has to make a substitution in (8.7) which shifts the spectrum to the left. If we seek the matrix B directly in terms of the matrix (8.8), we arrive at the "energy norm"

The space !/B gotten by completion of g'(B) relative to the corresponding norm will contain functions which do 110t lie in !z!;., in particular those for which the component Ul is a constant. In such a space the Cauchy problem for the system (8.7) will be uniformly correct. 3. Boundary problems Cor parabolic systems. Suppose that G is a bounded region of n-dimensional space with a sufficiently smooth boundary r. Consider the equation (8.24)

au/at

= A(x,D)u,

where v is an m-dimensional vector-function of t and x, (8.25)

A(x,D)

=

L

lal ;;;r

Aa(x)D',

169

§S. EXAMPLES

the Aa(x) being square matrices whose elements are sufficiently smooth functions of x defined in the closed region G. We shall call the differential expression A'(x,D) =

L

Aa(x)D«,

lal- r

containing only the terms of highest order, the principal part of A (x, D). Consider the corresponding polynomial matrix A'(x,p), p being a real vector. Suppose that r = 2s is an even number. If this matrix, for any fixed x E G and pERm generates in m-dimensional space a dissipative operator, so that (8.26)

Re(A' (x,p) u, u)m

<0

(u ~ 0),

then the operator A (x, D) is said to be strongly elliptic, and the system (8.24) strongly parabolic. We note that the elements of the matrix A'(x,p) are homogeneous functions of p of degree r = 2s, so that it follows from (8.26) that (8.27) where

-

~

=

max

Re(A' (x,p) u, u)m'

l:EG.lpl-l.lulm=l

If in ~(G) we consider the operator Ao generated by (8.25) on the smooth functions satisfying the conditions of the first boundary problem

au

U I r = an

(8.28)

I

r =

••.

=

as-1u

ans-1

I r

= 0,

then it satisfies Re(Aov,v)

= Re

fa

(A(x,D)v(x),V(x»mdx

~w L(U(X),U(X»mdx=w(U,U). The operator Ao admits a closure A, and the operator A - w[ will be a maximal dissipative operator. It therefore follows that the problem of finding a solution of equation (8.24) satisfying the boundary conditions (8.28) and the initial condition (8.29)

V(O, x) = I/l(x) E 9'(A)

is uniformly correct in 9'(A).

170

I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR

We note that the domain 9'(A) consists of all the vector-functions lying in W2and satisfying conditions (8.28) in the sense of the imbedding theorem. The semigroup corresponding to the problem (8.24), (8.28), (8.29) satisfies the condition I U(t) II ~ e..t• One can consider an analogous problem in the space ~ (G). In [235] it was proved that the closure A of an operator A' generated by the differential expression A (x, D) on the smooth functions satisfying (8.28) has a resolvent for which the inequality (8.30) holds in some sector I arg(~ - w) I ~ 7(/2. Thus the semigroup generated by the first boundary problem for a strongly parabolic system is analytic in ~(G). Inequalities of the type (8.30) have been established also for wider classes of boundary problems. To simplify the exposition we shall describe them for the case of one equation (8.31)

iJv/ iJt

= A (x, D) v,

where A (x, D) is a linear differential operator of order T with coefficients which are sufficiently smooth in G, and v(t, x) is the desired function. Suppose that the condition of strong ellipticity ,

ReA'(x,p)

<0

is satisfied for x E G and pER,.. It then follows that and that the equation

A' (X,p'

T

= 2s is even

+ !Po) - ~ = 0,

for Re ~ ~ 0, Po ¢ 0 and p' orthogonal to Po, has the same number of roots in the upper and lower halfplanes. Suppose given s boundary conditions (8.32)

(j = 1,2, •• • ,s),

where Bj(x, D) are linear differential operators of orders Tj with sufficiently smooth coefficients. Denote by BJ (x, D) the principal parts of these operators. Suppose that the following condition of Sapiro-Lopatinskii is satisfied: let x be a fixed point of the boundary r of G, n a unit vector normal to r at this point, and p' a vector orthogonal to n. Denote by rt= rt(x,p',>.) (k = 1,2,.· .,s) the roots of the equation

§S. EXAMPLES

171

A'(x,p' + tn) - A = 0, lying in the upper halfplane. CONDITION 1. The polynomials Bj(x,p' + rn) (j = 1,2, •. ·,s) in r are linearly independent relative to the modulus of the polynomial Ilk-l(r - rt) for each x E r, p' orthogonal to n, and A with ReA ~ O. For the closure A in the space 5fp(G) of the operator Ao generated by the differential expression A (x, D) on the smooth functions satisfying the boundary conditions (8.28), estimate (8.30) holds, so that the problem of finding a solution of equation (8.31) satisfying the boundary conditions (8.32) and the initial condition v(O, x) = I/>(x) E g(A) is uniformly correct. The semigroup corresponding to this problem is analytic. We note that the conditions imposed on the operators Bj are not effectively verifiable. Here no constructive process is set forth whereby, given any class of equations, we can construct operators B j satisfying Condition 1 for any region with a smooth boundary. We except from these remarks the first boundary problem, considered above, for which Condition 1 is always satisfied. In the case when A (x, D) is an elliptic operator of the second order (s = 1) with real coefficients, (8.33)

it has been proved that estimate (8.30) holds also for an operator generated by the differential expression (8.33) on the functions satisfying the second boundary condition:

av/a" -

11

(x) V = 0,

where l1(x) ~ o. 4. Symmetric hyperbolic systems. Consider the case when the operator A (x, D) in (8.24) is an operator of the first order of the form (8.34)

A(x,D)v

n

a" + B(x)",

= EAi(x) i-I

aXi

where Aj(x) and B(x) are matrices with sufficiently smooth coefficients, defined in a bounded region G. The matrices Ai(X) are supposed to be real and symmetric. Integrating by parts, we obtain

172

I. FIRST-ORDER EQUATIONS WITH CONSTANT Ol'ERATOR

+ BV'V) mdx = JG (EAi~V. i-I IIX.

(Av,v) = r

+

- r (",EA i

JG

i-I

!".) dx IIX. m

r((B-t:A~)V'V) JG .=I I1X• mdx+ Jrr (tAiniV,V) .-1 mds,

where the ni are the coordinates of the exterior normal to 1'. Hence (8.35)

Re(Av,v) = r((B+B*_E oAi\",,,) dx+ r(EAini"'V) ds. JG i=1 OXi / m Jr i-I m

If we suppose that

(8.36) and (8.37)

r ( t AiniV,V) ds

Jr

.-1

m

~ 0,

then the operator A is dissipative if it is regarded as defined on the set of functions where all the operations carried out above are legitimate. For a differential operator of the first order, it is natural to give the homogeneous boundary conditions in the form of equations (j = 1,2, •. . ,1),

where the Wj(x) are continuous vector fields given on the boundary I' of the region G. In other words the conditions may be treated in such a way that the vector u(x), for xE 1'. has to belong to some subspace N(x) of the entire m-dimensional subspace (N(x) is orthogonal to the vectors Wj{x». Here the subspace N(x) varies continuously as x varies. We make two hypotheses: 10 • The rank. of the matrix An{x) = "Lf_lAi(X) ni(x) does not change when x runs along the boundary 1'. 20 • The subspace N(x) is a maximal subspace on which the form of the matrix An(x) is nonpositive, (Anu, u)m ~ O. Now if an operator A is given by a differential expression (8.34) in ~(G) on the functions which are continuous in G, satisfy the boundary conditions (8.38)

u(x) IrE N(x) ,

173

§S. EXAMPLES

and have square-integrable first derivatives, then under conditions 1° and 2° this operator admits closure to a maximal dissipative operator. Therefore it follows that the problem of finding a solution of the system (8.39)

oV

- =

ot

" oV LA,;+ B(x)v, OXi

i-I

satisfying (8.38) and the initial condition v(O, x) = .(x) E 9J(A), if (8.36), 1° and 2° are satisfied, is uniformly correct in ~(G). The corresponding semigroup is contracting. If we do not assume that (8.36) is satisfied, but hold to condition (8.37), then we find from (8.35) that Re(Av, v) ~ ",(v, v),

where

'" = max :rEG

II B(x) + B*(x) -

i:.

OAi i-I OXj

II. m

This suggests that equation (8.39) reduces to an equation with a dissipative operator (see §4.4, Example 1). If, for example, one assumes that the matrices Ai(X) and B(x) are periodic in all variables and if one takes G to be the parallelepiped of periods, then for the functions u(x) satisfying the same periodicity conditions relative to the space variables,

5. The SchrOdinger equation. In quantum mechanics an essential role is played by the Schrodinger equation of the type (8.40) where x = (Xl, Xl!> xa) is a three-dimensional vector running through the entire space R a, d is the Laplace operator, and b(x) is a given function. Because of the physical meaning of the function "" it is natural to study this equation in the space ~(Ra). The differential expression - d + b(x) generates in a natural way an operator Ao on functions with compact support. This operator is symmetric. Under certain conditions on the function b(x) the closure A of Ao will be a selfadjoint operator in ~(Ra). Then equation (8.40)

174

I. FIRST-ORDER EQUATIONS WITH CONSTANT OPERATOR

generates an equation in the Hilbert space Y2(G) considered in Theorem 4.7. To it there corresponds a group U(t) of unitary operators in Y 2 (G). 6. Equations with retarded argument. We begin by considering the simplest equation of the form (8.41)

y' (t)

= ay(t -

1)

(0

~t

<

Q).

°

It is clear that in order to find a solution of this equation for all t > it is necessary to give the function y on the segment [- 1, 0]. Then y(t) may be sought successively on the segments [n, n + 1] from the integral equations (8.42)

y(t)

= y(n)

+ a itY(T -

(n ~ t ~ n

l)dT

+ 1).

We write xo(s) = y(s) (- 1 ~ s ~ 0). We suppose that this function lies in the space C[-I,O]. We consider the function x(t,s) =y(t+s) and regard it for each fixed t as an element of the space C"[- 1,0]. Thus the solutions of equation (8.41) with the initial conditions (8.43)

y(s)

= Xo(s)

~

(-1

8

~

0, Xo(s) E C[-l,O])

generate the operators U(t) xo(s)

=

x(t, s)

= y(t + s).

Obviously these operators form a strongly continuous semigroup of bounded operators in C[ - 1,0]. This semigroup satisfies the Cocondition. Indeed, it follows from (8.42), for example, that

max I U(t)xoll = O;liiI;lii1,-1;lii';liiO max Iy(t + s) I

0:;;1:;;1

~ IXo(O) I + a

l1 lxo

(T - 1) IdT

~ (1 + a) Ilxoll·

It is clear from (8.42) that the semigroup U(t) has retarded smoothness: for n < t ~ n + 1 the solutions y(t + s) have derivatives of the (n + l)th order. By a simple calculation we find that any differentiation operator, given on all functions x(s) which are continuously differentiable on [-1,0] and satisfy the boundary condition (8.44)

x' (0)

= ax( -

1),

is a generating operator for the semigroup U(t).

175

§S. EXAMPLES

The function x(t, s) satisfies the equation ox/ at = ax/as,

(8.45)

so that the problem (8.41), (8.43) may be treated as the problem of finding a solution of the hyperbolic equation (8.45), satisfying on the segment [- 1,0] the nonlocal boundary condition (8.44). As we have shown, this problem is uniformly correct in C[ - 1,0]. We note that the problem (8.45), (8.44) does not lie in the class of problems considered for hyperbolic systems in subsection 4. It is possible to generalize the considerations presen~d above. Consider the equation dy/dt = B(y(t + s»

(8.46)

(0 ~ t

<

(0),

where y is an m-dimensional vector-function, defined on [- 1, (0), and B is a linear operator mapping the space C [ - 1,0] of vector-functions continuous on [- 1,0] into the m-dimensional vector space Rm. In equation (8.46), for each t, the operator B operates on y(t + s) as on a function of s of C[-l,O]. In the example (8.41) B(x(s» = x(-l) (x E C[O, 1]). Just as in this example, the solution of (8.46) is uniquely determined, if one gives the conditions (8.47)

y(s)

= xo(s)

(- 1 ~ s

~

0, xo(s) E C[ - 1,0]).

The problem (8.46), (8.47) may again be considered as the problem of finding a solution of the hyperbolic system ox/ at = ax/as,

satisfying the boundary condition x' (t, 0)

= B(x(s»

x (0, s)

=

and the initial condition xo(s).

This problem is uniformly correct in C[ - 1,0].

CHAPl'ERII EQUATIONS OF THE FIRST ORDER WITH VARIABLE OPERATOR §1. Unbounded operators depending on a parameter 1. Operators with a constant domain, strongly continuous on it. Suppose that for each t E [0, T] there is defined a linear operator A (t) with a domain ~(A), not depending on t and dense in E. Suppose that A(t) admits closure. If B is an operator with the same domain ~(B) = ~(A), having a bounded inverse, then the operator A(t)B- I is defined everywhere, is closed, and accordingly is bounded (see Lemma 7.1, Chapter I). We shall say that the operator A (t) is strongly continuous on the set .L C~(A) if for any xE....It the function A(t)x is continuous. If the operator A(t) is strongly continuous on ~(A), then the bounded operator A(t)B- 1 is strongly continuous in t, so that in view of the Banach-Steinhaus Theorem it is uniformly bounded in t: "A(t)B-I~ ~ C. If the function f(t) (0 ~ t ~ T) takes on values of ~(A) and the function Bf(t) is continuous, then the function A(t) f(t)

=

A (t)B-IBf(t)

is also continuous. Now suppose that for each t E [9, T] the operator A (t) has a bounded inverse A -l(t). Then we may take as the operator B for example the operator A(O). From the foregoing it follows that the operator A(t)A -1(0) is bounded and strongly continuous in t on [0, T]. In particular, (O~t~T).

If we assume that A -let) is uniformly bounded on [0, T], it will follow from this that it is strongly continuous. Indeed, for x E E

IIA -let + ~t)x -

A -l(t)xll

= I A -let + ~t)

(A(t) - A(t

+ ~t»A -l(t)xll

~MII(A(t) -A(t+~t»A-I(t)xll--+O

as ~t-O.

In view of the same Lemma 7.1 of Chapter I, the operator A (O)A -l(S) will be bounded for each s. This operator is inverse to the strongly continuous bounded operator A(s)A -IA -1(0). If we assume that (1.1)

IIA(O)A -l(S) II ~ M 176

(0 ~ s ~ T),

flo

UNBOUNDED OPERATORS

177

then it will follow from the foregoing that the operator A (0) A -1(S) is strongly continuous for s E [0, T].linally we consider the bounded operator A (t) A -l(S). From the representation (1.2) it follows that the operator A(t)A -1(S) is strongly continuous as a function of the two variables s and t (0 ~ s, t ~ 1') Thus we have arrived at the following assertion. LEMMA 1.1. Suppose that the operator A (t) (0 ~ t ~ 1') with a constant domain g&(A) is strongly continuous on g&(A) and has a bounded inverse A -l(t) satisfying condition (1.1). Then the operator A (t) A -l(S) is bounded and strongly continuous relative to t and s in the square o ~ s, t ~ T. 2. Approximation by bounded operators. We say that the bounded operators A,.(t) (n = 1,2, •.. ) converge strongly and uniformly for t E [0, T] on 91 (A) to the operator A (t), if for each x E 91 (A) the function A,.(t)x converges to the function A (t)x uniformly relative to t E [0, T]. Suppose that the operator A (t) satisfies the conditions of Lemma 1.1. Then the bounded operators A,.(t) A -1(0) converge strongly and uniformly on [0, T] to the bounded operator A(t)A -1(0). Suppose that x is a fixed element of E. The function A(O)A -1(S)X is continuous, so that its values form a compact set in E. In view of Lemma 3.1 of the Introduction, the operators A,.(t) A -1(0) will converge to A(t)A -1(0) on this set uniformly relative to t and s. LEMMA 1.2. Suppose that the bounded operators A,.(t) converge strongly and uniformly relative to t E [0, T] to an operator A (t) which satisfies the conditions of Lemma 1.1. Then the bounded operators A,.(t) A -1(S) converge to the bounded operator A (t) A -1(S) strongly and uniformly relative to t and s in the square 0 ~ s, t ~ T.

3. Strongly continuously differentiable operators on g&(A). The operator A (t) (0 ~ t ~ 1') is said to be strongly continuously differentiable on ~(A) if the function u(t) = A (t)x for x E ~(A) has a continuous derivative u' (t) on [0, T]. Writing u'(t) =A'(t)x,

We obtain a linear operator A' (t), defined on ~ (A) and strongly continuous there. Suppose that B is a bounded operator on g&(A) with a bounded

178

II.

FIRST-oRDER EQUATIONS WITH VARIABLE OPERATOR

inverse B- 1• Then for any z E E the element x = B- 1z lies in ~(A), so that

(1/ at) [A(t + t.t)B-1z - A (t)B- 1z]-+A' (t)B- 1z. Thus the bounded operators on the left converge strongly to the operator A'(t)B- 1• It therefore follows that the operator A'(t)B-1 is bounded. From the foregoing it follows that it is strongly continuous on [0, T] and, in particular, is uniformly bounded. In other words, we have shown that the bounded operator A (t) B -1 is strongly continuously differentiable and [A (t)B- 1], = A'(t)B-1• It follows from Lemma 3.5 of the Introduction that the operator A(t)B- 1 is continuous in norm, and moreover satisfies the Lipschitz condition (1.3)

1.3. Suppose that the operator A (t) has a constant domain and is strongly continuously differentiable on it. Suppose that the operator B is defined on ~(A) and has a bounded inverse. Suppose finally that the function f(t) is continuously differentiable, and that the functions Bf(t) and BI' (t) are defined and continuous. Then the function A (t) I(t) is continuously differentiable and LEMMA

~(A)

[A (t) I(t)]' PROOF.

= A' (t) f(t)

+ A (t) I' (t).

First we establish the formula

(1.4)

[Bf(t)]' = BI'(t).

Since the operator B is closed, (1.5)

l'B{'(T)dT = B l'{'(T)dT = B[/(t) - 1(0)],

so that (1.4) is obtained by differentiation. Here we have [A(t) fCt)]'

=

[A (t)B-1Bf(t) ]'

= A' (t)f(t)

= A' (t)B-1Bf(t) + A (t)B-1B{,(t)

+ A(t)l'(t).

The lemma is proved. . REMARK 1.1. From the proof it is clear that it would suffice to have f(O) E ~(A) in place of the hypothesis that BI(t) is continuous. The remainder follows from (1.5). LEMMA 1.4. Suppose that the operator A (t) has a constant domain and is strongly continuously differentiable on it, and that the operator

§1. UNBOUNDED OPERATORS

179

Q(t) is bounded and strongly continuously differentiable. Then the operator C(t) = Q(t)A(t), defined on ,q(A), is strongly continuously differentiable there and (1.6)

C' (t)x

PROOF.

(1.7)

= Q' (t)A (t)x + Q(t)A' (t)x (x E ,q(A), t E [0, T]).

Consider the identity

(6Cj6t)x = 6Q(6A/6t) x

+ Q(t) (6Aj6t) x + (6Q/6t) A (t)x.

The first term on the right tends to zero as 6t~0, because of the continuity in norm of the operator Q(t) (Lemma 3.5 of the Introduction) and the fact that (6Aj M)x~A' (t)x. On passing to the limit in (1.7) as 6t~0, we obtain (1.6). The lemma is proved. REMARK 1.2. In what follows we shall often encounter a bounded operator depending on a parameter and strongly continuously differentiable on some set ,q dense in E. The semigroup operator U(t) corresponding to a uniformly correct Cauchy problem is an example. We may then consider the restrestriction A (t) of such an operator to ,q and then apply Lemmas 1.3 and 1.4 to it. N ow suppose that A (t) has a bounded inverse. Then we may take as the operator B, for example, the operator A (0), and arrive at the conclusion that the operator A (t)A -1(0) is continuous in the operator norm, satisfies the Lipschitz condition (1.8)

II (A(t) -A(r»A-l(O)11

~clt-rl,

is

strongly continuously differentiable, and that [A (t)A -1(0)]' = A'(t)A -1(0). In view of Lemma 3.8 of the Introduction the operator A(O)A -l(S), inverse to the operator A(s)A -1(0), will also be continuous in norm for 0 ~ s ~ T and strongly continuously differentiable relative to s, while [A (O)A -l(S)]'

= - A (O)A -l(s)A' (s)A -l(O)A (O)A -l(S)

(1.9) =

-

A(O)A -l(s)A'(s)A -l(S).

From the arguments just gone through and representation (1.2), one immediately obtains the following lemma. LEMMA 1.5. Suppose that the operator A (t) is s.troTtgly continuously differentiable on ,q(A) and has a bounded inverse A -let). Then 1°. The operator A(t)A -l(S) is continuous in the operator norm in the variables s and t taken together, where 0 ~ s, t ~ T, and satisfies a Lipschitz condition in each of them.

180

II.

FIRST-oRDER EQUATIONS WITH VARIABLE OPERATOR

2°. The operator A (t) A -l(S) is strongly differentiable relative to t and s, and the derivatives [A (t)A -l(S)]: = A' (t)A -l(S) and [A (t)A -l(S)]: = - A(t)A -l(S) A' (s)A -l(S) are strongly continuous as functions of the two variables sand t.

We note that it of course follows from (1.9) that the operator A -l(S) is continuously differentiable and that the following formula holds: [A -l(S)]' = - A -l(s)A' (s)A -l(S).

(1.10)

LEMMA 1.6. Suppose that the operators A(t), Adt) and A 2 (t) have domains independent of t. Suppose that A (t) is strongly continuously differentiable on 9'(A), that A 1 (t) satisfies the conditions of Lemma 1.5, and

(1.11) Then the operator A 2 (t) is strongly continuously differentiable on 9'(A 2) = 9'(A), and for x E 9'(A) A'(t)x = PROOF.

+ A 1(t)A2(t)x. , Ai1(t + ~t) to both

A~(t)A2(t)X

We apply the operator

sides of the

identity

After some transformation we get l~A

~Ai-x+Ai

(1.12)

~t

l()~A

t-x ~t

= -~A2 X + Ai 1(t) -~Al A 2(t) x + ~Ai 1~A1 A 2(t)x. ~t

~t

~t

In view of the continuity in norm of Ai1(t) and the existence of the limits of ~AI ~t and (M1I ~t)A2(t)X, the first term on the left and the last term on the right tend to zero as ~t --+ O. We note that equation (1.11) presupposes the inclusion ~(A2(t» C 9'(A 1). Then from (1.12) it follows that the derivative

181

§1. UNBOUNDED OPERATORS

exists. The lemma is proved. 4. Fractional powers of operators depending on a parameter. Suppose that the operator A (t) satisfies the conditions of Lemma 1.1. Suppose moreover that its resolvent satisfies the estimate

I RA(t) ( -

(1.13)

s)

II

~

M/(l + s)

(s ~ 0)

with a constant not depending on t. Consider the fractional powers A"(t) (0 < a < 1) of A(t). A"(t) has a wider domain than A (t), so that now we cannot guarantee the constancy of the domain of A"(t). We can prove the following weaker assertion. LEMMA

1.7. Suppose that for any to E [0, T] the element x lies in the

oomain of the operator AP(to). Then it lies in the domains of all the operators A "(t) for a < fJ and t E [0, T]. Moreover, the inequality

I A"(t) A -P(to) I

(1.14)

~

c

Iwlds with a c..onstant not depending on t and to-

Suppose that x E ~(AJI(to». Then x = A -Jl(to)x. Using formula (5.8) of Chapter I for a negative power of A(to), we get PROOF.

x

SinfJ7r!o'" =s-IIRA(to) ( - s)zds. r

0

We shall show that the operator sinarl'" a"- 1A (t) RA(t) ( - a)dl1 r

0

is applicable to x. We have "7A'. fZ

=

= SinarsinfJr!o" 2 11a-1A()R t A(t) ( -

11

)!o'" s -IIRA(to) ( -

r O O

SinarsinfJ r2 1r {

Jo

('"

11

,,-lRA(t) ( - 11)

("-IIA()AJ/ t to )A() to RA(to) ( -

+ .fo'" 11,,-lA(t)RA(t)(-I1) Hence

s)z ds d 11

1(

f:

s)dsdl1

S-IIRA(to)<-S)dsdl1} z.

182

II.

FIRST-oRDER EQUATIONS WITH VARIABLE OPERATOR

since a < p. We approximate the element z by a sequence of elements z"E.9J(A). Then A -P(to)z,. E .9J(A) and A -P(to)z,.-+A -P(to)z. In view of formula (5.13) of Chapter I ..7z,.= A"(t)A-P(to)z".

It follows from inequality (1.15) that ..7z,.-+..7z, i.e. Aa(t) A -P(to)z" -+..7z, so that, because the operator A"(t) is closed, the element A -P(to)z =xE.9J(A"(t» and ..7z = A"(t)A-P(to)z. Inequality (1.15) implies (1.14). The lemma is proved. Now we investigate the question of the smoothness of A.,(t). To this end we first recall estimate (5.15) of Chapter 1. For n = 0 and o < "y < 1 we have (1.16)

It is clear from the derivation of this estimate that in our case the constant c may be taken independent of t. From (1.14) and (1.16), for 0 < "y < Ii < 1 we get

I A y(t) RA(T) ( -

sH ~ I A'Y(t) A -'(r) IIII Ai (r)R A(.)( -s) I

(1.17) where c, does not depend on t, T and B. LEMMA 1.8. If the operator A (t) satisfies the conditions of Lemma 1.5, then the operators A "(t) A -P(to) (0 < a < p ~ 1) and A "(t) RA(t) ( - s) satisfy a Lipschitz condition relative to t. PROOF. We choose a number al so that a (5.13) of Chapter I we find that

(1.18).

'"

= small" lI"

aA"(t)A -"l(t)

< al < p. From formula

r s"RA(t+t.t) ( _ s)aA{t) A -l{t) A l-"l{t)RA(t) ( - s)ds.

Jo

183

§1. UNBOUNDED OPERATORS

Applying (1.16) for 'Y = 1 - ab we get

II.:1A"(t) A -"l(t) I ~ c r '"

Jo

s" 1+

(1+s)

a1

ds II.:1A (t) A -let)

I ~ cdl.:1A (t) A -let) II.

From estimate (1.8) it follows that (1.19)

so that in view of (1.14) (1.20) II.:1A a(t)A -1I(to)

I

~

II.:1A a(t) A -al(t) I I A "I (t) A -II (to) I

Further, suppose that a

~ Cal.:1tl.

< 'Y <~. Then

I Aa(t + .:1t)RA(t+4t) ( - s) - A a(t) RA(t) ( - s) I ~ II.:1Aa(t)A (to) I I A7(tO) RA(t) ( - s) I -7

+ IIA"(t+ .:1t)RA(t+4t)( - S) 1111.:1A(t)A -let)

I I A (t) RA(t) ( -s) II.

In view of (1.17), (1.19), (1.20) and (1.13) we obtain

I A a(t+ .:1t)RA(HAI)( - s) -

(1.21)

A "(t) RA(t) ( - S)

I ~ c l.:1tl/(1 + S)1-.,

where the constant c does not depend on t, s and .:1t. The lemma is proved. REMARK 1.3. From inequality (1.21) and the identity RA(t)( - s - .:1s) - RA(t)( - s) = - .:1sRA(t) ( - s - .:1s) RA (t) (

-

s)

it follows that the operator function (1 + s)PA"(t) RA(t) ( - s) is uniformly continuous for " < 1 - a for s and t in the semistrip 0 ~ t ~ T, O~s<(X). THEOREM 1.1. If the operator A (t) has a constant domain, is strongly continuously differentiable on 9'(A) and satisfies condition (1.13), then the operator A aCt) (0 ~ a < 1) is strongly continuously differentiable on 9'(AII(to» for each fl > a and to E [0, T], and the following formula holds:

PROOF.

We again take a1 so that a

.:1Aa(t) .:1t A

-"1 (t)z

-

sina'll"

r'"

-'/1"-)0

< a1 < fl. Using

(1.18), we find that

s"RA(t) ( - s)A' (t)A -a1(t) RA(t) ( - s)zds =

184

II.

FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

sinar =r

J:'" a[ 8

0

RA(IHt)(- S) - RA(t)(- S) ]

M xA -l(t) A l-a,(t) RA(t) ( -

(1.23)

~t

s)zds

A-I(t) -A'(t)A-I(t)] r Jo('" SaRA(I)(_S)[~A ~t

+ sinar

XAI-~(t)RA(I)(- s)zds

= Jlz+ J~.

Let us estimate the integral Jlz. As a consequence of (1.19) the operator (M/ ~t)A -let) is bounded uniformly in t and At. From this same inequality and (1.13) we get

I RA(IHt) ( -

s) - RA(t) ( - s)

I

~

(c/(l + s» IAtl·

Then, using (1.16), we find

I Jizil ~cd~tl

J:'" (l+s;)l+Q1dsllzll =c2111~tl,

i.e. as ~t-+O the operator J I tends uniformly to zero in the operator norm. In the integral J 2z the operators

~A ~t

A -let) _ A' (t)A -let) =

~ ('HI [A' (T) ~t

Jt

- A' (t)]A -l(t)dt

tend to zero strongly and uniformly in t, because of the continuIty of the functions A'(T)A -let) as a function of the two variables t and T. Further, in view of Remark 1.3, for a < a2 < al the function (1 + s) a2A l-al(t) RA(t) ( - s) is uniformly continuous in the strip 0 ~ t ~ T, o ~ s < 00,80 that the set of its values is compact in E. It follows from this that

II (1 + s) ~~ A -let) "2 [

may be made less than any in t, so that

E

A' (t)A -l(t)] A I-a l (t) RA(t)( - s)z

> 0 for a sufficiently small

~t,

I

uniformly

185

§1. UNBOUNDED OPERATORS

Thus the right side in (1.23) tends to zero uniformly in t, which means that (1.24) lim ~~ ..A -"1 (t) Z = sinar 6t-oO

~

1r'

Jor" s"RA(t) ( ~ s)A'(t)A -"1 (t) RA(t) ( -

s)zds,

the limit being uniform relative to t. Suppose that x E !?I(A"(to». Then x = A -"(to)u. In view of Lemma 1.7 the elements Z(T) = A"l(T)A -"(to)u, are defined, so that, in view of Lemma 1.8, they form a compact set for 0 ~ T ~ T. Hence ~Aa

~A"

lim-t-A -a1 (t)Z(T) = lim- A -a1 (t)Aa 1(T)A -P(to)u

M-+O ~

61-+0 ~t

exists uniformly in t and (Aa(t)x)'

T.

In particular, for

= lim ~Aa x = sinar M-+O

~t

1r'

r" saRA(t) ( -

Jo

T

=t

s)A'(t)RA(t)( - s)xds,

the limit being uniform for t E [0, T]. Since the functions ~A"x

---;;t = (1/ ~t) [A aCt + ~t) - A a(t) ]A -"(to) u

are continuous (Lemma 1.8) in t, their uniform limit (A
~ A"(t) A -a(s)

I

I

~ A (t)A -l(s)

I ".

Further, suppose that the operator A (t) satisfies the condition of Lemma 1.5 and that IIA'(t)A -l(T) I ~ C1. The operator A'(t) will be a symmetric operator on !?I(A), so that for y E !?I(A)

I (x, A -l(T)A'(t)y) I = I (A -l(T)x, A' (t)y) I =1(A'(t)A- 1(T)x,y)1 ~c11IxIIIIYII·

186

II.

FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

Hence IIA-1(T)A'(t)yll ~clilyll,

which means that the operator A -1(T)A' (t) admits closure to an operator defined throughout the space, T = A -l(T)A'(t). Also

M = I Til

(1.26)

~ IIA'(t)A -l(T) II.

Now we apply Theorem 7.1 of Chapter I to T. putting HI = H, A = A(s) and B = A(T). Let us verify the conditions of the theorem. The operator T is bounded. Further, for x E g&(A)

(1.27)

IIBTxl1 =IIA'(t)xll

~IIA'(t)A-1(S)IIIIA(s)xll

=M11IA(s)xll,

i.e. condition (7.5) is satisfied. In view of the assertion of Theorem 7.1

IIB"Txll = IIA -(1-a)(T)A'(t)xll

~ Ml-aMI'IIAa(s)xll.

Making the substitution Aa(s)x = z, we arrive at the inequality

IIA -(1-a)(T)A'(t)A -a(s)zll

~

Ml-aMl'llzll,

valid for z E g&(A I-a). Hence it follows that the operator A -U-a)(T)A'(t)A -"(s)

admits a bounded closure. Using inequalities (1.26} and (1.27), we arrive at the following assertion. LEMMA 1.9. If the operator A (t) satisfies the hypotheses of Lemma 1.5 and is a selfadjoint positive definite operator, then the operator A -(l-"')(T) xA'(t)A -"'(s) (0 ~ 'Y ~ 1) admits a bounded closure, and

(1.28) I A -(I-"')(T)A' (t)A -'Y(s) I

~

I A' (t)A -l(T) 1 1-"'11 A' (t)A -l(S) II"'.

Now we shall show that the operator given by formula (1.22) is defined on g&(Aa). To this end we consider the operator

Q = RA(t)( - s)A'(t)A -a(t) RA(t) ( - s). For x E g&(A) and y E H we have

I (Qx,y) I = I (A -(1-.,.) (t)A' (t)A -.,.(t) A 'I'-a(t) RA(t) ( - s)x, A l-"'(t)RA(t) ( ~ IIA'(t)A -let) II II A.,.-a(t) RA(t) ( - s)xIIIIA 1-"'(t)RA(t)( - s)yll· Putting

'Y

s) y) I

= (1 + a) /2, we find that

I (Qx, y) I ~ I A' (t) A -let) IIII A (1-a)/2(t) RA(t) ( -

s) x IIII A (l-a)/2(t) RA(t) ( - s) y II·

187

§1. UNBOUNDED OPERATORS

Then

!o'"

sal (QX,y) Ids

!o'" I [!o" s"ll

~ I A' (t)A -let) I

~ I A' (t)A -let) X

sail A (1-a)/2(t) RA(I) ( -

[!o s"ll

s)xlill A (l-a)/2(t) RA(t)( -

A (1-a)/2(t) RA(t) ( - s)xl12ds ]

A (l-a)/2(t) RA(I) ( - s) yl12ds ]

s) yll ds

1/2

1/2.

In order to calculate the integrals just obtained on the right we use the spectral resolution of the operator A (t). We have

l '" o

=

sa~ A (l-a)/2(t) RA(I) ( - s)xl12ds =

J:'" i'" >.+s

( >.l-a )2·d(EAx, x)ds

sa

0

1

r'" >.1-"Jor" (>.+s) Sa 2dsd(EAx, X) = r" ( 0'2 2dull x112. Jo 1+0')

Jl

Thus, for xE ~(A)

I ([A "(t)]' A -aCt) x,y) I

I!o"

s"(Qx,y)ds

I~ I

A'(t)A -l(t) I

!a '" (10'::)211 xliii yll·

Hence

I [Aa(t)],A-a(t)xll

(1.29)

~IIA'(t)A-l(t)IICaIIXIl·

It follows from (1.25) and (1.29) that (1.30)

I [A"(t)j'A-a(r)xll

~

CaIIA'(t)A-l(t)IIIIA(t)A-l(r)llallxll·

Now suppose that z E ~(Aa). Then z = A -a(T)x. Approximate the element x by a sequence Xn E ~(A). It follows from relations (1.25) and (1.30) that the sequence of functions Aa(t)A -"(r)Xn and their derivatives converge uniformly on [0, T], so that the function Aa(t)A -a(r)X = Aa(t)z is continuously differentiable and (A"(t)z)' .

= sin a'll" 'lI'

r" s"RA(I) ( -

Jo

s)A'(t)RA(I)( - s)zds.

We have arrived at the following assertion.

188

II.

FIRST-oRDER EQUATIONS WITH VARIABLE OPERATOR

THEOREM 1.2. Suppose that the positive definite selfadjoint operator A (t) has a constant domain and is 8trongly continuously differentiable on it. Then the operator A a(t) has a constant domain and is strongly continuously differentiable on it. The derivative [Aa(t)]' is given by (1.22).

§2. Equations with bounded operators 1. Evolution operator. Consider the differential equation (2.1)

dx/dt = A(t)x

(0 ~ t ~ T)

with a bounded linear operator A (t) strongly continuous in t. Since A(t) is strongly continuous it is uniformly bounded on [0, T]. Hence for equation (2.1) the Cauchy problem is solvable for any initial condition x(O) = Xo E E. This solution may be constructed by the method of successive approximations, applied to the Volterra integral equation x(t)

= xo+ Ia'A(T)X(T)dT.

The proof of uniqueness of the solution is obtained in the usual way. For equations with variable coefficients an essential role is played by the value of t for which the initial data are given. In connection with this it is usual to consider the generalized Cauchy problem with the condition x(s) = xo,

(2.2)

where s is some number on the segment [0, T]. The value of the solution x(t,s) of the problem (2.1)-(2.2) at the moment t is denoted in a natural way as follows: x(t,s)

=

U(t,s)xo

(s~t~T).

The operator U(t, s) is called the evolution operator. It is a bounded linear operator. It may be constructed by the method of successive approximations, starting from the Volterra integral equation (2.3)

U(t,s)

=

1+ f.t A(T) U(T,s)dT.

The solution of this equation is strongly differentiable in t and will be a solution of the Cauchy problem (2.4)

a U(t,s)/at

= A(t) U(t,s),

U(s,s)

in the space of bounded operators acting in E.

=I

§2. BOUNDED OPERATORS

189

If we consider the solution of the integral equation (2.5)

Vet,s)

=I

+

it

V(t,T)A(T)dT,

it will be a solution of the Cauchy problem iJV/iJs = - V(t,s)A(s),

Vet, t) = 1.

We have (iJ/iJS)[V(t,S)U(S,T)] = - V(t,S)A(S)U(S,T)

+ V(t,s)A(S)U(S,T) =0,

i.e. the operator Vet, s) U(s, T) does not depend on s. Hence, putting = T, we obtain

S

(2.6)

Vet, T) = U(t, T).

Replacing Vet, s) throughout by U(t, s), we arrive at the relations U(t, s) U(s, T) = U(t, T)

and iJU(t,s)/iJs = - U(t,s)A(s),

U(t,t) = I.

Equation (2.6) proves at the same time the uniqueness of the evolution operator U(t,s). It is obvious that in the case of a constant operator A (t) == A U(t, s) =

e(t-B)A.

We note that in the place of the condition of strong continuity of A (t) we could consider a less restrictive condition. The integral equation (2.3) has a solution U(t, s) if for example the operator A (t) is strongly measurable and its norm uniformly bounded on [0, T] (or bounded by a function summable on [0, T]). It then follows from (2.3) first that U(t, s) is uniformly bounded in t and s for O;;;! s ;;;! t ;;;! T. and then that it is continuous in t in the operator norm. If now we suppose that the operator is weakly continuous, then the integrand in (2.23) will also be weakly continuous, so that the operator U(t, s) will be weakly differentiable in t and will satisfy equation (2.4) in the weak sense. One can consider equation (2.5) in an analogous manner and show that its solution coincides with the solution of equation (2.3), if the Operator A (t) is weakly continuous. 2. Comparison of evolution operators. Suppose that in addition to equation (2.1) we are given the further equation

190

II.

FIRST-QRDER EQUATIONS WITH VARIABLE OPERATOR

dy/dt

= A(t)y

(0 ~ t ~ 1),

where A(t) is also strongly continuous in E. Denote by U(t,s) the corresponding evolution operator. We compute (a/Or)[U(t, r) U(r,s)]

= - U(t, r)[A(r) - A(r) ]U(r,s).

Integrating from s to t, we obtain (2.7)

U(t,s)

= U(t,s)

+ f.tU(t,T)

= U(t,s)

+ f.t U(t,T)[A(r) -

[A(r) - A(r)]U(r,s)dr.

Analogously (2.8)

U(t,s)

A(r)]U(r,s)dr.

From these identities there results an estimate for the difference of the evolution operators: (2.9)

I U(t,s)

- U(t,s)

I

~

MAl

sup O~T~T

IIA(r) -

A(r) I T,

where

M =

sup O~.~t~

T

I U(t,s) I

and

AI =

sup O~.~t~

T

I U(t,s) II·

It follows from (2.9) that if the operators A,,(t) converge uniformly on [0, T] then the corresponding evolution operators U,,(t,s) converge uniformly in the triangle 0 ~ s ~ t ~ T. In what follows, however, the condition of uniform convergence of the operators A,,(t) is often not satisfied, and the following conditional theorem is useful. THEOREM 2.1. Suppose that A,,(t) is a sequence of strongly continuous bounded linear operators on [0, T] and that U,,(t, s) is the sequence of evolution operators corresponding to them. Suppose that B(t) is a strongly continuous operator on [0, T] and that U,,(t, s) is the evolution operator corresponding to the operators A,,(t) = A,,(t) + B(t). If the sequence of operators U,,(t, s) is uniformly bounded, then the sequence U,,(t,s) is uniformly bounded as well. If the sequence U,,(t, s) converges strongly and uniformly relative to t and s in the triangle 0 ~ s ~ t ~ T, then the sequence all(t, s) also converges strongly and uniformly with respect to t and s in this triangle. PROOF.

have

For each n

= 1,2, .•• we may write the identity (2.8). We

62.

U.. (t,s) =

(2.10)

191

BOUNDED OPERATORS

U .. (t,s)

+

it

U .. (t, T)B(T) lJ.. (T,s)dT.

The operator B(T) is uniformly bounded in T, then

I B(T) I

~

C, and if

I U.. (t, T) n ~ M,

I U.. (t,s) I ~ M + MC itll U.. (T,S) IldT, so that

I U.. (t, s)11

~ Me McT•

The first assertion of the theorem is proved. For the proof of the second we consider the space G(E) consisting of all convergent sequences = (x.. } of elements of the space E, with the norm

x

II!II

sup

=

1~ ..

< ...

Ilx.. ll.

The operators U .. (t,s) generate in a natural wayan operator (fn(t,S) in G(E) according to the formula U(t,s)!

=

(Un(t,s)xn}.

In view of the strong convergence of the U,,(t, s) the sequence {U.. (t, s)x.. I E G(E). The boundedness of the operator fl(t,s) follows from the uniform boundedness of the operators Un(t,s). The operator OCt, s) is strongly continuous for 0 ~ s ~ t ~ T. Indeed, fix i E G(E) and put Xo = lim......... xn. Since the functions U.. (t,s)xo converge strongly to U(t,s)Xo, they form an equicontinuous family, i.e. for each E> 0, for' Ihl. Ikl < 6(t}

I U .. (t + h, s + k)Xo -

Un(t, s) Xo I

< E/3 (n = 1,2, ••. ). For sufficiently large n we have I x" - XoII < E/3M, so that from (2.11) (2.11)

for n

~

N we obtain

I U .. (t + h, s + k)x" -

U .. (t, s) x.. I x

< E. For n ~ N inequality (2.12) is valid for Ihi, Ik I < 61 ~ 6, because

(2.12)

of the strong continuity of each of the operators U .. (t,s). Thus, for

Ihl, Ikl < 61 IIO(t+h,s+k)!- O(t,s)!11

<E.

The operator B(T) is defined on the space G(E) in a natural way:

192

II.

FIRST-ORDER EQUATION.S WITH VARIABLE OPERATOR

E(T),t = {B(T)X,,}. It is obviously strongly continuous. In the space G(E) we consider the integral equation H(t,s) = U(t,s)

+

it

U(t,T)B(T)H(T,S)dT.

By the method of successive approximations it is possible to find a solution of this equation, continuous in the triangle 0 ~ s ~ t ~ T. Since the operators and 11 operate on each coordinate ,t independently, the operator II will have the same property, i.e. lI(t,s),t = (H"(t,s)x,,t. Obviously the operators H,,(t,s) will satisfy the integral equations (2.10), and in view of the uniqueness of the solutions of these equations, H,,(t,s) = a,,(t,s). Thus lI(t,s),t = (U,,(t,s)x,,} E G(E) for ,t E G(E). Suppose that x is any element of E. We choose ,t = {x Then from the continuity of the function II (t, s).t it follows that the functions H,,(t, s) x are equicontinuous in the triangle 0 ~ s ~ t ~ T. Further, since H(t,s)Xlies in the space G(E), the sequence H,,(t,s)x converges for each t and s. From these two facts it follows that the sequence H,,(t, s) converges strongly and uniformly for t and s in the triangle 0 ~ s ~ t ~ T. The theorem is proved. REMARK 2.1. The assertions of the theorem remain valid if the operator A,,(t) has the form A,,(t) + B,,(t), where B,,(t) ,is a sequence of strongly continuous operators converging strongly and uniformly on [0, T] to an operator B(t). REMARK 2.2. Under the conditions of Theorem 2.1 and Remark 2.1 it is possible to replace strong 'continuity of the operators A,,(t) and B,,(t) by weak continuity.

a

t.

§3. The uniformly corred Cauchy problem 1. The Cauchy problem. The evolution operator. We tum to the consideration of the differential equation (3.1)

dx/dt = A(t)x

(0 ~

t ~ T)

with an unbounded operator A(t). We suppose that for all tE [0, T] the operators A (t) have a common domain ~(A (t» = ~(A) which is dense in E and closed. DEFINITION 3.1. A solution of equation (3.1) on the segment [2, T] (0 ~ s < T) is a function x(t) with values in ~(A), having a strong derivative x' (t) and satisfying (3.1) on the segment [8, T]. By a Cauchy problem in the triangle T,,: 0 ~ s ~ t ~ T we mean the

fa.

UNIFORMLY CORRECT CAUCHY PROBLEM

193

problem of finding for each fixed s E [0, T] a solution x(t,s) of (3.1) on the segment [s, T], satisfying a given initial condition xes,s)

(3.2)

=

XoEg(A).

DEFINITION 3.2. The Cauchy problem (3.1)-(3.2) is said to be uniformly correct if 1) For each s E [0, T] and any Xo E g(A) there exists a unique solution x(t,s) of (3.1) on the segment [s, T] satisfying condition (3.2). 2) The function x(t, s) and its derivative x; (t, s) are continuous for s, t in the triangle Tb,. 3) The solution depends continuously on the initial data in the sense that if the xO,n E g(A) converge to zero then the corresponding solutions xn(t,s) converge to zero uniformly relative to t and s in T".. If the Cauchy problem is uniformly correct, then it is possible to introduce a linear operator U(t, s) (0 ~ s ~ t ~ T) which assigns to each element Xo E g(A) the value at the point t of the solution of the problem (3.1)-(3.2) on the segment [s, T]: x(t,s) = U(t,s)Xo. The operator U(t,s) is defined on ~(A). For fixed t and s it is bounded and therefore admits a continuous extension to the entire space E, which we shall denote also by U(t,s). The operators U(t,s) are uniformly bounded. In the contrary case there would exist a sequence Xo,n E g(A), converging to zero, and a sequence of pairs of numbers Sn, tn such that U(tm sn) Xo,n -+ 00 , which contradicts the uniform correctness of the problem. The operators U(t, s) map g(A) into itself. The following compositional property holds for them:

(3.3)

Indeed. if

U(t, T) U(r, s)

=

U(t, s)

xgE~(A)

the element U(T,s)XoE,q(A). On the segment [T, T] the functions U(t,s)xo and U(t, T) U(T,S)Xo are solutionsofequation (3.1) with the same initial values U(T,S)Xo. and therefore they coincide. It follows from the boundedness of the operators on both sides of (3.3) and the fact that their values coincide on the dense set g(A) that they are identical. The operator U(t, s) is strongly continuous in t and s in Tb,. as follows from the requirement 2) of uniform correctness and the uniform boundedness of U(t, s) in t and s. For x E .9:i(A) the function U(t, s) is differentiable in t for each s and (3.4)

aU(t,s)x/at = A(t) U(t,s)x.

194

II.

FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

In view of 2) the operator A (t) U(t, s) is strongly continuous in t and s on g-(A).

We shall investigate the properties of the function U(t, s) x for fixed t. In view of (3.3), for as > we have

°

~ as

[U(t,s

+ as) -

U(t,s)]x = _ U(t,s

+ as) U(s + as,s) as

I x.

°

For x E g-(A) and as ~ the limit on the right side exists and equals - U(t,s)A(s)x, from (3.4). Therefore the left side has a limit, i.e. the function U(t, s) x for t> s has a right derivative o+U(t,s)xjos = - U(t,s)A(s)x.

If we suppose that the operator A (s) is strongly continuous on g-(A) for s E [0, T], then the right derivative of the function U(t,s)x will be continuous on the semi-interval ~ s < t, so that U(t, s) x will be differentiable in s on [0, t). At the point s = t it will have a left derivative - A (t) x. In order to verify this)- we consider the functions U(T,S) for T> t. These functions are differentiable in s already on the interval [O,t]. As T~t they tend to the function U(t,s)x uniformly in s on [O,t], and their derivatives - U(T,s)A(s)x to the function - U(t,s)A(s)x. Here the uniformity of the convergence follows from the compactness of the set (A(s)x} for sE [O,t]. 'Thus for those s

°

(3.5)

oU(t,s)xjos

= - U(t,s)A(s)x.

In what follows the operator U(t, s) for a uniformly correct Cauchy problem is said to be an evolution operator, or evolutionary. If along with equation (3.1) one considers the equation (3.6)

dyjdt= [A(t) - >.oI]y(t),

then solutions of the Cauchy problem with one and the same initial condition are connected by the relation (3.7)

x(t, s) =e Ao (t-8)y(t, s).

Therefore if the Cauchy problem for one of equations (3.1), (3.6) is uniformly correct, it is uniformly correct for the other and the corresponding evolution operators U(t, s) and UAij(t, s) are connected by the equation (3.8)

U(t,s) = eAij(t-B) UAij(t, s).

§3. UNIFORMLY CORRECT CAUCHY PROBLEM

195

If all the operators A (t) have a common regular point and their resolvents at this point are uniformly bounded in t E [0, T], then after the substitution (3.7) we arrive at an equation in which the operators have inverses which are uniformly bounded relative to t E [0, T]. In what follows we frequently suppose at once that such a substitution has already been made. Thus we suppose that the operator A (t) (0 ~ t ~ T) is strongly continuous on ~(A) and has a bounded inverse such that condition (1.1) is satisfied. Supposing that the Cauchy problem is uniformly correct, we shall write out the properties of the evolution operator. Properties of the evolution operator of a uniformly correct Cauchy problem. 10. The operator U(t, s) is bounded in E uniformly relative to t and s:

II U(t,s) II

(3.9)

~

(0 ~ s ~ t ~ T).

M

2°. The operator U(t,s) is strongly continuous in the triangle T I1:

o ~s ~ t ~ T. 3°. The following identity holds: (3.10)

U(t,s) = U(t,T) UT,S), U(t,t) = I

4°. The operator U(t,s) maps the region (3.11)

Vet,s)

=

(O~S~T~t~T). ~(A)

into itself. The operator

A(t) U(t,s)A -1(S)

is bounded and strongly continuous in the triangle T I1• 5°. On the region ~(A) the operator U(t, s) is strongly differentiable relative to t and s, while (3.12)

iJU(t,s)/iJt = A(t) U(t,s) and iJU(t,s)/iJs = - U(t,s)A(s).

2. Nonhomogeneous equations. Consider the equation (3.13)

dx/dt

=

A(t)x + f(t) ,

where f(t) is a function continuous on [0, T]. The definition of a solution of this equation is quite analogous to Definition 3.1. Between the solutions of equations (3.1) and (3.13) there is a connection which we have already noted for equations with constant coefficients. THEOREM

3.1. If the Cauchy problem for the homogeneous equation

(3.1) is uniformly correct and the operator A (t) is strongly continuous on ~(A) for 0 ~ t ~ T, then every solution of equation (3.13) is represent-

196

II.

FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

able in the form (3.14) PROOF.

x(t,s)

=

U(t, s)x(s, s)

+ f.t U(t,,,)f(,,)d,,.

We apply the bounded operator U(t,,,) to both sides of the

identity dX(",s)/d"

= A (,,)x('"

s)

+ 1(,,).

Then, using (3.5), we obtain (iJ/iJT) (U(t, T)X(T,S»

=

U(t, T) f(T).

Integrating with respect to T from s to t, we obtain (3.14). The theorem is proved. For the proof of the existence of solutions of equation (3.13) we need to impose additional conditions on the operator A (t) and the function f(t). It is natural here to seek the solution in the form (3.14). Since the first term on the right in (3.14) is the solution of the homogeneous equation, it suffices to prove that the integral term is the solution of equation (3.13). THEoREM 3.2. If the Cauchy problem for equation (3.1) is uniformly correct and such that the evolution operator has properties 1°-50 , and the function f(t) is such that the function A (t) f(t) is defined and continuous, then the formula

(3.15) yields a solution of the nonhomogeneous equation (3.13).

It follows from property 4° of the evolution operator that the function PROOF.

is defined and continuous. For At> 0 we calculate 1 1 f,t+.1t -[y(t+At,s) -y(t,s)]=[U(t+At,T) - U(t,T)]t(T)dT At

At

+ U(t

+ At,t) At

,

I y(t,s)

+~ At

ft+.1t U(t, T)f(T)dT.

J,

§3. UNIFORMLY CORRECT CAUCHY PROBLEM

197

The first integral tends to zero because the operator U(t + J1t, T) - U(t,T) tends to zero strongly and uniforml~ on the compact ·set of values of the continuous function I(T). The second integral tends to A (t) yet, s) because yet, s) lies in the set ~(A) and because of property 5°. The last integral obviously tends to I(t). Thus the right derivative of the function yet, s) with respect to t exists: d+y/dt = A(t) yet,s)

+ I(t).

From what has already been said this derivative is continuous, which means that it coincides with the left derivative. The theorem is proved. As we already said in §6 of Chapter I, the condition that the values of I(t) lie in the domain of the operator A (t) is frequently unnatural, so that the following theorem is of interest. THEoREM 3.3. II the operator A (t) satisfies the conditions of the preceding theorem and if moreover it is continuously differentiable on ~(A), and the function f(t) has a continuous derivative, then formula (3.15) yields a solution of equation (3.13). PROOF. We transform formula (3.15), using the second of properties 5°. We have

yet,s) =

!.' •

U(t, T)A (T)A-l(T) I(T)dT = _f.'OU(t,T) A-l(T)f(T)dT • OT

= U(t, s)A -l(S) I(s) - A -let) I(t)

+ f.' U(t, T) :T [A -leT) I(T) ]dT.

The function f/>(T) = (d/dT) [A -leT) I(T)] has the property that the function A(T)f/>(T) = !'(T) - A'(T)A -leT) is continuous. Therefore the last integral z(t,s) has a derivative equal to A(t)z(t,s) + f/>(t). The terms outside the integral are also differentiable. One shows by a direct verification that yet, s) satisfies equation (3.13). The theorem is proved. 3. Perturbed equations. We shall suppose that in the equation (3.16)

dx/dt

= A(t)x + B(t)x

the operators B(t) and A (t)B(t)A -let) are bounded and strongly continuous for t E [0, T]. Suppose that Xo E ~(A). Write A(s)Xo = Yo and consider the integral equation

198

II.

FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

y(t,S) = V(t,S) Yo

+

I.t

V(t, T)A(T)B(T)A -l(T) Y(T,s)dT

-

with strongly continuous kernel. Introducing the function x(t, s) =A- 1 (t)y(t,s), we get x(t,s) = U(t,s)Xo+

I.t

U(t, T)A- 1 (T)A(T)B(T)A- 1(T) y(T,s)dT.

It follows from Theorem 3.2 that dx/dt = A (t)x(t,s)

+ B(t)A -l(t) y(t,s) = A(t)x + B(t)x.

Thus the solution of the Cauchy problem for equation (3.16) with the condition x(O) = Xo E ~(A) exists. It is unique, since it follows from Theorem 3.1 that every solution of the Cauchy problem for equation (3.16) satisfies the integral equation x(t,s) = U(t,s)xo

+

I.t

U(t, T)B(T)X(T,S)dT,

which has a unique continuous solution. It is also clear from this integral equation that the solutions x(t, s) depend continuously on the initial data. Finally, from the joint continuity of the function y(t,s) in the triangle TI!.o it follows that the functions x(t,s) = A -l(t) y(t,s) and x;(t,s)=y(t,s)+B(t)A- 1 (t)y(t,s) are continuous in T A • We have arrived at the following assertion. 3.4. If the Cauchy problem for equation (3.1) is uniformly correct, condition (1.1) is satisfied, and the operators B(t) and THEoREM

A(t)B(t)A -l(t) are bounded and strongly continuous in t, then the Cauchy problem for equation (3.16) is uniformly correct.

4. Stable approximations of the evolution operator. Consider the family of equations (3.17)

(n

= 1,2, ... ;

°

~

t ~ T)

with bounded strongly continuous operators. We suppose that the operators An(t) on ~(A) converge strongly and uniformly relative to t E [0, T] to an operator A (t) satisfying condition (1.1). In view of Lemma 1.2 the operators [A(t) - An(t)]A -l(t) will tend strongly to zero on the entire space E, uniformly relative to tE [0, T]:

§3. UNIFORMLY CORRECT CAUCHY PROBLEM

(3.18)

lim sup II [A (t) - A,,(t)]A -l(t)xII = 0

" ..... '" O~t::;;T

199

(xEE).

It follows from this in particular that

(3.19)

II [A (t) - A,,(t) ]A -let) II ~ M,

where M does not depend on n or t E [0, T]. We denote the evolution operators corresponding to equations (3.17) by U,,(t,s). LEMMA 3.1. Suppose that the evolution operators U,,(t,s) are uniformly bounded in n, t, and s. If condition (3.18) is satisfied and x(t, s) is any solution of the problem (3.1)-(3.2), then the following formula holds for it:

(3.20)

x(t,s)

=

limU,,(t,s}xo.

The limit exists uniformly in t and s in the triangle T A• Thus the solution of the problem (3.1)-(3.2) is unique. PROOF.

From the identity dx(t, s) Idt

=

A,,(t) x(t,s)

+ [A (t) -

A,,(t) ]x(t, s)

we get x(t,s) = U,,(t,s)xo+

f.t U,,(t,T)[A(T) -

A,,(T)]A-l(T)A(T)x(T,s)dT.

The operators [A(T) - A,,(T)]A -leT) converge strongly and uniformly in T to zero as n -+ co, on the entire space E. This means that they converge to zero on the compact set of values of the continuous function A(T)X(T,S). It follows from this that the functions [A(T) - A,,(T)]X(t,s)

converge to zero uniformly relative to 8 and T in TAO In view of the hypothesis of uniform boundedness of the operators U,,(t,s), it therefore follows that the integral terms converge to zero uniformly in t and 8. The lemma is proved. DEFINITION 3.3. If there e~ists a sequence of bounded strongly continuous operators A,,(t) for which condition (3.18) is satisfied along with the condition of uniform boundedness of the evolution operators, i.e.

200

II.

(3.21)

FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

\I U,.(t,s)

II

~

M

(M does not depend on n, t and s),

then we will stay that the operator A (t) is stably approximated by the operators A,.(t). The following proposition follows directly from Lemma 3.1. THEoREM 3.5. If the Cauchy problem is uniformly correct for the operator A (t) and it is stably approximated by the operators A,. (t), then the evolution operators U,.(t, s) converge strongly and uniformly in t and s to the evolution operator U(t, s). PROOF. It follows from Lemma 3.1 that U,.(t,s)x converges to U(t,s)x uniformly in t and s for x E ~(A). In view of the uniform boundedness of the operators U,.(t, s) this will also hold for any x E E. The theorem is proved. This theorem admits the following incomplete converse. THEoREM 3.6. Suppose that the operator A (t) is strongly continuously differentiable on ~ (A), has a bounded inverse and is stably approximated by operators A,.(t) which for each tE [0, T] commute with A(t) on ~(A). If the evolution operators U,.(t,s) converge as n---* 00 strongly and uniformly in t and s to an operator U(t, s), then the Cauchy problem lor equation (3.1) is uniformly correct and U(t,8) is the evolution operator corresponding to it. PROOF.

Consider the equations

(3.22)

It follows from the continuous differentiability of A(t) on ~(A) that the operator A'(t)A-1(t) is strongly continuous (Lemma 1.5). Denote by V,.(t, s) the evolution operators corresponding to equations (3.22). According to Theorem 2.1 the operators V,.(t,s) converge strongly and uniformly relative to t and 8 to a limit which we shall denote by V(t, s). The operator V(t, s) is strongly continuous for t and s in T",. We make the substitution A -l(t) y(t,s) = x(t,s) in (3.22). Then dyx/dt

= A -l(t)dy/dt - A -l(t) A' (t)A-1(t)y = A -1 (t)A,.(t)y = A,.(t)x.

In view of the uniqueness of the solution of equation (3.17) A -l(t) y(t,s) = x(t,s) = U,.(t,s)x(s,s),

so that y(t,s) = A(t) U,.(t,s)x(s,s)

= A(t) U,.(t,s)A -1(S) y(s,s)

§3. UNIFORMLY CORRECT CAUCHY PROBLEM

201

or, in other words, Vit,s)

= A(t) U,,(t,s)A -1(S).

Since the U,,(t,s) converge to U(t,s) and the V,,(t,s) to V(t,s), and since A (t) is closed, we have V(t, s) = A (t) U(t, s) A -1(S).

Suppose that Xo e g-(A) and A (s) Xo = y .. It follows from Lemma 3.1 that equation (3.1) can have only the unique solution defined by formula (3.20). We shall show that this formula indeed determines a solution of the problem (3.1)-(3.2). The function U,,(t,s)Xo satisfies the equation dU,,(t,s)Xo/dt

= A,,(t) U,,(t,s)xo = A,,(t) A -1(t) V,,(t,s)y•.

In view of (3.18) the operators A,,(t) A -1(t) tend strongly, and uniformly in t on Is, TJ, to the identity operator, so that the derivatives dU,,(t, s)Xo/dt converge uniformly in t to the function V(t, s) y. = A(t) U(t,s)xo. Since the differentiation operator is closed we have dU(t,s) Xo/dt

= A(t) U(t,s)Xo,

i.e. the function x(t, s) = U(t, s) Xo is the solution of equation (3.1) on [so T]. Obviously x(s,s) = U(s,s)xo = lim"..... ~ U,,(s.s) Xo = Xo, i.e. x(t, s) is the solution of the Cauchy problem. Properties 2) and 3) of Definition 3.2 follow from the boundedness and strong continuity of the operators U(t, s) and V(t, s). The theorem is proved. REMARK 3.1. The function V(t, s) is connected with the function U(t, s) by the integral relation V(t,s) = U(t,s)

+ f.t U(t,r)A'(r)A -1(r) V(r,s)dr.

Indeed, solving equation (3.22), considering in the process A' (t) A -1(t) y(t) as a known function, we obtain

or

202

II.

FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

Hence the required relation is obtained by passing to the limit. 5. Perturbed equations and improvement of the smoothness of the solution. We can approach the problem of the correctness of the Cauchy problem for equation (3.16) from another point of view. If the operator A (t) is stably approximated by the bounded operators An(t), then the operator A(t) + B(t), where B(t) is a bounded strongly continuous operator, is stably approximated by the operators An(t) + B(t). This again follows from Theorem 2.1 applied to the equations dx/dt = An(t) x

(3.23)

+ B(t)x.

Further, if Un(t, s) converges to U(t, s) strongly and uniformly, then the evolution operators Un(t, s) corresponding to equations (3.23) converge strongly and uniformly to U(t,s). Thus, we may directly apply Theorem 3.6 to equation (3.16), considering the sum A(t) + B(t) as one operator. In this way we arrive at the following assertion. 3.7. If the operator A (t) satisfies the conditions of Theorem 3.6, and the operator B(t) is bounded, strongly continuous for t E [0, T] and strongly differentiable on g-(A), then the Cauchy problem for equation (3.16) is uniformly correct. THEoREM

U sing Theorems 3.4 and 3.7 on the perturbed equation, one can obtain theorems on the existence of solutions of equation (3.1) with improved smoothness. THEOREM 3.8. Suppose that A (t) satisfies the conditions of Theorem 3.6. Suppose moreover that one of the two following conditions is satisfied: 10 • The operator A (t) A I (t) A -2(t) is defined, bounded and strongly continuous for t E [0, T]. 2 0 • The operator A (t) is twice continuously differentiable on 9' (A). Then for Xo E g-(A 2(s» 3) the solution of the problem (3.1)-(3.2) is twice continuously differentiable.

PROOF.

(3.24)

We apply Theorems 3.4 or 3.7 respectively to the equatior dy/dt = A(t) y

+ A' (t)A -l(t) y.

It follows from them that under the hypotheses 10 or 2 0 the Cauchy 3) We note that the squares of the operators A(s) can already have different domains for different s.

§3. UNIFORMLY CORRECT CAUCHY PROBLEM

203

problem for this equation is uniformly correct. Suppose that Xo E ,q(A2(S» for some sE[O,T]. Write Yo=A(s)xo. Obviously YoE9"(A). Construct the solution yet, s) of equation (3.24) with the initial value yes, s) = Yo, and write x(t, s) = A -let) y(t). Just as in the proof of Theorem 3.6, one verifies that x(t, s) is a solution of the problem (3.1)-(3.2). Further, dx(t,s)/dt = A (t)x(t,s)

= yet,s),

and since yet, s) is continuously differentiable in t, the function x(t, s) is twice continuously differentiable in t. We present one further assertion on the improvement of the smoothness of a solution, which we shall use in the following chapter. 3.9. Suppose that the operator A (t) is the same as in Theorem 3.8, with condition 10. Suppose that the operators B (t), A (t)B(t)A -let) and A 2(t)B(t)A -2(t) are defined, bounded and strongly continuous for t E [0, T]. Then the solution x(t, s) of equation (3.16) with the initial oolue Xo E 9" (A 2(S» has the property that the function A (t) x(t, s) is continuously differentiable. THEoREM

PROOF.

Set up the equation dy/dt

= A(t) y + A (t)B(t)A -let) y + A'(t)A -let) y.

From our hypotheses and Theorem 3.4 it follows that this equation has a solution which is continuously differentiable on [0, T] under the initial condition y (0) = Yo E 9"(A). Putting Yo = A (s) Xo for XoE 9" (A2(S), we find that yet) = A (t)x(t,s), so that the assertion of the theorem follows. The theorem is proved. 6. Equations with a generating operator for the contraction semigroup. In this subsection we indicate a class of operators for which it is possible to verify the conditions of Theorem 3.6. The following uniqueness theorem holds. THEOREM 3.10. If for the equation (3.1) the resoluent of the operator A (t) satisfies the condition

(3.25)

II RA(t) (>..) I ;;;; 1/>.. for >.. > 0,

then the solution of the Cauchy problem is unique and satisfies the following inequality:

(3.26)

I x(t, s) II

~

II xes, s) I

(O~s~t~T).

204

II.

FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

PROOF. It is obvious that the first assertion of the theorem follows from the second. In order to deduce (3.26) we note that by the definition of the solution

(x(t

+ f,S) -

x(t,s»/E~A(t)x(t,s) as

E ~o,

so that x(t

+ E,S) = (1 + EA (t»x(t, s) + 0(E)

= =

(1 - E2A 2(t» (1 - ~A (t» -1X(t, s)

+ 0 (E)

(1- EA(t»-1X(t,S) - E2A(t) (1- fA(t»-1A(t)x(t,s)

We will take

E

> O. Then in view of (3.25) I (1 - EA (t» -111 = (1/ E) I RA(I) (1M I

+ 0(E).

~ 1.

Hence

Ilx(t + E,S) I

~ ~x(t,sH

+ EIIA(t)R A(I)(1/E)A(t)x(t,s) I + O(E).

Further, the operators A(t)RA(I)(1/E) are bounded uniformly in E, tend to zero as E ~ 0 on the elements of ~ (A), and accordingly converge strongly to zero (for details see §2 of Chapter I). Thus (x(t

+ E,S) I -II x(t,s) II) IE ~ I A (t) R A(I)(1/E)A(t) x(t, s) I + O(E) /E~ o.

From this relation it follows that the upper right derivative of the function ,p(t) = I x(t, s) I is nonpositive. The function ,p(x) is continuous, . so that it is nonincreasing. Inequality (3.26) and the theorem are proved. REMARK 3.1. As is clear from the proof, we used only the fact that the right derivative of the function x(t) is equal to A(t)x(t). REMARK 3.2. If the operator A (t) satisfies the condition

I RA(I)(>') I then substitution of >.0 =

w

~

1/(>. - w) for >. > w,

in (3.7) leads to the estimate

I x(t, s) I

I

~ eW(t-B) x(s, s) II.

Supposing that (3.25) is satisfied, we construct the operators (3.27)

A,,(t)

=-

nA(t)RA(t)(n).

Obviously these operators commute with A(t) on ~(A). If A(t) is strongly continuous on 9(A), then for x E ~(A)

205

§3. UNIFORMLY CORRECT CAUCHY PROBLEM

11- nRA(I)(n)x - xii = IIRA(t)(n)A(t)xll

~ (lIn)

max IIA(t)xll-O

O:;;I:;;T

uniformly for tE [0, T] The norms of the operators - nRA(t)(n) are bounded by unity, so that in view of the Banach-Steinhaus theorem the operators - nRA(t)(n) tend to the identity operator strongly and uniformly in t. Therefore it follows that the operators AII(t) tend strongly and uniformly on 9'(A) to the operator A(t), i.e. condition (3.18) is satisfied. LEMMA 3.2. The operator A (t) AII(t) of the form (3.27).

is stably approximated by the operators

PROOF. It remained to be shown that the evolution operators UII(t, s) corresponding to the operators AII(t) are uniformly bounded relative to n,.s, and t. Consider the resolvents of the operators AII(t). From the identity

RAIIW(A)

= -

n: A

1+ (n:2 A)2 RAW (n ~ A)

and condition (3.25) we find that 1

n2

1

1

I RAII(t) (A) I ~ n + A + (n + >.)2 nA/(n + >.) = A

(>.> 0).

The estimate just obtained makes it possible in our case. to apply Theorem 3.10 to equation (3.17). It follows from this theorem that

I UII(t,s) I

(3.28)

~

1.

'rhe lemma is proved. In Theorem 3.6 we supposed that the operator A (t) has a bounded inverse operator. Therefore in what follows we replace condition (3.25) by the stronger (3.29)

I R AW (>') I

~ 1/(1

+ >.)

for >. ~

o.

We turn to the proof of convergence of the evolution operators U,.(t,s) corresponding to the operators A,,(t). To this end we suppose that A (t) is strongly differentiable in t on 9' (A).

First we argue as in the proof of Theorem 3.6. We consider equations (3.22) and the evolution operators VII(t,S) = A(t) U,,(t,s)A -1(8) corresponding to them. By the first part of Theorem 2.1, applied to

206

II. FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

equations (3.22). taking account of (3.21). we conclude that the operators Vn(t, s) are uniformly bounded relative to n, t, and s:

I Vn(t,s) I

(3.30)

I A (t) Un(t,s)A -1(S) I

=

~ K.

We subdivide the segment [0, T] into N equal pieces and put for

k-l

k

~T~t
(k=1,2 •.•• ,N).

For the equations dx/dt = An,N(t)x

we may construct evolution operators Un,N(t,s), which will be piecewise differentiable. If

i-1 i j-I j - - Ts,s<-T< ••• <--Ts,t<-T N N N N' then

In view of Theorem 3.10 the estimate

I Un,N(t, s) II ~ 1

(3.31)

will hold for the operators Un,N(t, s). Now we shall estimate the difference of the operators Un(t,s) and Un,N(t,s). Using identity (2.7) for x E9(A) we get

II fJn,N(t,s)x -

~

Un(t,s)xll

Ilit

Un,N(t, r) [An,N(r) - An(r) ]Un(r,s)xdr

I

~ T max II[An,N(r) - An(r)]A-l(r) II max II A(r) U(r,s)A -1(S) I O:;;T;;;T

O:;;T:;;T

I A(s)xll·

Let us estimate the magnitude of the second factor. Suppose that (k-1)T/N~r
[A~,N(r) -

An(r)]A -1(r) " =

I[

An (k

~ 1 T) -

An(r) ] A -1(r)

I~

§3. UNIFORMLY CORRECT CAUCHY PROBLEM

(3.32)

~ II nRA«(k-lJ/N)T)(n{ A(T) -A( k; 1 T) ]

~ II[ A(T) -

A (k

~ 1 T) ]

A-1(T)

207

A-1(T)nRA(T)(n)

I

I.

Further,

Thus

II u",N(t,s)x - Un(t,s)xll ~ Kc T2/NIIA(s)xll· It follows from this estimate that as N -+ 00 the operators U,.,N(t, s) converge strongly on g?(A) to the operator Un(t,s)x uniformly in n, s, and t. Since in view of (3.28) and (3.31) these operators are uniformly bounded in all the variables, the indicated convergence will hold for all x E E. N ow we consider what happens to the operators U,.,N(t, s) when n -+ 00 and N is fixed. On each segment «k - 1)/N) T ~ s, t ~ kT/N the operators U,.,N(t,s) = exp(t - s)An«k -1)/N) are semigroup operators constructed relative to the operators An (k ~

1) = _ nA(k ~ 1)RA«k_l)/N)(n).

Since the operators A «k - 1) / N) satisfy the condition (3.29), in view of Theorem 2.9 of Chapter I the operators Un,N(t, s) converge strongly and uniformly relative to t and s to the operators Un(t,s) = UA«k-l)/N)(t - s). For any t, s E Til the operator Un,N(t,s) is the product of a finite number of operators of the form exp[TAn«k -1)/n) J, so that (3.33) strongly and uniformly for t, s E Til' Now we shall prove that the sequence of operators Un(t,s) converges. For xEE we have

II Un(t,s)x - Um(t,s)xll ~ II Un(t,s)x - Un,N(t,s)xll + II Um(t,s)x - Um,N(t,s)xll + II Un,N(t,s)x - Um,N(t,s)xll· The first two terms may be made as small as desired for all t, s, and n

208

II. FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

by choice of a large N. After this, for fixed N, one chooses n and m so large that the second and third terms will become arbitrarily small for all t and s. The possibility of such a choice follows from (3.33). Thus, the sequence Un(t, s) converges to the bounded operator U(t, s) strongly and uniformly for t and s in the triangle T A• We have proved the following important theorem. THEOREM 3.11. If A (t) satisfies conditon (3.29) and is strongly continuously differentiable on .!i?I(A) , then the conditions of Theorem 3.6 are satisfied for it and the operators An(t) constructed according to formula (3.27). The Cauchy problem for equation (3.1) is uniformly correct. REMARK 3.4. It follows from the proof of Theorem 3.11 that the evolution operator U(t, s) may be represented in the form of a multiplicative integral

(3.34)

U(t,s)

=

lim UN(t,s)

N.......

in terms of semigroup operators corresponding to the operators A (t) for various t. Here the operator UN(t,s) = UA«{k-l)/N)fT)(t - s),

if (k - 1) T / N ~ S, t ~ kT/ N, and for other t and s it is defined using the equation (0 ~ s ~ r ~ t ~ T).

In (3.34) the limit exists uniformly for t; sETA. Theorem 3.8 on the improvement of the smoothness of solutions and Theorem 3.7 on the perturbation of equations carry over naturally to the class of equations considered in Theorem 3.11. The theorems on the improvement of the smoothness of solutions may be reformulated here as follows. THEOREM 3.12. Let the operator A (t) satisfy (3.29). be strongly continuously differentiable on ~(A) and satisfy one of the two following conditions: 1°. The operator A(t)A'(t)A -2(t) is defined, bounded and strongly continuous relative to t E [0, T]. 2°. The operator A (t) is twice continuously differentiable on ~(A). Then for Xo E 9(A 2(S» the solution of the problem (3.1)-(3.2) is twice continuously differentiable.

§4. WEAKENED CAUCHY PROBLEM

209

§4. Weakened Cauchy problem 1. Weakened Cauchy problem, correct on ~(A). Suppose that in equation (3.1) the operator A (t) has a domain ~(A) independent of t, on which it is strongly continuous. There exists a bounded inverse operator A -l(t). DEFINITION 4.1. A weakened solution of equation (3.1) on the segment [s, T] (0 ~ s ~ T) is a function x(t), continuous on [s, T], continuously differentiable and satisfying equation (3.1) on (s, T]. By a weakened Cauchy problem in the triangle Tt:.: 0 ~ s ~ t ~ T we mean the problem of finding for each fixed s E [0, T) a weakened solution x(t, s) of equation (3.1) on the segment [s, T], satisfying a given initial condition (4.1)

xes,s)

= Xo.

4.2. The weakened Cauchy problem is said to be correct on ~(A) if the following conditions are satisfied. 1) For any s E [O,T) and Xo E ~(A) there exists a unique weakened solution x(t, s) of equation (3.1) on the segment [s, T], satisfying condition (4.1). 2) The function x(t, s) is continuous for s, t E T 4 • 3) The derivative x{(t, s) is jointly continuous in t and s in the half-open triangle 0 ~ s < t ~ T. 4) The solution depends continuously on the initial data in the sense that if the Xo ... E ~(A) tend to zero then the corresponding solutions x,.(t, s) tend to zero uniformly in each region DEFINITION

t-s~6>0

(O~s~T).

For a weakened Cauchy problem, correct on ~(A), one may, just as for a uniformly correct Cauchy problem, introduce an evolution operator U(t,s) (0 ~ s < t ~ T). For all t and s with t> s it will be a bounded operator. For XoE ~(A) the function x(t,s) = U(t,s)xo will be a weakened solution of the problem (3.1)-(4.1). The uniqueness of the weakened solution implies the relation U(t, T) U(T, s)

=

U(t, s)

(O~S
Further, in each region t - s ~ 6 > 0 the operator U(t, s) is uniformly bounded, and since it is strongly continuous on ~(A) in the entire triangle T 4 • in this region it will be strongly continuous on the entire space E.

210

II. FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

The operator U(t, s) maps 9'(A) into 9'(A) and is strongly continuously differentiable on 9'(A) for t> s. If Xo E 9'(A) the function U(t, s) Xo satisfies the equation

= A (t) U(t, s) Xo (t> s) and the initial condition limt...... U(t,s)Xo = Xo. (4.2)

a U(t, s) xo/ at

I t follows from the properties just enumerated that the operator U(t,s)A -1(0) is strongly continuous for t, s E TI1 if we put it equal to A -1(0) for t = s. The operator A(t) U(t,s)A -1(0) is strongly continuous in t and s for t> s. Equation (3.1) is said to be abstract parabolic, if for each Xo E E and

s E [0, T] there exists a unique weakened solution of equation (3.1) for the segment [s, T], satisfying condition (4.1) and having properties 2)-4) of Definition 4.2. For an abstract parabolic equation the evolution operator U(t, s) is uniformly bounded and is strongly continuous in Tt>.. It is strongly continuously differentiable in t for t> s. The uniform boundedness and the strong continuity of the evolution operator U(t, s) still do not imply the uniform correctness of the Cauchy problem, since it is not clear whether the function U(t,s)Xo for Xo E 9'(A) has a derivative with respect to t which is continuous in the closed triangle T I1 • This will automatically hold when in addition the operator V(t,s)

= A(t) U(t,s)A -1(S),

V(s,s)

= I,

is strongly continuous in Tt>..' Indeed, suppose that Xo E 9'(A) and Xo = A -1(0) Yo. In view of (4.2) the derivative of the function U(t,s)xo, equal to A(t) U(t,s)A -1(s)A(s)A -1(0) Yo, will be a continuous function in TI1• Reasoning in the same way as in Lemma 1.2 of Chapter I, we show that the function U(t,s)xo has a derivative at zero equal to A(s)A -1(0) Yo = A (s)Xo. Thus, the function U(t,s)Xo is a solution of equation (3.1). The uniform correctness of the Cauchy problem is already a consequence of the properties of the operator U(t,s) indicated earlier. 2. Kernels with weak singularities. Suppose that the bounded linear operator K(t,s) acting in the Banach space E is defined in the triangle o ;a; s ;a; t ;a; T. We shall say that K(t,s) is an operator kernel with a weak singularitY of order IL, if the operator K (t, s) is strongly continuous in the variables t, s for t > sand

§4. WEAKENED CAUCHY PROBLEM

II K(t,s) II

(4.3) where O;::i! JL

<1

211

~ C/(t - S)".

and the constant c does not depend on t, s.

4.1. Suppose that K(t,s) is a kernel with a weak singularity of order JL and that ",(t, s) is a kernel with a weak singularity of order ". The integral equations LEMMA

W(t,s)

(4.4)

+i t W(t. T)K(T,S)dT

= ",(t,s)

and W(t,s) = ",(t,s)

(4.5)

+ itK(t'T)W(T,S)dT

have unique solutions which are operator kernels with weak singularities of order II.

Consider equation (4.4). We shall solve it by the method of successive approximations. The desired solution will be written in the form of a series PROOF.

m

(4.6)

W(t,s)

=

L W,,(t,s). ,,=0

where Wo(t,s)

(4.7)

W"+l(t,S)

=

= ",(t,s),

i t W,,(t, T)K(T,S)dT.

We estimate

II W 1 (t,s) II

~ //f.'",(t,T)K(T,S)dTII ~C2f.t(t_T)~(:_S)"·

Making the substitution (T - s)/(t - s) = r, we get

II W 1 (t,s) II ~C2(t_S)1-'-" (1

dr

Jo r'(l - r)"

c2 (t _ s) l_"r(l - v) r(l - JL) • (t-s)' r(2-v-JL)

Using (4.7), we conclude by induction that

212 (4.8)

II. FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

II W,,(t,s) II ~ cr(1- /I) -

(t - s)'

[cr(l - ,,) (t - s)1-I']". r(n(1- ,,) 1- /I)

+

Hence it is clear that for 0 ~ ~ ~ t - s ~ T the series (4.6) converges uniformly. The function W(t, s) is strongly continuous for t > s. Indeed, suppose that t - s ~ ~ > o. The function (E

< ~/2)

is obviously strongly continuous in the region norm of the difference dr r'(1- r)"

~ ~t -

s

~

(,16

+ Jo

T. The dr

)

r'(l - r)"

tends to zero uniformly in this region, which means that the function W1(s,t) is strongly continuous for t> s. In the same way we establish the strong continuity of all the functions W,,(t,s). From the uniform convergence of the series (4.6) it follows that W(t, s) is strongly continuous for t > s. Obviously W(t,s) satisfies equation (4.4). Finally, it follows from (4.8) that (4.9)

II W(t,s) I ~ cr(l -

i:.

/I) [cr(1- ,,) T 1 -"],, (t-s)' ,,_or(n(1-,,) +1-/1)

Cl

(t-s)··

The uniqueness of a solution of equation (4.4) which has the property (4.9) is, as usual, deduced from the estimate (4.8). Analogously one considers equation (4.5). The lemma is proved. REMARK. If for t> s the kernels I/>(t,s) and K(t,s) are continuous in the operator norm, then the operator W(t, s) is also continuous in the operator norm for t > s. 3. Integral equations for evolution operators. Along with equation (3.1) we consider the equation (4.10)

dx/dt

= A (to) x,

where to is some point of the segment [0, T]. This equation will be called an equation with frozen coefficients. If the weakened CauchY problem is correct on 9(A) for equation (4.10), then its solution may

§4. WEAKENED CAUCHY PROBLEM

213

be written in the form x(t) = UA(to)(t - s)Xo, where UA(to)(t) is the semigroup operator corresponding to this problem. It is natural to suppose that solutions of equations with frozen coefficients are in some sense close to the solutions of equation (3.1). The following considerations will be of a heuristic nature. Write equation (3.1) in the form dx/dt

=

+ [A (t) -

A (to) x

A (to)

lx.

Under well-known conditions there results the formula U(t, s) Xo

=

UA(~ (t -

s) Xo

+ I.t U A(to) (t -

(4.11)

T) [A (T) - A (to)] U(T,s)xodT,

•

where U(t,s) is the evolution operator for the problem (3.1)-(4.1). Equation (4.10) may in tum be transformed into the form dx/dt

=

A (t) x

+ [A (to) -

A (t) ]x,

and then, formally.

UA(~(t -

s)Xo

=

U(t,s)xo

(~lm

+ I.t U(t, T) [A (to) -

A (T)] UA(Io)(T- S)XodT.

•

Thus, if the evolution operator exists and has "nice" properties, the integral equations (4.11) and (4.12) hold for it. Now we shall attempt to employ these integral equations for the construction of the evolution operator. The function UA(~ (t) has a singularity at zero. A (to) UA(~(t) has an even greater singularity. Choose the point to in equations (4.11) and (4.12) so that the difference in square brackets may be used for weakening the singularity of the factor UA(~(t). To this end, we put to = t in equation (4.11) and to = s in equation (4.12). Then we obtain the equations (4.13) U(t,s)

=

UA(t)(t - s)

+ I.t UA(t)(t -

T) [A(T) - A(t}] U(T,s)dT

and (4.14) U(t, s) = UA(B)(t - s)

+ I.t U(t, T) [A (T) -

A (s)] UA(s)(T - S)dT.

In equation (4.13) we have put in the--closure sign in order to make it Possible to suppose that the kernel of (4.13) is a bounded operator for t:;r. T.

214

II. FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

Thus, suppose that the kernels ------~~------~

UA(t)(t - s), UA(.)(t - s), UA(t)(t - s) [A(s) - A(t)]

and [A(t) -A(s)]UA(.)(t-s)

are kernels with weak singularities. Then in view of Lemma 4.1 there exist solutions of equations (4.13) and (4.14). LEMMA 4.2. The solutions with weak singularities of equations (4.13) and (4.14) coincide.

PRoOF. Suppose that fJ(t,s) is a solution with a weak singularity of equation (4.13). Write

Z(t,s)

=

i t fJ(t,'r) [A(T) - A(s)] UA(.lT - S)dT

+ UA(.)(t -

s).

This integral converges absolutely, since both factors, by hypothesis, have weak singularities at the ,J>oints T = t and T = s respectively. Now we substitute in place of U(t,s) its expression found from equation (4.13). We have Z(t,s)

=

i'

UA(t)(t - T) [A(T) - A (s)]UA(.)(T - S)dT

+ i'i'UA(t)(t-IT)[A(IT) -A(t)]U(IT,T)dlT X[A(T) - A(s) ]UA(.)(T - S)dT

+ UA(.)(t -

s).

Changing the order of integration in both integrals, we get Z(t,s)

=

itUA(I)(t-T) [A(T) -A(s)]UA(.)(T-S)dT

+ i t UA(I)(t -

IT) [A (IT) - A(t)]

f.'

U(IT, T)

X[A(T) - A(s) ]UA(.)(T - S)dTdlT

+ UA(.)(t -

s)

= f.t UA(t)(t - IT) [A (IT) - A(s) ] U A(.) (IT - s)dlT

+

f.'

UA(t)(t - IT) [A (IT) - A(t) ]Z(a,s)dlT

+ UA(.)(t -

s).

§4. WEAKENED CAUCHY PROBLEM

215

The first integral on the right is a difference UA(t) (t - s) - U A(.) (t - s) (see formula (7.21) of Chapter I), so that Z(t,s) is a solution of equation (4.13). This solution also has a weak singularity. In view of the weakness of the solution Z(t,s) == aCt,s), i.e. U(t,s)

=

UA(.)(t - s)

+

it

a(t,T) [A(T) - A (s)]UA(.)(T - S)dT.

Thus aCt, s) is also a solution of equation (4.14). In view of the uniqueness of a solution of (4.14) with a weak singularity, the lemma is proved. 4. The operator Vet, s). We have seen already earlier the important role played by the operator (4.15)

Vet, s)

= A (t) U(t, s)A -l(S).

Consider the integral equation Vo(t, s)

= A (0) UA(t)(t -

s)A -1(0)

«4.16)

The operator A (0) UA(t)(t - s)A -1(0)

=

A(O)A -lUA(t)(t - s)A(t)A -1(0)

has a weak singularity of the same order Suppose that the operator A (0) UA(t) (t has a weak singularity. Then there exists tion (4.16) with a weak singularity of the

as the operator UA(t) (t - s). T) [A (T) - A (t) ]A -1(0) also a solution Vo(t, s) of equasame order as the operator

UA(t)(t - s).

N ow we tum to the consideration of the equation Qo(t, s)

(4.17)

=

UA(t)(t - s)A -1(0)

+ f.t UA(t)(t -

T) [A(T) - A(t) ]Qo(T,s)dT.

•

The term outside the integral has a singularity no worse than the kernel UA(t) (t - s), so that equation (4.17) has a solution. (In the lllajority of cases the kernel UA(t)(t - s)A -1(0) is continuous in T"" SO that the operator Qo(t, s) will also be continuous in T",.) Comparing equations (4.16) and (4.17), we arrive at the conclusion that (4.18)

A -1(0) Vo(t, s)

=

Qo(t, s).

Comparing equations (4.13) and (4.17), we get

216

II. FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

U(t, s)A -1(0) = QO(t, S).

(4.19)

It follows from (4.18) and (4.19) that the operator U(t,s) carries the domain q)(A) into itself and the operator (4.20)

Vo(t,s)

= A (0) U(t,s)A -1(0)

is a kernel with a weak singularity of the same order as the operator UA(t)(t - s). Obviously the- operator V(t,s) of (4.15) will have the same property. 5. Differentiability of the operator U(t,s) on q)(A). Write down a new integral equation:

(4.21)

Ro(t,s)

=

A -l(O)A(t) UA(.)(t - s)

+ f.t Ro(t, r) [A (r) -

A (s)] UA(.)(r - s)dr.

The term outside the integral on the right has a weak singularity, so that the solution of (4.21) exists and has a weak singularity. Supposing that Xo E q)(A) , we calculate

f.' Ro(t, s) %odt + A -1(0) Xo =

f.T A-1(0)A(t) UA(,)(t - s)xodt + f.T [ f ' Ro(t, r)dt] x[A(r) -

= f.0" A -1(0)A(s) UA(,)(t -

A(s) ]UA(.)(T - S)dT

+ A -1(0)Xo

+ f.' [ f ' Ro(t, T)dt + A -1(0)] X[A(T) - A(s) ] U A(.)(T - S)dT + A -1(0)Xo.

s)Xodt

The first integral on the right is equal to A -1(0) UA(.)(r - s)Xo - A -1(0) %0. Thus

f.' Ro(t,S)Xodt + A -1(0)Xo = A -1(0) UA(.)(r (4.22)

+

f.

T

S)%o

[f'R o(t,T)dt+A- 1(0)] [A(T) -A(s)]UA(,)(T-S)XodT.

This equality has been proved for Xo E ~(A), but since the operators on both sides are bounded, it is valid for any Xo E E. Comparing equation (4.22) with equation (4.14), we arrive at the conclusion that

217

§4. WEAKENED CAUCHY PROBLEM

f.~ Ro(t, s) dt + A -1(0) = A -1(0) U(r,s). On the other hand, comparing (4.21) with (4.14), we get Ro(t, s) = A -1(0) A (t) U(t, s),

i.e. A -1(0) U(r,s)

= f.~ A -l(O)A(t) U(t,s)dt + A -1(0).

We apply the operator A -1(0) on the right. Then A -1(0) U(r, s) A -1(0)

As

we

proved

in

= f.~ A -1(0) A (t) U(t, s) A -1(0) dt + A -2(0). the

preceding

subsection,

the

function

A(t) U(t,s)A -1(0) has a weak singularity for t = s, so that we can

pre-multiply both sides by the operator A (0) and obtain the relation U(r, s) A -1(0) =

f.~ A (t) U(t, s) A -1(0) dt + A -1(0),

from which it follows that (4.23)

iJU(r,s)A -l(O)/iJr = A(r) U(r,s)A -1(0).

Thus, for XoE 9'(A) the function x(t) = U(t,s)Xo is a solution of the weakened Cauchy problem (3.1)-(3.4). 6. Parabolicity. We shall make clear under which conditions the function U(t, s) Xo is differentiable with respect to t for any Xo E E. Suppose first that Xo E 9'(A) and Xo = A -1(0) Yo. Writing out the operator equation (4.14) on the element Yo and taking account of (4.20), we arrive at the relation

(4.24)

where t < 1$ ~ t - s is some number on the interval (0,1$). Suppose that the operator A (0) UA(t)(t - s) has a singularity of

218

II. FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

order'Y for t = s (usually 'Y (t - s) 'Y. Then

I (t -

~

1). We multiply both sides of (4.24) by

s)TA(O) U(t,s)XoII

~ ell XoII + (t - s)T f.'+·11 A (0) UA(I)(t -

T) [A(T) - A(t) ]1111 U(T,s>lldTII XoII

+ (t-s>"'f.:.IIA(O) UA(I)(t- T)[A(T) -

A(t)] A -1(0) I

X(T - s) -

I (T -

s)TA(O) U(T,s)XolldT.

If we suppose that the closure A(O)UA(I)(t- T)[A(T) - A(t)]

is an operator uniformly bounded in t and T for t - T ~ ~ > 0, then, since the kernel U(t,s) has a weak singularity, the first integral is majorized by the quantity elll %011. In the second integral the first factor has a weak singularity, and the second is bounded. Thus we arrive at the inequality

I 'If(t) I

~

t

e'll XoII + e" f.,+. (t -

1 T)"

I 'If(T) I dT,

where ,,< 1 and 'If(t) = (t - s)TA(O) U(t - s)Xo. Iterating this inequality, after a finite number of steps we obtain an inequality of the form

with a continuous kernel aCt, T). From this last inequality it will follow that

11'If(t)II =11(t-s)TA(O)U(t,s)XoII ;ii!MIlXoII· Hence the operator A (0) U(t, s) is uniformly bounded in the region t - s ~ ~ > 0, and since it is strongly continuous in t and s in this region on !?I(A), it will be strongly continuous also on the entire space E. It follows from equation (4.23) that for Xo E !?I(A) U(t,s)Xo - U(s

+ E,S)Xo = J,+. rt A(T) U(T,s)XodT.

Since the operators on both sides are bounded, this equation is valid

219

§4. WEAKENED CAUCHY PROBLEM

for any XoEE. In view of the continuity of A(T) U(T,S)Xo, aU(t,s)Xo/Ot = A(t) U(t,s)Xo.

Thus, under our hypotheses, for any Xo E E the function U(t,s)Xo satisfies equation (3.1) for t > s. We took this property as the basis for the definition of parabolicity. 7. Uniqueness, eorreetness. We have prepared all the material required for passing to the proof of the fundamental theorems. THEoREM 4.1. Suppose that for equation (4.10) with frozen coefficients for any to E [0, T], the weakened Cauchy problem is correct on ~(A). If the operator UA(t)(t - s)A -1(0) is strongly continuous for t and s in the triangle T41 the operator A -1(0) A (t) is defined and uniformly bounded on [0, T], and the operators

UA(t)(t - s), UA(.)(t - s), UA(t)(t-s)[A (s) - A (t)], [A (t) - A (s)] UA(.)(t - s) and A (0) UA(t) (t - T) [A (T) - A (t) ]A -1(0) are defined and are kernels with weak singularities, then the weakened solution of the problem (3.1)-(4.1) is unique.

Suppose that x(t,s) is a weakened solution of equation (3.1) on the segment [s, T]. For f> 0 the value xes + f,S) E ~(A). Construct on the segment [s + f, T] the function PROOF.

z,(t)

=

U(t,s

+ E)x(s + f,S) -

x(t,s).

In view of the foregoing this function is defined and continuous on [s + E, T]. For T > s + f the function i,(T) is differentiable, and its derivative is given by

or dz./dT = A (t)Z,(T)

+ [A(T) -

A(t)]A -l(T)A(T)Z,(T).

The function

+ f)A -l(O)A(O) xes + f,S) the semi-interval (s + E, T]

A(T)Z,(T) = A(T) U(T,S

- A (T)X(T,S)

is continuous in T on and has a weak singularity for T = S + E. In view of the remark to Theorem 6.1 of Chapter I and the fact that z,(s

we get

+ E) = r-+.+« lim U(T,S + E)A -1(O)X(S + E,S) -

xes

+ E,S) = 0,

220

II. FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

Z,(t)

=

f.'•+.

UA(t)(t - T) [A(T) - A(t) ]z,(T)dT •

The kernel

weak singularity. Then

(" < 1), from which it follows that Z.(T) = O. So, U(t,s

+ E)X(S + f,S) =x(t,s)

(s

+ E ;:i! t;:i! T).

Now let E tend to zero. The weakened solution x(t,s) is continuous for t = s, and the operator U(t, s + E) is strongly continuous for t > s. Therefore U(t,s)x(s,s)

= x(t,s),

ie. the weakened solution x(t,s) is uniquely defined. The theorem is proved. THEOREM 4.2. If the conditions of Theorem 4.1 are satisfied, then the weakened Cauchy problem for equation (3.1) is correct on ~(A). PROOF. Suppose that XoE ~(A). The function x(t,s) = U(t,s)Xo, because of (4.23), satisfies equation (3.1) for t> s. Further, because of the continuity of U(t,s)A -1(0) in Tb" the function x(t,s) = U(t,s)A -l(O)A(O)x is continuous in Tb, and xes,s) = liIru..... U(t,s) xA -l(O)A(O)Xo = Xo, i.e. the initial condition (4.1) is satisfied. The derivative (t, s) = A (t) U(t, s) A -1(0) A (0) Xo is continuous for t> s. Finally, the uniform boundedness of U(t, s) in every region t - s~ &> 0 (O;:i! s ;:i! T) implies continuous dependence on the initial data.

x:

The theorem is proved. Theorems 4.1 and 4.2, in spite of their somewhat awkward hypotheses, are convenient in that they reduce the whole problem to the consideration of the properties of solutions of equations with constant ("frozen") operators. In the following subsections we shall investigate the question of realization of the hypotheses of Theorems 4.1 and 4.2. 8. Properties of the solutions of equations with "frozen" coefficients. We suppose that the operator A (t) satisfies the condition (4.25)

I R A (,) (A) I

;:i!

M/(l

+ ITI)ll

(A

=

IT

+ iT, IT ~ 0),

221

§4. WEAKENED CAUCHY PROBLEM

where the constants M and {J do not depend on t, and 0 < (:J ~ 1. In view of Theorem 3.3 of Chapter I the weakened Cauchy problem for equation (4.10) is correct on ,q(A). The semigroup UA(I(J) (r) is strongly differentiable for r > O. The operator A (to) UA(I(J) (r) is bounded and

I A (to) UA(to)(r) II ~ c/r l /fI - l

(4.26)

(see Theorem 3.4 of Chapter I). From these conditions it follows that the operator A(ta) UA(to)(r) satisfies the same estimate, possibly with another constant. For the semigroup UA(to)(r) itself, the inequality

II UA(to)(r) I

(4.27) is satisfied for

(:J

< 1, and I UA(to)(r) I

~

~ c/r W

- l

cln(e + l/r)

for {:J = 1 (see Remark 3.2 of Chapter I). From the 'proof of Theorem 3.4 of Chapter I it is clear that the constants c in the last inequalities do not depend on the choice of the point ta. The correctness of the Cauchy problem on ,q(A) implies that the operators UA(ta)(t)A -leta) converge stronglY to the operator A -l(ta) as r-+O, so that the operators UA(to) (r)A -l(ta) are uniformly bounded. Moreover, if {J> 1/2, these operators are continuous in norm. Indeed,

II UA(to)(r + M)A -1 (to) ~

- UA(to) (r)A -l(ta) II

/ir '+
/

I~rl = c I ~rI2-l/fI, ~ C I~rll/ll-l

if r ~ 21 ~rl. But if r < 21 ~rl , the limits of the integral may be moved apart to 0 and 3~r, and again an estimate of type (4.28) is obtained. In what follows we shall suppose that (:J> 1/2. Now we are going to be interested in the question of the way in which the function UA(t)(r)A -1(0) changes when t changes. To this end we use the representation of the semigroup UA(t) (r) in the form of a contour integral (3.13):

where the contour rq has in our case the equation

CT

= - (q/M)

(1

+ ITj)fI

222

II. FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

and 0

< q < 1.

Therefore we obtain

[UA(tl)(r) - UA
r e~rRA(t1)(~) [A (to) - A (t1) ]RA(to)(~)A -1(0) d~. Jrq On the contour rq an estimate of type (4.25) holds, so that I [UA(Il) (r) - UA(to) (r)]A -1(0) I = - 21. rl

SC

r

- Jrq

~ Cl

e-(q/M)(l+iTi)/lr

1:"

(1

Id~1

+ ITi)2/1

II[A(to)-A(t)]A-1(to)11

(1!TT) 2/1 I [A (to) - A (t1) ]A -1 (to) I

1

~ c211 [A (to) -

A (tl ) ]A -l(ton •

In what follows we shall need to suppose that the following Holder condition is satisfied: HA(t) -A(s)]A- 1(0)11 ~clt-sl"

(4.29)

(0
Then

I [UA(t)(r)

(4.30)

-

U A(.) (r)]A -1(0) I ~

c31 t -

sl".

N ow we can prove the following assertion: LEMMA 4.3. If conditions (4.25) with fJ> 1/2 and (4.29) are satisfied, then the function UA(t) (t)A -1(0) is continuous in the operator norm in the triangle T 4.' PROOF.

I [UA(tHt)(t + .6.r) - UA(I)(t)]A -1(0) I ~ I [UA(IHI)(r + .6.r) - U A(I+4.tl(r)]A -1(0) ~ + I [UA(t+4.I)(r) - UA(t)(r)]A -1(0) I ~ c'(I.6.rI 2- 1//I + I.6.W)· The lemma is proved. Now we shall estimate the quantity (k = 0,1) for r > O. We have

I A k(O)[UA(tl)(r)

r

-

I Ak(O)[UA(It)(r)

- UA(to)(r) ]11

UA('o)(r)]11

IAIII+1

~ C Jrq exp[ - (q/M) (1 + ITi)fJr] (1 + ITj)2/1ldAII tl -

toll' ~

14.

223

WEAKENED CAUCHY PROBLEM

where the last step was made by substituting 8 = - (q/M) (1 + 1')/lr • Since (k + 2)/fJ - 3 > k -1 > -1, we finally have

t

Itl-tol"2

II

IIA (0) [UA(II)(r) - UA(/o)(r) ]11 ~ c (H2)//I

(4.31)

r

-

(k

= 0, I),

LEMMA 4.4. The function AII(O) UA(I)(r) (k = 0,1) is continUQus and r for r > 0 in the sense of the operator norm.

In

PROOF.

IIAIe(O)[UA(IHI)(r+ar) - UA(I)(r)]11

~

IIAk(O)A-Ie(t+

at)

i

rHr

at) UA{I+M) (s)ds

AII+1(t+

II

(

larl

+~A (0) [UA(I+M)(r) - UA(I)(r)]11 ~c r(Hl)//I-l

II

)atl" ) + r(H2)//I-2 •

The lemma is proved. It follows from the lemma that the operators UA(I)(t - s), AA(a)(t - s), A (0) UA(I)(t - s) and A (0) UA(a)(t - s) are jointly continuous in t and 8 for t > s in the operator norm. Then, in view of (4.29), the products [A (t) - A (s) ]A -1(0) A (0) UA(a) (t - s)

and A (0) UA(I)(t - s)[A(s) - A(t) ]A-l(O)

both have the same property. Further, it follows from (4.29) and (4.26) that

I [A(t)

- A(s)]A -I(O)A(O) UA(a)(t - s)

I ~ cit -

sl,,-2//l+1

and ~A(O) UA(I)(t - s) [A(s) - A (t)]A -1(0)

II

~ cit - 81,,-2//1+1.

224

II. FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

In order that the singularity be weak it is necessary that p - 2/fJ 1, i.e. p > 2/fJ - 2. Such a p < 1 exists if 2/fJ - 2 < 1, i.e.

+1> fJ> 2/3.

We have proved the following. LEMMA 4.5. Suppose that condition (4.25) with fJ> 2/3 and condition (4.29) with p > 2(1/fJ - 1) are satisfied. Then the functions UA(t)(t - s), UA(.)(t - s), [A(t) - A(s) ]UA(.)(t - s) and A (0) UA(t) (t - s) [A (s) - A (t)]A -1(0) are operator kernels with weak singularities, and when t> s they are jointly continuous in t and s in the operator norm.

Unfortunately, one of the hypotheses of Theorems 4.1 and 4.2 has far remained unverified. We have in fact not yet proved that the operator UA(t)(t - s) [A (s) - A (t)] admits a closure which is a kernel with a weak singularity. We shall impose an additional condition of the operator A (t): for Xo E 9'(A) the following inequality must hold: 80

(4.32)

IIA-1 (t) [A(s)

-A(t)]XoII ~clt-sIPII%o~.

It follows from this condition that

that the operator A -1(t)A(s) admits a closure A -1(t)A(s) uniformly bounded in T I1• Further, for Xo E 9'(A)

80

I [A -1(0) A (t + .:1t) - A -1(0) A (t) ]XoII ~ IA -1(0)A(t) [A -1(t)(A(t +.:1t) -

A(t» ]XoII ~

C1C

I.:1WII %011,

i.e. the function A 1(0)A(t) is continuous in the operator norm. Analogously one proves that A -1(t) A (0) is continuous in the operator norm. Then UA(t)(t - 8) [A(s) - A(t)] = A(t)A -1(0)A(0) UA(t)(t - s) xA -1(t)A(0) [A -1(0)A(s) - A -1(0)A(t)]

will be continuous in the operator norm for t > s and

Thus we have arrived at the following theorem.

225

§4. WEAKENED CAUCHY PROBLEM

4.3. Suppose that condition (4.25) with (J> 2/3, condition > 2(1/{J - 1) are satisfied. Then the ~akened Cauchy problem is correct on ~(A) for equation (3.1). The evolution operator U(t, s) corresponding to this problem has the properties: 10 • It is continuous in norm for t > s. 2°. The operator U(t,s)A -1(0) is continuous in norm in the closed triangle T4.: 0 ~ s ~ t ~ T. 3°. The operator A (t) U(t, s) A-I (s) is continuous in norm for t > s. 4°. The operator U(t, s)A -1(0) is differentiable in the operator norm for t> s. THEOREM

(4.29), and condition (4.32) with p

9. Abstract parabolic equations. In subsection 6 we found the following condition for parabolicity: THEoREM 4.4. In addition to the conditions of Theorem 4.1 let the following be also satisfied: the operator

A (0) UA(f)(t - r) [A (r) - A (t)]

is uniformly bounded for t - r

> o.

~ &

U(t, s) Xo satisfies equation (3.1) for t

Then for each Xo E E the function > s.

The hypotheses of Theorem 4.3 imply the hypotheses of Theorem 4.4. Indeed, IIA(O)UA(t)(t-r) [A(r) -A(t)]11

I

~ A (O)A -let) ~

cit -

rI

I I A 2(t) UA(I) (t -

P +l- 3//i

r) I

I A -1 (t)[A (r)

- A (t) ]11

~ C/&3//I-l- p •

THEOREM 4.5. Suppose that ~he hypotheses of Theorem 4.3 are satisfied. If moreover for each to equation (4.10) with "frozen" coefficients is abstract parabolic and the semigroup UA(/o)(r) is uniformly bounded relative to to, then equation (3.1) is also abstract parabolic. Its Cauchy problem is uniformly correct. PROOF.- For each XoEE the function x(t,s) = U(t,s).ro satisfies equation (3.1) for t> s. It follows from the integral equation (4.14) that the operator U(t, s) is bounded uniformly in t and B. On ~(A) we have lilllt_. U(t, s) Xo = x, which means that for each Xo E E the initial condition xes, s) = Xo is satisfied as well. The uniqueness of the solution x(t, s) was proved in Theorem 4.1. The remaining required properties of the solution x(t, s) follow directly from the properties of the operator U(t,s).

226

II. FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

Finally, the uniform correctness of the Cauchy problem, as we already noted in subsection 1, follows from the parabolicity and the uniform boundedness of the operator V(t, s), which is deduced from the uniform boundedness of the operator

A (0) UA(t)(t - s)A -1(0) = A(O)A -l(t) UA(t)(t - s)A(t) using the integral equation (4.16). The following is a special case of Theorem 4.5. THEOREM

4.6. Suppose that the operator A (t) satisfies the condition

(4.33) .

(ReA

~

0),

where M does not depend on t. If (4.29) is satisfied, and (4.32) with any p > 0, then the assertion of Theorem 4.5 is valid. Indeed, condition (4.25) is satisfied with {J = 1~ and the equation with "frozen" coefficients is abstract parabolic in view of Theorem 3.8 of Chapter I. The theorem is proved. REMARK. In a somewhat more complicated way it has been proved that the assertion of Theorem 4.6 is valid without condition (4.32) (see the References to the Literature). If the Cauchy problem is uniformly correct for an equation with "frozen" coefficients, and if its evolution operator is uniformly bounded, condition (4.25) may be replaced by the condition (4.26). Indeed, condition (4.25) was used in t~e estimate of the norms of the operators

[UA(t1)(r) - UA(to)(r»)A -1(0), A k(0)[UA(t1)(r) - UA(to)(t») and Ak+1(t) UA(t)(s) (k = 0,1). We shall show that analogous estimates may be obtained directly from (4.26) and (4.29). We begin with the last operator. From (4.26) we have IIA2(t) UA(t)(s) II = IIA(t) UA(t) (s/2) A (t) UA(t) (s/2) ~ ~ C1/ S·/1l -

2•

The estimates of the norms of the first two operators are more complicated. We use the identity

U1(r) - Uo(r) = [ U1(~) - Uo (~) ] Uo (~)

+A

1

Ul

(i) [ U (i) - Uo (i) ] Ail l

227

§4. WEAKENED CAUCHY PROBLEM

where we employ the shortened notations A (tJ = Ai and

UA(lj)

=

U j

(i=O,1).

N ow we apply the integral representation for the difference of semigroups (see (7.21), Chapter I). Then we get Ul (r) - Uo(r) =

!a (i (i) L (i r/2

+ AIUl

+ U l (i)

s) [Ao - Ad Uo

Ul

r/2

Ul

(~ -

s ) ds

s) [Ao - Al]AolUO(s)ds

(i) + Ul(r) [Ao -

[AI - Ao]AolUO

AlfAi)l.

From this idenity there directly follows the estimate

II UA(tl)(r)

- UA(to)(r)

II

~c {Lr/2(r/2+~)2/17_1 + (r/2~2//1-1·i +2} Itl-tol'

(4.34)

In order to estimate the operator [UA(t])(r) - UA(tol(r)]A -1(0) we may use a simpler identity: U l (r) - Uo(r)

L

r/2

=

(i - [Ao - Ad Uo (i + s) ds + U l (i) L7/2 U (i - s) [Ao - AdU1(s) ds, U1

S)

1

so that (4.35)

II [UA(ll)(r)

- UA(to)(r)]A-l(O)

II

~cltl-tolp·

Finally, from the identity Al[Ul(r) - Uo(r)]

= AIU1

(i) [ Ul (i) - Uo (i) ] + U l (i) [A

1 -

Ao]Uo(~) +

228

II. FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

and inequalities (4.34) and (4.29) it follows that II A (t1)[UA(tl)(r) - UA(to)(r)]11 (4.36)

Estimates (4.34)-(4.36) replace (4.28) and (4.31). Thus the following holds. 4.7. The assertion of Theorem 4.5 remains valid if condition (4.25) is replaced by condition (4.26) with {j> 2/3. THEOREM

§5. Abstract parabolic equations with operators having variable domains 1. Statement of the problem. Up to this point we have assumed that the domain of the operator A (t) does not depend on t. In addition we extensively used the fact that under these conditions the operator A(t)A -l(S) is bounded. We assumed and employed the strong or uniform continuity of that operator, its differentiability with respect to t and s and so forth. If the domain 9'(A (t» varies with t, then the operator A(t)A-l(s) may already fail to be defined in the whole. space or even on a dense set. The requirement of strong continuity or strong differentiability of the operator A (t) on ~(A) loses its meaning. However the case when in equation (3.1) the operator A (t) has a variable domain is an important one. Thus, for example, for partial differential equations it corresponds to the case when the boundary conditions of the problem depend on the time t. Let us suppose that the equation with "frozen" coefficients is abstract parabolic and that condition (4.35) is satisfied. The necessary assumptions of smoothness will be introduced in terms of the operator A -let) or of the resolvent RA(I) (>.) rather than in terms of the operator A (t) itself. Thus, we shall suppose two conditions satisfied: 10. The operator A -l(t) is continuously differentiable in t on [0, T], and

(5.1)

IldA -l(t)/dt - dA -l(s)/dsll ~ Kit - sla.

2°. There exist constants Nand (5.2)

p,

II (iJ / iJt) RA(t) (>.) II

0

~p

~ N/

< 1, such that

I>.1 1

-p

§5. ABSTRACT PARABOLIC EQUATIONS

229

in the sector IargAI ~ r - q, (q, < r/2). I t turns out that under these conditions one can also construct an evolution operator for equation (3.1). First we shall establish a series of auxiliary propositions. LEMMA 5.1. Under conditions (5.1) and (5.2) the derivative with respect to t of the resolvent of A (t) satisfies a HlJlder condition, and

(5.3) HiJ/iJt) RA(I) (A) - (iJ/iJr) RA(r) (A) II ~c[lt-rIIAlp+lt-rl"J. PROOF. We first estimate the increment in the resolvent. In view of (5.2), for t > r we have

(5.4) Then (5.5)

II RA(t) (A) - RA(r)(AH = IIi':s RA(.)(A)ds

I A (t) RA(t) (A)

"~NI:r~p,

- A (r) RA(r) (A) II

=IAI ~RA(t)(A) -RA(r)(A)11 ~N(t-r)IAlp·

We compute (iJ/iJt)RA(t)(A)

= A (t)RA(t)(A) (dA -l(t)/dt) A (t)RA(t)(A),

from which iJ iJ iJt RA(t) (A) - iJr RA(r) (A)

=

[A (t) RA(t) (A) - A (r)RA(r)(A) ]

(5.6)

+ A (r)RA(r)(A) [ + A (r) RA(r) (A)

dA -let) dt A (t) RA(t) (A)

dA -let) dt

dA -l(r)] dr A (t)RA(t) (A)

dA -l(r) dr [A (t)RA(t)(A) - A (r)RA(r)(A)].

(5.3) is a consequence of this identity and of (5.5) and (5.1).

The lemma is proved. 2. Properties of the solutions of an equation with "frozen" coefficients. Now we consider the function UA(I)(t - s), defined in the triangle T 4 : O~s ~ t ~ T. LEMMA 5.2. For t > s the function UA(t) (t - s) is continuous in t and s in the operator norm. For t ~ s this function is strongly continuous in t and s.

230

II. FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

PROOF. For the proof of the first assertion we show that for r> 0 the function UA(t)(r) is jointly continuous in t and r in the operator norm. We have

(5.7)

UA(t+lI)(r

+ k) =

UA(t)(r)

UA(t+h)(r+k) -

UA(t)(r+k)

+ UA(t)(r+k) -

UA(t)(r).

In order to estimate the first difference we use the integral representation of the semigroup UA(t) (r):

~

;= -

Assuming

(5.8)

Ikl

~r/2

271"£

r eA(r+k)[RA(t+h)(X) Jrq

RA(t)(X) ]dx.

and using (5.4), we get that

I UA(t+h)(r + k)

-

UA(t)(r

+ k) II

as h--+O. The second difference is easily estimated as follows: ~ UA(t)(r

+ k) -

=

UA(t)(r)

I

Ilir+kA(t) UA(t) (T) dT

II~c lirH~T I=c Iln(l+~)

I.

It also tends to zero as k --+ o. The first assertion of the lemma is proved. Now suppose that t = s. Condition (4.33) implies the uniform boundedness of the operators UA(t) (r) in t and r, so that it suffices to prove that (UA(t+Ia)(k) - I)Xo--+O on a set dense in E, for example on ~(A(t». If xoE ~(A(t», then Xo = A -l(t)yo and (UA(t+h)(k) - J)xo

=

[UA(t+Ia)(k) - J]A -l(t - [UA(t+la) (k) -

+ h) Yo

J] [A -l(t + h)

- A -l(t) ]Yo.

The first term is a quantity of order k in view of the uniform boundedness (relative to h) of the semigroup UA(t+h) (r). The second term is of order h for the same reason along with condition 10. Now we shall estimate the derivatives of the function UA(t)(t - s). For the derivative with respect to s we obtain (5.9)

I (iJ/as) UA(t)(t -

s)

I = II -

A(t) UA(t)(t - s)

II

~ c/(t - s).

231

i5. ABSTRACT PARABOLIC EQUATIONS

The situation is more complicated with respect to t. We have (5.10)

Using the integral representation of the semigroup UA(t)(T), we get (:t

+ :8) UA(t)(t -

8) = -

2~i

i,

eA(I-')

:tRA(t)(A)dA.

It follows from condition (5.2) that (5.11)

UA(t)(t-S) II::s:c r·e-(9/M)(l+r)('-.)~::s: I ( ~+..!.) at as - Jo IAll-~ - (t-8)~ Cl

•

From (5.9), (5.10) and (5.U) we obtain the estimate

I (a/at) UA(t)(t -

(5.12)

8) I ~ c/(t - s).

N ow we tum our attention to the fact, very important in the sequel, that the sum of the derivatives with respect to t and to s has for t = s a weak singularity, though each of them has a strong singularity. 3. Construction of the evolution operator. We seek to find the evolution operator in the form (5.13)

U(t,s)

=

UA(t)(t - s)

If we write formally au = at a UA(t)(t at

s)

+

i'

UA(I)(t - T)R(T,S)dT.

i'

a + • iitUA(t)(t -

T)R(T,S)dT

+ R(t,s)

and

then in order to satisfy the equation

aU/at = A(t) U,

(5.14)

it is sufficient to choose R(t, s) as a solution of the integral equation (5.15)

where

R(t,s)

= R 1 (t,s)

+

it

R 1 (t,T)R(T,S)dT,

232

II. FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

For the function Rl (t, s) we have in (5.11) already found the estimate (5.16)

LEMMA 5.3. The function Rl (t, s) for t> s is continuous in t in the operator norm, and even satisfia a Hiilder condition. PROOF.

Supposing r < t, we estimate the increment of the function

R1(t,s) in t. We have Rl(t,S) - R1(r,s)

= 21 . rJ

r

J r"

eA(I-.)

{!....RA(t)(A) -

at

~RA(r)(A)} dA

ar

(5.17) For the first integral in (5.3) we obtain the estimate

PII ~c ~C

1" [

e-(Q/M)(1+,)(I-')[(t-r)IAI'+(t-r)"]dT

t - r (t _ s)1+,

+

(t - r) "] t - s •

The second integral in (5.17) may be estimated in the form

i'-'J:

1. II = -2 ... , r-'

Ae A•

rq

a

ar RA(r) (A) dAdo •

-

'

Since in view of condition (5.2)

IIJ:

q

Ae Al :r RA(r)(A)dA

II ~ .J:" C

e-(q/M)(1+')'IAI'dT

~ 01:P'

we have

~ cl(r - s) -P

sJ [1 - rt_ s =

r

t Cl (r _ s)P(t _ s) •

Finally (5.18)

II Rl(t,S)

- R 1 (r,sn ~

C

t - r [ (t _ s)(r _

s)P

+ (tt --

r) s

"J

•

233

§5. ABSTRACT PARABOLIC EQUATIONS

The lemma is proved. Repeating the arguments presented in the proof of Lemma 4.1, we arrive at the conclusion that for each s there exists a solution R(t, s) of the integral equation (5.15) which is continuous in t for t > s in the operator norm and which has a weak singularity for t = s: IIR(t,s) II ~c/(t-s)p.

(5.19) LEMMA

5.4. For t> s the function R(t, s) satisfies a Holder condi-

tion relative to t. PROOF.

We calculate

R(t,s) - R(r,s)

=

RI(t,s) - RI(r,s)

+

it

RI(t,T)R(T,S)dT

+ f.I'[RI(t,T) -R I (r,T)]R(T,s)dT=I+II+III.

For the term I estimate (5.18) holds. For II, it follows from (5.16) and (5.19) that

111111 5. cft

(5.20)

dT < ,(t - T)'(T - s)P

-

CI

(t - r)l-p. (r - s)P

Finally, for III it follows from (5.18) and (5.19) that

111I1115.Cf.'[ -

(5.21)

•

t-r (t - T)(r - T)P

+ (t-r)"'] t- T

5. c(t _ rP'f.'

dT • (r - T)p+'r(T - s)P

-

~ C1

[

(t - r)'Y (r _ s)2p+'Y- I

dT (t - s)P

+ c(t _

r)a-.f.' dT • (r - T) I-'(T - s)P

r)a-.]

(t -

+ (r -

s)P-' '

where l' and E > 0 are chosen so that p + 'Y < 1 and E < a. I ) Estimates (5.18), (5.20) and (5.21) for t - s ~ fJ/2 and r - s ~ fJ/2 I)

More precise arguments [105] lead to the estimate

I III I ;a c

{

t- r (t-a)(r-a)2p-l

(t - r)l-'(r _ a) 1-,

+- - --t-a +

(t_r)l-, (t - a)'

+

(t-r)"[ (t - a)'

t-a

Int - r

]}

+1

.

234

II. FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

lead to the inequality ~ R(t,s) - R(r,s)

(5.22)

II

~ c(~) (t - r)",

where «= minh, a - E}. The lemma is proved. N ow we can pass to the proof of the fact that the operator U(t,8) defined by formula (5.13) satisfies equation (5.14). To this end we introduce the functions W,,(t,s) =

I.-"

UA(t)(t - T')R(T',s)dT'.

In view of estimate (5.19) the functions W,,(t,s) converge uniformly in t to the function W(t,s) =

I.

UA(t)(t - 1') R(T,S)dT'.

For the derivative with respect to t for t > 8 we get

aw" UA(,,(h)R(t - h,s) ijt=

+ it-II (:t +

:J

- UA(,,(h)R(t,s)

UA(t)(t - T)dTR(t,s)

+ UA(,,(t-s)R(t,s).

In view of estimates (5.22), (5.12) and (5.11) all the terms on the right have a limit as h ~ 0 uniformly in any segment s + ~ ~ t ~ T, so that the derivative oW i t 0 lit = • aTUAW(t -

T')[R(T,S) - R(t,s) ]dT

- i t R 1 (t, T)dTR(t,s)

+ UA(,,(t -

s)R(t,s)

exists. Hence

(5.23)

-f.

t

R 1 (t,T')dTR(t,s)

+ UA(,,(t -

s)R(t,s).

§5. ABSTRACT PARABOLIC EQUATIONS

235

Further, A(t) U(t,s)

= - -a as

UA(I)(t - s) - f.' -a UA(I)(t - T) [R(T,S) - R(t,s) ]dT • aT

- R(t,s)

+ UA(I)(t -

s)R(t,s).

Thus, in view of the integral equation (5.15), au TtA(t) U(t,s)

= -

Rl(t,S)

+ R(t,s)

- f.'R1(t,T)[R(T,S) -R(t,s)]dT- f.'R1(t,T)dTR(t,s) =0.

It follows from equation (5.13) that the operator U(t, s), just as the operator UA(,)(t - s), is strongly continuous in t for s ~ t ~ T and U(s, s) = I. Further, from formula (5.23) we obtain the estimate Ilau/atil ~c/(t-s).

We note that the derivative a U(t, s) / at differs from the derivative aU A (,)(t - s)/os by a term having a weak singularity for t = s. 4. Properties of the operator U(t,s) as a function of s. We introduce the operator U(t, s) defined by the formula U(t,s)

=

UA(.)(t-s)

+ f.'Q(t,T)UA(,)(T-S)dT,

where the kernel Q(t, T) is found from the integral equation (5.24)

in which Ql (t, s)

= (a lOt

+ iJ / iJs) UA (,) (t -

s).

Just as above one shows that Ql (t, s) is continuous in s for s < t in the operator norm and

I Ql(t,S) II

~ c/(t - s)P.

Hence it follows that the integral equation (5.24) has a solution having a weak singularity. There are propositions analogous to Lemmas 5.3 and 5.4 which assert that the functions Ql (t, s) and Q(t, s) satisfy Holder conditions in s for 8 < t. Finally one shows that for s < t the operator U(t, s) has a derivative with respect to s in the sense of the operator norm, that the operator U(t,s)A(s) admits a bounded closure

236

II. FIRST-ORDER EQUATIONS WITH VARIABLE OPERATOR

and (5.25)

iJU(t,s)/iJs

=-

U(t,s)A(s).

The estimate

II iJU(t,s)/asll ~ c/(t -

s)

holds. It follows from equations (5.14) and (5.25) that (5.26)

U(t, s)

=

U(t, s).

Indeed, the derivative of U(t, T) U(T,S) with respect to l' for 0 ~ s ~ T < t ~ T is equal to zero, so that this function does not depend on T. Using the strong continuity in T of both factors, putting T = t and T = s, we arrive at (5.26). Property (5.25) implies the uniqueness of the solution of the weakened Cauchy problem for equation (3.1). Indeed, suppose that X(T) is a weakened solution of (3.1) on the segment [s, T]. Then dx/dT - A (T)X(T)

=0

(s

< T ~ 1').

Applying the operator U(t, T) to both sides with t> T, we get (a/aT) [U(t, T)X(T)] = 0,

i.e. U(t, T)X(T)

Letting

T

= const.

tend to s and to t, we have x(t)

= U(t,s)x(s)

as we were required to prove. 5. Nonhomogeneous equations. Consider the equation (5.27)

dx/dt

= A(t)x + f(t).

We shall show that in the case when the function f(t) satisfies a Holder condition (5.28) there exists for each Xo E E a solution of the weakened Cauchy problem for equation (5.27), defined according to the formula (5.29)

x(t, s)

= U(t, s) Xo +

f.t U(t, u)f(u) duo

237

§5. ABSTRACT PARABOLIC EQUATIONS

It follows from the properties of the operator U(t,s) that the first term is a solution of the weakened Cauchy problem for the homogeneous equation.. Let us study the function (5.30)

y(t,s)

=

f.t U(t, u) I(u)du.

Reasoning just as in the deduction of formula (5.23), we calculate dy f.ta -d = - U(t, u) [/(u) - I(t) ]du t • at + f.t [:t U(t, u) - :t UA(t) (t - u) ] I(t) du + f.t (~+~) UA(t) (t - u) I(t) du • at au

+U

A(I)

(t - s) I(t).

Here the flrst integral converges absolutely in view of condition (5.28), and the second because, as we have noted, the operator in square brackets has a weak singularity. The third converges because of (5.11). Then dy a U(t, u)f(u) dt = f.t • [ at

+ aua UA(t) (t -

u) I(t) ] du

+ UA(t) (t -

s) I(t)

(5.31) = A(t) f.' (U(t, u) I(u) - UA(t)(t - u) I(t) ]du

+ UA(t)(t -

s) I(t).

Further,

f. ,

(5.32)

•

UA(t) (t - u) I(t) du

=

-

f.' - a UA(t) (t - u) A -l(t) I(t) du • au

= UA(t)(t - s)A -1(t) I(t) - A -1(t) I(t).

It follows from (5.31) and (5.32) that the integral (5.30) lies in and

~(A(t»

dy/dt= Ay+/(t).

We have arrived at the following assertion: THEOREM 5.1. Suppose that conditions (4.33), (5.1), (5.2) and (5.28) are satisfied. Then lor each Xo E E and s E [0, T] there exists a unique solution of the weakened Cauchy problem for equation (5.27) on the segment [s, T], defined by formula (5.29).

CHAPTER

III

EQUATIONS OF SECOND ORDER

i1. The hyperbolic ease. The Cauchy problem 1. Equations with constant operator. Consider the differential equation (1.1)

(0 ~ t ~ T),

d 2u/dt2 = B 2u

where B is an unbounded linear operator with a domain ~(B) dense in E, having regular points. DEFINITION 1.1. A solution of equation (1.1) is a function u(t) with values in ~(B2), twice continuously differentiable and satisfying equation (1.1) on the interval [0, T]. We impose one further condition, condition B), on the solution of equation (1.1): the function u' (t) takes on values from ~(B), and the function Bu' (t) is continuous on [0, T]. It follows from condition B) that (1.2)

dBu(t)/dt = Bu'(t).

Indeed, the operator B may be applied to both sides of the identity u(t) = u(O)

+ !o' u'(T)dT.

Because B is closed and because of condition B), B may be carried under the integral sign. Then Bu(t) = Bu(O)

+ !o' BU'(T)dT,

so that we obtain (1.2) by differentiation. By the Cauchy problem for equation (1.1) we mean the problem of finding a solution satisfying the given initial conditions (1.3)

u(O)

=

flo and u'(O)

= u6.

It follows from the definition of a solution that necessarily Uo E ~(B2). If the solution has property B), then u6 E ~(B). Suppose that u(t) is the solution of equation (1.1). Write 238

239

§1. HYPERBOLIC CASE

(1.4)

U'(t)

=

U'(O)

+ !ot u"(T)dT = U'(O) + B !ot BU(T)dT.

We suppose that u' (0) is representable in the form (1.5)

U'(O) = Bvo

and write (1.6)

v(t)

=

Vo

+ !ot BU(T)dT.

From (1.4) and (1.6) we get (1.7)

du/dt = Bv and dv/dt = Bu.

We introduce the new functions (1.8)

x= u

+v

and y = u - v.

It follows from (1.7) that these functions satisfy the equations

(1.9)

dx/dt

= Bx and dy/dt = - By.

The function u (t), by defmition, is twice continuously differentiable. If moreover it satisfies condition B), then in view of (1.2) and (1.7) the function v(t) will also be twice continuously differentiable. Thus, if the solution u(t) satisfies conditions B) and (1.5), then, in accordance with formulas (1.6) and (1.8), it generates twice continuously differentiable solutions of equation (1.9). Conversely, suppose that x(t) and y(t) are twice continuously differentiable solutions of (1.9). Then, since B is closed, (1.10)

d 2x dt 2

= dBx = B dx = B 2x and d 2y = _ B dy = B2y dt

dt 2

dt

dt

•

It therefore follows that the function u(t)

= t[x(t)

+ y(t)]

is a solution of equation (1.1). This solution has property B), since it follows from (1.10) that the function Bu'(t)

= id 2x/dt 2 - id 2y/dt 2

is continuous. N ow suppose that the Cauchy problem for equations (1.9) is UnIformly correct. For Uo E !?)'(B 2) and ucS E !?)'(B) n ~(B) we put

240

III. EQUATIONS OF SECOND ORDER

%0 = Uo + Vo and Yo = Uo - vo, where Vo is one of the solutions of the equation Bv = u{,. Obviously %0, Yo E !?J(B 2 ). Denoting the semigroups generated by the first and second equations of (1.9) by U+(t) and U _(t) respectively, we construct the solutions of these equations:

x(t)

= U + (t)(uo + vol and y(t) = U _ (t)(Uo - vo)'

These solutions will be twice continuously differentiable on [0, T]. Then the function (1.11)

u(t) =t[U+(t)

+ U_(t)]Uo+t[U+(t) -

U_(t)]vo•

is a solution of the Cauchy problem (1.1)-(1.3) satisfying condition B). This solution does not depend on the choice of the solution Vo of the equation Bv = u~. Indeed, if Vo is another solution of this equation, then B(vo - Vo) = 0, so that U + (t)(vo -

Vo) = Vo - Vo

and U - (t)(vo -

Vo) = Vo - Vo,

from which it follows that [U+(t) - U_(t)]vo= [U+(t) - U_(t)]vo-

The Cauchy problem (1.1)-(1.3) cannot have other solutions satisfying condition B). In the contrary case we would arrive at the conclusion that for at least one of the equations (L9) the condition of uniqueness of the solution of the Cauchy problem would be violated. Thus, we have proved the sufficiency of the conditions in the following theorem. THEOREM 1.1. For the Cauchy problem for equation (1.1) to have a unique solution with the initial data

(1.12)

u(O)

= Uo E !?J(B 2) and u' (0) =

~ E !?J(B)

n ~(B)

satisfying condition B). it is necessary and sufficient that the Cauchy problem for equations (1.9) be uniformly correct. In order to prove the conditions of the theorem necessary we suppose that %0, Yo E !?J(B 2) and construct the elements Uo

=

(%0

+ Yo) /2,

Vo

=

(%0 - Yo) /2, u~

= Bvo.

Obviously Uo E !?J(B 2 ) and Uo E !?J(B) n !II (B). Construct a solution u(t) of equation (1.1) satisfying the conditions (1.12) and B). Using formulas (1.6) and (1.8), we obtain twice continuously differentiable solutions x(t) and y(t) of equations (1.9).

241

§1. HYPERBOLIC CASE

We shall show that the solutions of equations (1.9) just constructed are unique. Suppose that i(t) and yet) is a further pair of twice continuously differentiable solutions of equations (1.9) with the initial conditions i(O) = Xo and yeO) = Yo. As we have shown the average u(t) = (i(t) + y(t»/2 will then be a solution of the Cauchy problem for equation (1.1) satisfying condition B) and taking on the same initial values as u(t), i.e. u(t) u(t). Write f = x - i and J = y - y. To these solutions of equations (1.9) there correspond solutions il, f} of equations (1.7), with a = O. But then f} const = v(O) = 0, and in view of (1.8) of J O. The uniqueness is proved. The final assertion of the theorem follows from Theorem 2.13 of Chapter I. As we showed in §3 of Chapter I, the correctness of the Cauchy problem for both equations (1.9) implies that the operators

=

=

= =

U(t)

=

{U+(t) for t ~ 0, U_(t) for t ~ 0

form a group. Hence Theorem 1.1 may be alternatively stated in the following form: THEOREM 1.1'. For the Cauchy problem (1.1)-(1.2) to have a unique solution satisfying condition B), it is necessary and sufficient that the operator B be a generating operator of a strongly continuous group of operators U(t) (< t < IX». The solution of the Cauchy problem is given by the formula IX)

u(t)

(1.13)

= HU(t)

+ U( -

t)]Uo

+ HU(t) -

U( - t) ]vo,

is any solution of the equation Bvo = 14. It is not clear from formula (1.13) that the solution u(t) depends continuously on the initial data Uo and u~. If there exists a bounded operator B-t, then the solution where

(1.13)

Vo

u(t)

=

i [U(t)

+ U( -

t)]Uo

+ I [U(t) -

U( - t) ]B-IU~

obviously depends continuously on Uo and u~, this dependence being uniform on each interval [0, T]. As we shall prove below, this also holds in the general case. DEFINITION 1.2. By a weakened solution of equation (1.1) we mean a function u (t) satisfying the following conditions: 1) u (t) is con-

242

III. EQUATIONS OF SECOND ORDER

tinuously differentiable on [0, T] and twice continuously differentiable on (0, T]; the function Bu(t) is defined and continuous on [0, T]; 3) the function Bu'(t) is defined and continuous on (0, T]; 4) u(t) E ~(B2) for t E (0, T]. If u(t) is a weakened solution of equation (1.1), then u'(t)

As

E -+

°

=

U'(E)

+

I'

B 2u(T)dT.

the term outside the integral has a limit. so that

u' +(t)

=

u'(O)

+ (t

J+o B u(T)dT = 2

u'(O)

+B

Jo(' BU(T)dT.

This equation coincides with (1.4). Repeating the arguments, we again arrive at system (1.9). The difference consists in that now the solutions x(t) and yet) constructed according to formulas (1.6) and (1.8) will be twice continuously differentiable only on the semi-interval (0, T]. If now the weakened Cauchy problem has a solution for all Uo and u~ satisfying condition (1.12). then equations (1.9) will have solutions for all %0, Yo E ~(B2), these solutions being twice differentiable on (0, T]. In §3 of Chapter I we investigated the connection between true and weakened solutions of an equation of the first order. It follows from this investigation that, for equations (1.9) under the conditions described above, the weakened Cauchy problem will be solvable for any Xu, Yo E ~(B). Repeating the same arguments as in the proof of Theorem 1.1. we obtain the. following assertion. THEOREM 1.2. For there to exist on [0, T] a unique weakened solution of equation (1.1) for any uo, u~ satisfying conditions (1.12), it is necessary and sufficient that there exist unique weakened solutions of equations (1.9) for any initial conditions from ~(B).

2. Perturbed equations. Consider a more general equation than (1.1): (1.14) where Q is a bounded operator in E. We suppose further that the operator Q maps the domain ~(B) of the operator B into itself. Under these conditions the operator B + Q, closed on ~(B), is closed, and maps the domain ~(B2) into ~(B). Therefore the operators B2, B(B + Q), and (B + Q) 2 have the same domain. The definitions of the solution of equation (1.14) and of property B) remain the same as for equation (1.1).

243

§1. HYPERBOLIC CASE

Assuming that u'(O) = u6E 9£'(B)and repeating the of subsection 1, we arrive at the system of equations (1.15)

du/dt

=

arguments

Bv and dv/dt = (B + Q)u.

After the substitution (1.8) we obtain a system of differential equations for the functions x(t) and y(t): dx/dt

= Bx +!Qx + !Qy,

dy/dt = - By - !Qx - IQy.

This system may be considered as one differential equation in the product space ExE with operators given by the matrices

E = (~ _~) Writing (1.16)

x = g},

and

S = I (_~ ~).

we obtain the equation dx/dt = Ex + Sx.

Every solution of equation (1.14) satisfying condition B) for which u' (0) = Bvo generates, according to the formulas v=vo+ Lt(B+Q)U(r)dr, x=u+v, y=u-v,

x={~}

a twice continuously differentiable solution of equation (1.16). Conversely, every twice continuously differentiable solution f = {;} of equation (1.16) provides us with a solution u = (x + y) /2 of equation (1.14). This solution satisfies condition B), since the function Edf/dt = d 2 f/dt 2

-

is continuous. The initial conditions ~(B2) and u' (0) = B(x(O) - y(O» /2 our construction one finds that u(t) == 0, tions (1.15) that vet) == Vo and Bvo = o.

Sdf/dt

are u(O)

E

~(B)

=

(x(O)

n 9£'(B).

+ y(0»/2 E

under then it will follow from equaThus, the relation u ~ f can fail to be one-to-one only because of a possible nonunique choice of the solution of the equation u' (0) = Bvo in the case when the operator B has zeros. If B is given by a generating operator of a strongly continuous group, then the same is true of the operator B. In view of Theorem 7.5 of Chapter I the operator B + S will also be a generating operator of a group which is strongly continuous in E xE. If the initial conditions If

244

III. EQUATIONS OF SECOND ORDER

.to = {~J lies .....in ~<'B2), and, in view of the property of the operator Q, XoE P}«B + 8)2). There-. fore the corresponding solution of equation (1.16) will be twice continuously differentiable on [0, T]. Then the function Xo,yoE P}(B 2), then the initial vector

u(t)

= i[x(t) + y(t)]

will be a solution of equation (1.14) satisfying condition B). We have amassed all the preparatory material for the proof of the following theorem. THEOREM 1.3. For the Cauchy problem for equation to have a unique solution satisfying condition B) for any initial data (1.12), it is necessary and sufficient that the operator B be a generating operator of a strongly continuous group.

The proof of Theorem 1.3 is quite analogous to the proof of Theorem 1.1. We suppose that the operator B in equation (1.1) has a regular point >.0. Then that equation may be transformed in the following way: (1.17)

where Bo = B - >.01. The operator Bo has a bounde4 Inverse, so that Sif(Bo) = E. Putting Q = 2>.01 + ~RB(>.o) and applying Theorem 1.3, we obtain the following sharpened version of Theorem 1.1: THEOREM 1.1". If the operator B is a generating operator of a strongly continuous group, then the Cauchy problem for equation (1.1) has a unique solution for any Uo E P}(B 2 ) and ufJ E ~(B). This solution depends continuously on the initial data, uniformly on [0, T].

Various equations may be brought into the form (1.14). For example consider the equation (1.18)

d 2u/dt 2 = B 2u

+ Cu,

where the operator C is defined on ~(B). If the operator RB(>.o) C is a bounded closure, then equation (1.18) may be written in the fonn

+ 2>.01 + ~RB(>.o) + RB(>.o) C) u. Q = 2>.01 + ~RB(>.o) + RB(>.o) C is bounded,

d 2u/dt 2 = Bo(Bo

The operator that Theorem 1.3 may be applied. Obviously Q ~(B) C ~(B). We have arrived at the following theorem:

so

245

§1. HYPERBOLIC CASE

THEOREM 1.4. If in equation (1.18) the operator B is a generating operator of a strongly continuous group, and the operator C is defined on ~(B) and has the property that the operator RB(>.o) C has a bounded closure, then for any Uo E ~(B2) and u6 E ~(B) there exists a unique solution of the Cauchy problem for equation (1.18) satisfying condition B).

3. Nonhomogeneous equations. Consider a nonhomogeneous equation of the type (1.19) where f(t) is a continuous function. For simplicity of formulation we suppose that B has a bounded inverse. Acting in the same way as in subsection I, we can reduce the problem of solving equations (1.19) to that of finding twice continuously differentiable solutions of the nonhomogeneous system of equations dxjdt = Bx

(1.20)

+ B-1f(t),

dyjdt = .- By - B-1f(t).

Supposing that B is the generating operator of a strongly continuous group U(t), we may write the solutions of equations (1.20) in the form x(t) = U(t)Xo yet)

= U( -

+

t)yo -

l' l'

U(t - s)B-1f(s)ds, U(s - t)B-1f(s)ds.

If Xo, Yo E ~(B2), then the first terms on the right are twice continuously differentiable. In view of Theorem 6.5 of Chapter I, the integral terms have continuous derivatives. We calculate

~:

= U(t) BXo + B-1f(t)

dy dt

=

-

+ l' U(t -

U( - t)Byo - B-1f(t)

Introducing now the function u(t) =

s) f(s) ds,

+ Jo(' U(s - t)f(s)ds. (x(t) + y(t»j2, we get

du 1 1 1 (' dt = 2 U(t) BXo - 2 U( - t) Byo + 2 Jo [U(t - s) - U(s - t) ]f(s)ds. ( 1.21)

For this function to have a continuous derivative on [0, T], it is sum-

246

III. EQUATIONS OF SECOND ORDER

cient in view of the same Theorem 6.5 of Chapter I that one of the two following conditions be satisfied: the function f(t) is continuously differentiable on [0, T], or the function Bf(t) is defined and continuous on [0, T]. If one of these conditions is satisfied, then u(t) is a solution of (1.19). Moreover, in this case we may apply the operator B to the integral term in (1.21), obtaining as a result a continuous function. This means that the solution u (t) satisfies condition B). THEoREM 1.5. If in equation (1.19) the operator B has a bounded inverse and is a generating operator of a strongly continuous group U(t), and the function f(t) has one of the properties: 1) f(t) is continuously differentiable on [0, T], 2) the function Bf(t) is defined and continuous on [0, T], then for any Uo E !?J(B2) and ut, E !?J(B) there exists a unique solution of the Cauchy problem for equation (1.19) satisfying condition B). This solution is given by the formula

u(t) =

~ [U(t) + U( -

+~

i

t)]Uo + ~ [U(t) - U( - t) ]B-1ut,

[U(t - s) - U(s - t) ]B-1f(s)ds.

We note that Theorem 1.5 is valid also for the more general equation in which B2 is replaced by B(B + Q), just as in equation (1.14). 4. Equations with variable operators. Now we shall consider equations with variable operators: (1.22) We shall reduce this equation also to a system of equations of the first order, for which the leading terms contain the operator coefficients B(t) and - B(t). We shall find the conditions for the solvability of the resulting system on the basis of the theorems of §3 in Chapter II. Therefore we suppose that both B(t) and - B(t) satisfy the conditions of Theorem 3.6 of Chapter II. We recall these: 1) For all t E [0, T] the operators B(t) have one and the same domain !?J(B), and the operators B-1(t) are bounded (0 ~ t ~ T). 2) The operator B(t) on !?J(B) is strongly continuously differentiable relative to t E [0, T]. 3) The operators B(t) and - B(t) admit stable approximation by hounded operators B;t(t) and B;;(t) such that the evolutionary operators

247

§1. HYPERBOLIC CASE

U:(t,8) and U;(t,8) corresponding to the operators B:(t) B;(t) converge strongly and uniformly in t and 8.

and

We note further that in view of Theorem 3.11 of Chapter II the condition B) is satisfied if: 3') The operator B(t), for each tE [0, T], is the generating operator of a strongly continuous contraction group of operators. Suppose that u(t) is the solution of (1.22). Write du/dt=v and B(t)u(t)=w(t).

It follows from (1.22) that (1.23)

dv/dt

=

B2(t)U

=

B(t)w.

In view of the continuity of B2(t)U(t) and of B-1(t), the function wet) will be continuous on [0, T]. Moreover, if the solution satisfies condition B), then (see 11 of Chapter II) the derivative

= B(t)du/dt + B'(t)u = B(t)v + B'(t)B-1(t)w is continuous on [0, T]. Introducing the functions x(t) = wet) (1.24)

dw/dt

yet) = wet) - vet) and f(t) =

I;tz},

+ v(t) ,

we obtain in the product space

ExE the equation (1.25)

df/dt

= B(t)f + S(t)f,

where the operators B(t) and Set) are given by the matrices -B(t)

""" Set)

=

(B(t)

0

0 ) _ B(t) ,

1 (B'(t)B-1(t) B'(t)B-1(t»)

= 2 B'(t)B-1(t) B'(t)B-1(t) •

Thus every solution of (1.22) satisfying condition B) generates a solution of equation (1.25). Conversely, suppose that f(t) is a solution of (1.25), continuously differentiable on [0, TJ. Then the functions wet) = t [x(t) y(t)] and vet) = t [x(t) - y(t)] will be continuously differentiable solutions of equations (1.24) and (1.23). Now consider the function u(t) = B-1(t)w(t). From (1.24) we have

+

du/dt

= - B-1(t)B'(t)B-1(t)w + B-1(t)dw/dt = u.

Further, from (1.23) d 2u/dt 2 = du/dt

= B(t)w = B2(t)U.

248

III. EQUATIONS OF SECOND ORDER

Finally, the function B(t)du/dt = B(t)v = dw/dt - B'(t)B- l (t)w is continuous. This last equation shows that the function u(t) is a solution of equation (1.22) satisfying condition B). We may directly apply Theorems 3.4 and 3.7 of Chapter II to equation (1.25). Its Cauchy problem is uniformly correct if the operator B(t)S(t)B- l (t) is defined and strongly continuous for tE [0, T] or if for those t the operator S(t) is strongly continuously differentiable on 9(B). This leads us to the following theorem.

1.6. Suppose that B(t) satisfies conditions 1)-3) and moreover that it satisfies one of the two following conditions: 4) The operator B' (t) B- 1 (t) is strongly continuously differentiable on 9(B) for t E [0, T]. 4') The operator B (t) B' (t) B- 2 (t) is defined, bounded and strongly continuous in t on [0, T]. Then there exists a unique solution of the Cauchy problem for equation (1.22) for any Uo E g-(B 2(0» and u;, E 9(B). THEOREM

One may analogously treat equations of type (1.14) with variable operators. 5. Equations in Hilbert space. Throughout the foregoing we have assumed that the operator appearing in the equation has as leading term an operator B2(t) , where B(t) is an operator with given properties. If we consider an equation of the form (1.26) then there arises the question as to what conditions make it possible to represent the operator A (t) as the square of an operator B(t) satisfying conditions 1) -3). In the Banach space case we do not possess a satisfactory answer to this question even in the case of a constant operator A. In Hilbert space there is an important special case when this problem is easily solved. In fact, suppose that the operator A (t) is selfadjoint and negative definite, and that it has a domain 9(A) not depending on t. Suppose further that it is strongly continuously differentiable with respect to t. Then in view of Theorem 1.2 of Chapter II the operator B(t) = i ( - A (t» 1/2 will have a constant domain and on that domain it will be continuously differentiable with respect to t. For each t the operator B(t) will be a generating operator for a group of unitary operators. Analogously, B(t) satisfies conditions

249

§2. ELLIPTIC CASE

1)-3) of the preceding subsection. Moreover, from Lemma 1.6 of Chapter II it follows that

B(t)B'(t)B- 2 (t)

= B'(t)B- 1 (t) - A' (t)A- 1 (t)

so that conditon 4) of Theorem 1.6 is satisfied. From what has been said there follows the following proposition. THEOREM 1.7. Suppose that A (t) is a selfadjoint negative definite operator with a domain 9,(A) which is constant relative to t. Suppose that A (t) is strongly continuously differentiable on 9'(A). Then the Cauchy problem for equation (1.26) has a unique solution for any Uo E g?(A) and uO E A)1/2).

g?« -

§2. The elliptic case. Boundary problems In this section we shall study equations of second order with constant operator, (2.1)

(O~t~T),

under the hypothesis that the operator A has the property that for all ). ~ 0 there is a resolvent R().) of the operator A and ().~O).

(2.2)

This case is called elliptic. 1. The general solution of the homogeneous equation. Consider the homogeneous equation (2.3)

(O~t~T).

The definition of a solution of equation (2.3) will be the same as for equation (1.1): DEFINITION 2.1. A solution of equation (2.3) is a function u(t) with values in g?(A), twice continuously differentiable and satisfying equation (2.3) on the interval [0, T]. As we showed in §5 of Chapter I, it is possible to define fractional POwers for the operator A, and, in particular, the operator A -1/2. Consider the function (2.4)

v(t) = A -1/2du/dt.

Differentiating it with respect to t and using (2.3), we get (2.5) du/dt = A -1/2d 2u/dt 2 = A 1/2U. Equations (2.4) and (2.5) may be written in the form of a system:

250

III. EQUATIONS OF SECOND ORDER

du/dt = Al/2V, dv/dt = Al/2U. We make the substitution z = t (u - v), w = t (u + v). Then for the functions z(t) and wet) we obtain the equations

dz/dt = .,- Al/2Z;

(2.6)

dw/dt = Al/2w.

The operator - A 1/2 is a generating operator for an analytic semigroup Vet) satisfying the Co-condition. Thus for the first of equations (2.6) the Cauchy problem is uniformly correct, and for the second the inverse Cauchy problem is uniformly correct. We have

z(t) = V(t)Zo and wet) = veT - t)WT.

(2.7)

For z(t) and wet) to generate the solution of equation (2.3) it is necessary that they be twice continuously differentiable. This will be the case if Zo, wTE ~(A). However for equations (2.6) the solutions for < t < T will be infinitely differentiable for any Zo and WT. In this connection we introduce the following definitions. DEFINITION 2.2. The function u (t) is said to be a weakened solution of equation (2.3) if: 1) it is continuous and has a continuous first derivative on the interval [0, T] and a second derivative on (0, T); 2) its values lie in ~(A) for < t < T, and the function A 1/2U(t) is continuous on the entire interval ~ t ~ T; 3) u,(t) satisfies equation (2.3) in the interval (0, T). DEFINITION 2.3. The function u (t) is said to be a generalized solution of equation (2.3) if: 1) it is continuous on [0, T] and has a continuous second derivative· on (0, T), and the function A -1/2U (t) has a continuous first derivative on [0, T]; 2) the values of the function u(t) (0 < t < T) lie in ~(A), and 3) they satisfy equation (2.3) in the interval (0, T). If Zo, WT E ~(A 1/2), then the function

°

° °

u(t) = V(t)Zo + VeT - t)WT

(2.8)

is a weakened solution of equation (2.3). In fact,

A 1/2U(t)

=

du/dt =

V(t) A 1/2Zo + VeT - t)A 1/2WT' -

V(t) A 1/2Zo + VeT - t)A 1/2WT (0

~

t

~

T)

and

d 2u/dt 2 = A V(t)Zo + Av(T - t)WT = Au

(0

< t < 71.

§2. ELLIPTIC CASE

251

The continuity of the functions A 1/2(t) and du/dt on the closed interval [0, T] and that of the functions d 2u/dt 2 on the open interval (0, T) follow from the properties of the semigroup V(t). Analogously one verifies that the function (2.8) is a generalized solution of equation (2.3) for any Zo, WT E E. Conversely, suppose that u(t) is a generalized solution of (2.3). Then the function

v(t) = d(A -1/2u )/dt is continuous on [0, TJ

and satisfies equations (2.4) and (2.5) for z(t) and w(t) are weakened solutions of the direct and inverse Cauchy problems for equations (2.6), and so representable in the form (2.7), where

0< t < T. This means that the functions

= z(O) = Hu(O) - (A -1/2U )'(0», WT = Hu(T) + (A -1/2U)'(T». Zo

Now if u(t) is a weakened solution of (2.3), then Zo

= t (u (0) - A -1/2U ' (0» E 9(A 1/2),

WT=i(U(T) +A- 1/2U'(T» E 9(AI/2). Thus we have arrived at the following assertion. THEOREM

2.1. Every generalized solution of equation (2.3) has the

form (2.8), and, conversely, a function of the form (2.8) is a generalized solution of (2.3) for any Zo, WT E E. For the generalized solution (2.8) to be a weakened solution, it is necessary and sufficient that Zo, WT E 9(A 1/2). Every generalized solution of (2.3) is an analytic function of t for 0 < t < T. In what follows we shall consider only weakened or generalized solutions of equation (2.3). 2. Boundary value problems. Consider the system of boundary conditions

(2.9)

LI (u) = allUO + al2 uO+ i311UT + i312 u r = fl> ~(u) =

a21UO + a22uO + i32IUT+ i322ur = f2,

where aij> {Jij (i, j = 1,2) are complex numbers, fl and 12 given elements oftheBanachspaceE, uo=u(O),uo=u'(O), UT=U(T) and ur=u'(T). The forms LI (t) and L 2 (u) are assumed linearly independent. DEFINITION 2.4. If the weakened solution u (t) of equation (2.3)

252

III. EQUATIONS OF SECOND ORDER

satisfies the boundary conditions (2.9), then we will call it a weakened solution 01 the boundary value problem (2.3)-(2.9). Writing A 1/22'0 = gl and A 1/2WT = g2, we may bring any weakened solution of (2.3) into the form (2.10)

where V 1(t) = V(t) A

-1/2

and V 2(t)

= V(t) = V(T - t)A -1/2,

and gl and g2 are certain elements of the space E. Substituting (2.10) into the boundary conditions (2.9), we arrive at a system of operator equations:

+ Lt ( V2)g2 = ft, L 2( V1)gl + L 2( V 2)g2 = 12 Ll ( V 1)gl

(2.11)

for the determination of the elements gl and g2. In expanded form this system has the following form:

+ fJn V(T)A -1/2 - fJl2 V(T)]g1 + [au V(T)A -1/2 + a12 V(T) + fJl1A -1/2 + fJl2I]g2 = ft, [a21A -1/2 - a22I + fJ21 V(T)A -1/2 - fJ22 V(T)]gI + [a21 V(T)A -1/2 + a22 V(T) + fJ21A -1/2 + fJ22 I ]g2 = 12' [anA -1/2

-

al2I

All the operators entering into the system (2.11) are linear combinations of the bounded operators I, A -1/2. V(T) and V(T)A -1/2, commuting with one another. Therefore the system (2.11) may be solved just as in the scalar case. Here an important role is played by the operator determinant of the system: (2.12)

D =

I

L 1 ( VI) L 1 ( V 2 )

L 2 ( VI) L 2 ( V2 )

I ,

called the characteristic determinant. Solving the system (2.11) by the elimination of unknowns, we arrive at the equations (2.13)

Dg1 = L 2( V 2 ) 11 - Ll (V2) 12, Dg2 = L 1(V1)f2 - ~(Vl)fl'

Applying the bounded operator D to (2.10) and replacing Dg1 and

§2. ELLIPTIC CASE

253

Dg2 according to formulas (2.13), we obtain the relation

(2.14) where (2.15)

= Vl(t)~(V2) - V2(t)~(Vl)' S2(t) = - V 1 (t)L 1 (V2) + V 2(t)L l (V1).

SI(t)

DEFINITION 2.5. Every function continuous on [0, T] and satisfying relation (2.14) is said to be a generalized solution of the boundary value problem (2.3)-(2.9). If we suppose that the operators D-ILj(~) ('i, j = 1,2) are defined in the entire space E, then (2.13) yields elements gl and g2, the substitution of which into (2.10) gives a solution of the boundary value problem. If now we only suppose that the operators D- 1S1 (t) and D- 1S2 (t) are defined everywhere for t> 0, then all we can find from (2.14) is the generalized solution of the boundary value problem

(2.16) Thus we have arrived at the necessity of investigating the question of the existence of an inverse operator D- 1 and its connection with the operators Li ( ~). The different possibilities which arise here are closely connected with the various properties of the boundary value problem (2.3)-(2.9). 3. Homogeneous boundary value problems. The question of the existence of the operator D- 1 is naturally connected with the solvability of the homogeneous boundary value problem for the equation (2.3), i.e. the problem of finding a weakened solution of (2.3) satisfying the boundary conditions (2.17)

Lt(u) = L 2 (u) = O.

THEOREM 2.2. The homogeneous boundary value problem (2.3)-(2.17) has a nonnull solution if and only if the characteristic determinant D,

as an operator in the space E, does not have an inverse. NECESSITY. Suppose that the problem (2.3)-(2.17) has a nontrivial solution u(t) ¢. O. Then, as is every weakened solution of equation (2.3), it is representable in the form (2.10), where at least one of the elements gt. g2 is nonzero. Equations (2.13) for the homogeneous problem take the form

254

III. EQUATIONS OF SECOND ORDER

i.e. D vanishes on at least one nonzero element. And this means that it does not have an inverse operator. SUFFICENCY. Suppose that D does not have an inverse. Then there exists a nonzero element h E E such that Dh = o. Using the element h we may construct a nonzero solution u(t) of the homogeneous problem. To this end it suffices to select elements gh g2 E E, and formula (2.10) gives the desired solution. The basic difficulty here is that the solution must be constructed so that it satisfies the boundary conditions (2.17), i.e. so that the following relations are satisfied:

+ L 1(V2)g2 = 0, ~(Vl)gl + L 2(V2)g2 = o.

L1(V1)gl

(2.18)

Two cases may occu"r here: a) all the coefficients of the system (2.18) carry the element h into zero; b) one of the coefficients on the element h differs from zero. In the first case it is sufficient to take gl = g2 = h, and the boundary conditions are satisfied. In the second case we need to choose appropriate cofactors as the solutions of the system. Suppose for example that ~(V2) ~O. Then we put gl=~(V2)h, g2= -L2(V1)h. We shall verify that the elements thus constructed are a solution of (2.18): Lt(Vl)~(V2)h - L 1(V2)L2 (V1)h

== Dh = 0, L2(Vl)L2(V2)h-L2(V2)~(Vl)h= O. To make the proof complete we need also to verify that if gl ~ 0 or then u(t) = V l (t)gl + V 2(t)g2 ~ 0 on [0, T]. This means that the particular solutions Vl(t) and V 2(t) of the homogeneous differential equation are in a certain sense linearly independent. Suppose then the contrary, i.e. that

g2 ~ 0,

(2.19)

V(t) A

-1/2gl

+ V(T -

t)A -1/2g2

== 0 (0

~ t ~

T).

Differentiating this relation and applying the operator A -1/2 to the result, we obtain (2.20)

- V(t) A -l/2gl + V(T - t)A -1/2g2 == 0

(0

~ t ~

T).

On adding (2.19) and (2.20) we get V(T - t)A -1/2g2 = 0, and on sub-

255

§2. ELLIPTIC CASE

traction V(t) A -1/381 = O. In particular, for t = T and t = 0 we have A -1/282 = A -1/281 = 0, or, after application of the operator A 1/2, gj = 82 = O. The theorem is proved. It is convenient to employ Theorem 2.2 in the following formulation. THEOREM 2.3. The operator n- 1 exists if and only if the homogeneous boundary problem (2.3)-(2.17) has only the trivial solution u (t) == O.

4. The dual boundary value problem. Suppose that E' is the Banach space dual to E, and A' the operator dual to A! acting in E'. By hypothesis, the domain of A is dense in E, so that the operator A' exists. Suppose further that yet) is a function with values in E', defined on the segment [0, T]. The differential operator (2.21)

is said to be the dual to the operator L(u) == d 2u/dt 2 - Au. If we understand as the product (y, u) the result of applying the functional y E E' to the element u E E, as in the scalar case, integration by parts gives us the following "Green's identity":

LT[(Y,L(U» - (L'(y),u)]dt= Q(y,u),

(2.22)

where Q(y, u) is the bilinear form (2.23)

Q(y,u) == [(y,u') - (y',u)Jll = (YT,UT) - (yhUT) - (Yo,uQ)

DEFINITION

(2.24)

+ (yo,uo).

2.6. Boundary conditions of the form LHy) == 'Yl1YO

+ 'Y12YO + c5

L2(y) == 'Y21YO + 'Y22YO +

11

YT+

c521 YT

c5 12 YT

+ c5

22

= 0,

YT = 0,

where 'Yij, c5ij are certain complex numbers, are said to be dual to the original boundary conditions (2.17) if for any pair of functions yet) and u(t) satisfying the relations yet) E P.)(L') , Lay) = LHY) = 0 and

u(t) E P.)(L) , Ldu)

=

L 2 (u) = 0,

the identity Q(y, u) = 0 holds. We have been assuming that all the points of the nonpositive real

256

III. EQUATIONS OF SECOND ORDER

axis lie in the resolvent set of the operator A. This assertion remains valid also for the operator A'. Moreover, inequality (2.2) implies an analogous inequality for the operator A I:

(2.25) Suppose further that 9(A') is dense in E'. This hypothesis is automatically satisfied for reflexive Banach spaces; see [1]. If it holds, the assertions proved for the original problem carry over to the dual boundary value problem L'(y) == d 2y/dt 2 - A'y = 0, LHy) = L;' = O.

(2.26)

The same role is played for example by the determinant D(A') set up for (2.26). The characteristic determinant of the original boundary problem will now be denoted by D(A). The connection between these two determinants is made clear by the following theorem. In connection with this statement, one should note that the dual boundary conditions are not defined uniquely. THEOREM 2.4. In the class of dual boundary conditions (2.24) there exist those for' which the characteristic determinant D(A'), set up for these conditions as an operator in E', is dual to the operator D(A):

(2.27)

D(A') = [D(A)]'.

PROOF. Write out the matrix of the coefficients of the original boundary value problem:

. (an au I1n l1u).

(2.28)

a21

a22 1121 1122

The minor of this matrix composed of the ith and jth columns (i <j) will be denoted by djj • If now the determinant D(A) of the system (2.11) is represented in the form of a sum of determinants anA -1/2

l a21 A -1/2

an V(T)A -1/21 a21 V(T)A -1/2

,

anA -1/2 a12 V(T) 1 , ••••

l a21 A -1/2 a22 V(T)

we obtain the following expression for D(A): D(A)

(2.29)

= - duI + (d 14 - d 23 )A -1/2 + d 13 A -1 + 2 (d12 + d 34 )A -1/2V(T)

+ [duI + (d

14 -

d 23 )A-

1/2

- d13 A -

1]

V(2T).

Since the domain 9(A') is by hypothesis dense in E', the family of operators V' (t) coincides with the semigroup W(t) generated by the

257

§2. ELLIPTIC CASE

operator - (A') 1/2. Hence it follows that the operator dual to D(A) will be [D(A')]' = - d24I'

+ (d14 -

c4a) (A') -1/2 + dJ.'j(A,)-1

+ 2 (d12 + d34 ) (A') -1/2W(T)

(2.30)

+ [~I' + (d14 -

c4a) (A') -1/2 -

d I3 (A') -1]W(2T).

N ow suppose that dij are the minors of the matrix of coefficients of the dual problem: ( 1'11 1'12 ~21 ~12). 1'21 l' 22 ~1 ~22

(2.31)

Write out the determinant D(A') in expanded form, for which it suffices in formula (2.29) to replace A by A' and the dij by dij : D(A')

+ (d14 - d (A') -1/2 + d + 2(d12 + d 34 ) (A') -1/2W(T) + [d24 I' + (d14 - d (A') d

= -

(2.32)

d24I'

I3 (A,)-1

23 )

-1/2 -

23 )

I3 (A') -1 ]W(2T).

Comparison of expressions (2.30) and (2.32) shows that for the proof of the theorem it suffices to verify the equality of the coefficients of the operators in corresponding terms. Here we have to consider several cases: 10. d12 ~ O. Up to the order of rows, (2.28) is equivalent to the matrix

(1o 01 ~llP21

(2.33)

~12). P22

and the condition Ll (u) = L 2 (u) = 0 may be written in the form Uo

= - ~11UT - §12Ut,

u6 = -

§21 U T

-

P22 U

t.

SUbstituting these expressions into the form Q( Y, u) in (2.23) yields Q(y,u)

= y~t- YtUT- Yo( - §21UT- §22Ut) + Yo( - §llUT - §12 Ut) = (§21YO - §11YO - yt)UT+ (P~Yo - §12YO + YT)Ut'

Hence it is clear that Q(Y, Y) vanishes, if one takes /

with the matrix

== §21YO - §nYo - YT = 0, LHY) == fJ~Yo - §12YO + YT = 0

LHY)

258

III. EQUATIONS OF SECOND ORDER

0 - 01) .

( ~21

- ~ll fJ22 - fJ12 1

(2.34)

The explicit form of the matrices (2.33) and (2.34) makes it possible

to compare the coefficients in (2.30) and (2.32):

du =

- P12 = d'J,4; d23 = P22 + Pll = d14 - ~; d13 = P21 = d13;

d14 d12 +da.=

It =~: I I~ -~ I I~ ~ I +

=

+

I~: ~: I= d12 + da..

Thus for the first case the theorem is proved. 2°. d13 ¢ o. The matrix (2.28) is equivalent to the following (1

(2.35)

o

a12 0 ~12), a22 1 fJ22

or, in the form of boundary conditions, , ;; , Uo = - a12UO - I-'12 UT,

-

UT = - a22u6 - (3~UT. For Q(Y, u) we obtain

Q( y, u)

= (- Yo -

a12Y6 + a22Yt) U6 +

(- tI12YO + YT + P22Yt) Ut.

As the dual boundary conditions we put

LJ.(y) = - P12Y6 + YT+ P22Yt = 0, LHY) = - Yo - a12Y6 + a22Yt = 0 with the matrix (2.36) Comparison of the minors of (2.35) and (2.36) proves the theorem in this case as well. The remaining four cases d14 ¢ 0, d23 ¢ 0, d'J,4 ¢ 0, and da. ¢ 0 are considered analogously. COROLLARY. If ~(A') is dense in E', then for the homogeneous dual boundary problem to have a nonzero solution it is necessary and sufficient that the range .!iR(D) of the operator D not be dense in the space E.

§2. ELLIPTIC CASE

259

Indeed, the last condition is satisfied if and only if there exist nonzero solutions of the equation D'y = O. But D' = D(A'), and the assertion follows from Theorem 2.3.

5. Uniformly corred boundary value problems. Regular boundary conditions. DEFINITION 2.7. The boundary value problem (2.3)-(2.9) is said to be uniformly correct if for every flo 12 of E there exists a unique generalized solution of this problem, continuously depending on fi> f2 E E in the norm of the space C(E). From what was said in subsection 3 one immediately deduces the following. THEOREM 2.5. For the problem (2.3)-(2.9) to be uniformly correct, it is necessary and sufficient that the operators D- 1S 1 (t) and n- 1S 2 (t) be uniformly bounded on [0, T].

It turns out that for a uniformly correct problem an essential role is played by the regularity property of the boundary conditions. 1) In our case the boundary conditions will be regular only when one of the following conditions is satisfied:

1°. du ¢ O. 2°. d u = 0, but la121 + 1.8121> 0 and d23 - du• ¢ O. 3°. a12 = .812 = a22 = .822 = 0, but dIS ¢ O. We shall deal with these cases in detail. 1 0. d u ¢ O. Put d u = - 1. Then the characteristic determinant D, according to formula (2.29), may be represented in the form n = I - Rh where Rl is a bounded operator (a linear combination of the bounded operators A -112, V(T) and their products). If we suppose that unity lies in the resolvent set of the operator Rh then the operator D will have a bounded inverse. Therefore it follows that the operators n- 1s1 (t) and D- 1S2 (t) will be uniformly bounded on [0, T]. 2°. d u = 0, 1a121 + 1.8121 > 0 and d23 - d 14 ¢ O. Suppose that d23 - d 14 = - 1. Under these conditions the matrix (2.28) can be represented in the form (2.37)

1) M. A. Naimark, Linear differential operators, transl., Ungar, New York, 1967.

GITI'L, Moscow, 1954; English

260

III. EQUATIONS OF SECOND ORDER

and the determinant D, by formula (2.29), takes the form D = A -1/2(1 - R2 ),

where R2 is a bounded operator. Thus, even if the operator (1 - R 2 ) -I is bounded, the operator D- 1 = (1 - R 2 ) -IA 1/2 is nevertheless unbounded. However the operators SI(t) and S2(t) have as factor the operator A -1/2, so that the products D- 1S1 (t) and D- I S2 (t) will be uniformly bounded if the operator (1 - R 2 ) -1 is bounded. 3°. al:l = fh2 = a22 = fJ22 = 0, but dl3 ¢ O. These conditions are equivalent to the first boundary problem Uo = fl> UT = f2 with the matrix

( 1 0 0 0) o0 1 0 •

(2.38)

For the determinant D we obtain the formula D

=

A-I[I - V(2T)],

i.e. the operator D-l = [I - V(2T)]-IA, ifit exists, is unbounded because of the factor A. However, in view of the fact that the boundary conditions do not . contain derivatives, the operators L;(u) have the factor A -1/2, so that the operators SI (t) and S2(t) have the factor A -I. This factor "kills" the unboundedness of the operator D- 1• THEOREM

2.6. Suppose that the boundary conditions of the problem

(2.3)-(2.9) are regular. Then the operator D may be represented in one of

the following three forms: 1°~D=c(1-R),

2°. D = cA -1/2(1 - R), 3°. D = cA -1(1 - R), where R is a bounded operator (particular to the case) and c is a constant. If unity is not a point of the spectrum of R, then the boundary problem (2.3)-(2.9) is uniformly correct on the segment [0, T]. All the generalized solutions of the problem are generalized solutions of equation (2.3).

In the proof we need only the last assertion. I t follows from the fact that in the cases at hand the generalized solution has the form u(t)

=

V(t) hi + V(T - t)h 2 ,

where hi and h2 are certain elements of E. Let us elucidate further the question as to when under regular boundary conditions the generalized solutions of a uniformly correct

261

§2. ELLIPTIC CASE

problem are weakened solutions. For this it suffices that it be possible to determine the elements gl and g2 from equations (2.13). Consider these equations in all three cases of regularity of the boundary conditions. In case 10 the operator D- l is unbounded, so that gl and g2 may be found for any 12 E E. In case 20 we will suppose that the matrix (2.28) has the form (2.37). Then the boundary condition L 2 (u) = a22Uo + P2lUT determined by it does not contain derivatives, so that the operators L 2 (Vj) contain the factor A -1/2. Therefore it follows that the operators D- l L 2 ( Vj) are bounded. The operators D-lL I ( l-j) are unbounded and have the form ~A 1/2, where ~ is a bounded operator. Hence gl and g2 cannot be determined for any arbitrary element f2. It is required that '2E 9(Al/2). Finally, in case 3° with the matrix (2.38), all the operators D -1 L j ( l-j) have the form cA 1/2, so that equations (2.13) are solvable for III 12 E 9(A 112).

'1,

THEOREM 2.7. Under the hypotheses 01 Theorem 2.6 the generalized solution will be a weakened solution lor arbitrary III 12 E E in case 10. In order that it be weakened in the remaining cases, it suffices that fl, 12 E 9l(A 1/2) •

8. Incorrect boundary value problems. For nonregular conditions (2.9) there remains only the possibility that (2.39)

d 24 =

0, la121

+ IP121 > 0,

d23 - d 1•

=

o.

The determinant D takes the form D=A-l[dI3I+2(d12+d34)A1I2V(T) -d13 V C2T)].

For analytic semigroups the operator A 1/2 V(t) is bounded, so that the operator in square brackets is also bounded. Suppose that d13 ~ o. Then D = A -1(d13 I - R.), where R. is a bounded operator. If dIS does not lie in the spectrum of the operator R., then D- l is proportional to A, with a bounded operator coefficient. At the same time, in view of the condition Ia121 + IP131 > 0 the operator Ll (u) contains a differentiation, which "kills" the factor A -1/2 appearing in the functions Vj(t) (i = 0, 1,2). As a result the operator

_I A -1/2V(t) S2(t) -

LI(Vl )

A -1/2V(T - t) 1 L l (V2)

contains as a factor only A -1/2, and the product D- l S2 (t) turns out to be an unbounded operator of the type RA 1/2. This means that problem (2.3)-(2.9) for the boundary conditions in question,

262

III. EQUATIONS OF SECOND ORDER

(~11 a12 fJ:l (312),

(2.40)

a21

0

f321

0

la121 +1f3121

>0,

d24 = d'}JI - d14 = 0,

is incorrect in the large. But if we suppose that 12 E 9(A 1/2), then the element D- 1S 2 (t)12 will be completely defined. Further, the expression for SI(t) contains the factor A -1. Accordingly, the operator D- 1S1 (t) will be bounded, and a generalized solution may be found from formula (2.16). In this connection a boundary problem of the type (2.40) will be called semicorrect. We note that for each f2 E E the function D- 1S2(t) f2 will be defined for any tE (0, T), since for these t the operators Al/2 V(t) and A 1/2 V( T - t) are bounded. However this function, and thus the function V(t) as well, may increase unboundedly as t ---+ 0 or t ---+ ex>. If we nevertheless consider the function u(t) in some sense to be a solution of the problem, then that solution will not depend continuously on f2 in the norm of C(E) , but will depend on it continuously for each fixed t E [0, T]. Suppose finally that we add to the conditions (2.38) one further one, dIS = O. Then

so that D- 1 = (l/2(d12 + d34 » A 1/2V- 1(T).

The operator D- 1, . generally speaking, is not applicable to any of the terms on the right of (2.14). The boundary problem (2.3)-(2.9) turns out to be "strongly incorrect". The general form of the corresponding boundary conditions is as follows~

aU' (0) - f3u' (T) = flo aU(O)

+ f3u(T)

= f2'

For f3 = 0 we obtain the conditions of the Cauchy problem. THEOREM

2.8. The Cauchy problem is incorrect for equation (2.3).

The Cauchy problem for equation (2.3) naturally generates a Cauchy problem for the system (2.6) with the initial conditions z(O)

= Huo -

A

-1/2 U

6) and w(O) = Huo + A

-1/2 UO)'

§2. ELLIPTIC CASE

263

For the first equation (2.6) the Cauchy problem is uniformly correct, 8Ild for the second it is incorrect. However, inasmuch as the operator - A 112 is a generating operator for an analytic semigroup, the Cauchy problem for it is correct in the class of bounded solutions (see §3 of Chapter I). Thus, for the second equation the Cauchy problem will be correct in the class of bounded solutions w(t). We have arrived at the following assertion: 2.9. The Cauchy problem for equation (2.3) is correct the class of solutions u(t) for which the function u(t) + A -lU' (t) uniformly bounded on [0, T] by some constant M. THEoREM

m lS

If a solution of the Cauchy problem for equation (2.3) exists, it is unique. COROLLARY.

7. The case of a completely continuous operator A -1. For regular boundary conditions Theorem 2.6 reduces the question of the uniform correctness of the problem (2.3)-(2.9) to the question of the existence of a bounded inverse to the operator I - R. Such an operator certainly does not exist if D vanishes on some nonzero element. In view of Theorem 2.3 this will be the case if and only if the corresponding homogeneous boundary problem has nontrivial solutions. However the operator D may be unbounded in other cases as well. Suppose for example that A has a point a in its continuous spectrum. Consider the regular boundary conditions defined by the matrix

( Va 1 0 0). o 0 Va 1 Then D = (aI - A) A -1[1 - V(2T)] and I - R = (I - A/a) [I - V(2T)].

If the factor 1- V(2T) has a bounded inverse, then 1- R does not have a bounded inverse because of the factor (I - A/a). The difficulty just indicated is removed on considering a special but important class of operators A, those having a completely continuous inverse. The theory resulting in this case is analogus to the theory of scalar boundary value problems. An example is as follows: 2.10. Suppose that the boundary conditions (2.9) are regular and the operator A -1 completely continuous. For the boundary problem (2.3)-(2.9) to be uniformly continuous, it is necessary and sufficient that the corresponding homogeneous boundary problem have only zero solutions. THEOREM

264

III. EQUATIONS OF SECOND ORDER

If the domain of the dual operator A' is dense in E', then these same conditions are necessary and sufficient for the uniform correctness of the dual problem (2.41)

d2yjdt Z = A'y; Ll(y) = t/Jl; LHy) = t/Jz.

PROOF. If the operator A -1 is completely continuous, then the same is true of the operator A -1/2. Further, the operator

V(t)

=

Al/ZV(t)A-l/Z

is for t > 0 completely continuous as a product of a bounded operator by a completely continuous one. By Theorem 2.6, under the regular boundary conditions

where a takes one of the values 0,l,1, and R is some polynomial in the completely continuous operator A -1/2 and V(T). Accordingly, it is a completely continuous operator. But then the point 1 of the complex plane either is an eigenvalue of R, or else lies in its resolvent set. In the first case the determinant D does not have an inverse, and the homogeneous problem has a nontrivial solution. In the second case the operator (l - R) -1 exists, and is defined and bounded on the whole space. Accordingly, in view of Theorem 2.6, the boundary value problem (2.3)-(2.9) is uniformly correct. By Theorem 2.4 the determinant D(A') of the dual boundary problem, as an operator in E', is dual to the determinant D (A). The spectra of the dual operators D(A) and D(A') coincide. Moreover, since R is completely continuous, the operator R' is too. Therefore all that we have just said about the problem (2.3)-(2.9) remains valid for the nonhomogeneous dual problem (2.41). Thus the proof of the theorem is complete. 8. Boundedness toward infinity of the solution. Suppose that u (t) is a generalized solution of equation (2.3), defined on [0, CD). Suppose that it is bounded: (2.42)

sup O~t<",

II u(t) I <

CD.

The functions u (t) and A -l/Zu are continuous at zero. In view of Theorem 2.1, for any T> 0, for 0 ;;;i! t;;;; we have the representation

r

(2.43)

u(t)

= 1V(t) [u(O)

- (A -1/2 U ') (0)]

+! V(T -

t)[u(T)

+ (A -l/ZU)'(T)].

265

§2. ELLIPTIC CASE

Putting t = 0 and then t = T in this identity, we get u(O)

(2.44)

+ (A- 1/ U)'(0) = 2

V(T)[u(T)

+ (A-

1/ 2U)'(T)],

u(T) - (A- 1/ 2 U)'(T) = V(T)[u(O) - A- 1/2 U)'(0)].

Excluding the terms with (A -1/2U )'(T) from these equations, we arrive at the relation (2.45)

u(O)

+ (A-

1/ 2U)'(0)

=

2V(T)u(T) - V(2T)[u(0) - (A- 1/ 2 U)'(0)].

It follows from Theorem 5.4 of Chapter I that for the resolvent of the operator A 1/2 the following estimate holds:

I R A l/ 2 (X) I

~

M/(l + I xl>

I Vet) I

(2.46)

~

(X ~ 0),

Me -I.

Letting T tend to infinity in equation (2.45), and taking account of (2.42) and (2.46), we arrive at the conclusion that (2.47)

u(O)

+ (A -1/2U )' (0) = o.

Now using the fact that the analytic semigroup Vet) cannot vanish on a nonzero element for any t, we find from the first equation in (2.44) that (2.48) u(T) + (A -1/2U)'(T) = O. Equations (2.47), (2.48) and (2.43) lead to the formula (2.49)

u(t)

= V(t)uo.

We have proved the following assertion: THEOREM 2.11. For each Uo E E there exists a unique generalized solution of equation (2.3), bounded on the semiaxis [0, (Xl), satisfying the initial condition u(O) = uo. This solution is given by the formula (2.49).

9. Nonhomogeneous equations. Now we turn to the consideration of the nonhomogeneous equation (2.50) with the same operator A and a function f(t) continuous on [0, T]. The definitions of solution, weakened solution, and generalized solution for equation (2.50) remain the same as for the homogeneous equation (2.5). Making the substitution v = A -1/2du/dt, z = Hu - v), w = Hu + v) reduces equation (2.50) to the form

266

III. EQUATIONS OF SECOND ORDER

dz/dt = - A 1/2Z dw/dt = A 1/2W -

(2.51)

+ fA -1/2f(t) , fA -l/'1(t)

(O~t~T).

The solution of the first equation with the initial condition z(O) be written in the form (2.52)

z(t)

=

V(t)zo

+ ~ fot V(t -

= Zo

may

r)A -1/2f(r) dr.

For the second equation one solves the inverse Cauchy problem with the terminal condition w(T) = WT: (2.53)

w(t) = V(T - t)WT+

~ i T V(r -

t)A -1/2f(r) dr.

For the general solution u(t) of equation (2.50) we obtain the formula (2.54) u(t)

=

1 (T V(t)Zo+ V(T - t)WT+ 2 V(I t - rl)A -l/2f(r) dr.

Jo

The first two terms in (2.54) are the general solution of the homogeneous equation, which we have already studied. We shall occupy ourselves in more detail with the functions V.(t )=tV(lt- I)A_1/2={tV(t-r)A-1/2 for t~r, o ,r r lV(r-t)A- 1/2, for t~r.

From the definition of the function Vo(t, r) one perceives directly that it has the following properties characterizing the fundamental solution of the differential equation (2.50): (I) Vo(t, r) is an operator function of two variables, defined and strongly continuous in the square 0 ~ t, r ~ T. (II) As a function of t it is strongly continuously differentiable in each of the intervals [0, r) and (r, T] and twice continuously differentiable in the intervals (0, r) and (r, T), while for any x E E lim

~[Vo(t, r)x] -

t~T-oat

lim

~[Vo(t, r)x] =

t_T+Oat

x.

(III) The function A l/2Vo(t, r) is defined and strongly continuous on the entire square 0 ~ t, r ~ T, and the function A Vo(t, r), considered as a function of t, is defined and strongly continuous in each of the intervals 0 < t < rand r < t < T. (IV) As a function of t, Vo(t, r) satisfies the homogeneous differential equation a 2Vol at 2 = A Vo in each of the intervals 0 < t < rand T
267

§2. ELLIPTIC CASE

(2.55) We shall determine under what conditions it is a weakened solution of equation (2.50). We have (2.56)

11t

-~ = - dt

2

Vet - T) f(T)dT

0

liT

+ -2

VeT - t) f(T)dT.

t

It is clear from formulas (2.55) and (2.56) that the function get) is continuous and has a continuous first derivative on [0, T]. The function . A1!2g(t)

=} lT V(lt -

TI)f(T)dT

is also continuous on [0, T]. However, generally speaking, the func-

tion get) may fail to have a second derivative, and its values may fail to lie in 9"(A). On the basis of §6 of Chapter I both of these properties are satisfied for 0 ~ t < T if the function f(t) satisfies a Holder condition. By differentiation one verifies that then formula (2.54) yields a solution of equation (2.50). N ow consider, for equation (2.50), the boundary value problem with the conditions (2.9). We have already studied this problem for the homogeneous equation, and so, in view of the linearity of equation (2.50), it suffices to study the problem with the homogeneous boundary conditions (2.57) We represent the solution (2.54) in the form (2.58) where VI (t), V2 (t) , gl and

g2

are the same as in (2.10), and

(2.59) Substituting the solution into condition (2.57), we arrive at a system for the determination of gl and g2 analogous to (2.13): (2.60)

Dg1

=

-

+L (V )L (g) + L

L 2 (V2) L1(g)

Dg2 = - L I

1

2

1 (V2 )L2 (g),

1)L1(g).

2 (V

The function get), being a particular weakened solution of (2.50), is continuously differentiable on the segment [0, T], while

268

III. EQUATIONS OF SECOND ORDER

: =

J:

[:t VO(t, r) ] f(r) dr.

T

Hence Lj(g) = flLj(Vo),f(r) dr, where the index t indicates that the operator L j operates on Vo(t, r) as on a function of t. Applying the operator D to (2.58) and using (2.60), we get (2.61)

Du (t) =

J:

T Go(t,

r)f( r) dr,

where Go(t, r) = VI (t)[ - L 2 ( V 2)L1 (Vo) t

+ Ll (V )L 2

2(

Vo),]

- V 2 (t)[L 1 ( V 1)L2 ( Yo), - L 2 ( V 1)L 1 ( Yo),]

+ DVo(t, T).

Every continuous function u(t) satisfying (2.61) w.ill be called a generalized solution of the problem (2.50) -(2.57). The function Go(t, T) may be written in the form of a determinant VI (t)

(2.62)

Go(t, r) =

V 2 (t)

Vo(t, r)

Ll (VI) Ll (V2 ) Ll (Vo) t. V 2 ( VI) L 2 ( V 2 ) L 2 ( yo) t

As is clear, the question of the solvability of the problem (2.50)-(2.57) reduces to the question of the existence of an operator D- 1 and its connection with the operator Go(t, T). The problem (2.50)-(2.57) is said to be uniformly correct if a generalized solution exists for each continuous function f(t) and depends continuously on it in the metric of the space C(E). For a uniformly correct problem (2.50)-(2.57) it is necessary and sufficient that the integral operator Gf= D-

1J:T Go(t,r)f(T)dr

be a bounded operator in C(E). Suppose that the boundary conditions are regular. Then according to Theorem 2.6 the operator D is representable in one of the forms 1°-3°. Suppose that 1 is not a point of the spectrum of R. In the case 1° the operator D- 1 is bounded and the operator Go(t, T) is bounded uniformly in t, T on the square 0 ~ t, T ~ T, so that the operator G is bounded in C(E). In case 2° the operator D- 1 contains an unbounded factor A 1/2. However the operator L 2 (u) in this case does not contain derivatives, so that all the operators in the bottom row of the determin-

269

§2. ELLIPTIC CASE

ant have the factor A -112. The operator n-IGo(t,T) is uniformly bounded relative to t, T. Analogously, in case 3° n- 1 has the factor A, and the operators of the second and third rows of the determinant (2.62) have the factor A -l/2~ The operator n-IGo(t, T) is again uniformly bounded. THEOREM 2.12. If the boundary conditions are regular and unity is a regular point of the operators R (see Theorem 2.6), then the problem (2.50)-(2.57) is uniformly correct. If the function f(t) satisfies a Holder condition, then the generalized solution is a weakened solution.

It is not hard to verify that in the case of equation (2.50) a boundary value problem can be uniformly correct even though it was semicorrect for nonhomogeneous boundary conditions. Let us treat at more length the function G(t,T) = n-IGo(t,T). It is natural to call this the Green's function, since it yields a solution of the nonhomogeneous differential equation u H = Au + f(t) under the homogeneous boundary conditions LI (u) = L 2 (u) = 0 in the form u(t)

=

foT G(t,T)f(T)dT.

The notation for the Green's function is quite analogous to the scalar case:

(2.63)

G(t, T) =

n-

I

LI (VI)

Ll ( V 2 ) Ll ( Yo) t •

L 2 ( VI) L 2 ( V 2 ) L 2 ( Yo) t

and furthermore it also recalls the Green's function for the scalar case by its properties: (I) G(t, T) is a fundamental solution of the equation (2.50). (II) For each fixed T (0 ~ T ~ t ), as a function of t satisfies the homogeneous boundary conditions L1(G)t = L 2 (G)t = o. Both properties follow directly from the representation (2.63). Indeed, from (2.63) it follows first that G(t, T) is a linear combination, with bounded operator coefficients, of particular solutions VI (t) and V 2 (t) and a fundamental solution Vo(t, T) of the homogeneous differential equation, i.e. that it is itself a fundamental solution; and, second, after applying the operator Li (i = 1,2), we get Li(VI ) L i (V2 ) Li(G)t=

n-

I

Li(VO)t

LI(VI ) L I (V2 ) LI(VO)t L 2 ( V 2 ) L 2 ( V 2 ) L 2 ( yo) t

270

III. EQUATIONS OF SECOND ORDER

The first row of this determinant coincides either with the second (for i = 1) or with the third (for i = 2). I.e. the determinant vanishes, and this means that G(t, T) satisfies the homogeneous boundary conditions. We present some examples of the Green's function. 10 • The first boundary value problem:

o.

u(O) = u(T) =

The Green's function has the form G(t, T) =

if V( It -

TI) - [/ - V(2T) ]-I[V(t + T)

+ V(2T - t - T) - V(2T + t -

T) -

V(2T - t

+ T) ]} A -1/2.

2°. The second boundary value problem:

u' (0)

=

u' (T)

=

o.

The Green's function is: G(t,T) =

+ [/ - V(2T)]-I[V(t+ T) V(2T- t - T) + V(2T+ t - T) + V(2T- t+ T)]}A -1/2.

if V(lt +

TI)

3°. The problem of periodic solutions: u(T)

= u(O), u'(T) = u'(O).

The Green's function is: G(t,T) =i{V(lt- TI)+ [/ - V(T)]-I[V(T+t-T)

+ V(T-t+T)]}A- 1/ 2•

§3. The Cauchy problem for the complete equation of the second order 1. The complete equation or the second order. The behavior of the solutions and the character of the problems which it is natural to pose for the complete homogeneous equation of the second order

(3.1)

d 2u/dt 2 = A (t)du/dt

+ B(t)u,

where A(t) and B(t) are linear operators with domains which are dense in D, depend on which of the terms on the right is the leading term. If the second term is leading, then the equation can by hyperbolic or elliptic depending on the properties of the operator B(t). If the first term is leading, then the properties of the solutions of (3.1) are closely connected with the properties of the solutions of the equation of first order with the operator coefficient A (t). This section is devoted to just this question. In the process of the investigation we determine when the first term on the right may be regarded as leading.

271

§3. CAUCHY PROBLEM

Just as in the preceding sections, we introduce the concept of solutions of equation (3.1). DEFINITION 3.1. A solution· of equation (3.1) is a function u (t) , twice continuously differentiable on the segment [0, T], for which the functions A(t)dujdt and B(t)u are defined and continuous on [0, T] and which satisfies equation (3.1) on the segment [0, TJ. DEFINITION 3.2. A weakened solution of equation (3.1) is a function u(t) having the following properties: 1) u(t) is continuously differentiable on [0, TJ and twice continuously differentiable on the semi-interval (0, T]. 2) The function A(t)u(t) is defined and continuous on [0, T], and the functions A(t)dujdt and B(t)u(t) on (0, T]. 3) The function u(t) satisfies equation (3.1) on (0, T]. As usual, by a Cauchy problem (weakened Cauchy problem) we mean the problem of finding a solution (weakened solution) satisfying given initial conditions u(O) = Uo,

(3.2)

u'(O)

=

u6.

2. Reduction to a system of equations of the first order. We suppose that the operator A(t) has a domain ~(A), independent of t, and that it is strongly continuous on it and has a bounded inverse operator A -l(t).

Suppose that u (t) is a solution of equation (3.1) such that u (0) E ~(A). From the continuity of the function A (t)dujdt and the continuity of the operator A(O)A -l(t) it follows that the function A(O)dujdt is continuous. Since the operator A (0) is closed, (3.3)

fotA(O)

:~ dT = A (0) J:':~ dT = A(O) [u(t)

- u(O)].

By hypothesis, u(O) E ~(A), so that it follows from (3.3) that u(t) E ~(A) and the function A (0) u (t) is continuously differentiable on [0, T], while (3.4)

d(A (0) u(t»jdt

=

A(O)du/dt.

N ow we make the substitution (3.5)

dujdt

= v(t)

A(O)u(t) = w(t).

Both the functions v(t) and w(t) will be continuously differentiable on [0, T], and, in view of (3.1) and (3.4), they will satisfy the equations

272

III. EQUATIONS OF SECOND ORDER

dv/dt = A(t)v

(3.6)

+ B(t)A -l(O)W,

dw/dt = A(O)v,

The system (3.6) may be considered as one equation in the product space ExE: dx/dt=~(t)x+ 5S(t)x,

(3.7) where the operators

~ (t)

and 5S (t) are given by matrices

~(t) = (1~~) ~)

and 5S(t)

=

(~ B(t)~-1(0»).

The domain of the operator ~ (t) may be taken in a natural way to be the set 9'(A) xE, and the domain of 5S (t) to be the set Ex 9'(B(t)A -1(0». We have shown that every solution of equation (3.1) with the condition u(O) E 9'(A) generates in accordance with formulas (3.5) a solution x(t) = {~~~d of equation (3.7). Conversely, suppose that x(t) = {':n%} is a solution of equation (3.7). Consider (3.8) Since the functions v(t) and w(t) are continuously differentiable on [0, T] and satisfy equations (3.6), the function u(t) is twice continuously differentiable on [0, T]. Further, it follows from the second of equations (3.6) and the fact that the operator A (0) is closed that w(t) =

fol

A (O)v(T)dT

+ w(O)

=

A(O)u(t).

We then find from the first of equations (3.6) that the function u(t) satisfies equation (3.1). Finally, from the continuity on [0, of the function dw/dt = A(O)v(t) follows the continuity of A (t)du/dt

=

A (t)v(t)

=

TJ

A(t)A -l(O)A(O)v(t),

and then the continuity of dv/dt implies the continuity of B(t)A -l(O)W(t) = B(t)u(t) on [0, T]. Thus the function u(t) of (3.8) is a solution of equation (3.1). Obviously u(O) E 9'(A). N ow we turn to the consideration of the mutual relation between the weakened solutions of equations (3.1) and (3.7). As a consequence

§3. CAUCHY PROBLEM

273

of the continuity of A (t) u (t), the identity (3.3) remains in force for a weakened solution. From this (3.4) once again follows, and the substitution (3.5) leads to the equation (3.7). It is not difficult to verify that the function x(t) =I:'i!n, defined by formula (3.5), will be a weakened solution of equation (3.7). Conversely, suppose that x(t) = {~2} is a weakened solution of equation (3.7). Introduce the function (3.8). From the properties of v(t) it immediately follows that the function u(t) is continuously differentiable on [0, T] and twice continuously differentiable on (0, T]. Further, from the continuity of A (0) v(t) on (0, T] it follows that the function A(t)v(t) = A (t)du/dt is continuous on (0, T]. Integrating the last equation in (3.6) between the limits E and t, we find that w(t) - W(E) =

f.

A(O)V(T)dT= A (O)[u(t) - U(E)].

In view of the continuity of w(t) on [0, T] the right side tends to the limit w(t) - w(O) as f-O. The function u(t) is also continuous on [0, T], so that u(t) - u(e) - u(t) - u(O) = u(t) - A -1(0) w(O). Since the operator A(O) is closed it follows that u(t) - A -1(O)W(O) E 9(A), which means that u(t) E 9(A), and A(O)u(t) = w(t) is continuous. Hence it follows immediately that A (t)u(t) is continuous on [0, T] and that u(t) satisfies equation (3.1) on (0, T], from which it is clear that the function B(t)u(t) is continuous on (0, T]. Thus we have established a one-to-one relation between the solutions (weakened solutions) of equation (3.1) satisfying the condition u(O) E 9(A), and the solutions (weakened solutions) of equation (3.7). In this sense equations (3.1) and (3.7) are equivalent. 3. Uniformly correct Cauchy problems. In addition to the hypotheses made earlier, we will suppose that the operator A (t) is strongly continuously differentiable on 9(A) and such that the Cauchy problem for the equation (3.9)

dv/dt = A(t)v

is uniformly correct on [0, T]. We denote by U(t, s) the evolution operator corresponding to it. We consider the Cauchy problem for equation (3.7) in the triangle O~s~t~T in the case when ~ =0. If Xo={~}E9(~),then from the first equation in (3.6) we find that v(t, s) = U(t, s) Vo. Substituting this expression into the second equation, we get

274

III. EQUATIONS OF SECOND ORDER

(3.10)

w(t,s) =

I.t

A (0) U(T, s) VOdT

+ WOo

The operator A(O) U(T,S) is generally speaking unbounded, so that it is not clear whether w(t, s) may be represented by a bounded operator in terms of Vo and WOo However by integrating by parts we may transform (3.10) into the form w(t,s) = A(O)A -l(t) U(t,s)vo - A(O)A -l(S)VO

+

I.t

A(O)A -l(T)A'(T)A -l(T) U(T,s)vod".

+ WOo

Here already the operators applied to the elements Vo and Wo are uniformly bounded relative to t and s. We have arrived at the following conclusion: if the operator A (t) is strongly continuously differentiable on 8I(A) and the Cauchy problem is uniformly correct for equation (3.9), then for the equation (3.11)

dxjdt

=

~x

the Cauchy problem is also uniformly correct. Equation (3.7) may now be considered as a perturbed equation (3.11) and the theorems of §3 of Chapter II applied to it. If the bounded operators An(t) converge to A (t) strongly and uniformly rela· tive to t, then the operators . ~ (t) n

=

0)

(An(t) An(O) 0

have the same property relative to the operator ~ (t). Suppose that the operators An(t) constitute a stable approximation to the operator A (t) and that they are strongly continuously differentiable. The evolution operators corresponding to the operators ~n(t) will be constructed from the evolution operators corresponding to the An(t) according to the formulas vn(t, s) = Un(t, s) Vo. wn(t,s)

= An(O)A,;-l(t) Un(t,s)vo -

An(O)A,;-l(s)vo

+ Sot An(O)A,;-l(T)A~(T)A,;-l(T) Un(T,s)vodT + WOo If we assume that the operators An(O)A,;-l(t) and A~(t)A,;-l(t) are bounded uniformly in nand t, then the operators ~n(t) constitute a

275

§3. CAUCHY PROBLEM

stable approximation to the operator ~(t). The operators ~,,(t) commute if the A,,(t) commute. Thus the operator ~ (t) will satisfy all the conditions of Theorem 3.6 of Chapter II, except that it does not have an inverse. However this may be achieved by adding to it an operator proportional to unity. We note that in the case when the operators A,,(t) are constructed according to formulas (3.27) of Chapter II, i.e. A,,(t) = - nA(t)RA(I)(n),

(3.12)

then the uniform boundedness of the operators A,,(O)A;l(t) and is automatically satisfied if the resolvent of the operator A (t) satisfies the estimate

A~(t)A;l(t)

(~> w).

Indeed, A,,(O) A;l(t)

=

I

+ nRA(o) (n)

- nRA(o) (n) A (0) A -l(t) ,

so that

I A,,(O)A;l(t) I

~ 1

+ (Mn/(n - w»

(1

+ max I A (O)A -l(t) II) ~ C1.

Further, IIA~(t)A;l(t)

I

=

I -

nRA(I)(n)A'(t)A -l(t)

I

~

C2·

N ow suppose that 58 ~ O. If we use Theorem 3.7 of Chapter II, we come to the conclusion that under the conditions of boundedness and continuous differentiability of the operator B(t)A -1(0) the Cauchy problem for equation (3.7) is uniformly correct. Hence the theorem on the solvability of the Cauchy problem for eauation (3.1) will result. Let us summarize our investigation. THEOREM 3.1. Suppose that the operator A (t) satisfies the conditions of Theorem 3.6 of Chapter II relative to the operators A,,(t) of type (3.12). If the operator B(t)A -1(0) is bounded and strongly continuously differentiable on [0, T]. then the Cauchy problem for equation (3.1) has a unique solution for any Uo, u6 E E&(A).

We may restrict the formulation of the theorem, but make it more effective, if we use Theorem 3.11 of Chapter II. on

THEOREM 3.2 If the operator A (t) is ~(A) and satisfies the condition

II RA(t) (X) II

~ 1/(1

strongly continuously differentiable

+ X)

for ~ ~ 0,

276

III. EQUATIONS OF SECOND ORDER

and the operator B(t)A -1(0) is bounded and strongly continuously differentiable relative to t E [0, T], then the Cauchy problem for equation (3.1) has a unique solution for any uo, Uo E 9(A). We could have tried to apply to equation (3.7) Theorem 3.4 of Chapter II, which does not require the differentiability of the perturbation operator. It is not hard to verify that this leads to the requirement of boundedness and continuity of the operator A(t)B(t)A -1(t), which is substantially coarser than the requirement of boundedness of the operator B(t)A -1(t). Analogously one considers the Cauchy problem for the nonhomogeneous equation d 2u/dt 2 = A (t)du/dt + B(t)u

+ f(t).

The reader will easily apply Theorems 3.2 and 3.3 of Chapter II to obtain criteria for solvability of this problem. 4. Abstract parabolic equations. We consider the weakened Cauchy problem first for equations with constant coefficients: (3.13) We pass again from this equation to an equation of the type (3.7). In the case of a constant operator A the semigroup operator U (t) corresponding to the operator ~ is expressed very easily in terms of the semigroup operator U(t) of the operator A: .

U (t)

=

0) .

(U(t) U(t) _ I I

Therefore it follows immediately that for the abstract parabolic equation (3.9)

du/dt

=

Au

the corresponding equation (3.11) will also be abstract parabolic. If the operator BA -1 and thus also the operator 58 is bounded, then

from what has been said it follows that the weakened Cauchy problem for equation (3.7) is solvable, and accordingly that this problem is solvable for equation (3.13). However in the situation at hand it is natural to try to dispense with the requirement of boundedness of the operator BA -1 and to apply to equation (3.7) Theorem 7.2 of Chapter I on perturbed abstract parabolic equations, where unbounded perturbations are admitted.

277

§3. CAUCHY PROBLEM

N ow something arises which interferes with doing this directly. The resolvent of the operator ~ has the form

If now we calculate the operator 58( ~ _ AI)

58(~

- ")../) -1, then we get

-1 = ((1/A) BA -1~RA (A)

- (1/A~BA

-1) ,

from which it is clear that in the case of an unbounded operator BA- 1 the resolvent of the operator ! does not kill the unboundedness of the operator 58. This is connected with the presence on the diagonal of the matrix (! - AI) -1 of an operator proportional to unity. In order to get rid of this, we introduce a new function z(t) = v(t) - w(t)

and transform the system (3.6) into the form dv/dt = Av dz/dt

+ BA -IV -

BA -IZ,

= BA -IV - BA -IZ.

We write this as one equation in ExE: (3.14)

dS/dt

= !oS + 58 oS,

where

!o=

(~

1:),

58 0 =

(=1: ~),

Ao= -BA-l.

We study the operator !o. Its domain is the collection of elements f = I~} for which v E 9'(A) and z E 9'(Ao). If the operators A and Ao are closed, then the operator !o is closed. Now suppose that for some A the operators A and Ao have the resolvents R(A) and RO(A)

respectively. Then the operator !o has the resolvent

(~ _ AI) -1 = (R(A) - R(A) AoRo(A) )' 0

o

RO(A)'

For the operator 58 0 to be defined on 9'( !o) we need to require that 9(Ao) ~ 9'(A). Then we calculate (3.15)

58 0(!0 _ ")../) -1 =

AoR(A) AoR(A) AoRo(>") ) - AoR(>..) AoR (>..) AoRo(>") •

(-

278

III. EQUATIONS OF SECOND ORDER

If Ao is closed and 9'(Ao) => 9(A), then in view of Lemma 7.1 of Chapter I the operator AoR(A) is bounded, so that the resolvent of ~o kills the unboundedness of SB o. However, success in this direction is not achieved without corresponding sacrifice. Now on the diagonal of the operator in the leading part of equation (3.14) there appear the operators A and Ao = - BA -1, and in order to study the properties of the equation (3.16)

dyldt

=

!loY

we have to know the properties of equations of the first order with the operator coefficients A and Ao. LEMMA 3.1. If the operators A and Ao are generating operators of analytic semigroups, then ~o also has this property. PROOF.

By hypothesis II R (x) II ~

Mil A- wi

(ReA> w)

and (3.17)

II Ro(A) II ~ Mol IA - wol

Then for ReX>

w=

(ReA> wo).

max(w,wo) we have

11(~o-Al)-111 ~max{1 A-W M 1'1 Mo I' A-Wo

(3.18)

(1

+

Ixl Mo ) M} IA-wol lA-wi

Mo ;------=---;=IX-wl'

.<

The theorem is proved. 3.3. If the operators A and Ao = - BA -1 are generating operators of analytic semigroups and Ao is fully subordinate to A, then for any uo E 9(B) n 9(A) and u6 E 9(A) there exists a unique solution of the Cauchy problem for equation (3.13). For any uo E 9(A) and u6 E E there exists a unique solution of the weakened Cauchy problem, analytic in some sector containing the positive real semiaxis. THEOREM

PROOF. We shall show that the operator ~o + SB o is a generating operator of an analytic semigroup. We have

II (~o + SSo - AI) -111 ~ II (~o - xI) -111 HI + SB o( ~o - AI) -1]-111· If we show that the operator SBo(~{l - AI) -1 is small in norm for sufficiently large lxi, then an estimate of type (3.18) will hold for the resolvent (~o + SB o - XI) -1. In the proof of Theorem 7.2 of

279

§3. CAUCHY PROBLEM

Chapter I it was" shown that if the operator Ao was fully subordinate to the operator A, the norm of the operator AoR (A) becomes arbitrarily small for large IAI (ReA> w). The norm of the operator AoRo(>,) for large IAI and ReA> w is uniformly bounded. It then follows from (3.15) that II~o(!o-A1)-111_0 as IAI-CD (ReA>max(w,wo». Thus equation (3.-13) is abstract parabolic, and all of its weakened solutions are analytic. If Uo E ~(B) n ~(A) and u6 E ~(A), we put Vo = u6 and Zo = Uo - Auo. Then, since 9'(Ao) ~ ~(A), the element AoZo = Aouo - Buo, is defined, i.e. Zo E ~(Ao). Construct the solution z(t) = {~l:l} of the Cauchy problem for equation (3.14) with the initial data Zo = {~}. Then the function

u(t) = ltV(T)dT+UO will be a solution of the problem (3.12)-(3.2). For any uoED(A), uoEE we construct a weakened solution of the problem (3.13)-(3.2) in the same way. The uniqueness of the corresponding solutions follows from the equivalence of equations (3.13) and (3.14). The theorem is proved. N ow consider the equation

(3.19)

d 2u/dt 2 =Adu/dt+EBu

(E>O).

To it there corresponds the equation

dx/dt = !l.x + ~.x, where the operator Ao has been replaced throughout by the operator A. = - EBA -1 = EAO' For the resolvent R.(A) of the operator A, we have IIR,(A)II =

II~Ro(~) "~~IA/E~wol ~IA~wol'

where Wo is equal to the maximum of EWo over all the values of E being considered. Therefore it follows that the resolvent of the operator !l satisfies estimate (3.18) with constants not depending on E. Further, the operator A.R.(A) will be bounded for large IAI with ReA> Wo uniformly in E and A. If we now assume that Ao is subordinate to A, then the operator "58 (!l -"AI) -1 = •

•

AoR(A) AoR("A)A.R.(A») - AoR("A) AoR("A)A.R.("A)

E (-

280

III.

EQUATIONS OF SECOND ORDER

may be made as small in norm as desired for large I~ I because of the factor E. Repeating the arguments in the proof of Theorem 7.3 of Chapter I, we arrive at the following assertion. THEOREM 3,4. If under the conditions of Theorem 3.3 the operator Ao = - BA -1 is simply subordinate to the operator A, then the assertion of the theorem remains valid for equation (3.19) for sufficiently small E. REMARK 3.1. The assertion of Theorem 3.4 is valid also for the equation (3.20)

In fact, the substitution T = ET reduces this equation to (3.19). REMARK 3.2. The operator Ao is subordinate (fully subordinate) to the operator A, if the operator B is subordinate (fully subordinate) to the operator A 2. 5. Weakened Cauchy problem. Now we suppose that the operators A and Ao have the properties which guarantee for the corresponding differential equations of the first order that the weakened Cauchy problem is correct on 9'(A). In fact we suppose that for some f3 and f30 (0 < f3, f30 < 1) the inequalities

I R(~) I

~ M(l

+ I~I) -II;

(3.21)

I Ro(~) I

~ Mo(1

+ I~I) -flo

(ReX> w)

are satisfied. Then

so that for the resolvent of

~o

I (~o - AI) -111

we also have the estimate ~

Mt/(1 + I~I )1I+fJo- 1.

If f3 + f30 - 1 > 0, then this estimate, in view of Theorem 3.3 of Chapter I, guarantees that the weakened Cauchy problem for equation (3.16) is correct on 9'(~0). In order to obtain an analogous estimate for the operator ~o + 58 0 , it suffices to impose on 58 0 conditions such that (3.22)

§3. CAUCHY PROBLEM

281

Now we may use Lemma 7.8 of Chapter I. If Ao is subordinate to A with order a, then in view of this lemma

I AoR(X) I

(3.23)

~

Cj(l

+ IXI)II-a

and

I A oR (X) A 0R0 (X) I

<

= (1

C

+ IXI) 11-"

[ 1+

(1

Mo IXI ]

+ IXI) 110

Thus, if a < fl + flo -1, then (3:22) is satisfied. If a = fl + flo - 1, then it follows from (3.23) and (3.22) that the operator 5B o( ~o - AI) -1 is bounded. Repeating the arguments of the preceding subsection, we therefore find that

I m.(~. - xl) -111 ~o as

E ~ 0 uniformly in X for sufficiently large I XI. We have arrived at the following theorem.

THEOREM 3.5 If the operators A and Ao = - BA -1 satisfy conditions (3.21) with fl + flo > 1, and the operator Ao is subordinate to A with order a < fl flo - 1, then for any Uo E ~(B) n ~(A) and Uo E 9'(A) there exists a unique weakened solution of the problem (3.13)-(3.2). This solution is an i!1-finitely differentiable function for t > o. If Ao is subordinate to A with order a = fl flo - 1, then the preceding assertion is valid for equations (3.19) and (3.20) for sufficiently small E.

+

+

The investigation of the weakened Cauchy problem presented!- in the last two subsections may be carried out for equations (3.1) with variable coefficients as well. Here it is convenient to consider the functions vet)

=

dujdt and z(t)

=

dujdt - A(t)u.

Then for the vector-function x(t) = I~!:l} we obtain an equation of type (3.14) with a supplementary term ~(t), where

~(t) = (-A'(t)~-I(t) A'(t)~-I(t»)· Under our hypotheses the operator A'(t)A -I(t) is bounded, so that the supplementary term ~ (t) x does not change the essential properties of the equation under conditions of specific smoothness of ~ (t) as a function of t.

282

III. EQUATIONS OF SECOND ORDER

As an example we formulate the following assertion. THEOREM 3.6. Suppose that the operators A(t) and Ao(t) = B(t)A ~l(t) have domains which do not depend on t, and for each t satisfy conditions (3.17) with constants independent of t. Suppose that the operators A' (t)A -l(t), Ao(t) AOl(O) and Ao(t) A -"(0) are bounded for some a E [0,1] and satisfy for that a some Holder condition relative to t. Then for each Uo E ~(A) n ~(B) and u~ E ~(AP) for p > a there exists a unique weakened solution of the problem (3.1) - (3.2).

CHAPrERIV ASYMPTOTIC METHODS §1. Equations with a small parameter on the highest derivative 1. Equations with a constant operator. We suppose that for the equation x' = Ax the Cauchy problem is uniformly correct and that the semigroup U(t) corresponding to it is of negative type:

II U(t) II

(1.1)

~

(a> 0).

Ce- oJ

The operator A then has a bounded inverse A -I. Consider the equation (O<E~fO)'

fdxjdt = Ax+ f(t)

(1.2)

where f(t) is a continuous function. We shall be interested in the behavior of its solution as f - O. The operator f-IA will be the generating operator of a semigroup U.(t) , with U,(t) = U(tje). Therefore

I U, (t) I

(1.3)

~

Ce -oJ/,.

If x,(t) is the generalized solution of equation (1.2) with the initial

condition x,(O)

= Xo

not depending on

x,(t) = U,(t)xo+1 f

i'

e,

then

U.(t-s)f(s)ds

0

or (see (6.7) ofChapterI) x,(t) (1.4)

=

U,(t)Xo

ll'

+ -E 0 U.(t - s) fries) - f(t)]ds + U,(t) A -If(t) - A -If(t).

We shall show that the integral term tends to zero as on [0, T]. We have

II~

It u.(t -

s) [f(8) - f(t) ]d8

II ~ f lle-IIT/'II f(t 283

E-

0, uniformly

r) - f(t) I dr

284

IV. ASYMPTOTIC METHODS

=

C

!a e-aull r

o

f(t - EU) - f(t) I du

2C

+ -a

max I f(s) I e- ar

O~B~T

for any r > O. We take r so large that the second term on the right is quite small. Then for fixed r we make E so small that the oscillation of the function on any segment of length Er, and along with it the first term on the right, is also small. For these values of E the integral term in (1.4) will be sufficiently small. From the arguments just presented and from estimate (1.3) it follows that (1.5)

X,(t)

-+ -

A -1f(t)

uniformly on each segment [Il, T], where 0 < Il < T. This convergence will be uniform on [0, T] only for that particular solution of equation (1.2) for which x,(O) = - A -1f(0). Indeed, in this case U,(t)xo

+ U,A -1f(t) =

U,(t) A -1[f(t) - f(O)].

On a sufficiently small segment [0, Il] this function is small because of the continuity of f(t) and the uniform boundedness of U,(t) , and on the segment [Il, T] it is small for sufficiently small E in view of (1.3). For any Xo E E the character of the convergence in (1.5) may be made more precise. We introduce the following definition: DEFINITION 1.1. A family of functions c/J,(t) (0' < E ;;:£ EO) on the interval (0, T] converges almost uniformly to the function c/J(t) if for each 1J there exists an r(1J) and E1(1J) such that

I c/J,(t)

- c/J(t)

I

;;:£ 1J

for all E < E1 (1J) and t E [r(1J) E, T]. The simplest example of a family of scalar functions converging almost uniformly to zero on [0, T] is the family c/J,(t) = e- atl '. From almost uniform convergence on (0, T] it follows that the convergence is uniform on each segment [Il, T] C (0, T]. The converse is not always so. Thus, for example, the family e-at/'/E converges uniformly to zero on any [Il, T] but does not converge almost uniformly. It follows from the above arguments and from estimate (1.3) that the sequence x.(t) converges almost uniformly on (0, T] to the function A -1f(t). N ow we suppose that f(t) is continuously differentiable. Then (see (6.8) of Chapter I)

§1. SMALL PARAMETER ON HIGHEST DERIVATIVE

x.(t)

=

285

U.(t)XO+ l tU.(t-S)A- 1{'(S)dS

(1.6)

+ U,(t) A -If(O) -

A -If(t).

If x,(t) is the solution of equation (1.2), then we apply the operator A to both sides of (1.6), with the result (1.7)

AX,(t)

=

U.(t)AXo+ ltu,(t-S)I'(S)dS+ U,(t)f(O) -f(t),

from which it follows that the functions Ax, (t) converge almost uniformly on (0, T] to the function - f(t). This assertion may be given the following meaning. Consider the equation (1.8)

Ax

+ f(t)

=0.

The residual of this equation corresponding to the function y(t) is the function A(t) = I Ay(t) + f(t) II. It follows from the above that the residual A,(t) = I AX,(t) + f(t) I converges almost uniformly to zero on (0, T]. Finally let us consider the derivatives of the solutions x, (t). From equation (1.2) and (1.7) we obtain the relation dxd,(t)

(1.9)

=~

U,([Axo

+ f(O)] +.! (t U,(t -

t E E

+ U,(t)A- f'(t) 1

Jo

s) [fl (s) - I' (t) ]ds

- A- 1{,(t) ,

from which it is clear that the derivatives dx, (t) j dt of the solutions tend to the derivative of the solution of the equation (1.8) dxjdt = - A-II' (t) uniformly on each segment interior to [0, T]. The derivatives of the particular solutions for which x,(O) = - A -If(O) tend to dxjdt almost uniformly on (0, T]. Let us summarize. THEOREM 1.1. If the Cauchy problem for the equation x' = Ax is uniformly correct and is of negative type, then the generalized solution x,(t) of equation (1.2) with the initial values x,(O) = Xo E E converges almost uniformly on (0, T] to the solution x(t) = - A -If(t) of equation (1.8). If the function f(t) is continuously differentiable and Xo E 9'(A) , then the residuals of equation (1.8) relative to the functions x.(t) converge to zero almost uniformly on (0, T], and the derivatives dXJdt tend to dx/dt uniformly on each segment interior to [0, T].

286

IV. ASYMPTOTIC METHODS

For the particular solutions x?(t) with the initial values x?(O) = - A -If(O) there is uniform convergence to i(t) on [0, T], uniform convergence of the residuals on that segment to zero, and almost uniform convergence of the derivatives to di/dt on (0, T]. REMARK 1.1. It is clear from (1.7) that the functions AX,(t) are uniformly bounded in E and t. 2. Equations with variable operator. A quite analogous situation holds also for the equation (1.10)

Edx/dt = A(t)x

+ f(t)

with variable operator, if we suppose that the operator A (t) has a domain ~(A) independent of t, is strongly continuously differentiable on it and that for the equation Edx/dt = A(t)x

the Cauchy problem is uniformly correct for any EE (0, EO], and for the corresponding evolution operator U,(t, s) the estimate

II U,(t,s) II

(1.11)

~ ee-a(t-s)/,

holds. For the generalized solution x,(t) of equation (1.10) with the initial value x,(O) = XQ, we have in analogy to (1.4) X,(t)

=

U.(t,O)Xo

+! Jor U.(t, s)f(s) ds E

J

= U.(t, O)Xo +! (1.12)

E

t

Jor U.(t, s) (f(s) -

+ Jo U.(t,s) + U, (t, 0) A

f(t) ]ds

dA -l(S)

-1

ds

f(t)ds

(0) f(t) - A-I (t) f(t).

Here we have used property (3.12) of Chapter II for the evolution operator, and have integrated by parts. From the estimate (1.11) and the continuity of f(t) it again follows that the integral terms tend to zero uniformly on [0, TJ, which means that the functions x,(t) tend to the function i(t) = - A -l(t)f(t) almost uniformly on (0, T]. Here i(t) is a solution of the equation (1.13)

A (t) i(t)

+ f(t) = o.

If f(t) is differentiable, then in analogy with (1.6)

§1. SMALL PARAMETER ON HIGHEST DERIVATIVE

X,(t) = U,(t,O)Xo

(1.14)

287

r u,(t,S) dA -1(s)/(s) ds ds

+ Jo

+ U. (t, 0) A

-1 (0) 1(0)

- A -1(t) I(t) .

N ow we introduce the operator V,(t,s) = A(t) U.(t,s)A -1(S).

If this satisfies the estimate

(1.15) then, applying the operator A(t) to both sides in (1.14), for Xo E 9(A) we obtain the estimate A(t)x,(t) = V,(t,O)A(O)Xo

(1.16)

+ !at V, (t, s) [ -

A' (s) A -1(S) I(s)

+ V, (t, 0) 1(0) -

I(t),

+ I' (s) ]ds

from which it follows that the residues of equation (1.13) corresponding to the functions x,(t) converge almost uniformly to zero. Finally, transforming the integral term in (1.14), as we did in (1.12), we have X,(t) = U.(t,O)Xo

r

+ Jo E

+ !at U,(t,s) [~ (A -1(s)/(s» U.(t,s)

dA -1(S) ds

[ddt (A

-1 (t)/(t»

! J--

(A -1 (t)I(t» ] ds

ds_

+ EU.(t,0)A- 1(0) :t (A- 1(t)/(t» - EA- 1(t) + U.(t,0)A- 1(0)/(0) -A- 1(t)/(t).

:t

(A- 1(t)/(t»

Applying the operator E- 1A(t) to both sides and adding E- 1/(t) , we arrive at the expression dx, dt

=

1 )A) E V,(t,O (0 Xo

+ -1E (1.17)

it 0

V,(t, s) [- A' (s)A -1(S) I(s)

+ f' (s)

+ A(s)A -1(t)A'(t)A- 1(t)f(t)

- A(s)A -1(t)f'(t) ]ds

+

288

IV. ASYMPTOTIC METHODS

+

.c

V.(t,s)A'(s)A-l(s) :t (A-l(t)f(t»ds

- v. (t, 0) dd lA -l(t) f(t» - ~ (A -let) f(t» t dt

~

V.(t, 0) f(O),

E

from which it is clear that the derivatives dX,(t)/dt tend uniformly on each segment interior to [0, T] to the derivative dx/dt = - (d/dt) x (A-l(t)f(t» of the solution of equation (1.13). It follows from formulas (1:14), (1.16), and (1.17) that the particular solutions x?(t) with initial values x?(t) = - A -1(0) f(O) tend uniformly on [0, T] to the function x(t), the residuals corresponding to them in equation (1.13) tend to zero uniformly on [0, T], and the derivatives dX,/dt converge to dx/dt almost uniformly on (0, T]. THEOREM 1.2. Suppose that the operator A (t) has a domain independent of t and is strongly continuously differentiable on it. If for any E E (0, EO] the Cauchy problem for the equation EX' = A (t) X is uniformly correct and the estimates (1.11) and (1.15) hold, then all the assertions of Theorem 1.1 remain valid for the equations (1.10) and (1.13) as well. If the operator E -1 A (t) satisfies the hypotheses of Theorem 3.6 of Chapter II, then inequality (1.15) follows from inequality (1.11). Indeed, in view of Remark 3.1 to that theorem, V.(t,s)

U.(t,s)

=

so that

I v.(t,s) I ~

+ f.t U.(t,T)A'(T)A-l(T) ;'(T,s)dT,

it

+ C e-a(t-T)/'II A'(T)A -leT) IIII V.(T,S) I dT,

ee-a(t-s)/,

or

From this integral inequality it follows that the inequality

Ie

at /,

V. (t, s)

I

~ ee aB /'e C2 (t-S) ~

Ce C2TeaB /'

=

Cle as /,

holds for the function eat/, V. (t, s) of t, and this inequality is equivalent to (1.15). Inequality (1.1) holds also for operators satisfying the hypothesis of Theorem 3.11 of Chapter II: (1.18)

I

RA(t)

(A) I ~ 1/ (1

+ A)

for A ~ O.

§1.

SMALL PARAMETER ON HIGHEST DERIVATIVE

289

Indeed, in this case

II RA(t)f, (A) I

I

= f RA(t) (fA)

I

~ f/ (1

+ fA) =

1/ (A

+ 1/E) •

From Remark 3.3 of Chapter II we then obtain the estimate

II U.(t, s) II i.e. (1.11) with a

~ e-
= 1.

THEOREM 1.3. If the operator A (t) satisfies the hypotheses o[ Theorem 3.11 of Chapter II, then the assertions of Theorem 1.1 are valid for equations (1.10) and (1.13). REMARK 1.2. Theorems 1.2 and 1.3 could have been proved for the more general equation Edx/dt

=

A(t,E)X

+ f(t) ,

if the estimates (1.11) and (1.15) are valid for the operators E- 1A(t,E) and the operators A (t, E) converged strongly on 9(A) to an operator Ao(t) in such a way that the functions A -1(t, E) and A{(t) A t- 1(t, E) converged strongly and uniformly on [0, TJ to the functions A01(t) and AtS(t) A01(t). Here the role of equation (1.13) is played by the equation Ao(t)i(t) + f(t) = O. 3. Boundary layer. For smoothing the difference between the solutions x,(t) and i(t) of equations (1.10) and (1.13) which arises near zero because of inconsistency in the initial conditions, one introduces further auxiliary functions, which are found from equations simpler than (1.10). Consider along with (1.10) the equation with constant operator

(1.19)

fdx.ldt = A(O)x,

and its solution X, (t) for which For the difference z.(t)

=

Z,(t) = -1 E

x,(t) - x.(t)

+ [A (0) -

fdz,/dt = A (t)z,

so that

l'

Xo

= x, (0).

we obtain the equation

A (t)

Jx, + [(0)

- f(t),

U.(t,s) [A (0) - A(s) Jx.(s)ds

0

1.£1

+E

in

x, (0) =

+ f(O)

u.(t, s) [/(0) - [(s) Jds.

0

It follows from Remark 1.1 that the Aox,(s) are bounded uniformly E and s and the operator A(t)A -1(0) is continuous in norm, so that

290

IV. ASYMPTOTIC METHODS

the quantity II [A (0) - A(s)]A -l(O)A(O)x,(s) II will be small uniformly in E for sufficiently small s. Put t = Er and make the substitution s = EU in the integral. Then we obtain Ilz,(Er) 11;;[: c I'll [A (0) - A (EU) ]A -1(O)A(O)X,(EU) II du

+ c I'll f(O)

- f(EU) II du.

I t follows from this estimate that for arbitrary r > 0 and arbitrary 11 > 0 there exists an E2(r,11) such that for E;;[: E2 (r, 11) (1.20)

II x,(t) - x,(t) II ;;[: 11 for

0;;[: t;;[:

Er.

This property characterizes the asymptotic closeness of the solutions x,(t) and x,(t) as t-+O. Now consider four functions: the solution x,(t) of equation (1.10), the solution i,(t) = - A -l(t)f(t) of equation (1.13), the solution x,(t) of equation (1.19) and the function x = - A-1(0)f(0). It follows from the preceding subsections that x, (t) -+ i(t) and x. (t) -+ ~ almost uniformly on (0, T]. Then v,(t)

=

x,(t) - i(t) - x,(t)

+i

tends almost uniformly to zero on (0, T]. This means that for any 11 there exist r(11) and El (11) such that II v,(t) II ;;[: 11 for r(11) E.;;[: t ;;[: T and all E< El(11). Further, it follows from (1.20) that for E;;[: f2(r(11), 11/2) we have Ilx,(t) - x,(t) II ;;[: 11/2 for 0;;[: t;;[: Er(11). Finally, from the continuity of the function A~I(t)f(t) we find that Ili(t) -xii <11/2 for E ;;[:Eg(11). Then Ilv,(t) II ;;[: 11 on the entire segment [0, T] for E;;[: min(Eh E2, Eg). Thus the functions v, (t) converge uniformly to zero on [0, T]. THEOREM 1.4. Under the hypotheses of Theorem 1.2 or 1.3 the solution (1.10) is for sufficiently small E uniformly close to the function

y,(t)

= -

A -1(t)f(t)

+ x,(t) + A -1(O)f(O),

where x,(t) is the solution of equation (1.19) with constant operator.

We note that the function y, (t) may be expressed explicitly in terms of the semigroup of the operator A (0). We have x,(t)

so that

=

UA(O) (tiE) Xo

+

UA(O) (t/E) A -1(0) f(O) - A-I (0) f(O),

§1. SMALL PARAMETER ON HIGHEST DERIVATIVE

+

y, (t) = - A -l(t) I(t)

UA(O)

(tj E) Xo

+

UA(O)

291

(tjE)A -1(0) 1(0).

The two last terms "rectify" the behavior of the function i(t) = ..- A -1(t)/(t) near zero (in the boundary layer). 4. Perturbed equations. Consider equation (1.10) with a perturbation of a special sort: dy Edt =A(t)y+

(1.21)

Jot B(r)y(r)dr+/(t),

where the operator B(t) has the property that B(t)A -1(t) is bounded and strongly continuous in t. Along with equation (1.21) we will consider the degenerate equation (1.22)

A(t)y(t)

+ ltB(r)y(r)dr+/(t) =Q.

We make the substitution A (t) y(t) = v(t) in it. Then (1.23) Differentiating, we obtain dvjdt

= - B(t)A- 1(t)v - f'(t).

This equation with bounded strongly continuous operator B(t)A -l(t) has a solution. Weare interested in its solution v(t) satisfying the condition v(O) = - 1(0). Obviously it will also be a solution of equation (1.23). The function y(t) == A ~1(t)V(t) will then be a solution of equation (1.22). This solution is differentiable: dyjdt

= - A -l(t)A'(t)A -1(t)V(t)

+ A -1(t)dvjdt,

and therefore the function A (t)dyjdt = - A'(t)A -1(t)V(t)

+ dvjdt

is continuous. Finally, y(O) = A -1(0)/(0). Suppose that y,(t) is a solution of equation (1.21) with the initial value y,(O) = Xo E g(A). For the difference z.(t) = y(t) - y.(t) we obtain the equation E

dz, dt

= A (t)z, +

Jot B(r) Z,(r) dr + EdYjdt.

Recalling that z,(O) = - A -1(0)/(0) - Xo, we get

IV. ASYMPTOTIC METHODS

292 Z,(t)

(1.24)

= - U,(t,O) (Xo + A -1(0) t(O»

Changing the order of integration and using the fact that -1 [ E

U, (t, s)gds = - U, (t, s) A -1 (s) g 'F

I' + [ .,.

1 U. (t, s) dAds (s) gds,

T

we get z.(t)

=-

+

U,(t,O) (Xo

J:

+ Jof'

+ A -1(0) t(O»

U,(t, T)A -l(T)B(T)Z.(T)dT -

It T

U,(t,s)

.r

A -l(t) B(T)Z.(T)dT

dA -l(S) ds B(T)Z.(T)dsdT

+ Jof' U,(t,s)dy ds

ds.

Apply the operator A (t) to both sides. We have A(t)z.(t)

= - V.(t,O) (A (O)x. + teO»~

+

l' V.(t'T)B(T)A~l(T)A(T)Z.(T)dT

_it -J:

B(T)A -1.(T) A (T)Z.(T) dT

[V.(t,S)A'(S)A- 1 (S)B(T)A- 1 (T)A(T)Z.(T)dsdT

+ Jor V.(t,s)A(s) flY sd ds. The first term on the right, in view of the estimate (1.15), is uniformly bounded in E and t and converges almost uniformly on (0, T] to zero as E -+ O. The last term, in view of the continuity of the function A (t)dyjdt, tends to zero uniformly on [0, TJ as E-+O. Then for the functions II A (t) z. (t) II we obtain an integral inequality of the form (1.25)

II A (t)z.(t) I

~
where the functions
11.

SMALL PARAMETER ON HIGHEST DERIVATIVE

293

on E. From this inequality we find in the first place that the functions A(t)z,(t) are uniformly bounded in E and t. Further, for any ,,> we choose r1 = r(1//2e CT ) and E1 = E(1//2eC7')in such a way that tP.(t) < T//2e CT for Er1 ~ t ~ T. Then we find an E2 so that for E < E2

°

This may be done in view of the uniform boundedness of the functions in the integrand. Then IA(t)z.(t) II

~ ~+Ci'~A(T)Z'(T)lldT e

for

"1

Erl~t~T.

Hence it follows that

II A (t)z.(t) I

~ (T//eC7')e C(I-.r1) ~ T/

for

Er1

~ t ~ T.

We have proved that the functions A (t) z. (t), and thus the functions z. (t) , tend almost uniformly to zero on (0, T]. THEOREM 1.5. Suppose that the hypotheses of Theorem 1.2 or 1.3 are satisfied. Suppose that the function B (t) A -1 (t) is bounded and strongly continuous, and that the function f(t) is continuously differentiable. Then the solution y.(t) of equation (1.21) with the initial value Xo E ~(A) converges almost uniformly to the solution yet) of equation (1.22). In addition the residuals of equation (1.22) corresponding to the functions y.(t) are uniformly bounded and converge almost uniformly to zero. If fo = - A -l(O)f(O), then the y.(t) converge to yet) and the residuals converge uniformly to zero on [0, T].

In the proof we can restrict our attention to the last two assertions. The first of them follows from the fact that the uniform boundedness and almost uniform convergence to zero on (0, T] of the functions A(t)z.(t) implies the uniform convergence to zero on [0, T] of

Sot B(r)A -l(T)A (T)Z.(T) dT. If Xo = - A- 1 (0)f(0), then in (1.25) the function tP.(t) converges to zero uniformly on [0, T], from which the last assertion of the theorem follows. 5. Equations ofsecond order. Now we turn to the practically important case of an equation of second order with a small parameter on the

294

IV. ASYMPTOTIC METHODS

second derivative: (1.26)

Ed 2u/dt 2 = A (t) du/dt

+ B(t) u + f(t).

We have investigated the solvability of the Cauchy problem for such an equation in §3 of Chapter III. Consider the degenerate equation (1.27)

A (t) du/dt

+ B(t) u + f(t) = o.

The substitution A (t) u = w(t) reduces it to the equation dw/dt = [A'(t) - B(t)]A -1(t)W - f(t).

Under the hypotheses of the preceding section this will be an equation with bounded operator, which has a solution w(t) satisfying the initial condition w(O) = A(O)u(O). We then write u(t) = A -l(t)W(t). It is easily verified that u(t) will be a solution of equation (1.27), while the functions A(t)du/dt and B(t)u(t) are continuous. Now we return to equation (1.26) and write du/dt = y(t). For the function y(t) we obtain the equation (1.28)

Edy/dt=A(t)y+B(t)

5otY (T)dT + f(t) +B(t)u(O),

similar to equation (1.21). Repeating the arguments of the preceding subsection, we find that it is necessary to investigate the integral -1 E

50 [J.(t,s)B(s) 50' z,(T)dTcis, t

0

0

which under the hypothesis of strong continuous differentiability of the operator B(t)A -1(0) may be transformed into the form

50

t

[J.(t, T)A -1(T)B(T)Z,(T)dT -

- 50

t

r

J:

A -l(t)B(t)Z.(T)dT

[J.(t,s)A- 1(s) [A'(s)A- 1(s)B(sA- 1 (t) + B' (s) A -1(T)]A (T)Z.(T) dsdTo

Thus we again obtain an inequality of type (1.25). Thus we arrive at the conclusion that the solution y. (t) of equation (1.28) with initial value y, (0) = ub not depending on E converges almost uniformly as E -+ 0 to the solution y(t) of the equation

§1. SMALL PARAMETER ON HIGHEST DERIVATIVE

A(t)y

+ B(t)

.f

y(r)dr

295

+ f(t) + B(t) Uo = O.

The solution of this equation is obviously a function of du/dt. Further, the functions A(t)Y,(t) are uniformly bounded and converge almost uniformly on (0, T] to the function A (t) du/dt. If we now denote by u,(t) the solution of the Cauchy problem for equation (1.26) with the initial data u,(O) = Uo and u:(O) = uO, then it follows from the foregoing that the du./dt are uniformly bounded and converge almost uniformly on (0, T] to au/dt, and A (t) du./dt to A(t)du/dt. We note that it therefore follows that the functions u,(t) converge uniformly on [0, T] to u(t) and A(t)u,(t) to A(t)u(t). The first is obvious, and the second follows from the identity A(t)u,(t)

= A (0) Uo

+ !at [A(r)~:' + d(A(r~~-I(O» A(O)u,(r) Jdr. We have arrived at the following theorem. THEOREM 1.6. Suppose that the operator A (t) satisfies the hypotheses of Theorem 1.3, and that the operator B(t) is strongly continuously differentiable on ~(A). Suppose further that f(t} is continuously differentiable, and that the initial data Uo, uO E ~(A) for the Cauchy problem for equation (1.26) do not depend on E. Then the solution u,(t) of the Cauchy problem has the properties: 1. The functions u, (t) converge uniformly on [0, T] to the solution u(t) 0/ equation (1.27) with the initial value u(O) = Un. 2. The functions A(t)u,(t) converge uniformly on [0, T] to A (t)u(t), and hence also the B(t) u.(t) converge uniformly there to B(t) ;;(t). 3. The derivatives dU,/dt are uniformly bounded and converge to du/dt almost uniformly on (0, T]. 4. The residuals in equation (1.27) corresponding to the functions u, (t) are uniformly bounded and converge to zero almost uniformly on (0, T]. For particular solutions with the condition uO = - A -1 (0) f(O) -A- 1 (0)B(0)Uo in properties 3 and 4 above, the convergence is uniform on [0, T]. REMARK 1.3. The existence of the solutions u,(t) under the hypotheses of Theorem 1.6 follows from the analog of Theorem 3.2 of Chapter III for the nonhomogeneous equation (1.26).

296

IV. ASYMPTOTIC METHODS

As is clear from the preceding discussion, for an equation of second order the inconsistency of the solutions of equations (1.26) and (1.27) happens because of different initial values of the derivatives of these solutions. One may pose the question of uniform approximation on [0, T] of the solution u.(t) and of its derivative by simpler functions. Here we may as in subsection 3 apply the method of boundary layer theory and construct the function ~(~ =~(~ +~(~ -~(~,

where a.(t) and a(t) are solutions of the equations td 2u/dt 2

=

A(O) du/dt

+ B(O)u + 1(0)

and A (0) du/dt

+ B(O) u + 1(0) =

0

respectively, satisfying the initial conditions il.(O)

= /lo, a;(O) = ufJ,

11(0)

= /lo.

The difference u. (t) - v. (t), along with its first derivative, tends uniformly to zero on [0, T]. . 6. Homogeneous equations with small parameters. In what follows we shall need to consider further the case of the equation (1.29)

tdy/dt = A(t) y

+ tB(t)y

and compare its solution with the solutions of the equation (1.30)

tdx/dt = A(t)x.

Suppose that B(t) is a bounded operator, strongly continuously depending on t. If the Cauchy problem for equation (1.30) is uniformly correct, then in view of Theorem 3.4 of Chapter II the Cauchy problem for equation (1.29) will also be uniformly correct, given that the operator A(t)B(t)A -l(t) is defined, bounded and strongly continuous in t. Suppose that this condition is satisfied. Then for the solution of equation (1.29) with the initial data y(O) = Xo one can write the integral equation y(t) = U.(t,O)Xo+ LtU,(t'T)B(T)Y(T)dT'

§2. EVOLUTION OF SUBSPACES

297

The first term is the solution x(t) of equation (1.30) with the sarne initial date x(O) = Xo. Therefore yet) -x(t) =

(1.31)

If the operator gral inequality

J:

U.(t,r) B(r)y(r) dr.

U. (t, s) satisfies condition (1.11), then from the inte-

it follows that the functions yet) are uniformly bounded in E and t, and then it follows from (1.11) and (1.31) that the difference yet) x(t) tends uniformly (with order E) to zero as E-- O. Further, since A (t)B(t)A -let) is bounded and continuous, A(t)y(t) - A (t)x(t) =

J:

V.(t,r)A(r)B(r)A -1(r)A(r)y(r)dr.

It follows from estimate (1.15) that the functions A (t)y(t) are bounded uniformly relative to E and t, and that as E --+ 0 the difference A (t) yet) - A (t) x(t) tends to zero uniformly on [0, T]. We have arrived at the following assertion. THEOREM

1.7. Suppose that the Cauchy problem for equation (1.30)

is uniformly correct and that inequality (1.11) holds. Suppose that the operator B(t) satisfies the hypotheses of Theorem 3.4 of Chapter II. Then as E --+ 0 the difference yet) - x(t) of equations (1.29) and (1.30) with the same initial conditions, not depending on E, will tend uniformly on [0, T] to zero with order O(E). In addition, under these hypotheses A(t) [yet) - x(t)] also tends uniformly to zero.

§2. Evolution of subspaces in Banach space 1. Statement of the problem. Suppose that pet) is a projection operator, continuously depending on t in the operator norm. For each t the range of the operator pet) forms a closed subspace .!ift = pet) E of E. As t changes this subspace varies in some way in the space E. Since the operator pet) is continuous in the norm, the dimension of the subspace .sf, does not change under this variation. We shall prove this. Suppose that t and r are such that

298

(2.1)

IV. ASYMPTOTIC METHODS

II P(t)

- P(T>iI

< 1.

Choose any element x E Y. and construct the element z = (P(T)

+ P(t» -lX,

which exists because of condition (2.1). Write y = P(t)z. Then yE Y t• Further, x = (I - P(T) + P(t»z and therefore x = P(T)X = [P(T) - P2(T) + P(t)]z = P(T)Y. Thus for each x E Y. we have found ayE Y, such that P(T)Y = x, i.e. under projection onto Y. the image of Yt covers all of y.. The dimension cannot increase under projection. Hence dim!£; ~ dimYt. In view of the symmetry of (2.1) in t and T we therefore have dim!£; = dimYt. Because P(t) is uniformly continuous in norm on each finite segment, it follows that the last assertion is valid for all t and T. We have shown that under condition (2.1) P(T)Yt = Y.. However, for other t and l' this relation may not be true. For example, over some interval of time from T to t the subspace Yt might become orthogonal to !£;, i.e. P(T)Yt = o. In various problems of spectral theory it is important to know how to construct operators Q(t, T), defined on the whole space. E, realizing an isomorphic mapping of the space !£; onto the subspace Yt and depending in a sufficiently smooth manner , on parameters t and T. DEFINITION 2.1. We shall say that an operator Q(t,s) (0 ~ s ~ t ~ T), acting in the entire space E, bounded and strongly' continuous in t and s, follows the evolution of the subspace Yt, if (2.2)

Q(t, s) Q(s, T) = Q(t, T)

and (2.3)

P(t)Q(t,T)P(T)

= Q(t,T)P(T).

We say that it realizes the evolution of the subspace Yt if moreover it is defined for all 0 ~ s, t ~ T and (2.4)

Q(T,t)Q(t,T)P(T) = P(T).

Denote by 1t the complement to Yt, orthogonal in the sense of the projection P(t) of the subspace, and by p(t) the operator of projection onto it: p(t) = I - P(t). The entire space E decomposes into a direct sum E = Yt +1,. If the operator Q(t, T) follows the evolution of Yt as well as the evolution of 1" then in addition to (2.3) the identity

(2.5)

§2.

EVOLUTION OF SUBSPACES

299

is satisfied as well. Hence it follows that P(t)Q(t,T)P(T) = O.

Combining this equation with (2.3), we obtain the relation (2.6) which contains (2.3) and (2.5). DEFINITION 2.2. We shall say that the operator Q(t, s) (0 ~ s ~ t ~ T) follows the evolution of the direct sum E = Sfe 1{t if (2.2) and (2.6) are satisfied, and realizes the evolution of the direct sum E = Yt Jil if moreover it is defined for all t E [0, TJ and

+

(2.7)

Q( T, t) Q(t, T)

=

+

I.

2. Construction of an operator realizing the evolution of a subspace. Since when t is close to T the operator P(T) realizes a mapping of Yt onto ~, it is natural to attempt to construct the operator Q(t, T) as a product of operators P(t) with slightly different arguments. Denote by q a subdivision of the segment [T, tJ into pieces by the points T = to < tl < ... < tn = t. Construct the operators Qq(t, T) = P(tn) P(tn-l) •.• P(t1 ) P(to)

and Qq( T, t) = P(to) P(t1 )

The operators P(t) (0

(2.8)

~

t

~ T)

1 ~ I P(t)

II

~

•••

P(t,._l) P(tn).

are uniformly bounded:

M

(O~t~T).

However it does not in general follow from this that the operators are uniformly bounded relative to all subdivisions q. We can only assert that ~(t, T)

(2.9)

II Qq(t, T) II

~ Mn+l

(0 ~ T < t ~ T).

By imposing additional requirements on the dependence of the operators P(t) on t, we may achieve uniform boundedness of the operators Qq(t, T) relative to q. Suppose that the operator P(t) is of bounded variation in t in the sense that N

(2.10)

V

=

sup L II P(Tk) - P(Tk-l) " N

<

CX>,

k=l

Where the supremum is taken over all possible finite choices of points

o ~ Tl < T2 < . .. < TN ~ T.

300

IV. ASYMPTOTIC METHODS

LEMMA

2.1. Under the hypothesis (2.9)

the operators Qq(t, T)

and

Qq (T, t) are uniformly bounded relative to t, T, and q. PROOF.

We write P(tk ) = P" for short. We represent the operator

Qq(t, T) in the form Qq(t, T) = PnPo+ Pn(PI - Pn)Po+

(2.11)

...

or, using the identity

Pn(P" - Pn) P"-I = Pn(P" - Pn) (Pk- I - P,,), in the form n

(2.12)

Qq (t, T)

=

PnPo+ 2: P"(P,, - Pn) (Pk - I - Pk) P"-2 ••• Po. II-I

Write

Mn = sup II P(t,.) ••• P(to) I = sup II Qq(t, T) II , where the supremum is taken over all collections of points {til}: T ~ to ~ ••• ~ t,. ~ t. Obviously Mo = M ~ Ml ~ ... ~Mn ~ M n+1' It follows from (1.29) that Mn ~ Mn+1. Denote by w(t, T) the oscillation of the function P(s) on the interval [T, t], .i.e.

w(t, T) = sup II P(s) - P(s') I . • ;;;;.,s';;;;1

Then n

I Qq(t, T) II ~ II Pnll I Poll

+ 2: I Pnlill P" "~1

Pnll I Pk - 1 -

Pilil II Pk - 2 •• • Poll

n

~ M2+ MwM"-l2:IIP"-l-

Pilil

~ M2+ MM"_I Vw(t,T).

II-I

We shall suppose the segment [T, t] sufficiently small that (2.13)

Then

Hence

W(t,T)

~

a/MY,

o
§2. EVOLUTION OF SUBSPACES

~

M,.

301

M2 + aM,._l'

Applying this relation successively, we obtain

M,.

~

M 2 /(1_ a).

In view of the uniform continuity of the operator P(t) we may divide the segment [0, T] into No pieces so that in each of them (2.13) is satisfied, for example, with a = i. Then

Qq(t, T)

~

[2M2]No

=

C

for all t, T and q. Analogously one proves the uniform boundedness of Qq(T,t). The lemma is proved. THEOREM 2.1. If the number of points in the subdivision q is increased unboundedly in such a way that the diameter d(q) of the subdivision tends to zero, then there exists a limit in the operator norm:

Q(t,T) = lim Qq(t,T) and lim Qq(T,t) = Q(T,t). d(q) .....O

d(q)--+O

Suppose that the number 6 = 6(E) is such that on any seg· ment of [0, T] of length 6(t) the oscillation of the function P(t) does not exceed E. Suppose that ql and q2 are two subdivisions whose diameters do not exceed ~. As usual, we first suppose that the subdivision q2 is a refinement of the subdivision ql' Suppose that ql is constructed by means of the points T = to < tl < . .. < tj < tj+l < . .. < tn = t, and q2 with the points T = TO < Tl < . .. < Tkl = tl < . .. < Tk2 = ~ < ... < Tkj = tj < ... < Tk,. = t. Then the operator ~(t.T) has the form PROOF.

Qq2(t,T) = R,.R,._l··· R1P(T), where

The operator Q~ (t, T) may be represented in the form

Qql(t,T) where

= R~R~_l'"

R~P(T),

302

IV. ASYMPTOTIC METHODS

Let us estimate the difference of the operators Rj and RJ. To this end we again use an identity analogous to (2.12). We have

II Rj

II

- RJII =

kj-kj_l-I

~

P(Tkj) [P(Tkj-8) - P(Tk)]

x [P(Tkj-.-I)

- P(Tkj-') ]P(Tkr.- 2 )

•••

P(Tkj_I)

II

k.

t

~ MCE

II P(Tj-I)

- P(Tj) II.

i-kj_I+!

We then obtain for the difference of the operators Q'I2(t,s) and QqI(t,S) the quantity n

I Qql (t, T)

- Qq2(t, T)

I

~

L I Rn - •. Rj+! (Rj -

Rj ) RJ-I .•• RW( T)

I

j-I

n

n

~ C2 LII RJ -

Rjll

~ MC 3

j-I

EL L I P(Ti-l) -

j-l

P(Tj)

I

~ MC3 VE.

j-I i-kj_l+!

For arbitrary qi and q2 with diameters not exceeding 6 we obtain

I Qql(t,T)

- Qq2(t,T)

I

~ 2MC3 VE.

The theorem is proved. 3. Properties of the operator Q(t, T). All the properties of the operator Q(t, T) are deduced from the properties of the operators Qq(t, T) by passing to the limit. We shall find them. 10. Obviously P(t) Qq(t, T) = Qq(t, T) P(T) = Qq(t, T), so that (2.14)

P(t) Q(t, T) = Q(t, T) P( T) = Q(t, T).

2 0 • We shall show that Q(t, T) Q(T, t) = P(t) and Q(T, t) Q(t, T)

(2.15)

=

PH.

To this end we transform P(T) - Qq(T, t) Qq(t, T) = Po Po - POP1

Pn-IPnPn-I • _. Po

•••

n-l

=

L

Po' •. Pk(Pk

-

PHI) Ph' •• Po-

k-o

Using the identity Pk(Pk - P k+!) Ph

we estimate

=

Pk(Pk

-

PHI) 2 P k

303

§2. EVOLUTION OF SUBSPACES n-1

I P(T)

~

I

- Qq(T, t) Qq(t, T)

L I PO'"

Pk(Pk - P kH )2Pk

• ••

Poll

k-O n-l

~ (;2£ L II P k

-

PkHl1

~ C2E V.

k=O

Letting d(q), and hence also E, tend to zero, we obtain the second of identities (2.15). The first is established analogously. 3°. The identity (2.16)

Q(t, s) Q(s, T)

=

Q(t, T)

for T ~ S ~ t obviously follows from the construction of the operator Q(t, T). It is valid also for any relations between T, s, and t, as follows from the foregoing and properties 1° and 2°. lim Q(t,T)

,-.

=

(2.17)

P(T).

From identity (2.12) it follows for t >

T

that

n

~ I (Pn - Po) Poll

+L

I Pn(P" -

P n) (Pk -

P k)P"-2'"

1 -

Poll

1<=1

~

Mil P(t)

- P(T)

I

+ MCw(t, T) V.

Passing to the limit as d(q)

I Q(t, T)

- P(T)

I

--+ 0,

~

we get

Mil P(t)

- P(T)

I + MCw(t, T) V,

from which (2.17) follows. Analogously one verifies the case t < T. 5°. The function Q(t, T) is continuous in t and T. Indeed, in view of (2.14) and (2.16)

I Q(tlt T)

- Q(t, T)

I

=

I (Q(t h

t) - P(t» Q(t, T)

I

~

GIl Q(tlt t)

- P(t)

I --+ 0

t1 --+ t. Analogously one verifies continuity in T. It follows from properties 1°-5° that the operator Q(t, T) realizes the evolution of the subspace Y,.

as

THEOREM 2.2. If the operator P(t) is strongly continuously differentiable in t, then the operator Q(t, T) is also strongly continuously differentiable m t and satisfies the equation

(2.18)

dQ(t, T) Idt = pI (t) P(t) Q(t, T).

304

IV. ASYMPTOTIC METHODS

PROOF.

From identity (2.12) we find that Qq(t, '1) - P(T) _ P(t) - PCT) P(T) t-T t-T

so that '1) - P(T) I Qq(t, t-T

_ P(t) - P(T) P(T) t-T

II ~ MCw(t, '1) V. t-T

Passing to the limit as d (q) ---+ 0 yields (2.19)

II Q(t, '1)t --

P(T) _ P(t) - P(T) P(T) '1 t - '1

II ~ MCw(t,'1) V. t - '1

The operator P(t) satisfies a Lipschitz condition, so that V ~ KI t - '11, and thus the right side of (2.19) tends to zero. Therefore it follows that the derivative of the function Q(t, '1) exists at the point t = '1: (2.20) Now we calculate the derivative for any t. We have (Q(t + I1t, '1) - Q(t, '1» /l1t

=

«Q(t + dt, t) - P(t}) /l1tQ) (t, '1)

Formula (2.18) follows from this and formula (2.20). The theorem is proved. 4. The operator realizing the evolution of the direct sum. Relative to the projection operator P(t) = I - p(t) we construct in the way described above the corresponding operator Q(t, '1), realizing the evolution of the subspace ..L,. Obviously (2.21)

Q(t, s) Q(s, '1) = Q(t, s) P(s) P(s) Q(s, '1) =

o.

Put (2.22)

R(t, '1)

= Q(t, '1)

+ Q(t, '1).

This operator has properties which follow immediately from the properties of the operators Q(t, '1) and Q(t, '1). 10 • The identity (2.6) holds: P(t)R(t,T) = R(t,T)P(T).

Indeed, in view of (2.14)

305

§2. EVOLUTION OF SUBSPACES

P(t)R(t, T)

+ P(t) Q(t, T)

= P(t) Q(t, T) = Q(t, T)

+ P(t) P(t) Q(t, T) =

Q(t, T)

and

+ Q(t,T)P(T)

R(t,T)P(T) = Q(t,T)P(T) = Q(t,T)

+ Q(t,T)P(T)P(T) =

Q(t,T).

2°. The operators R(t, T) and R(T, t) are mutually inverse. Indeed, in view of (2.15) and (2.22) R(t,T)R(T,t)

= P(t)

+ P(t) =

1.

3°. It follows from (2.16) and (2.21) that R(t,s)R(s,T)

=

R(t,T).

4°. It follows from (2.17) that lim

R(t, T)

= 1.

t-+~

5°. The operator R.(t, T) is continuous in t and T. It follows from properties 1°-5° that the operator R (t, T) realizes the evolution of the direct sum E = Y t 1 t • 6°. If the operator P(t) is strongly continuously differentiable, then the operator R(t, T) is as well, and

+

(2.23)

dR/dt

= (PI (t) P(t)

+ pI (t) P(t» R.

Indeed, in view of (2.18) and (2.14) dR/dt = pI (t) P(t) Q(t, T)

= (P'(t)P(t)

+ pI (t) P(t) Q(t, T)

+ PI(t)P(t»R(t,T).

For the case of a differentiable operator P(t) the operator R(t,T) may be directly constructed as an evolution operator for the differential equation (2.23) with a bounded operator coefficient. We may consider the more general case when the space E is decomposed into the direct sum of subspaces 5L"~k) (k = 1, 2•• · .,N) such that mutually orthogonal operators Pk(t) correspond to them. If we introduce the operators Qk(t, T) corresponding to the Pk(t), then the operator N

R(t, T) =

L k-l

Qk(t, T)

306

IV. ASYMPTOTIC METHODS

will have properties 10 -6 0 • Here property 10 is written as follows: P,,(t)R(t,T) = R(t,T)P,,(T),

and the differential equation (2.23) takes the form dR N ----d = ~ Pk(t) P,,(t) R.

(2.24)

t

k-1

The function R(t, T) is its solution with the initial value R(r, T) = I. We note that from the identity

there follows the relation N

N

~ PW)Pk(t)

= - ~ Pk(t)Pk(t),

k-1

"-1

so that equation (2.24) may be rewritten in the form (2.25)

dR -= -

dt

N.

~Pk(t)PW)R. k-1

The operator R(t, T) has the same smoothness as "the operators Pl. If the operators P,,(t) admit analytic continuation into some region of the complex plane of t, then the operator R(t, T) also admits analytic continuation into the same region. 5. Hilbert space. In Hilbert space the case in which the space H decomposes into the orthogonal sum of spaces Y 1") is of interest. In this case it is natural to regard the operators P,,(t) as orthogonal projection operators. THEOREM 2.3. Suppose that the space H is decomposed into the orthogonal sum of subspaces Y lk) (k = 1,2, ... , N). Suppose that the orthogonal projection operators are continuous in norm relative to t and that they are of bounded variation. Then there exists a unitary operator R(t, T) realizing the evolution of the orthogonal decomposition H =

Y:

ffiY1N). In the proof we need only to know that the operator R(t, T) is unitary. From the construction of the operators Qq(t, T) from the selfadjoint projection operator Pk(t) it follows that ffi· ••

307

§2. EVOLUTION OF SUBSPACES

Then the same relation will hold for the operator Q(t, T) and thus for R(t, T). Hence R'(t,T)

=

R(T,t)

=

R-1(t,T).

The theorem is proved. 6. Evolution of invariant subspaces. Suppose that A (t) is a closed operator with a domain ~(A) dense in E and independent of t. Suppose that A(t) is strongly continuously differentiable on ~(A) and has a bounded inverse A -l(t). Suppose that A1(t) is a closed portion of the spectrum A(t) of the operator A(t) having a closed complement A(t) - A1(t). Suppose that r 1(t) is a smooth arc separating Al (t) from A(t) - Al (t). If for any to E [0, T], for t sufficiently close to to, Al (t) coincides with the portion of the set A(t) remaining inside the region bounded by the contour fl (to), then we shall call Al (t) an isolated branch of the spectrum of the operator A (t). We suppose that the spectrum of A (t) decomposes into the sum of isolated branches A(t) = Ao(t)

+ A1(t) + ... + An(t)

such that the sets A1(t)," ',An(t) are bounded. We denote by fk(t) the smooth simple closed curve isolating Ak(t) from its complement. As we pointed out in §4 of the Introduction, the operators Pk(t) = -

2~i

f

RA(t) (x)

dX

(k

=

1, •.. , n)

fk(t)

will for each t E [0, T] be pairwise orthogonal projection operators commuting with the operator A (t). In the invariant subspaces !zflk) = Pk(t) E the operator A (t) is bounded and has a spectrum coincident with the set Ak(t). The operators Pk(t) are strongly continuously differentiable. This is connected with the fact that for a t close to to we may, since the branches Ak are isolated, calculate Pk(t) according to the formula Pk(t)

== -

2~i

J fk(tO)

RA(t)

(x) dx.

308

IV. ASYMPTOTIC METHODS

Then the strongly continuous differentiability of P.(t) follows directly from the strongly continuous differentiability in t of the resolvent RA(t)

(X).

Define the operator Po(t) = 1- .EZ-IPk(t). It will have the same properties as the operators Pk(t), and will be orthogonal to them. Relative to the system of operators PO(t),PI(t)," .,p.. (t) we construct an operator R(t, T), which is an evolution operator for the equation

t

dR pf.(t) Pk(t) R. dt k-O

(2.26)

THEOREM 2.4. The operator R(t, T) leaves invariant the domain of the operator A(t). The operator A (t)R(t, T)A -leT) is bounded and strongly continuously differentiable in t. PROOF.

(2.27)

Consider the differential equation dS = dt

±

PHt)Pk(t)S

+

k-O

t

Pk(t)A'(t)A -l(t)Pk(t)S.

k-O

Denote by Set, T) its solution satisfying the initial condition SeT, T) = 1. This solution exists and is strongly continuous since all the operator coefficients in equation (2.27) are bounded and strongly continuous. Write V(t,T) = A-I(t)S(t,T). Then

(2.28)

.. + A -let) .E Pk(t) A' (t)A -l(t)Pk(t)S . .10-0

Consider the operators Bk(t) = A -1(t)P;,(t)

(2.29)

= A -1(t)Pk(t)

+ A-l(t) Pk(t)A' (t) A -let) - Pk(t)dA -l/dt.

Here we are using the fact that p.(t) A -let) = A -l(t)P.(t).

Differentiating this identity in t, we get Pk(t)dA -l(t)/dt + Pk(t)A -let)

= (dA -l(t)/dt)Pk(t)

+ A -1(t)Pk(t).

Expressing the first term in terms of the others, and substituting into (2.29). we get

309

§3. SPLITTING AN EQUATION

Hence n L Bk(t)Pk(t) k-o

=

n dA -l(t) L PW)Pk(t)A -l(t) - ------'~

dt

k=O

The differential equation (2.28) takes the form dV n -d =LPW)Pk(t)V t k=O and coincides with equation (2.26). Weare interested in the solution of it which satisfies the initial condition V(r,r) = A -l'
so that S(t,r) = A(t)R(t,r)A -l(r).

The theorem is proved. §3. Splitting of an equation into equations in subspaces

1. Solution of an equation in a subspace. Consider the differential equation (3.1)

dx/dt = A(t) x,

where A(t) is closed, has a domain ~(A) not depending on t, and is strongly continuous on it. We suppose that for equation (3.1) the Cauchy problem is uniformly correct. We denote by U(t, s) the corresponding evolution operator. Suppose that the space E is decomJ!.,osed into the direct sum of subspaces Yand .At. We denote by P and P the ~ojection ~erators generated by this decomposition: PE = Y, PE = .At, PP = O. We are interested in the question as to when the subspaces .Y and .At remain invariant under the transformation U(t,s).

The property of invariance for a subspace may be written (see (2.6» in the form

(3.2)

PU(t, s)

= U(t, s) P.

Now we suppose that the projection operator P has the property that

310

IV. ASYMPTOTIC METHODS

P~(A) C ~(A). Then, if (3.2) is satisfied identically m t' and s, then for Xo E ~(A) the equation

(3.3)

PU(t,s)Xo = U(t,s)PXo

may be differentiated with respect to t. We get PACt) U(t,s)xo

=

A,(t) U(t,s)PXo.

N ow putting s = t, we have (3.4)

(XoE ~(A».

PA(t)Xo = A (t) PXo

, Conversely, if (3.4) is satisfied, then the function yet) satisfies the differential equation dyjdt

=

PACt) yet) = A (t)Py(t)

=

= PU(t, s) Xo

A (t)y(t)

and the initial condition yes) = PXo. In view of the uniqueness of the solution of the Cauchy problem for equation (3.1), yet) = U(t, s) PXo, and we have arrived at the fact that (3.3) is satisfied on the elements Xo E ~(A). Since ~(A) is dense in E and the operators U(t, s) and P are bounded, (3.2) follows. Thus in order that under the condition' P g-(A) C g-(A) the subspaces Y and 1 remain invariant it is necessary and sufficient that the projection operator P commute with the operator A (t) on its domain. If we are interested only in the solution with initial conditions from Y, and (3.4) is satisfied, then instead of equation (3.1) in the whole space E we may consider the equation dxjdt = PAx in the subspace Y. 2. Evolution operators following the direct sum. N ow suppose that we are given a variable direct decomposition E = Y, + 1, of the space E with the corresponding projection operators pet) and P(t): pet) E

= Yh

pet) E

= 1"

pet)

+ p(t) =

I.

We pose the question of finding conditions under which the evolution operator U(t, s) of equation (3.1) follows the direct sum E = y, + ...Itt. We shall find these conditions by the same method as we used above, supposing that the operator pet) is strongly continuously differentiable in t and pet) ~(A) c ~(A). Suppose that the operator U(t, s) follows the direct sum, i.e. that (3.5)

pet) U(t, s)

=

U(t, s) pes)

(0 ~ s ~ t ~ 1').

Applying the operators on the left and right of this equation to the element Xo E ~(A), we get

311

§3. SPLITTING AN EQUATION

PI(t) U(t,S)Xo

+ P(t)A(t) U(t,S)XO =

A(t) U(t,S)P(S)Xo.

Putting s = t, we have (3.6)

[A (t) P(t) - P(t) A (t) ]Xo = pI (t) Xo

(Xo

E

~(A».

Conversely, if (3.6) is satisfied, then the function y(t) satisfies the equation dy/dt

(3.7)

=

P(t) U(t,s)Xo

= pI (t) U(t, s)Xo + P(t)A (t) U(t, s)Xo = A (t) P(t) U(t, s) Xo = A (t) y

and the initial condition y(s) = P(s)Xo. Hence, just as above, we get (3.5). Thus condition (3.6) is necessary and sufficient for the operator U(t, s) to follow the direct decomposition E = Y t -+- .Lt. If identity (3.6) is satisfied, the following problem arises: if we are interested only in those solutions x(t) of (3.1) for which the initial data lie in Y" then is it possible to find these solutions from a simpler differential equation? This question is particularly important in the case when Y is finite dimensional. Equation (3.6) shows that the corresponding equation must be written in the form (3.8)

dx/dt

= A(t)P(t)x.

This equation is simpler than the original equation (3.1), particularly when P(t) is finite dimensional. However it has the defect that it cannot be considered as a differential equation in some subspace, since the values of its solutions, along with the subspace Y" might wander all over the space E. In order to reduce equation (3.8) to an equation in a subspace, for example in the subspace Yo, we may use the operator R(t, r) realizing the evolution of the subspaces Y t and .Lt. If we introduce the function z(t) = R(O,t)x(t), then under the condition Xo E Yo its values already lie in the subspace Yo for all t. Let us set up the differential equation for the function z(t). We have dz/dt = (dR(O,t)/dt)x(t)

Recalling that R(O,t) (3.9)

dz/dt

=

=

+ R(O,t)A(t)x(t).

R- 1 (t,0), we obtain the equation

[(dR- 1 (t, 0) /dt) R(t, 0)

+ R-

1

(t, 0) A (t) R(t, 0) ]z,

and the initial condition z(O) = x(O) = Xo. U sing the differential equation for the operator R (t, 0), write equation (3.9) in the form (3.10)

dz/dt = R(O, t) [ - pI (t) P(t) - pI (t) P(t)

+ A (t) ]R(t, O)z

312

IV. ASYMPTOTIC METHODS

or in the form dzldt

=

R(O, t) [pet) P' (t)

+ pet) P' (t) + A (t) ]R(t, O)z.

VVe note that the operator S (t) = - P' (t) pet) - P' (t) pet)

(3.11)

+ A (t) ,

in view of condition (3.6), commutes with the operator pet) on ~(A). Indeed, using the identity P' P = P P' ,we get pet) Set) = - pet) P' (t) pet) - pet) P' (t) pet)

+ pet) A (t)

= pet) P' (t) [P(t) - P(t)] + A (t) pet) - P' (t) =

P'(t)P(t)

-

+ A (t)P(t) =

S(t)P(t).

Hence in particular it is immediately clear that the operator coefficient in equation (3.10) commutes with the operator P(O): P(O)R(O, t)S(t)R(t, 0)

R(O,t)P(t)S(t)R(t,O)

=

= R(O,t)S(t)P(t)R(t,O) =

R(O,t)S(t)R(t,O)P(O).

It is easy to verify that equation (3.11) with the operator Set) commuting with pet) implies condition (3.6). VV e formulate our conclusions in the form of a theorem. THEOREM

3.1. Suppose given a direct decomposition of the space E: E

=

Y, -+- .L,. Suppose that the projection operators pet) and p(t) generated by this decomposition are strongly continuously differentiable with respect to t. Suppose further that the domain of A (t) remains invariant. For the evolution operator U(t, s) of equation (3.1) to follow that decomposition, it is necessary and sufficient that A (t) be representable in the form

A (t)

=

P' (t) pet)

+ P' (t) pet) + S (t),

where the operator Set) is defined on ~(A) and commutes there with the operators pet) and P(t).

In order to find solutions of equation (3.1) with initial conditions from Yo we may solve the simplified equation (3.8). After the substitution z = R(O, t)x the equation for z reduces to two independent equations dzldt

and

=

P(O)R(O,t)S(t)R(t,O)z

13.

SPLITTING AN EQUATION

313

dz/dt = P(O)R(O,t)S(t)R(t,O)z,

each of which may be solved in its corresponding space Yo or Ato• N ow we suppose that in the spectrum of the operator A (t) there is an isolated portion Al(t). As we pointed out above, there corresponds to it a projection operator pet) and an invariant subspace Y, = pet) E. Suppose that A (t) is strongly continuously differentiable on 9J(A). Then the operator pet) will be strongly continuously differentiable. The operator A (t) commutes with P(t), so that relation (3.6), as a rule, is not satisfied. It can be be satisfied identically only in the case of a constant operator P(t). Thus, the evolution operator U(t, s) as a rule does not follow the invariant subspaces of A (t). Here lie the fundamental difficulties of the theory of linear equations with variable coefficients. 3. Splitting of the evolution operator. For the case just considered, when the operator pet) corresponds to an isolated portion of the spectrum of A (t), we shall attempt to represent the evolution operator in the form of the product of two operators, one of which follows the evolution of the subspace Y" and one of which is derived from Y ,. lt would be desirable that both of these operators be found from equations simpler than (3.1). Consider the auxiliary equation (3.12)

dv/dt

= (A (t)

+ P' (t) pet) + P'(t) P(t» v.

This equation may be considered as a perturbed equation relative to equation (3.1), with the bounded perturbation operator B(t) = P' (t) pet) + P' (t) P(t). If the Cauchy problem for equation (3.1) is uniformly correct, then from Theorem 3.4 of Chapter II the Cauchy problem for equation (3.12) will be unformly correct if the operator A(t)B(t)A -let) is defined and strongly continuous in t. Differentiating the equation A (t) pet)

= pet) A (t)

with respect to t, we get A' (t)P(t)

+ A(t)P'(t) =

P' {t)A (t)

+ P(t)A'(t).

Using this relation, we make the transformations A{t)B{t)A -let)

= A (t)P'(t) P(t) A -let) - A(t) P' (t) P(t) A -let)

+ P(t)A'(t) - A'{t)P(t) ][P(t) A -let) - P(t) A -l(t)] = P'(t)P(t) + P'(t)P(t) - P(t) A' (t)A -1(t)P(t) - P(t)A'(t)A -1(t)P(t).

= [P'(t)A(t)

314

IV. ASYMPTOTIC METHODS

Hence it is clear that the operator A(t)B(t)A -l(t) is strongly continuous. Thus, for equation (3.12) the Cauchy problem is uniformly correct. Denote by U1 (t, s) the evolution operator corresponding to it. The operator A1(t) = A(t) + B(t) satisfies the condition (3.6), so that the operator U1 (t, s) follows the evolution of the direct sum E = Y t +'1t • If we are interested only in the solutions of (3.12) with initial data in Y., then they may be found from the equation (3.13)

dv/dt = A1(t)P(t)v = [A (t)P(t)

+ pI (t)P(t) ]v.

We wish to find solutions of equation (3.1) with initial data from U(t, 0) P(O) in the form of the product of the operator U1 (t, 0) with a "rectifying operator" Y(t):

Yo. To do this we attempt to represent the operator

(3.14)

U(t,O)P(O)

= Y(t) U1(t,0)P(0).

Now we try to select Y(t) in such a way that the relation (3.15) (dY /dt) P(t) = A (t) Y(t) P(t) - Y(t) A (t) P(t) - Y(t) pI (t) P(t)

is satisfied. If this is done, then, in view of (3.13) and (3.15), for Xo E 9&'(A) the function z(t) = Y(t) U 1(t, 0) P(O) Xo satisfies the equation dz/dt

= (dY/dt) U1(t,0)P(0)Xo

+ Y(t)[ A (t) P(t) + pI (t) P(t) ] U

1 (t,

0) P(O) Xo

= A(t) Y(t)P(t) U1(t,0)P(0)Xo = A (t)z(t). Thus the function z(t) is the solution of equation (3.1), and if one supposes that Y(O) = P(O), then z(O) = P(O)xo, so that z(t)

= Y(t) U 1(t, 0) P(O) Xo = U(t, 0) P(O) Xo,

i.e. (3.14) is satisfied. In order to find an operator-function Y(t) satisfying (3.15), we may consider the differential equation /

(3.16) dY/dt

= A (t) Y(t) - Y(t) A (t) P(t) - Y(t)(PI (t)

+ pI (t) P(t».

By multiplying on the right one easily verifies that each solution of equation (3.16) satisfies (3.15) as well. A solution satisfying the initial condition Y(O) = P(O) has the property that (3.17)

Y(t)P(t)

=

Y(t).

315

§3. SPLITTING AN EQUATION

Indeed, if we differentiate the left side, we see easily that it satisfies the same differential equation (3.16) and the same initial condition as the function Y(t). We shall not concern ourselves with investigating the existence of a solution of equation (3.16). In this regard the assertion of the following theorem is of a formal character. We note only that for the case of a finite-dimensional operator P(t) this is not hard to check, considering the equation as a perturbed equation (3.1). THEOREM 3.2. Suppose that Xo E Yo n 9(A). Then the solution x(t) of equation (3.1) may be represented in the form

x(t)

= Y(t) V(t, 0) Xo,

where V(t, s) is the evolution operator corresponding to the equation (3.13) and Y(t) is the solution of the differential equation (3.16) with the initial condition Y(O) = P(O).

Of course it is hard to claim that equation (3.16) has a simpler structure than the original equation (3.1). However in certain cases one may suppose that the leading part of the evolution operator U(t, s) is described by the operator U I (t, s) and that the operator Y(t) consists of the operator P(t) and a component which may be found by approximation. We consider such a situati()n in the following subsection. 4. Equations with a small parameter on the derivative. Suppose that the original differential equation contains a small parameter on the derivative: (3.18)

Edxjdt = A(t)x

In order to apply Theorem 3.2 we set up differential equations analogous to (3.12) and (3.15): (3.19)

Edvldt = A(t)v

+ E(P'(t)P(t) + P'(t)P(t»v

and (3.20)

E(dY Idt) P(t) = A (t) Y(t) P(t) - Y(t) A (t) P(t) -

E

Y(t) P' (t) P(t).

If E is small, then in equation (3.20) we may drop terms containing E, and then put, approximately,

Y(t) == Yo(t) = P(t).

316

IV. ASYMPTOTIC METHODS

If Xo E 9,(A) function

n YO,

then we find in accordance with (3.14) the

v?(t) = P(t) U1 (t, 0) Xo = U1 (t, 0) Xo,

the last equality holding in view of the fact that U1 (t, s) follows the evolution of Y t • In order to compare the solutions of equations (3.18) and (3.19) we may apply Theorem 1.7, which brings us to the following assertion: THEOREM 3.3. Suppose that A (t) satisfies the conditions of Theorem 1.7. ut x(t) be a solution of (3.18) with the initial data x(O) = Xo E 9(A) n Yo. ut v?(t) be a solution of the equation E

dv/dt = A(t)v + EP' (t) P(t) v,

lyin& in the space Y t and having the same initial data. Then the difference x(t) - v?(t) converges to zero uniformly on [0, T] as f -+ 0, with order O(E). In addition the functions A (t)[x(t) - v?(t)] also converge to zero uniformly on [0, T]. REMARK. We call v?(t) a null approximation to the solution of equation (3.18).

N ow we tum to the construction of further approximations to the solution of (3.18). We shall try to satisfy equation (3.20) up to terms of order E2. Put (3.21)

Y(t) = P(t)

+

f

Y1(t) ,

substitute this expression in (3.20) and drop the terms of order Then we come to the equation (3.22)

A (t) Y1 (t) P(t) - Y1 (t) A (t) P(t)

=

E2.

P(t) P' (t) P(t).

In view of condition (3.17) it is natural to seek Y1 (t) so that the following relation is satisfied: (3.23) ,/

The proof of existence of a solution of (3.22) is based on the following lemma. LEMMA 3.1. Suppose that El and E2 are two Banach spaces. Let Al and A2 be two operators in these spaces. Let T be a bounded operator acting from E2 into E 1 • Finally suppose that the spectra of the operators Al and A2 in the extended complex plane do not intersect. Then there

317

§3. SPLITTING AN EQUATION

exists a unique bounded operator X acting from E2 into EI and satisfying the equation (3.24) PROOF. We suppose first that Al and A2 are bounded. Suppose that the regions Gi contain the spectra of the operators Ai and do not intersect. Let r i be their boundaries. Consider the bounded operator acting from E2 into El defined by the formula

2

X = -

(3.25)

411"2

r rR

Jr2 Jrl

Al (>.)

TR4#L) d>.d#L.

A - #L

•

Calculate

and

_~ r r #LRAl (>.) TRA2 (#L) d>.d#L. J P2 J r l

h

>. - #L

The first integrals in both equations are equal to zero in view of the analyticity of the integrand functions inside one of the contours of integration. Then

AlX - XA 2 =

-

~ 411"

r RAl (>.) d>. T Jr2r RA2 (#L) dy.. = Jrl

T.

We shall prove the uniqueness of the solution. If AlX - XA 2 = 0, then (AI - >./)X - X(A 2 - >./) = O. Applying the resolvents on left and right, we get

Hence

318

IV. ASYMPTOTIC METHODS

N ow we turn to the general case. If the spectra of the operators Al and A2 do not intersect, then these operators have at least one regular point >.0. Set up the equatiQn (3.26)

RAI (>.0) X - XR A2 (>.o) = - RAI (>.0) TR A2 (>.o).

By what we have said, it has a solution which will also be a solution of equation (3.5) in the sense that equation (3.24) becomes an identity on each element u E ~(A2)' Indeed, represent the element u in the form u = R A2 (>.o) , and apply (3.26) to v. Then RAI (>.0) Xv - Xu = - RAI (>.0) Tu.

It follows from this equation that the element Xu E ~(AI)' so that the operator Al - >.oj can be applied to both sides. On doing this, we get X(A 2 - >.oJ)u - (AI - >.oJ)Xu = - Tu

or

The lemma is proved. Now we turn to equation (3.22). Applying the operator P(t) to both sides of this equation, we obtain the equation A (t) P(t) YI (t) P(t) - P(t) YI (t) A (t) P(t) = 0,

which we may satisfy by putting P(t) YI (t) == O. Taking account of (3.23), we have Y I (t)

(3.27)

= P(t)

Y I (t) P(t).

Now applying the operator P(t) to (3.22), we arrive at the equation P(t)A(t) YI(t)P(t) - P(t) YI(t)A (t)P(t)

=

P(t)pI (t)P(t).

Using (3.27) and the permutability of the operators A(t), P(t) and '" \ P(t) , we finally obtain (3.28)

P(t) A (t) YI (t) - Y I (t) A (t) P(t)

= P(t) pI (t) P(t).

This equation may be considered for each t as an equation of the type (3.24), putting

EI = 1t, E2 = Y A2

=

t

and Al = P(t) A (t),

A (t) P(t) and T

=

P(t) pI (t) P(t).

319

§3. SPLITTING AN EQUATION

The conditions of Lemma 3.1 are satisfied, so that there exists a solution of equation (3.28) defined uniquely on Y, and mapping Y, into .L,. If we complete the definition of the operator Y1 (t) by putting it equal to zero on .L, and making it linear on E, we obtain a solution of (3.22) which satisfies the condition (3.27). It is clear from formulas (3.25) and (3.28) that this solution will be strongly continuously differentiable in t if the operator A (t) is twice strongly continuously differentiable on ,q(A). Hence we have found Y1(t). According to formula (3.14), for Xo E ,q(A) n!2fo we construct the function

w!l) (t) =

(P(t)

+

f

Y 1(t» U 1(t, 0) Xo

=

(I

+

f

Y 1(t» U 1(t, 0) Xo.

This function satisfies the equation Edw/dt

=

A(t)w

+f

2 (dYddt)P(t)

U1(t,0)Xo,

which differs from equation (3.18) by terms of the second order of smallness. However the function W!l)(t) has one defect-its initial valu_e differs from the initial value of the function x(t) by terms of the' first order of smallness: w!l) (0) = Xo + f Y 1(0) Xo. We note that the component Y 1 (t) Xo already lies in the space .Lo• In order to remove this discrepancy in the initial conditions we construct the null approximation to the solution of the equation (3.18) with the initial data Y1(0)Xo. We have

aw) (t) =

U 1 (t, 0) Y 1 (0) Xo.

Then the function v!l) (t) = W!l) (t) - dv!O) (t) will satisfy the initial condition v!l) (0) = Xo and the equation fdv!l)/dt

= A (t)v;(l)

+ f 2[(dYddt)P(t) U1(t,0)Xo - pI (t)P(t) U1(t,0) Y1(0)Xo].

(3.29) Denote by g2(t) the function in square brackets. It is clear from the foregoing that this function is uniformly bounded in f and t. Solving equation (3.29), we get (3.30)

V!l) (t) = x(t)

+

f

l' u.

(t, S)g2(S) ds,

from which there follows the estimate

II V!l)(t) REMARK.

- x(t)

II

~ C2e2•

If instead of the estimate (1.11) the operator

U. (t, s) satisfies

320

IV. ASYMPTOTIC METHODS

only the inequality

I U,(t,S) I ~ M, that I V!l)(t) - X(t) I

(3.31)

then from (3.30) we find ~ C~E. i.e. in this case the difference between the exact solution and its first approximation will converge to zero uniformly on [0, T]. It is essential that in seeking the first approximation v!l) (t) we have had to solve equation (3.19) not only in the subspace Y, but also in the subspace ..Itt. THEOREM 3.4. If A (t) is twice strongly continuously differentiable on g-(A) and satisfies the conditions of Theorem 3.6 of Chapter II, then the first approximation to the solution x(t) of equation (3.18) with the initial value Xo E g-(A) n Yo may be constructed according.to the formula

v!l) (t)

=

(l

+

E

Y1 (t» U1 (t, 0) Xo -

E

U1 (t, 0) Y1 (0) Xo,

where Y1 (t) is the uniquely defined solution of equation (3.22) satisfying the condition (3.23).

If inequality (1.11) holds, then the estimate

I x (t)

- V!I) (t)

I

~

C2E2

holds. If only inequality (3.31) holds, the estimate

Ilx(t) -

V!l)(t)

I

IS

~ C~E.

5. Approximations of higher orders. The process of construction of approximate solutions of equation (3.18) may be pushed forward so as to satisfy equation (3.20) with accuracy to terms of higher orders in E. Consider in detail the construction of the second approximatIon. Put (3.32) We substitute this expression in equation (3.20), drop the terms of order E3 and-equate the coefficients of the same power of E on left and right. Then we obtain equation (3.22) and the equation (3.33)

A(t) Y 2 (t)P(t) - Y 2 (t) A (t)P(t) = (d(Y1 (t)P(t»/dt)P(t).

We wish the operator Y 2 (t) also to map the subspace Y, into the subspace ..It,. Then we would have P(t) Y 2 (t) = 0, and it would follow from equation (3.33) that P(t) (d (Y1 (t) P(t» / dt) P(t) ==

o.

321

§3. SPLITTING AN EQUATION

This, however, in no way follows from "equation (3.22), from which the function Y1 (t) was found. For this reason the procedure of seeking the second approximation must become more complicated. We use the fact that the evolution operator Wet, s) of the equation (3.34) EdW/dt = A(t) W

+ E(P'(t)P(t) + P'(t)P(t»

W + E28 1 (t) W

will, in view of Theorem 3.1, follow the evolution of the subspaces y, and .L, for any operator 8 1 (t) commuting with the operator P(t). If now we attempt to represent the evolution operator U,(t,s) on the subspace Y, in the form (3.35)

U,(t,O)P(O)

=

Y(t)W(t,O)P(O),

then for yet) we obtain the equation E(dY/dt)P(t)

= A(t) Y(t)P(t) -

- Y(t)A(t)P(t)

E2Y(t)8 1 (t)P(t)

- EY(t) P' (t)P(t).

If we substitute yet) from (3.32) into this equation and equate coefficients of E2, we obtain an equation analogous to (3.33) with the supplementary term A(t) Y 2(t)P(t) - Y 2(t) A (t) P(t) - P(t)8 1 (t)P(t) = (d(Y1 (t) P(t»/dt) P(t).

(3.36)

N ow we choose the operator 8 1 (t) so that (3.37)

8 1 (t) = P(t)8 1 (t)P(t) = - pet) (dY1 (t)P(t)/dt) pet).

Obviously the operator 8 1 (t) commutes with P(t). Then the equation obtained from (3.36) by application of the operator pet) will be identically satisfied if pet) Y 2 (t) == o. Thus we may seek a solution of equation (3.36) satisfying the condition (3.38)

Applying the operator (3.39)

pet)

P(t)A(t) Y2(t) -

to (3.36). we obtain Y 2(t)A(t)P(t)

=

pet) (d(Y (t)P(t»/dt)P(t). 1

A solution of this equation satisfying the condition (3.38) exists and is unique on the basis of Lemma 3.1. It will be continuously differentiable with respect to t if the function Y 1 (t) is twice continuously differentiable, and this in its turn will hold if the operator A (t) is three times strongly continuously differentiable on 9{A). The function

322

IV. ASYMPTOTIC METHODS

V!2)(t)

(l

=

+

E

Y1 (t)

+ E2Y 2(t» W(t, O)Xo

will satisfy equation (3.18) with accur.acy to terms of order E2. N ow again there is a complication connected with the initial value of the function V!2)(t):

V!2) (0) = Xo + E Y 1 (0) Xo + E2 Y 2 (0) Xo. In order to remove the discrepancy in the initial values one may again apply the method of successively rectifying them with the aid of functions satisfying condition (3.18) with accuracy to terms of order E3. We construct the function

V!2)(t) = (I + EY1(t)

+ E2Y 2(t» W(t, 0) Xo

+ EY1(t» U1(t,0) (Y1(0)Xo + tY2(0)Xo) + U (t, 0) Y(0) (Y (0) Xo + Y 2 (0) Xo),

(3.40)

- E(I E2

1

1

E

1

where Y1 (t) is an operator constructed relative to P(t) in the same way as the operator Y1 (t) was with respect to P(t). Obviously V!2) (0) = Xo, and the function V!2) (t) satisfies equation . (3.18) with accuracy to terms of order E3. We have arrived at the following assertion. THEOREM 3.5. If under the hypotheses of Theorem .3.4 the operator A(t) is three times continuously differentiable on 9(A), then. one may construct by means of formula (3.40) a second approximation V!2) (t) to the solution x(t) of equation (3.18), with initial value Xo E 9(A) n Yo, having the property that

if (1.11) is satisfied and ~x(t)

- V!2)(t) II

~

qE 2

if only (3.31) is satisfied. /

In the construction of further approximations there are no new difficulties in principle. We shall sketch this process in the following subsection for a more general equation. . 6. Equations with small perturbations. General asymptotic decompositions. We pass from equation (3.18) to a more complicated equation of the type (3.41)

Edx/dt = A(t)x + EB(t,E)X,

323

§3. SPLITTING AN EQUATION

where the operator B(t, E) is expanded in a power series: B(t,E)

'"

= LBk(t)E k, k-O

with bounded coefficients Bk(t) (k = 0,1, ... ) which are strongly continuously differentiable in t. Again we shall attempt to represent the evolution operator of equation (3.41) on the subspace Yo in the form of a product U(t, 0) P(O) Xo

= Y(t, E) W(t, 0, E) P(O) Xo.

For the operator W(t, 0, E) we set up the equation (3.42)

Edvjdt = A (t) v

+ E(PI (t) P(t) + pI (t) p(t» v + ES(t, E)V.

We shall choose the operator S(t,E) so that the evolution operator W(t, s, E) of this equation follows the evolution of the subspaces

Y t and ..,Itt. In view of Theorem 3.1 and the commutativity of the operators A (t) and P(t) this is so if and only if the operator S(t, E) commutes with P(t). We will suppose that the operator S (t, E) is also expanded in a series a>

(3.43)

S(t, E)

= L Sk(t) l. k-O

For the operator Y(t, E) we now obtain the equation (3.44)

E(dYjdt)P

= AYP -

YAP - EYSP - EYP' P+ EBYP.

We seek Y(t,E) in the form of an expansion Y(t)

= P(t)

'"

+L

Yk(t)E k •

k=1

Substituting into (3.44) and comparing the coefficients of the same powers of E, we arrive at the system of equations

(Yo(t)=P(t). k=1,2, .. ·).

We suppose that the operators Yh · · · , Y k already been found, and are such that (i

1

and So,·· .,Sk-2 have

= 1,2, •.. , k - 1)

324

IV. ASYMPTOTIC METHODS

and (j=0.1 •...• k-2).

We shall seek the operators S"-1 and Y". So that Y. will operator in the subspace ..It" we put (3.46)

S"-1

== PSk- I P = - P

Applying the operator

P to

(3.47)

dY"-IP dt P

"-1

+ •~-1 PB"-i-l YiP .

(3.45) and supposing that Y k = PY"p.

we obtain (3.48)

This equation has a unique solution satisfying condition (3.47). as follows from Lemma 3.1. . Thus the formal process of determining the operators SIc and Y" may be continued indefinitely. In order to accomplish N steps we need to have the operator A (t) N times strongly continuously differentiable on ~(A). and the operators Bk(t) (k = 0, ••• , N - 1) continuously differentiable N - k - 1 times. For YN(t) to have a continuous derivative, the smoothness of the operators A(t). and Bk(t) has to be increased by unity. The operators So(t) , •• • ,SN-1(t) will also be continuously differentiable. If we keep the terms up to k = N - 1, then the Cauchy problem is uniformly correct for the equation N-l

(3.49)

Edv/dt

=

A(t)v

+ E(P'(t)P(t) + P'(t)P(t»v + E L

Sk(t)ekV

1-0

if the operator E -1 A (t) satisfies the conditions of Theorem 3.7 of Chapter II. The corresponding evolution operator will be denoted by W!N)(t,s)./ This operator follows the evolution of the spaces 5zft and ..It" so that for initial data from 5zft or ..Itt equation (3.48) reduces to an equation in a subspace. If now for Xo E 9(A) n Yo we construct the function z!N)(t) =

(I + f:.

E"'Y,,(t) ) W!N)(t,O)Xo.

k-l

then it will satisfy equation (3.41) up to terms of order

E

N+l

•

325

§a. SPLITTING AN EQUATION

The function z!N) (t) does not satisfy the required initial conditions. Here we have to apply the procedure of successive rectification of the initial conditions which we described 41 the preceding subsections, or directly seek the solution v!N)(t)

(1 + fEkYk(t») W~)(t,O)Xo(d

=

k-1

(3.50)

where the elements Xo(E)

=

Xo

+ EX1 + ... + ENXN,

(3.51) are chosen so that the difference of the initial values v!N)(O) - Xo =

(1 + f Ekyk(O) ) Xo(E) k-l

(3.52)

is a quantity of order EN+!. It is easily verified that because the operators Yl°) act from Yo into ..Lo and the operators Y. from ..Ld into Yo, the elements Xl,"', XN, UO, •• " UN-l are uniquely determined. Here x. E Yo and Ui E ..Lo• Thus the procedure of finding the Nth approximation reduces to the following: 1) One solves successively N equations of the type (3.48), as a result of which one finds the operators Yb~··' Y N and, by formulas (3.46), the operators SO,"" SN-I' 2) One successively solves the N - 1 analogous equations with P replaced by P and conversely, as a result of which one finds the operators Yl>"" YN-l and SN-2' 3) One finds the elements Xo(E) E Yo and Uo(t) E..Lo of the type (3.51) by equating the coefficients of the powers of E up to the Nth to zero in equation (3.52). 4) One solves the equation

So, ... ,

E

d

N-l

dt

k-O

~ = A (t) pet) v + EP' (t) pet) v + E L

with the initial condition v(O) the equation

=

Sk(t) EkV

Xo(E) in the subspace

Yo, and then

326

IV. ASYMPTOTIC METHODS

dv t-d -

t

,....,

,. . ,."

= A (t) P(t) v

N-2,....,

+ tPI (t) P(t) v + t L f"!t,.I

k=O

..

Sk(t) tkV

with the initial condition v(O) = UO(E) in the subspace ..,/to. 5) The solutions W!N)(t,O)x,(O) and W!N)(t, 0) u,(O) just found are substituted in formula (3.50) for the Nth approximation. We may pose the problem of finding solutions of equation (3.41) for arbitrary initial data Xo E g-(A). Then Xo = PXo + PXo, and the Nth approximation to the solution may be written in the form v;«t)

=

U(t,s)Xo

=

(1 + ~tkYk(t)

+(1 + where the element Xo(t) = Xo +

is a quantity of order

)

W~N)(t,O) PXo(t)

~tkYk(t»)

W.N)(t,O)Px.(t),

EXl

+ •.. + ENXN is chosen so that

EN+!.

The procedure just described of asymptotically splitting the equation into equations in subspaces easily generalizes to the case when the spectrum of the operator decomposes into the sum of isolated branches Ao(t) , A1(t),·· ·,A,,(t) with corresponding operators Po(t), Pl(t)," ·,P,,(t). In this case, the process of construction of the complete solution requires the solution of equations of type (3.48) and differential equations in subspaces of type (3.49), differing in that in the place of the operators P(t) and P(t) there appear in them all possible pairs of operators Pj(t) and Pk(t) with i ~ k. 7. Nonhomogeneous equations. In §1 we considered the equation (3.53)

E dx/dt

=

A (t)x

+ f(t)

(0 ~ t ~ T)

and' found conditions under which its solution tended to the function - A -l(t)f(t). Using the methods of this section, it is possible to find asymptotic formulas of various orders of accuracy for the solution of equation (3.53). We shall seek a solution in the form of a series (3.54)

x(t)

=

L

tkhk(t).

k=O

The formal substitution of (3.54) into (3.53) and the comparison

327

§3. SPLITTING AN EQUATION

of coefficients of the same powers of

E

0= A (t)ho(t)

leads to the equations

+ f(t)

and (k = 1,2,·, .).

We suppose that A (t) has a bounded inverse. Then we obtain the null approximation x!O)(t) = ho(t) = - A- 1 (t)f(t), to which, as we already said, all solutions converge almost uniformly. Further, if the operator A -l(t) and the function f(t) are continuously differentiable, then we may construct the first approximation X~I)(t)

=

ho(t)

+ Eh (t) 1

= -

A -1(t)f(t) - EA -1(t)dA- 1(t)f(t)/dt.

In order to find the Nth approximation we need to require that the operator A -l(t) and the function f(t) be N times differentiable. Then x!N) (t)

(3.55)

= -

f

ek [A -l(t) dd

t

k-O

J k

(A -1(t) f(t».

For this function to be differentiable, it is necessary that the smoothness of A -l(t) and f(t) go up by one. Then the function X!N)(t) satisfies the differential equation dX(N) 1 1 cit = -; A(t)x!N)+~ f(t) -

J

d [ d N EN dt A -1(t) dt (A -1(t)f(t».

Solving this differential equation, we obtain x!N)(t) = U.(t,O)x!N)(O)

+-1 !at U.(t, s)f(s) ds E

- EN

Jot

°

d [ A -1(S) ds dJN (A -1(S) f(s» ds. U.(t, s) ds

The sum of the first two terms on the right is a particular solution x.(t) of equation (3.53). If the operator A -1(t) and the function f(t) are continuously differentiable N + 1 times, then the last term on the right is of order EN+!. If we use the identity

![

A -1(S)

!J

N

= _

A-1(s)A'(s)

[A-

1(s)

~]N +A-l(S) !22 [A-1(s) !]N-l

328

IV. ASYMPTOTIC METHODS

and suppose that inequalities (1.1) and (1.15) are satisfied, then one may arrive at the conclusion that (3.56) and

II dx!N)(t)/dt - dX.(t)/dtll

=

O(fN).

We have proved the following assertion. 3.6. Suppose that the operator A -I(t) and the function f(t) are N + 1 times strongly continuously differentiable and that the conditions of Theorem 1.2 are satisfied. Then, using formula (3.55), one may construct a function X!N) (t) differing from some particular solution x.(t) of equation (3.53) by a quantity of order fN+!. Here relations (3.56) and (3.57) are satisfied. THEOREM

The problem under consideration becomes less trivial in the case when the operator A (t) may fail for some t to have a bounded inverse. It is natural to call this the case of resonance. We select the most restricted possible isolated portion Al (t) of the spectrum of A (t) which contains the point>. = 0 for those t when A (t) does not have a bounded inverse defined over the entire space. Suppose that P(t) is the projection operator corresponding to the branch Al (t). We. again apply the method of splitting the equation. We shall seek the solution x(t) of equation (3.53) in the form (3.58)

x(t)

=

Y(t)v(t) +h(t,f),

where v(t) is a solution of the equation fdv/dt

=

A (t) pet) v

+ fP' (t) P(t)v + ES(t, f)V + get, E),

taking on values v(t) E!/" S(t, f) an operator commuting with P(t), and get, E) a function with values in !/t: ;

P(t)g(t, f)

=

g(t, f).

Substituting (3.58) into equation (3.53) and equating separately the terms containing v and the free terms, we get the two equations (3.59)

E(dY/dt)P+ Y(A+EP'P+ES)P=AYP

and (3.60)

fdh/dt

+ Yg =

Ah

+ f.

329

§3. SPLITTING AN EQUATION

Equations of the type (3.59) were studied in detail in the preceding subsections. Expanding the functions Y and S in series, we may for sufficiently smooth A (t) and f(t) find their solutions to arbitrary orders of accuracy in E. If the operator Y has been found, then we can find the functions g and h from (3.60), supposing them also expanded in series in powers of E: m

h(t,E)

=

~

Lhk(t)i and g(t,E) k=O

=

Lgk(t)E k• k-O

We begin with the null approximation. We recall that Yo(t) == P(t). Comparing the terms in (3.60) which are free of E, we have P(t)go(t) = A (t) ho(t)

(3.61)

+ f(t).

We will suppose that Pho(t) == O. Then, applying the operator P(t) to (3.61), we obtain

In the invariant subspace 1, the spectrum of A (t) does not contain the point >. = O. Denote by A-1 (t) the operator inverse to A (t) in the subspace 1 l' We now apply the operator P(t) to (3.61). Then

o = A (t) P(t) ho + P(t) f(t) , so that ho(t) = P(t) ho(t) = -

A-1(t) P(t) f(t).

Thus, in order to obtain the null approximation to the solution of equation (3.53), we need to solve the equation (3.62)

E

du/dt = A (t) P(t) u

+ EP' (t) P(t) u + P(t) f(t)

under some initial condition from ~(A) function x!O)(t)

= u(t) -

n .st"o.

and to set up the

A- 1(t)P(t)f(t).

The function x!O) (t) satisfies an equation differing from (3.53) by terms of order E. If we solve this differential equation for x!O) (t). we arrive at the relation (3.63)

x}O)(t) = x.(t)

+ [U.(t,S)P'(S)P(S)u(s)ds +~

L'U.(t,S)r/J(S)dS,

330

IV. ASYMPTOTIC METHODS

where x,(t) is some particular solution of equation (3.53), and c/J(s) is a known function which is continuous when A (t) and f(t) are smooth. Since we are allowing the point zero to belong to the spectrum, generally speaking the hypothesis that the estimate (1.11) is satisfied for the operator U.(t,s) is not natural, though there may be such cases. Condition (3.31) is more natural. We note further that by passing from equation (3.62) to an integral equation one may show that the function v(t) is uniformly bounded relative to E under condition (1.11), and is of order 1/E under condition (3.31). Thus the integral terms in (3.63) will be bounded if (1.11) is satisfied, and of order l/E if only (3.31) is satisfied. This says that in the presence of resonance the null approximation nowhere reflects the behavior of the solutions of equation (3.53). N ow we pass to the construction of the first approximation. From what was said in subsection 4, the operator Yl (t) is found from (3.28). In order to find the functions hI (t) and gl (t) we equate the terms of first order of smallness in equation (3.60). We get (3.64) We require that PhI (t) == O. Since PY1 (t) == 0, it suffices for this that gl (t) = P(t)gl (t) = - P(t) dho/dt.

N ow that gl (t) has been chosen, we can find hI (t) from equation (3.64): hI (t) = A-I P(t) dho/ dt

+ A-I Y (t) P(t) f(t). 1

The first approximation to the solution of equation (3.53) will be the function x?)(t)

where the function E dw/dt

=

=

(l

w(1)

+

E

Y 1 (t»w(1)(t)

+ ho(t) + Eh (t) , 1

(t) is a solution of the equation

A (t) P(t) w

+ EP' (t) P(t) w + P(t) f(t) +

Egl (t)

with iniiial data fom 2J(A) n Yo. A simple calculation shows that the first approximation satisfies the relation

where x,(t) is some particular solution of equation (3.53). Here under

A3.

331

SPLITTING AN EQUATION

condition (1.11) we may guarantee that the difference x, (1) (t) - i. (t) converges uniformly to zero with order E. Under condition (3.31) one can only show that it is bounded. We will present the formulas for the second approximation. The operator Y 2 (t) is found from (3.39). Further, g2(t) = P(t)gz(t) = - P(t) dhddt, hz(t) = P(t)hz(t) = A-1P(t)dhddt+ A-1Y 2 (t)P(t)f(t)

+ A-1Y1(t)gl(t).

Finally, X~2)(t)

=

(I

+ EY1(t) +

E2

Y 2(t»W(2)(t)

+ ho(t) + Eh1(t) + E2hz(t),

where the function W(2)(t) is the solution of the equation (3.66)

+ fP' (t)P(t)w + E2S1(t)W + P(t)f(t) + ~l(t) + E2g2(t) ~(A) n Yo, and the operator S1 (t)

Edw/dt = A (t)P(t)w

with initial data from is defined by formula (3.37). Arguments analogous to those presented above allow us now to obtain the estimate Ilx!2)(t)

-i.(t)11

~ CE

in the case in which (3.31) is satisfied as well. THEOREM 3.7. Suppose that the operator A(t) is twice continuously differentiable on ~(A) and that the function f(t) is three times continuously differentiable. In the case of resonance one may then still construct, using formula (3.65), a function differing under condition (3.31) from some particular solution of equation (3.53) by a quantity of order E, uniformly on [0, T]. For its construction we need to solve the operator equations and the differential equation (3.66) in an invariant subspace of the operator A (t) in which its spectrum may contain the point >. = o.

The reader may be surprised that we have called the case at hand the case of resonance, which is usually connected with the presence of oscillatory terms in the equation. This question will be clearer if we consider the equation (3.67) where Q(t) is a sufficiently smooth scalar function. The substitution

332 y(t)

IV. ASYMPTOTIC METHODS

= eQ(t)/'x(t) reduces this equation to the form (3.53) with the

operator A(t) = Ai(t) -,,(t)I,

where ,,(t) = dQ/dt. The function ,,(t) plays the role of oscillation frequency, and we see that in the case when the values of the frequency ,,(t) fall into the spectrum of the operator Ai (t), the operator A (t) will not have a bounded inverse defined on the entire space E. The formula y!2)(t) = x!2)(t)e Q (t)/· makes it possible to construct an approximation to a particular solution y. (t) of equation (3.67). Under the condition ReQ(t) ~ 0 we may prove that Y.(t) - y!2)(t) converges uniformly to zero.

CHAPI'ERV FINITE-DIFFERENCE METHODS §1. Factor-method of solution of operator equations 1. Factor-space_ Here we recall a number of known facts connected with the concept of a Banach space E. Suppose that El is a subspace of a Banach space E. For any element u E E the set of all elements u + x, where x E Eh is called a residue class u of the element u relative to the subspace E 1• The collection of all residue classes, in which it is natural to introduce the operations of addition and multiplication by a scalar, is a linear space and is called the factor-space E/ El of the space E by the subspace E 1• A norm may be introduced in the factor-space E/ El by putting

I uI E/El =

(1.1)

inf I vii· uEii

In this norm the space E/El is a Banach space (see for example [4]). In what follows we shall consider other norms in E/ El as well. In contrast to them, the norm (1.1) will be called the natural norm. The relation is a homomorphism of E onto E/El whose kernel is E 1• This homomorphism will be denoted by tPl: tPIU = u. In the natural norm the linear operator tPl has norm equal to 1. In what follows we suppose that with any norm introduced into E / El the homomorphism of E onto E / El is continuous. 2. Sequences of factor-spaces; factor-convergence. Now suppose that a sequence of subspaces E" has been distinguished in the space E. We construct the factor-spaces E/ E" and the homomorphisms tP" and norms I I E/E" corresponding to them. If the norms in the factor-spaces have the property that for each element vE E

u-u

(1.2)

sup I tP"v I E/E < "",

"

"

then in view of the principle of uniform boundedness, applied to the functional I tPnvll E/E", there exists a constant c such that (1.3)

II tP"V I E/E" ~ c I v I E· 333

v. FINITE-DIFFERENCE METHODS

334

In particular, relation (1.2), which means (1.3) as well, will be

satisfi~d

if

(1.4) DEFINITION 1.1. Suppose given a sequence {un l such that Un E E/ E". We shall say that the sequence {un l factor-converges to the element U of E if

lim ~ rl>nu - Un II E/Er, =

(1.5)

n.....

~

o.

Of course, with an arbitrary choice of the subspaces En and norms in the factor-spaces E/ En the concept of factor-convergence may fail to correspond to any natural concept of closeness of the elements u and Un. In what follows we shall impose a number of restrictions on this choice. 1.1. Suppose that either condition {1.4) or the condition En+l C E", n:_lEn = {O l is satisfied, and that the norms in the factorspaces E/ En are the natural ones. Then a sequence {un} cannot factorconverge to two distinct limits. LEMMA

PROOF.

Suppose that u and u' in E are such that

I rI>"U -

Un I E/En --+0

and

I rI>"U'

- Un I E/En--+0.

Then

I rl>n(U -

u') II E/En ~ II rl>n U - u,,11 E/En + I U,. - rl>nu'IIE'En --+ o.

If (1.4) is satisfied, then it immediately follows that U = u'. If En+l CEil and the norm in E / En is the natural one, then the sequence

I rl>n(u -

u') I E/Er,

does not decrease, so that

IlrI>,,(u This means that U=

1l -

u') liE/En

=

0 for all n

=

1,2,.··.

u' E E" (n = 1,2, ... ), and since

n :=lE,. = {o},

u'.

The lemma is proved. The connection between approximation to an element in the sense of factor-convergence and in the usual sense is established by the following assertion:

335

§1. OPERATOR EQUATIONS

LEMMA 1.2. Suppose that a natural norm has been introduced into each of the factor-spaces and that the sequence I u,,} converges to the element u E E. Then from each class u" one can select as a representive a u" E E such that (1.6)

""",-

PROOF. Suppose that w is any element of the class u".· If w runs through that class, then the difference u - w runs through the entire class c/>"u - u". In view of the naturality of the norm in EI E" it therefore follows that there exists an element w = u" E u" such that

(1. 7) II u - u,,11 E ~ II c/>" (u - u,,) II EIE"

+ lin =

II c/>"u - u,,11 E/E" + lin.

(1.6) follows from (1.7) and (1.5).

The lemma is proved. 3. Modification of the principle of uniform boundedness. Principle of Uniform Factor-boundedness. Suppose that in each space EI Em n = 1, 2, ... , the natural norm has been introduced. Suppose further that for each n there is a functional of>,,(u) on EI E" having the properties (a> 0)

and Iof>,,(u + V) I ~ I of>,,(u) I + I of>,,(u) I.

Suppose finally that for each u E E (1.8)

suplof>"(c/>,,u) I

<

00.

" Then there exists a constant k independent of n such that for all

uEEl E".

I of>,,(u) I ~ kll uIIE/E". In view of the linearity and continuity of the homomorphism of>"(c/>,,u) satisfies on E the principle of uniform boundedness. Therefore there exists a constant k such that PROOF.

!/>.., the functional

Iof>,,(u)I

=

Iof>"(c/>,,u) I ~klluIIE.

Taking the infimum on the right over all representatives of the class UEEl E", we obtain (1.8). The assertion is proved.

336

V. FINITE-DIFFERENCE METHODS

From the principle of uniform factor- boundedness, as usual, one deduces a number of consequences concerning operators defined on the factor-spaces E / En. "'-rIJ THEOREM 1.1. Suppose that in each space E/ En there has been introduced a natural norm, and that a linear bounded operator An has been defined mapping the space E/ E" into a Banach space E~. If on each element u E E

sup I A"4J,,U I

<

00,

n

then the norms of the operators An are uniformly bounded:

I A"II E/E,.-E;, ~ k. The proof follows directly from the principle of uniform factorboundedness applied to the functionals ~n(u) = I An u I E,,'

N ow suppose that F is another Banach space, Fn (n = 1,2, ... ) subspaces of F, F / F" factor-spaces, and '/In the corresponding homomorphisms. We suppose that the operators An are bounded operators acting from E / En into F/ Fn. DEFINITION 1.2. A sequence of operators A" factor-converges if for each vEE the sequence A n4J"v factor-conve~ges to some element of F. THEOREM 1.2. Suppose that the norms in the spaces E/ E" are natural, and that in the spaces F / Fn the condition

(1.9)

is satisfied. If the bounded linear operators An, acting from E / En into F / Fm factor-converge, then their norms are uniformly bounded. PROOF. Suppose that v is any element of E and g that element of F to which the sequence A,,4JnV factor-converges. Then

I An4JnV -

'/I,.g I F/Fn --+ O.

Therefore it follows that

I A n4J"V I F/Fn ~ I A,,4Jn v - '/I,.g I F/Fn + I '/I,.g I FlF" ~ I A,,4JnV - '/I,.gll F/F" + Cligll F <

00.

337

§1. OPERATOR EQUATIONS

It follows from Theorem 1.1 that IIA"IIE/E"--->F/F,, ~ k.

The theorem is proved. THEOREM

(1.10)

1.3. If the norms m the spaces F / F" satisfy the condition limll~,.gIIF/F" = IlgilF

,,-'"

(gEF),

then the factor-converging sequence of operators A" defines m the limit

a bounded linear operator acting from E into F. PROOF. For any vEE we denote by g = Av that element of F to which the sequence of elements A"4>,,v factor-converges. This element is uniquely defined in view of condition (1.10) and Lemma 1.1. Obviously the operator A defined in this way is linear. We shall show that it is bounded. In the proof of the preceding theorem it was shown that the functionals II A"4>,,V II are uniformly bounded on each element vEE. From the ordinary principle of uniform boundedness it then follows that

where C does not depend on nand v. Further, from the definition of the element Av it follows that II A"4>,,V - ~"AvIIF/F" ---+0, which means, from (1.10), that IIAvIIF= n-Q) limll~"AvIIF/F" = A_co limIIA"4>"vIIF/F,, ~ CIIvIIE. The theorem is proved. 4. Basic concepts of the factor-method. (1.11)

Consider

the

equation

Tu=f,

where T is a linear operator defined on the linear set 9'(T) of a Banach space E and acting into the Banach space F. Suppose that in E and F there have been distinguished sequences of subspaces En and F" and that the factor-spaces E/ E" and F / F" have been constructed with the corresponding homomorphisms 4>" and ~". On a linear subset of the space E/ E" containing the image of the domain 9'(T) of the

v. FINITE-DIFFERENCE METHODS

338

operator R, we define an operator T,. approximating T in some sense, and consider the approximate equation (1.12)

T"u"=1/;,,f.

A solution U" E E/ E" of (1.12) is regarded as approximating the solution of equation (1.11). We note that this approximation does not lie in the same space as the one in which we are" seeking the solution itself. The passage from equation (1.11) to equation (1.12) will be called the factor-method of approximate solution of equation (1.11) The basic characteristics of the factor-method are the properties of approximation and stability. DEFINITION 1.3. One says that the operators T" approximate the operator T at the element v E g( T), if (A)

limll1/;"Tv - T"cf>"vIIF,F" = 0, •

".......

i.e. if T,.cf>"V factor-converges to Tv. DEFINITION 1.4. The factor-method is said to be stable if for all n beyond some no there exist bounded linear inverse operators T;;l defined on the spaces F/ F,. and such that

II T;;lll FIF,.--.EIE,. ~ k

(S)

(n ~ no),

where the constant k does not depend on n. 5. Convergence and stability of the factor-method. The following simple assertion is fundamental. THEOREM 1.4. Suppose that u is a solution of (1.11). If the operators T,. approximate the operator T on the solution u, and the stability condition (S) is satisfied, then the approximate solutions factor-converge to the exact solution u.

u,.

PROOF.

From the stability condition, we have for n

~

no that

11cf>"u - u"IIE'E,. = II T;;lT"cf>"u - T;;l1/;"fhIE" (1.13)

= I T;; 1 (T"cf>"u -1/;"Tu) I EIE" ~ k II T"cf>,.u -1/;,..1. U I FIF,,· 'T'

It follows from the approximation condition that the right side tends to zero as n---+ 00. The theorem is proved. It also follows from inequality (1.13) that the order of convergence

§1. OPERATOR EQUATIONS

339

of the approximations to the solution coincides with the order of approximation to T by the operators Tn at the solution u. For convergence to zero of the left side in (1.13) it is not necessary that the norms II T,;-lll be uniformly bounded. They could generally speaking increase as n increases but in such a way that the order of their growth is less than the order of the approximation. This raises the question as to how necessary the condition of stability is for convergence. The answer to this question is given by the following. THEOREM 1.5. Suppose that a natural norm has been introduced in FI Fn. and that the norm in EI En satisfies condition (1.3). Suppose also that for all n ~ no there exist bounded operators T,;-l such that for any f E F the approximate solutions T,;-11/lnf converge. Then the factormethod is stable.

The proof follows directly from Theorem 1.2, on taking as the operator An the operator T,;-l and exchanging the roles of the spaces E and F. Suppose that G is a subspace of the space F. We suppose that factorconvergence of the approximate solutions T,;-11/l~ holds not for all the elements of F but only for the elements g of the subspace G. We shall see how to modify Theorem 1.5 in this case. Write Gn = G n Fn and form the factor-space G/Gn. Obviously this factor-space is algebraically isomorphic to the image 1/InG of the subspace G. For the natural norms in G/Gn and in 1/InG we have the relation (1.14)

If we denote by 1',;-1 the restriction of the operator T,;-l to 1/I"G, then 11',;-1} (n ~ no) may be considered as a family of operators given on G/Gn and operating into E/ En. It follows from inequality (1.14) that each of these operators is bounded, and accordingly that they satisfy the conditions of Theorem 1.5. We have arrived at the following assertion. COROLLARY 1.1. Suppose that the hypotheses of Theorem 1.5 are satisfied, and that the approximate solutions T,;-l,p,.g factor-converge for all g in the subspace G. Then there exists a constant kl'such that

(1.15) where G" = G n Fm the operator T,;-l is the restriction of T,;-l to 1/InG = G/G", and the norm in G/G" is the natural one.

340

v. FINITE-DIFFERENCE METHODS

It is clear from (1.14) that, as is to be expected, condition (1.15) is weaker than the requirement (8) of stability of the factor-method. 6. Convergence of the factor-method and the solvability of equation (1.11). The factor-method may serve as a means of proof of the existence of solutions of equation (1.11). THEOREM 1.6. Suppose that the norm in Ej E,. is the natural one, and that condition (1.10) is satisfied in Fj F,.. If the operator T is defined on the entire space E and the operators T,. are bounded and approximate T on any element vEE, then it follows from the factor-convergence of the solutions u,. to an element uE E that that element is a solution of (1.11). PROOF., From the approximation condition it follows that the operators T" factor-converge to T. As we saw in subsection 2, the relation (1.10) implies the relation (1.9), and, in view of Theorem 1.2,

(1.16) Then from (1.12) and (1.16) we find that 111f,,(Tu - f) IIFIF" ~ 111f"Tu - T"t/>"uIIFIF" + I T"t/>"u -

,p"fllm'..

~ 111f"Tu - T"t/>"uIIFIF" + kllt/>,,~ - u"IIEIE". The first term tends to zero in view of the approxllnation condition, and the second because of the convergence of the approximate solutions. This means that

I Tu - fll F =

lim 111f,,(Tu

,,-~

-

f) I FIF"

=

0,

i.e. Tu = f. The theorem is proved. §2. Finite-difference factor-method for evolution equations 1. Description of the method. In a Banach space E, consider the differential equation (2.1)

dxjdt = A (t) x + f(t)

(O~t~T),

where A (t) is a closed linear operator with a domain ~(A) which is dense in E and does not depend on t. We suppose that A (t) is strongly continuous on ~(A) and has a bounded inverse A -l(t). The function f(t) is supposed continuous. If one conceives as a model for equation (2.1) a partial differential

341

§2. EVOLUTION EQUATIONS

equation, then the process of setting up a difference scheme for the approximate solution of this equation may be broken into two stages. The first is the replacement of the differential operator A (t) connected with the space coordinates by a finite-difference expression. The second is the replacement of the derivative with respect to the time t by a difference quotient. In our abstract scheme these two stages will be taken into account. We divide the segment [0, T] into n equal pieces of length J1,.t = Tin by points tk = kJ1,.t (k = 0,1, .. . ,n). Generally speaking we should write ttl. We however drop the (n) to shorten the notation. We shall find the approximate values x(tk ) of the solution x(t) at the points tk. We replace the derivative at the point tk by the simplest difference quotient (x(tk+1) - x (tk) ) I J1,.t. In the space E we select a subspace izf", form the factor-space EI izf" and replace the operator A (t j ) approximately by the bounded operators A,,(tj ) acting in the factor-space EI izf". In place of equation (2.1) we consider the system of equations (2.2)

IA t (i·.(") ,+1 - x!n) ,~

= An (t·) x!n) +,f n) "

(i

= 0" 1 •.• ,n

- 1) ,

where flit) = cjJ"f(O, cjJ" being the natural homomorphism of E onto EI izf". The solution of this system will be a collection of n + 1 elements {x~"), .•. , x~")} of the space EI izf". We replace the Cauchy problem of finding the solution of equation (2.1) satisfying the initial condition (2.3)

x(O)

=

Xo E ~(A)

by the problem of solving the system (2.2) under the initial condition (2.4)

On finding the solution of the problem (2.2)-(2.4) we will suppose that

DEFINITION 2.1. We will say that the approximate solution converges to the function x(t) if

max

k~O.l ••••• n

as n--+

00.

II cjJ"x(tk )

-

xln ) I

E;.5!"" --+ 0

xl")

342

V. FINITE-DIFFERENCE METHODS

We consider the space C(E) of continuous functions with values in E, provided with the usual norm

and the space F(E)

Ex C(E) with the norm

=

I (Xo, I(t» I F(E) =

max { I Xo I E,

I III e(E) }.

We denote by 9(1') the set of all functions x(t) of C(E) which are continuously differentiable on [0, T] and such that the functions A (t)x(t) are defined and continuous on [0, T]. Each function of ~(T) is a solution of the problem (2.1)-(2.3) for some Xo E ~(A) and f E C(E). On ~(T) we define an operator T by the formula Tx

=

{x(O), dx/dt - A(t)x}.,

The linear operator T will act from ~(T) into the space F(E). Now we consider the operator generated by the problem (2.2)-(2.4). In the set of all collections a = {iio, ... , u" } of n + 1 elements of E/.Y" we introduce the norm

I aI = ,...max I udl ElY,,' 0, .. ·,n We define an operator T" in this set by means of the formula

(2.5) Let us find the inverse operator T;;I. We. have

T"a

=

f == {io,to, •• ·.t,,-d·

Hence

Uo = io, (Uk+l - Uk) / ~t = A,,(tk) Uk + tk Solving for

Uk+h

we obtain

(2.6) or finally i-I

(T;;l/)i

(2.7)

= U;

II (I + d"tA,,(t)uo j=O

(k=O, •• ·,n-l).

12. EVOLUTION EQUATIONS

343

In view of the boundedness of the operators A,. (tj ) the operator T;;l will be bounded. Now we shall give the facts just described another interpretation. Denote by C,.(E) the collection of all those functions u(t) of C(E) for which u(tk) E Y,. for all k = 0,1, •. " n. Since Y,. is closed it follows from the form of the norm in C(E) that C,.(E) is a subspace of C(E). We note that two functions u (t) and vet) of C(E) fall into one residue class relative to C" (E) if and only if for all k = 0, 1, •.• , n the equation rP"u (tk)

=

rP"v(tk)

holds. Hence to each residue class of the space C(E) relative to C,,(E) there corresponds a collection {rP"u(O), rP"u(~t), •• ',rP"u(T)} of n + 1 elements of E/ !/~. It is not hard to verify that if the natural norm is introduced in the factor-spaces E/ Y", then the natural norms in the spaces C(E)/C,,(E) will be calculated by the formula (2.8)

114>"u I C(E)/C,,(E) = . max.• ra I rP"U (til I ElY,., J-O,l,.~

where 4>" is the natural homomorphism of C(E) onto C(E)/C,,(E). We may consider the factor-spaces C(E)/C,,(E) and E/ Y" as well. However we will always suppose that the norms are consistent with equation (2.8). Thus, one may consider the factor-space C(E)/C,,(E) as the set of all collections of n + 1 elements of the space E/ Y,. with a norm equal to the maximum of the norms of the components. We may distinguish in the space F(E) the subspace F,,(E) consisting of the pairs (v, u(t» such that v E Y" and u(tk) E Y,., k = O,l, ••• ,n-l. The factor-space F(E) / F,,(E) is also isomorphic to the space of all collections of n + 1 elements of the space E/ Y". We introduce a norm in F(E) / F,,(E) by a formula analogous to (2.8):

I \}I,,(v,u) I F(E)/F,,(E) = maxI I rP"vIIE/Y", I rP"u (0) II ElY", ••• ,II rP"U(t,,_l) I ElY,,}' where \}I" is the natural homomorphism of F(E) onto F(E) / F,,(E). The operator T" defined by formula (2.5) will now be considered as a bounded operator acting from C(E)/C,,(E) into F(E)/F,,(E). It has a bounded inverse T;;\ found from formula (2.7). Thus we may treat the passage from the problem (2.1)-(2.3) to the problem (2.2)-(2.4) as the passage from the equation

(2.9)

Tx=/,

344

V. FINITE-DIFFERENCE METHODS

where x E ~(T), f

= (Xo,f(t» E

F(E), to the equation

1',n,t(n) =

(2.10)

fn) ,

where ,t(n) E C(E)/C,,(E) and f") = \IInf E F(E)/ Fro (E). We have found the factor-method for the approximate solution of equation (2.9). Our immediate problem is the investigation as to what properties of the operators An (t) are reflected in the properties of the factor-method. 2. Stability of the factor-method. We recall that the stability property of the factor-method consists in the uniform boundedness of the norms T;;l. THEOREM 2.1. For the stability of the factor-method it is sufficient that the condition

(2.11)

II

n

(l

J-k

+ ~ntAn(tj»

I EIYn ~ M

(0 ~ k ~ i ~ n - 1)

be satisfied, where M is a constant not depending on n, i and k, and the same condition is necessary with k = O. PROOF.

Suppose that (2.11) is satisfied. Then from formula (2.7)

we obtain i-I

I (T;;l/;) I ElY" ~ Mil uoll ElY" + ~"tM L Illkll E/Yn~ M(l + T) I {II F(E)/F,,(E)' k~O

Hence the stability condition follows: (2.12) where K = M(l + T). Conversely, if (2.12) is satisfied, put {= I uo, 0, •.• ,0 }. We then have

I T;;l{11 C(E)IC,,(E) = max {lluoIIEIY",

i!~.".II[t (l + ~"tA,,(t)uo IIEIY) ~ KlliloIIEIY".

Hence (2.11) results for k = 0 and M = K. It is difficult to verify the stability condition (2.11). Therefore frequently one applies the simpler sufficient condition:

§2. EVOLUTION EQUATIONS

345

THEOREM 2.2. For the stability of the factor-method it,s sufficient that the condition

III +
(2.13)

be satisfied, where a PROOF.

~

0 does not depend on nand j.

It follows from (2.13) that

I II (l +
j=k

ElY"

~ (1 + a
The theorem is proved. Theorem 1.5 of the preceding section makes it possible to establish the following proposition: (2.14)

dx/dt = A (t)Xk

THEOREM 2.3. Suppose that the natural norms have been introduced in the spaces E/5/'". A necessary condition for the approximate solutions Xknl of the homogeneous equation

dx/dt = A(t)x

(2.14)

to converge to some continuous function for every initial value Xo E E is that the stability condition (2.11) be satisfied for k = O.

The convergence of the approximate solutions of the homogeneous equation is in view of Definition 2.1 equivalent to the factorconvergence of the operators T;;l on the set of all elements of the form I xo, O} (Xo E E). This set obviously forms a closed subspace G of the space F(E)/F,,(E). It then . follows from Corollary 1.1 that the norms of the restrictions of the operators T;;l to the subspaces "'"G are uniformly bounded. But these last subspaces are isomorphic to the subspaces E/5/'n. As was proved in the demonstration of Theorem 2.1, condition (2.11) therefore follows with k = O. The theorem is proved. 3. Approximation and convergence. We turn to the derivation of conditions under which the operators Tn approximate the operator T. We suppose that the operators A,,(t) approximate the operator A(t) uniformly in t on its domain, i.e. that the condition PROOF.

(2.15)

lim max ~t/>nA(tJuo - A"(ti)t/>,,UoIIEIY,,== 0 ft_CD

i=O,l,.· ',n

346

V. FINITE-DIFFERENCE METHODS

is satisfied for any Uo E ~(A). Suppose that Vo is any element of E. Then Uo max II (cP"A (t i )

=

A -1(0) Vo E ~(A) and

A,,(tj) cP,,) A -1(0) Vo II ElY" --+ 0

-

(vo E E).

i-O,l,· .. ,1l

I t follows from the principle of uniform boundedness that . max II (cP"A(t j) -A,,(tj)A- 1 (0)vo IIEIy,,~KllvoIIE'

(2.16)

' .... O,1, •.• ,n

LEMMA 2.1. Suppose that the function v(t) is such that the function A(O)v(t) is defined and continuous. Then

max IlcP"A(tj)v(tJ - A"(tj)cP,,v(tj) II ElY. --+0

; ..... 0,1, ... ,"

as n --+

uniformly relative to

ex>,

PROOF.

n

i=

0, 1, •. " n.

Construct a subdivision of the segment' [0, T] sufficiently

fine that

II A (0) v(t j )

A (0) V(~k) II E ~ E/2K, where K is the constant from (2.16) and h is the point of the subdivision closest to t j • In view of condition (2.15) there exists. an no such that for n ~ no and all i = 0, 1, •. " N and k = 0, 1, ... ,N -

IlcP"A(tj)v(~k) - A"(tj)cP,,V(~k) II ElY"

< E/2.

Then

II cP"A (t j) v(tj )

-

A" (t j) cP"v(t j) II ElY"

~ II cP"A (tj)v(h) - A"(tj)cP,,V(~k) II ElY"

+ II (cP"A (t j) -

A" (tj)A -1(0) [A (O)v(tj) - A (0) V(~k) ]11 ElY"

The lemma is proved. N ow suppose that x(t) is the solution of equation (2.1). Then Tx = (Xo'/(t»

and ~"Tx = (cP"Xo, cP"f(to), •• ',cP"f(t,,-I»'

By the construction of the operator Tn T

",,x

=

(

cP"Xo, cP"

x(t1) - Xo tl."t - A"(to)cP,,Xo,

x(t2 ) cP" cP"

-

tl."t

x(t1 ) A"(t1)cP,,X(t1 ),

"',

x(T) - X(t,,-I) ) tl."t - A"(t"_I)cP,,X(t"_I) .

< E.

347

§2. EVOLUTION EQUATIONS

Therefore we get

I T",,x (2.17)

I F(E)IF,,(E) =. max I tP,,(X(ti+1) I-O,·· .. tIJ-l \}I" Tx

-

x(ti

»/A,t -

A,,(ti ) tP.,.x(ti) - tP"t(ti) II·

For the estimation of this norm we note that in 'View of equation (2.1)

IIX(ti+1~~X(t;)

Ik = lI~tLti+l[XI(S)-XI(ti)JdsIlE'

-A(ti)x(tj ) -t(ti )

We suppose that in the factor-space E/!£" the norm has been introduced in such a way that (2.18) I tP"zll E/Y" ~ GIl zll E (z E E). Then

IltP" X(ti+1).1~ x(t;)

~

- A"(ti)tP,,X(ti) - tP"t(t;)

IltP" X(ti+l~ ~ x(t;)

ILSf"

- tP"A (ti) x (ti) - tP"t(ti)

"

II ,ElY"

+ I tP"A (t;) x(ti ) -

~ C /1_1 t i+1[Xl (s) A,tJ.

A,,(ti) ifJ"x(ti ) I ElY" x' (ti)]ds

II

E

+ I ifJ"A(ti)x (ti) -

A"(ti)ifJ,,x(tj ) I ElY,,·

The solution x(t) is continuously differentiable, and the function continuous. The function A(O)x(t) = A (0)A-1(t)A (t)x(t) is continuous as well. Therefore the first term becomes arbitrarily small uniformly in t as n ---+ en because of the uniform continuity of x' (t) on [0, TJ. The second term becomes arbitrarily small in view of Lemma 2.1. We then obtain from (2.17) the following: A (t)x(t)

(2.19) as n ---+ 00. Conversely, suppose that (2.19) is satisfied for any solution of equation (2.1). Choose Xo E .9?'(A) and put x(t) == Xo and t(t) = - A (t) Xo. Then IlifJ"A(tj)Xo - A"(ti)ifJ"XoIIE/5o!,,

= l/ifJ"

Xo.1~ Xo -

~ " T",,x -

i.e. (2.15) is valid.

A"(t;)tP,,xo - ifJ"t(tj ) Il/5o!"

\}I" Tx"

F(E)/F,,(E) ---+ 0,

348

V. FINITE-DIFFERENCE METHODS

We have arrived at the following conclusion: THEOREM 2.4. Suppose condition (2.18) is satisfied. A necessary and sufficient condition that the operators Tn approximate the operator T on the set ~(T) of all solutions of equation (2.1) is that the operators An(ti) satisfy condition (2.15).

A direct consequence of Theorems 1.2, 2.1 and 2.4 is the following: THEOREM 2.5. Suppose that condition (2.18), the approximation condition (2.15) and the stability condition (2.11) are satisfied. Then the approximate solutions iln) converge to an exact solution of the problem (2.1)-(2.3) if such a solution exists.

We note that Theorem 2.5 is proved without any a priori assumptions about the solvability of the Cauchy problem. However the very constructibility of the sequence of operators An having the stability and approximation properties contains in itself a great deal of information on the properties of the Cauchy problem (2.1)-(2.3). N ow we suppose that the Cauchy problem for equation (2.14) is uniformly correct. We denote by U(t, s) the corresponding evolution operator. Every solution of the problem (2.1)-(2.3) is given by the formula (2.20)

x(t)

=

U(t, 0) xo +

.C

U(t, s) f(s) ds. '

According to Theorem 3.9 of Chapter II formula (2.20) will give a continuously differentiable solution of equation (2.1) if Xu E §(A) and the function A (t) f(t) is defined and continuous on [0, T]. For any Xo E E and f E C(E) formula (2.20) yields a continuous function x(t), called the generalized solution of equation (2.1). Every generalized solution may be approximated arbitrarily closely in C(E) by pure solutions of equation (2.1). Indeed, the element Xu E E may be approximated arbitrarily closely by an element Xo of ~(A), and the function f(t) uniformly on [0, T1 by a piecewise linear function ((t) with values in ~(A). Then the function A (O){(t) will be continuous, which means that the function A (t) ((t) = A (t) A -1(0) A (0) f(t) is continuous. Relative to Xo and f(t) we construct a solution x(t) of the problem (2.1) - (2.3). Obviously Ilx(t) -x(t) I ~M(IIXu-xoll

+ maxllf(t)-f(t)II>· O;;;t;;;T

349

§2. EVOLUTION EQUATIONS

Now we can show that under the conditions of Theorem 2.5, for any Xo E E and f E C(E) the approximate solutions x1n) converge to a generalized solution of the problem (2.1)-(2.3). In fact, (2.21)

xln) -

rJ>nX(tk )

= (lln) -

rJ>"f(tk

»+ (xl") -

it»

+ rJ>,,(f(t

k) -

x(tk

».

The last term will be arbitrarily small because of the construction of f(t) and property (2.18). The second term may be written in the form

xl") - i k = (T;;l(f-I>h By the construction

I f - fli F(E) =

I f - fli e(E) } is arbitrarily small. In view of property (2.18! I f - ~I F(E)IF,,(E), and as a consequence of the stability liT;; 1 (f - f) I C(E)/q, will also be max { I XO

-

fo I E,

(E)

uniformly small relative to n. Finally, the first term in (2.21), for a :fixed solution f(t), is arbitrarily small as n--+ because of the convergence of the approximate solutions to an exact solution, in view of Theorem 2.5. IX)

THEOREM

2.6. Suppose that the Cauchy problem for equation (2.14)

is uniformly correct and that the hypotheses of Theorem 2.5 are satisfied. Then the approximate solutions xl") converge to a generalized solution of the problem (2.1) -(2.3) for any Xo E E and continuoCls function f(t).

From Theorems 2.6 and 2.3 we may draw the following further conclusion. 2.7. Suppose that the Cauchy problem for the homogeneous equation (2.14) is uniformly correct. Suppose that the norms in the spaces E/5/" are the natural ones and that the approximation condition (2.15) is satisfied. A sufficient condition for the approximate solutions xl") to converge to a generalized solution of the Cauchy problem for equation (2.1) for any Xo E E. f E C(E) is that the stability condition (2.11) be satisfied, and the same condition is necessary with k = O. THEOREM

REMARK 2.1. Sometimes the construction of the difference scheme is not explicitly divided into two stages, and one directly writes the system (2.6) in the form of a recurrence relation -(,,) -(,,) (-(,,) (2 .22) Xo = rJ>"Xo, Xk+l = C"tk) Xk d"tfk (k = O. 1, ... , n - 1),

+

where C,,(tk) is a bounded linear operator in E/ .5£".

350

V. FINITE-DIFFERENCE METHODS

Naturally this system may be transformed into the fot~ (2.6) using the operator An = (Cn(tk )

-

I) /l1,.t.

Then the stability condition (2.11) takes the form (i=I, •.. ,n),

where M does not depend on n, k, and i. The approximation condition (2.15) may be written in the form lim max II tl>nA (ti) Uo - (1/ ~nt) [Cn(ti)tI>nUo - tl>nUo]11 E/~ = 0

A-GO io::::::O,lt··~,n

for any Uo E 9(A). In this form the approximation condition is sometimes called the consistency condition (see [65]). 4. The special case of an implicit scheme. The theory presented above includes also the special case when all the spaces Y,. = {o}. Then the operators A,.,. go into the identities. The stability condition takes the form

II~k (l + ~,.tAn(tj»

(2.23)

liE ~

M:

where M does not depend on i, k, and n. The approximation condition reduces simply to the condition that the operators A,.(tj ) converge strongly and uniformly in j on g-(A) to the operators A (tj). Suppose that A (t) is for each t E [0, T] a generating operator of a semigroup with the Co-condition. Then ss an example one may consider the operators approximating it, which we have already used repeatedly. Put

Then I

+ I1,.tAn(t)

=

-

(1/l1,.t)

RA(t) (1/

~..t).

Condition (2.23) then takes the form

\\ gRA(?)(b) II ~ M(~nt)i-k+1.

351

§2. EVOLUTION EQUATIONS

This condition may be replaced by a somewhat more rigid one:

IlilRA(~)(A)"~ A;~H

(2.24)

(A!; >.0).

In the case of a constant operator the condition takes the form

II R~(A) I

(2.25)

~ M/A;

and coincides with the condition of uniform correctness of the Cauchy problem for the equation (2.26)

dx/dt

=

Ax.

We have arrived at the following assertion: THEOREM 2.8. Suppose that for each t the operator A (t) is the generating operator of a semigroup with the Co-condition. Then, if condition (2.24) is satisfied, the approximate solutions, constructed according to the formuLas

(2.27)

xg»

=

Xo. X~n~1

= -

(1/ ~t) RA(tk) (1/ ~t) xln)

+ ~tf(tk) (k=O.l, ••.• n),

converge to a solution of the problem (2.1)-(2.3) if such a solution exists. Condition (2.24) is satisfied automatically if the operator A (t) is constant or satisfies the condition

II RA(t) (A) I

~ 1/>-

(A> 0).

For the case of a homogeneous equation the relation (2.27) may be further rewritten in the form (xl~1 - xln) / ~t

=

A (tk) xl~h

and we arrive at a scheme which is usually called implicit. If we write the analogous scheme for the case when the operator A (t) is factorapproximated by the operators An (t), then we get (2.28)

(il~1

- iln) /Ant =

An(tk)i1~1.

This scheme may be transformed to the form (2.22) with the operator Cn(tk) = - (1/ dnt) (An(tk) - (1/ dnt) I) -1 = - (1/ ~t) RAn(tkJ (1/ ~t).

The stability condition for a constant operator A takes the form

I R~n (1/ dnt) I

~ M (dnt);

352

v. FINITE-DIFFERENCE METHODS

or, in more rigid form,

I R~,. (A) I

~ M /A;

We have seen that these conditions are very close to the conditions of uniform correctness of the Cauchy problem for the original equation. This explains the fact that usually, with a reasonable approximation of the operator A by operators A,., the stability conditions are automatically satisfied for an implicit scheme (2.28). 5. Smooth solutions; improving the convergence. The passage from the space E to the factor-space E / Y,. corresponds in finite-difference methods to the passage from functions given in a region to functions given at the nodes of a net. However if the basic space E is a space of type L p , in which, incidentally, the parabolic equations are very well studied, then for functions of the space their values are not defined at the nodes of a net. In connection with this we introduce into Our abstract scheme one further space (in the applications the space C), in which it is possible to carry out a factorization. We first explain our new scheme for a homogeneous equation with constant coefficients, (2.26). Suppose a Banach space El is imbedded in E in such a way that (2.29) Suppose that El contains the domain of some power A k (k an integer) of the operator A: ~(A k) C Eh and (2.30)

IlxllEl ~ c211AkxIIE

(xE ~(Ak».

We shall be interested only in k + 1 times continuously differentiable solutions of equation (2.26), for which

dk +1 X /dtk+ 1 = A k+1 X • The values of these solutions lie in Eh and it follows from (2.30) that the function x(t) and its derivative Ax(t) are continuous in the norm of E 1• For the time being we may forget the existence of E and construct a difference SCheme starting from the space El and its subspaces Y,.. The construction of the operators T,. remains the same, the only difference being that the space E is replaced throughout by E 1• These operators will act from C(E1)/C,.(E1) into the space F(E1)/F,.(E1 )· One alters the stability condition correspondingly.

353

§2. EVOLUTION EQUATIONS

We now consider the approximation condition. We suppose that it is satisfied in the following form: (2.31)

lim II cf>"Auo - A"cf>"uo II ElIY" = 0

,,~~

for any Uo E ~(A k+l). Here cf>" is the natural homomorphism of El onto EdY". Replacing Uo by the expression A -(k+!lvo, we arrive just as in (2.16) at the conclusion that

Analogously to what was done in Lemma 2.1, we show using (2.30) that the relation (2.32)

II cf>nAv(t) - A"cf>"v(t) II EllYn - 0

therefore follows uniformly for t E [0, T], for each function v(t) for which the function Ak+!v(t) is defined and continuous. Under the condition (2.33) we may, using relation (2.32), reduce the verification of the approximation condition to the proof that

II (x(ti +!) - x(t j» / ~"t - Ax(tJ II El -

o.

In view of (2.30) it suffices to show that

II A k[X(tj+~t- x(tj)] =

A k+!X(t;)

liE

II~t iti+! [Ak+lX(S) -

Ak+!x(tj ) ]ds

11-0.

This last follows from the continuity of the function Ak+!x(t). We have arrived at the following assertion: THEOREM 2.9. Suppose that (2.33), the approximation condition (2.31) and the stability condition

(i

= 1,2, •.. , n)

are satisfied, where M does not depend on i and n. Then for each k + 1 times continuously differentiable solution x(t) of equation (2.26), the

354

v. FINITE-DIFFERENCE METHODS

approximate solutions

Xkn)

E/Yn, &-e.

/

conuerge to x(t) in the norm..J.}f the spaces u

(2.34) 2.2. In the case of a uniformly correct Cauchy problem, its solution is k + 1 times continuously differentiable if Xo E 9'(A kH). In this case, repeating the argument of (2.21), we can prove that if a solution is k times differentiable the approximate solutions converge. 6. Improving the convergence: variable operator. In attempting to carry Theorem 2.9 over to equation (2.14), one comes across the essential difficulty that the powers of A (t) may fail to have a common domain. In this connection we shall restrict ourselves to just the case k = 1. Thus, suppose that the space El is such that (2.29) is satisfied, ~(A) eEl and REMARK

.

(2.35)

IlxllEl ~ c21IA(0)xIIE ~ c21IA(t)xIIE (xE 9'(A) , 0 ~ t

~ T).

We will suppose that A (t) satisfies either the conditions of Theorem 3.4 or the conditions of Theorem 3.8 of Chapter II. In both cases, for Xo E 9'(A 2(0», the solution of the problem. (2.14), (2.3) is twice continuously differentiable, and the functions A (t) x' (t). and A 2(t) x(t) are continuous. We will write the stability condition in the form

lIn

(2.36)

j-k

(l

+ a"tAn(tj»

I

EJ.ly"

~M

(0

~ k ~ i ~ n -1).

We shall try first to introduce the approximation condition in the form, analogous to (2.31), (2.37) E

~(A2(0».

for any

Uo

(2.38)

I (4)n A (tj) -

This implies the inequality

An(tj)4>n)A -2(0)voll EllYn ~ K

I voll E

(Vo E E)

with a constant K not depending on j and n. Unfortunately this inequality still does not make it possible to prove the analog of Lemma 2.1, since, for example, it is not clear whether the operator A2(0) is defined at a solution x(t) with Xo E ~(A2(0». This difficulty doe~ not arise if one assumes in addition

355

§2. EVOLUTION EQUATIONS

that the operator A 2(t) has a domain independent of t and is strongly continuous on it. LEMMA 2.2. Under the additional hypothesis just made it follows from condition (2.37) that

(2.39) 00, uniformly in j = 0, 1, •• " n, for each solution of equation (2.14) with the initial value Xo E g-(A 2(0».

as n -

PROOF. It follows from the continuity of the function A2(t)X(t) that the function

is continuous. Then, repeating the proof of Lemma 2.1, we get II4>nA (tj)x(t) - An(t)4>nx (t) II EllYn (2.40)

~1I4>nA(t)x(h) -An(tj)4>nx(~k)IIEvYn

+ II [4>nA (tj)

- An(t) 4>nJA -2(0)[A2(0) x (tj) - A 2 (0)x(h) JIlEl/Yn '

The second term may be made arbitrarily small because of (2.38) and the uniform continuity of A 2 (0)x(t). The second can be made arbitrarily small because of (2.37) and the fact that x(h) E ~(A2(0». The lemma is proved. N ow we can prove that for the operators Tn the approximation condition is satisfied. To this end we need to prove that

II (x(ti +1)

(2.41)

- x (ti

»/.:l,.t -

A (ti) x (ti) II El - O.

Here we are supposing that condition (2.33) is satisfied. It suffices to show that IIA(O)

[X(ti+1~n~ x(O

(2.42)

=

- A (ti)X(t;) ]

II~ ('i+1 [A (0) x' (s) .:l,.tJ~

liE

A (0) x' (tj) Jds

IE

o.

As we saw, the function A(t)x'(t) is continuous, which means that A(O)x'(t) = A(O)A-l(t)A(t)x'(t) is continuous as well. Hence (2.42) follows, and thus also (2.41). The approximation condition for the operators Tn has been obtained. We note that at the last stage of the

356

V. FINITE-DIFFERENCE METHODS

proof the additional assumption of the constancy: of ,q)(A2(t» was not used. From the arguments presented above it is already easy to obtain the following. THEOREM 2.10. Suppose that the operator A (t) satisfies the conditions of Theorem 3.4 or 3.8 of Chapter II. Suppose moreover that the domain ,q)(A 2(t» is independent of t and that the operator A 2(t) is strongly continuous on it. If conditions (2.33), (2.35), (2.36) and (2.37) are satisfied, then the approximate solutions converge in the norm of Ed Y" to an exact solution of the problem (2.14)-(2.4) for any XoE ,q)(A(O».

For Xo E ~(A 2(0» the assertion of the theorem follows directly from the foregoing and Theorem 1.4. Suppose that Xo E ~(A (0». We approximate the element A (0) Xo by the element YoE~(A(O» and write A-I(O)yo=,fo. Then fo E ~(A2(0» and

I Xo - fo I EI ~ c211 A(O) Xo - A (0) foil E = c211 A (0) Xo - Yo I E ~ C2 E• Denote by x the solution of the Cauchy problem for equation (2.14) with the initial value x(O) = xo. Then, analogously to (2.21),

(2.43)

we obtain

I xl") -


+ I X},n) -

i£n)~ EllYn

+ I
»I EllYn'

X(tk

The last term will be small as a consequence of the uniform correctness of the Cauchy problem and (2.33). The second term will be small in view of (2.43) and the uniform boundedness of the operators T;;l in the norm of Ed Y n and (2.43). The first term is small because of the convergence proved above of the approximate solutions for solutions in ~(A2(t». The theorem is proved. We recall that in the reduction of a hyperbolic equation of second order in Hilbert space (§1.5 of Chapter III) to an equation of the first order we encountered an operator, there denoted by B(t), for which all the conditions of Theorem 2.10 were satisfied. In connection with this we have included this case in our considerations. To prove the convergence theorem in a more general case we make the approximation condition more stringent. In place of strong convergence to zero of the operators [..p"A(t) - A,,(t)
§2. EVOLUTION EQUATIONS

357

equivalent to condition (2.37), we require that the operators

[4>nA (tj) - An{t) 4>n]A -2 (tj) tend to zero in norm. More precisely, we suppose that (2.44)

as n -

The following lemma then holds.

<Xl.

2.3. If the approximation condition (2.44) ooids, then (2.39) is valid for any solution of equation (2.14) with the initial value Xo E LEMMA

~(A2(o». PROOF.

From (2.44) we have

II4>nA (tj) x(tJ - An(tj)4>nx(t) I El/~n ~ Pnll A2(t) x (tj) I E ~ cPn-O, where c is the maximum of the continuous function IIA2(t)x(t) liE' The lemma is proved. The remaining verification of the approxim,ation condition remains the same. Thus the following is valid.

2.11. Suppose that the operator A (t) satisfies the conditions of Theorem 3.4 or 3.8 of Chapter II. If inequalities (2.33) and (2.35) are satisfied, as well as the stability condition (2.36) and the approximation condition (2.44), then the approximate solutions converge in the norm of Ed!zfn to the exact solution of the problem (2.14)-(2.4) for each Xo E THEOREM

~(A(O».

We have considered the question of improving the convergence for the homogeneous equation. Under certain conditions on the function f(t), analogous facts may be established for the nonhomogeneous equation as well. REMARK 2.3. In practice the spaces E/5Ln are most often finite dimensional. In this case all the norms in the space E/!z!n are equivalent. However the constants in equivalence relations may depend essentially on n. Sometimes it is convenient to verify the stability condition in the one norm I I ~ l5&'n and the approximation condition in another, I liE lY'n' In this case the norms of the operators II T;;ll1 EllYn = 'Yn may grow unboundedly. If this growth is sufficiently consistent with the speed of decrease of the constants Pn in the approximation condition (2.44), then the convergence of the finite-difference method in the norm 1111 EllYn may hold on sufficiently smooth solutions.

358

V. FINITE-DIFFERENCE METHODS

7. Example: a partial dift'erential equation of p~abolic type. We shall illustrate our ideas by the example of the unifonnly correct problem considered in §8.3 of Chapter I. This was the homogeneous first boundary problem for the equation (2.45)

au/at = Yu,

u(0,9) = uo(9),

where Y is an elliptic differential expression of order 2m with sufficiently smooth coefficients. defined in an s-dimensional region G with sufficiently smooth boundary r. The domain ~(A) of the operator A generated by the expression Y consists of all functions u(9) E W!"'(G) satisfying the boundary conditions

(2.46)

u = 0, au/an = 0",· .,am-1u/anm - 1 = 0 on

r.

Without loss of generality we may assume that the operator A has a bounded inverse in Lp. Suppose that Kn is some division of s-dimensional space into cubes, and An a finite-difference operator constructed relative to the decomposition Kn and defined on those of its nodes which lie in G. The problem (2.45), (2.46) is replaced by the system, of finite-difference equations (2.47)

(V(tHb 9,) - v(tk , 9,» / ~nt

= Anv(tk, 9,),

v(O, 9,)

= uo(9,).

Here the point 9, runs through all the nodes of Kn lying in G, and tk = k~t, k = 0,1, .. " n. The boundary condtions (2.46) are taken into account in the construction of the operator An. The problem (2.45)-(2.46) is uniformly correct in the space Lp(G). Here we are taking E = Lp (G). As the space El we take the space C(G) of all functions continuous in G in the usual norm. Obviously relation (2.29) is fulfilled. For the operator A the following coercive inequality holds:

II u II w!p

~ c(l) II Au 11w,1-2m p

(l> 2m)

with a constant c not depending on u E 9(A) n W1. Hence it follows that for u E 9(A k) the following inequality holds: (2.48)

II u II ~ ~ c(2km) p

••• c(2m) II Aku II £p'

In view of an imbedding theorem of S. L. Sobolev, for kp > s/2m the space W;km( G) is imbedded in the space C( G), so that it follows

359

§2. EVOLUTION EQUATIONS

from (2.48) that II ull c ~ C211 Aku IILp

(kp

> 812m),

i.e. inequality (2.30) is satisfied. The collection of all the functions of C(G) which vanish on all the nodes of K" which lie in G forms a subspace !£". The factorspace C(G)/!£" = Ed!£" may be considered as a collection of functions given on the nodes of K". Here the natural norm will be (2.49)

the maximum being taken relative to the values of the function u(91) at the nodes of K" in G. Moreover, we introduce the norm (2.50)

where h" is an edge of an elementary cube of the decomposition K", and inside the summation sign there appear all the values of the function at the nodes of K" lying in G. For the norm (2.49) property (2.33) is obviously satisfied with constant 1. Further, (2.51)

where G" is the region consisting of all cubes intersecting g. Accordingly. for h ~ ho (2.33) will also be satisfied for the norm (2.50). We note further that (2.52) IluIIEl/!/"= maxlu(91r ) I

~ h~/q ~h~Llu(91r) Iq "

=

;./q

"

Ilull~ll!i'".

The verification of the stability condition (2.36) for the operators A" is frequently a complicated algebraic exercise. A large literature is devoted to the verification of stability conditions and the construction of stable schemes. The elucidation of this question is not one of the problems we discuss here, and we will suppose that the stability condition is satisfied. Let us clear up the approximation question. In many cases, usually with the help of Taylor's formula, it is easy to estimate the difference between values at the nodes of finite-difference and differential operators on sufficiently smooth functions. We shall say that the operators A" satisfy a natural approximation condition if A "u converges to .Yu uniformly for each function having continuous derivatives of order

360

V. FINITE-DIFFERENCE METHODS '-(§l

2m and satisfying the boundary conditions (2.46). If now uo(.9) E 9(Ak+l), then uo(.9) E W!(k+l)m(G), and under the condition kp > s/2m the derivatives of order 2m of the function uo(.9) will be continuous.

Then from the natural approximation condition it follows that (2.53)

I A,.rj>,.uo -

rj>,.Auoll El/5&',. = maxi A,.uo(.9 r) - Auo(.9r) 1---+0

as n ---+ ex> , i.e. the approximation condition (2.31) is satisfied. It then follows from Theorem 2.9 that the unite-difference method converges under the conditions of stability and natural approximation for any solution of the problem (2.5)-(2.41) with an initial function uo(.9) lying in W!km(G) for which the functions uo, YUo, •• • ,yk-1uo satisfy the boundary conditions (2.46). The choice of k is determined by the inequality k > s/2mp. We note that by increasing p we may always arrange that convergence will hold for k = 1, i.e. for all the solutions of the problem (2.45) -(2.46). It follows from (2.51) and (2.53) that the approximation condition will be satisfied for all norms I I ~/5&',.. Therefore in the norm in which one has succeeded in verifying the stability condition (most often this is the norm with q = 2), the convergence of the process will be proved as well. We note that for. parabolic equations of the second order (m = 1), because of the maximum principle, it is possible to verify the stability condition also in the most rigid norm I I E/5&',.. If the stability condition is verified in the norm I I ~/5&',. with .some q, then the norms 'Y,. of the operators T;;l relative to the norm (2.49) in the space EdY,. may increase like I/h:/q in view of (2.51) and (2.52). In order to cancel this growth one might strengthen the approximation condition. We shall say that the operators A,. approximate A with order 1+ a if for each function uo(.9) satisfying the boundary conditions (2.46), having derivatives of order 2m and lying in the Holder space C' (G), the inequality (2.54)

maxi A,.uo(.9 r ) r

-

Auo(.9) I ~ Nh~+all uoll c2m+I+a(G)

holds. We shall show that then we may verify a condition of type (2.38). We have

I (A,.rj>,. If l + a

< 2m -

rj>,.A) A - 2voll E 1/5&',. ~ Nh~+" I A - 2voll c2m+l+a(G)·

sip, then the space W;m, to which the function

§2. EVOLUTION EQUATIONS

361

A -2vo belongs, is imbedded in the space C2m+I +a , so that

I (An4>n -

4>nA) A 2voll EllYn ~ Nlh~+all A -2vollwr (G) ~ N2h~+a I voll 0.'

from (2.48). Thus, condition (2.38) is satisfied with Pn = N2h~+". This makes it possible to apply Theorem 2.11 to the proof of convergence of the finite-difference method for equation (2.45) with coefficients which are twice continuously differentiable variable functions of t. Moreover, we may establish theorems on uniform convergence in all variables of the approximate solutions to an exact solution when the stability condition is satisfied in one of the weak norms I 11~/.Yn.

REMARKS AND REFERENCES TO THE LITERATURE

Chapter I

The terms correct, uniformly correct and weakened Cauchy problem for the equation x' = Ax, which were studied in Chapter I, were introduced by the author in [4], Chapter III. §1. The investigation of the correct Cauchy problem is carried out here for the first time, although in large part it is based on well-known results presented in the book [2]. This relates to the assertions on boundedness, strong continuity and growth of a semigroup, and on the connection between semigroup and resolvent realized by the Laplace transform. Theorem 1.4 is formulated here for the first time. The method of construction of solutions of the Cauchy problem using the inverse transform was systematically applied by Hille [2]. In explicit form Theorem 1.5 was formulated in the paper of Ju. I. Ljubic

[71]. The concept of correctness set of an operator and Theorems 1.5-1.7 were considered by the author in [58]. ' §2. Here again a significant portion of the results is a methodological reworking of the theory of semigroups satisfying the Co-condition, as presented in [2]. Theorems 2.2-2.5 relate to this. Theorems 2.6-2.8 were formulated by P. E. Sobolevskii and the author in the paper [63] for the case of Hilbert space. However the proofs carryover without change to Banach space. We note that the proof of uniqueness in Theorem 2.7 is taken from a paper of E. Hille [32]. The sufficient condition (2.18) for uniform correctness for a Cauchy problem was found in 1948 independently by E. Hille (see [2]) and K. Y osida [107], and in the literature it is called the Hille-Y osida condition. The necessary and sufficient conditions (2.17) were found by Feller [21], Miyadera [76] and Phillips [83]. The proofs of Theorems 2.9 and 2.10 are carried out according to a scheme of Yosida [107]. The results of subsection 3 on the connection between solvability, uniqueness and correctness (Theorems 2.11-2.13) are due to Phillips [82]. Theorems 2.14 and 2.15 are published with proofs for the first time (see [58]). 362

363

CHAPTER I

§3. The uniqueness theorem for the weakened solution was obtained by Ju. I. Ljubic in 1960 (see [71]). Later it was generalized in the papers of L. I. Prokopenko [190] and S. Agmon and L. Nirenberg [116J. Theorem 3.2 also is due to Ljubic [71]. Theorems 3.3,3.5, and Remarks 3.1-3.4 are special cases of the results of E. Hille (see [2], Chapter XII, §2). We note that semigroups generated by operators whose resolvents satisfy condition (3.18) only on some line parallel to the imaginary axis were studied in a paper of M. A. Evgrafov [132]. Remark 3.6 is due to Ju. I. Domslak. Theorem 3.9 basically is due to K. Y osida [110]. The fact that it is not so much the behavior of the resolvent, but rather the location of the region of regular points, that governs the smoothness properties of the solutions of equation (1.1), was systematically exploited by S. Agmon and L. Nirenberg [116]. The phenomenon of "retarded" smoothness was also investigated by K. V. Valikov [210]. The theorem that the solutions of equation (1.1) lie in the quasi-analytic classes was obtained by O. I. Prozorovskaja [87]. The investigation of the correctness of the inverse Cauchy problem in the class of bounded solutions was carried out in the paper [152] of the author in 1957, and then in a paper of O. I. Prozorovskaja and the author [62], in which Theorem 3.12 was obtained. A number of closely related assertions, based also on various "convexity" properties of the solutions, are contained in the paper [116] of Agmon and Nirenberg. It should be noted that the bulk of these results relate to equations with constant operator A. Only for equations in Hilbert space with variable selfadjoint operator in [152], and then in [116] and [216], were some results obtained on the correctness of the inverse problem. If in an inequality of type (3.41) one fixes = to and varies T, then it is possible to obtain a lower estimate for the quantity I U(T)XoII as T --+ ex> , and in particular estimates of the possible speed with which the solutions of equation (1.1) tend to zero. For various classes of semigroups such estimates were obtained in this way by Prozorovskaja [191J. Another method of obtaining estimates from below for the solutions of differential equations and inequalities in Hilbert space was proposed earlier by P. D. Lax [57]. Later these questions were developed in the papers [116] and [139 ]. §4. Semigroups of bounded selfadjoint operators and their spectral resolutions had been already considered in the papers of B. Sz.-N agy and E. Hille in 1938 (see [2}, pp. 587-593).

r

364

REMARKS ON THE LITERATURE

Theorem 4.2 was established by V. E. Ljance [66}. The theory of maximal dissipative operators (Theorems 4.3-4.5, 4.7) was constructed by R. S. Phillips [85}. Here we have presented an exposition of it differing somewhat from the original. Theorem 4.6 is due to J. L. B. Cooper [223}. [224]. We note that the concept of dissipative operator was generalized to the case of Banach spaces in which the norms were differentiable in the sense of Gateaux in the paper of V. G. Maz'ja and P. E. Sobolevskii [176] and later in the paper [77] of E. Nelson. G. Lumer and R. S. Phillips [166] constructed a theory of such operators and the contraction semigroups corresponding to them in Banach space using the so-called semi-inner product introduced by Lumer [165] (see also the paper of M. Hagesawa [142]). Differential equations in Hilbert space with an unbounded Hamiltonian operator were investigated in the papers of V. I. Dergusov and V. A. Jakubovic [124]-[126]. The results of subsection 5, and also Theorems 4.13 and 4.14, apparently are due to the author [158]. We note that the method of introducing a new scalar product in the proof of Theorem 4.8 was taken from M. G. Krein (see [3]). For the case when the operator A is bounded, it was proved in [3] that the norm obtained in this way is equivalent to the original norm (see §8). S. R. Foguel [229], answering a question of B. Sz.-Nagy, constructed an example of an operator U for which the semigroup of powers Un was uniformly bounded, but the operator was not contractive in any equivalent Hilbert norm (see also [238]). Apparently, an analogous situation may hold also for a strongly continuous semigroup of operators. In connection with these questions see also [135]. The consideration of operators generated by forms was initiated in the papers of K. O. Friedrichs. The content of Theorem 4.15 basically coincides with the content of the so-called Lax-Milgram lemma [64]. Numerous variants of similar considerations are in the book of J. L. Lions [5]. Operators generated by regularly dissipative forms were investigated by T. Kato [47]. We note that precisely these operators were called regularly dissipative by Kato. In the text we use this concept in a wider sense. The hyperbolic equations considered in subsection 8, and the concepts connected with them, were studied by Ju. L. Daleckii [18]. The theorem on the representation of operators satisfying (4.65) in the form of the sum of an operator of multiplication by a function and a differ-

CHAPTER I

365

ential operator, (4.63), is a generalization of the theorem on the representation of operators satisfying the indeterminacy relation of quantum mechanics AX- XA = - iI (see [31], [240] and the literature referred to therein). §5. The semigroup of fractional powers of a bounded operator in Banach space was apparently first studied by Hille in 1939 [26]. He considered this question from the point of view of the possibility of embedding a bounded operator into an analytic semigroup (see/[25], p. 495). He proved the uniqueness of such a semigroup under certain conditions on its growth. The method of fractional powers of operators as a tool for the investigation of the solution of linear, and particularly of nonlinear differential equations in Hilbert and Banach spaces, has been systematically developed since 1956 in the papers of M. A. Krasnosel'skii, p, E. Sobolevskii, and the author (see the survey articles [51], [52], [95]). Fractional powers of a generating operator of a semigroup with the GJ-condition and the semigroups corresponding to them were first investigated by S. Bochner [221] and R. S. Phillips [80]. M. Z. Solomjak [99], [101] used formula (5.29) for the representation of fractional powers as given in [2] to obtain estimates of the products of fractional powers of the generating operator of an analytic semigroup by a semigroup (see §7.3). The construction and study of fractional powers of operators satisfying the condition (5.4) was carried out by M. A. Krasnosel'skii and p, E. Sobolevskii [53] (in their terminology an operator with the condition (5.4) was called weakly positive), and independently under the more general condition (5.30) by A. Balakrishnan [10]. In the paper of V. 1. Macaev and Ju. A. Palant [167] fractional powers of a bounded dissipative operator were introduced using the same formulas of type (5.8) as in [53] and [10], and the spectral properties of these powers were studied in detail. A uniqueness theorem was proved. These results were generalized to unbounded maximal dissipative operators by H. Langer [155]. The moment inequality (5.16) with the constant C(a,/3,'Y) == 1 was established in [225] for a positive selfadjoint operator in Hilbert space by the method presented in subsection 9, and for a weakly positive operator in Banach space in [53]. Theorem 5.4 is due to T. Kato [45]. Semigroups constructed rela-

366

REMARKS ON THE LITERATURE

tive to fractional powers were studied by Balakrishnan [10]. The theorem on raising powers to powers was obtained by J. Watanabe [106]. Theorem 5.6 was proved by K. Yosida [112]. In presenting the theory of fractional powers of operators with unbounded inverses (subsection 8) the author followed Kato's paper [47]. We note that the theory of fractional powers of operators is presented in the book [56], and for the case of generating operators of semigroups in the book [S]. However, neither in these books nor in the present one is there any treatment of the profound investigations of fractional powers of dissipative and regularly dissipative operators carried out by Kato in [47] and [4S] (see also J. L. Lions [161]). The theory of fractional powers is also being intensively worked out at the present time (see [147], [19S]). §6. Theorem 6.1 in the present formulation is new, although its proof follows an idea of Dunford. Theorem 6.4 is close to a result of S. Ja. Jakubov [35], [36]. Theorem 6.5 was established by R. S. Phillips

[SI]. The study of the nonhomogeneous abstract parabolic equation in Hilbert space was carried out by M. A. Krasnosel'skii, P. E. Sobolevskii and the author [55], and Banach space (Theorem 6.7) by M. Z. Solomjak [101]. Theorem 6.8 is basically due to S. Ja: Jakubov [35], [36]. Theorems 6.9 and 6.10 were obtained by the author and O. I. Prozorovskaja in the process of working on the book. We note that a number of theorems on the solvability and the smoothness of solutions of the nonhomogeneous equation are contained in

m

[72]. §7. Lemma 7.1 is contained in Kato's paper [42]. The connection between inequalities of the type (7.3), or, what is the same thing, (7.9), and questions as to the subordination of a fractional power of an operator was discovered for the case of a selfadjoint operator by P. E. Sobolevskii and the author in [63] and used for the study of fractional powers of differential operators by V. P. G lusko and the author in [22]. For generating operators of analytic semigroups in Banach space it was studied by M. Z. Solomjak [201] and in the general case by M. A. Krasnosel'skii and P. E. Sobolevskii [53] (see [56], and also the paper [115] of K. Yosida). Lemma 7.3 was proved in [53]. Theorem 7.1 was first found by E. Heinz [24], though it must be said with an imprecise constant in inequality (7.6). It was established

CHAPTER I

367

with a more precise constant by Kato in [41]. There later followed other proofs (J. Dixmier [127], P. S. Bullen [212]). In the form of certain interpolation theorems it was proved by J. L. Lions [256] and the author [57]. The proof presented in the text is due to the author. Kato proved in [48] that the basic assertion of the theorem remains valid also for maximal dissipative operators in Hilbert space. On the problem of carrying over the Heinz inequality to other functions of operators see [230]. Assertions analogous to Lemmas 7.4 and 7.5 on fully subordinate operators are contained in a paper of V. P. G lusko and the author [233]. Theorem 7.2 is apparently new. In a slightly different form it was proved in the graduation thesis of Ju. B. Savcenko. Theorems 7.3 and 7.4 are generalizations of theorems of S. Ja. Jakubov [34], [38]. Theorem 7.5 is due to R. S. Phillips [81]. In a paper of H. F. Trotter [209] the case was considered in which the operators A and B are generating operators of contraction semigroups, with ~(B):::> ~(A). Then for sufficiently small E the operator A + BE is a generating operator of a semigroup with the Co-condition. In a paper of E. Nelson [77] it was proved that in the case when II Bxll ~ IIAxll+bllxll (xE~(A» this will hold for O~E
368

REMARKS ON THE LITERATURE

coefficients, in order to make it possible to apply the results of Kato (see §3 of Chapter II) a variable differential norm was introduced in a clever way, making the basic operator dissipative. The idea of such a construction goes back to J. Leray. Strongly elliptic operators were studied by M. I. ViBik. The semigroup approach to the first boundary problem for strongly parabolic equations and systems was applied by Lax and Milgram [64] and V. E. Ljance [66]. As we noted in the text, estimate (8.30) for the first boundary problem in the· spaces Yp was obtained by Solomjak [201]. Under general boundary conditions it was investigated in the papers of Agmon [217], Agmon and Nirenberg [116], and M. S. Agranovic and M. I. Visik [217]. For the second boundary problem for an equation of second order an estimate of the type (8.30) in Yp was obtained in a paper of Maz'ja and Sobolevskii [176]. Symmetric hyperbolic systems were studied from the point of view of the theory of differential equations in Banach space in the papers of Lax [248], Phillips [85], [263], and Lax and Phillips [249]. A treatment of equations with retarded argument as equations in Banach space was given by N. N. Krasovskii in the book [242]. The description found there of the generating operator of the corresponding semigroup was done there with insufficient rigor. This question was treated more precisely by Ju. G. Borisovic, Ju. S. KolesoV', K. V. Valikov and others Chapter II §§1 and 2 basically contain preparatory material, to be found in the papers of many authors. Theorem 1.2 is due to Ju. L. Daleckii [11], and Theorem 2.1 to J. Kisybski [50]. §3. The definition and investigation of the properties of the correct Cauchy problem for an equation with a variable operator is carried out here for the first time (subsections 1-3). The corresponding theorems were obtained earlier under hypotheses on the operator A (t) which guaranteed uniform correctness of the Cauchy problem. Thus, for example, the analogs of Theorems 3.1 and 3.2 are found in [42], of Theorem 3.3 in [43] and [54], of Theorem 3.4 in [16]. Lemma 3.1 was established by P. E. Sobolevskii [i9]. The central results of the section are the fundamental theorems of uniqueness, Theorems 3.10 and 3.11, both of which are due to Kato [42]. The proof of Theorem 3.11 which we present is due to J. Kisybski

CHAPTER II

369

l50J. An analysis of Kisynski's proof led the author to the conditional Theorem 3.6 on uniform correctness. Theorem 3.8, under Kato's hypotheses, was proved in [16] (see [17}). A nonessential generalization of Kato's theorem to the case when condition (3.35) is satisfied in a norm which is variable in t and remains equivalent to the original norm was carried out by Kisynski in [50]. In a number of papers, proofs of the uniform correctness of the Cauchy problem for an equation with variable operator have been carried out by the method of Y osida: the consideration of an equation with a bounded operator A,,(t) = - nI - n 2 R(n), and then a passage to the limit (see K. Yosida [114], [215], J. Elliot [131] and E. Heyn [143]). Here one lowers the requirements either on the smoothness of the operator, or on the domain, or on the behavior of the resolvent. One should, however, note that these proofs were carried through under assumptions which appear to us to be quite difficult to verify. We note in this direction a quite recent paper of Ja. B. Lopatinskii [164]. §4. Abstract parabolic equations with variable selfadjoint operators in Hilbert space were first considered by Sobolevskii [90]. He introduced the integral equations (4.11) and (4.12) connecting the volution operator of an equation with variable operator with that of an equation with "frozen" constant operat~ Some time later H. Tanabe [102], using an idea of Lewy, proposed a new method of proving existence theorems for the evolution operator under the hypotheses of Kato, also basing himself on the consideration of integral equations of the type described above (see equations (5.13), (5.15». After Solomjak [99] considered the abstract parabolic equation with constant operator which is a generating operator of an analytic subgroup, both Sobolevskii [92] and Tanabe [103] independently carried over their methods to equations with such variable operators. Later Mamedov and Sobolevskii [73] noted that in the investigation of the corresponding integral equations an important role was played by the mutual connection between the estimate of the derivative of the semigroup of the equation with "frozen" coefficients and the estimate of the increments of the functions A(t)A -1(0) and. A -l(O)A(t). Supposing that instead of (4.26) the estimate I A (to) UA(to)(r) II ~ c/w1(r) holds, and instead of (4.29) and (4.32) the estimates

370

REMAKRS ON THE LITERATURE

they proved under hypotheses connecting the functions "'1 and "'2 that the evolution operator exists. We note that they assumed here that the "frozen" Cauchy problem was uniformly correct. The method of proof adopted in §4 is close to that of [73]. However in our special case we dropped the condition of boundedness of the semigroup UA(to) (r) at zero. Apparently more general conditions connecting the functions "'I. "'2, and "'0 can be formulated, where

II UA(to)(r) I

~ c/"'o(r).

Just as in [73], we imposed the condition (4.32) on A(t), which makes the proof easier but seems rather restrictive. To verify it, we must establish an inequality of the type (4.29) for the dual operator A *(t). In the" original papers of Sobolevskii and Tanabe the existence of the evolution operator was proved without this assumption. E. T. Poulsen [S6] obtained an analogous result under the hypothesis that UA(to) (r) satisfies the Co-condition. and that A (t) satisfies the estimate (4.25) with f3> 5/6. Already in his report to the International Congress of Mathematicians in Amsterdam [lOS] K. Yosida posed the question of removing the requirement of constancy of -the domain of A (t). For partial differential equations this requirement means that the coefficients of the boundary conditions are independent of t. For abstract parabolic equations the first essential step in this direction was taken by Sobolevskii in [91]. For an equation with selfadjoint operator in Hilbert space he constructed an evolution operator under the hypothesis that the fractional power of the operator has a constant domain. The naturalness of this hypothesis for differential operators was investigated before that by O. M. Kozlov [14S], and then by Sobolevskii [91], the author [57], Kato [47] and others. Shortly thereafter Sobolevskii [90] and Kato [46] carried these results over to Banach space. H. Tanabe [105], for the case when the operator has a variable domain, imposed conditions (5.1) and (5.2) on the derivative of the resolvent, and also constructed an evolution operator. These conditions did not include the requirement of constancy of the domain of the fractional power of the operator, but rather require a smoother dependence of the resolvent on t, i.e. differentiability instead of a Holder condition. Our exposition in §5 follows that of Kato and Tanabe [49]. The conditions imposed on an operator A (t) with variable domain

CHAPTER III

371

are usually verified by considering the quadratic form corresponding to the operator (see [91], [46], [49] and others). In a recent paper of Sobolevskii [200] he obtained a new theorem of existence of the evolution operator for equations in Banach space, in which the hypotheses are directly formulated in terms of the properties of the form of the operator and its dual. In the applications this theorem yields an existence theorem for solutions of parabolic equations with common normal boundary conditions in Yp-spaces. We note that in all the papers indicated above with respect to abstract parabolic equations, the questions considered were to a greater or lesser degree those of improving the smoothness of the solutions and on their analyticity, topics not reflected in our exposition. Chapter III In §§1 and 3 we investigate the strong solutions of certain classes of differential equations of the second order. Generalized and weak solutions of various types of differential equations of the second order in Hilbert space were studied in the papers of Visik [268] and Ladyzenskaya [244]-[246] already mentioned, and then in the papers of Lions (see his monograph [5]). §1. The Cauchy problem for equation (1.1), and for the more general equation x(n) = Bnx , was investigated by Hille and Phillips. The results of their investigations were presented in the book [25], pp. 617-633 (where references to the original papers are made). Among these results in particular are Theorems 1.1 and 1.1'. These authors do not appear to have carried out investigations for wider classes of equations. The hyperbolic. equation (1.25) with variable selfadjoint operator in Hilbert space was studied by Ju. L. Daleckii in 1957. Later these results, and, in particular, Theorem 1.7, formed a part of his doctoral thesis [16]. They were improved and developed in a paper of V. A. Pogorelenko and P. E. Sobolevskii [189]. Hyperbolic equations of second order in Banach space were studied in the papers ofS. Ja. Jakubov [33], [34]. He obtained existence theorems under more stringent hypotheses than Theorem 1.6. We note also the work of Ja. D. Mamedov, devoted basically to the study of nonlinear equations, but containing certain a priori estimates as well for linear equations, from which there result a number of consequences on the behavior of the solutions [168]-[171]. §2--. This section basically presents results of G. I. Laptev and the author, obtained in the papers [59]-[61]. Exceptions are Theorem 2.9,

372

REMARKS ON THE LITERATURE

contained in the dissertation of O. I. Prozorovskaja, and Theorem 2.11, due to A. V. Balakrishnan [10]. We should note that in [10] this theorem was proved under weaker restrictions, when the operator A may fail to have an inverse, and instead of the estimate (2.2) we have only the estimate II RA(t)(s) I ~ Mjs. The availability of formulas for the Green's function makes it possible to construct a theory of eigenvalue problems. This was done in the papers [59] and [156 ]. It is natural to ask whether the results of this section can be carried over to equations with variable operator. In a simple case this question was considered by V. A. Trenogin [207]. Deeper results were obtained recently by G. I. Laptev. §3. Strong solutions of the Cauchy problem for equations of second and higher order (the type (3.27» were investigated in a paper of B. S. Mitjagin [74] in connection with the problem of equations with a small parameter on the highest derivative. He solved the problem, interesting in itself, of conditions under which the operator E d 2 jdt 2 + A djdt + B can be decomposed into a product E(djdt - C1) (djdt - C2 ) of operators of the first order. It turns out that if E is sufficiently small this may be done if the operator A -IB is bounded. For an equation of higher order a decomposition is obtained under more special restrictions on the operators and subordinate terms. The basic results on the solvability of the Cauchy problem for equations of second order with variable operator were obtained by Sobolevskii [97], [981 (Theorems 3.2-3.4 and 3.6 in our text). S. Ja. Jakubov in the papers [34], [35] and [37] worked out a very simple new method of proving existence theorem, which is presented in its essentials in §3. This made it possible for him to generalize the results of Mitjagin (see §3.6) and to free himself from superfluous restrictions on the smoothness of operators in equations with variable coefficients. Moreover, Jakubov considered the case of a non-uniformly-correct problem, with a result close to Theorem 3.5. Chapter IV §1. Apparently the first paper in which was considered the question of the behavior of the solutions of differential equations of the second order with unbounded operators in the presence of a small parameter on the second derivative was the paper [74] of Mitjagin already mentioned in Chapter III. Then there followed a paper of Sobolevskii

CHAPTER IV

373

[96], where the same question was considered by another method and under more general assumptions. The problem of removing a discrepancy in the boundary conditions was considered by B. I. Panoioti [78]. where also he obtained and proved the formula (1.29) and investigated the question of almost uniform convergence of the solutions. In the papers referred to the subject was equations of the second order, but all the difficulties in fact appeared in the investigation of the evolution operator U,(t,s) for equations of the first order. In connection with this we proposed to present the entire theory for equations of the first order, and, as is clear from the exposition, its application to equations of the second order does not lead to new difficulties. Along the way it was possible to reduce by one the smoothness requirements imposed on the operator A(t) in the papers [96] and [78]. We have considered the equation of second order under the hypothesis of uniform correctness of the Cauchy proble:m. For the abstract parabolic equation of second order one may carry out analogous considerations also in the case when the operator B is unbounded (see §3.4 of Chapter III). This was done in the papers [97] and [98] of Sobolevskii. It appears that one may investigate in the same way the case when the weakened Cauchy problem is correct (see §3.4 of Chapter III). §2. The material presented in the second section is of an auxiliary character. However, it seems to us to be of interest in itself. The problem of construction of smooth operators realizing the evolution of a direct decomposition of a space is important in various problems of the spectral theory of operators depending on a parameter, for example in the problem of reduction of a matrix depending on a parameter to quasi-diagonal form using a transformation having the same degree of smoothness as the original matrix. The theorem of existence of an operator realizing the evolution of a subspace was obtained by Daleckii and the author in [20] under the hypothesis that the projection operator P(t) is differentiable. Later Kato [23I} obtained independently an analogous result using the same method. Further on Daleckii [12] succeeded in generalizing that result to the case of an operator realizing the evolution of the decomposition of a space into the direct sum of several subspaces, and in replacing the differentiability condition by the condition of bounded variation. This last required an essential change in the method of proof. Daleckii [14] also proved Theorem 2.4 by another method. I t is interesting to note that under the hypothesis that P(t) is only

374

REMARKS ON THE LITERATURE

continuous no one has succeeded in proving the existence of an operator realizing the evolution of the corresponding subspace. It is possible, however, that the continuity in norm of P(t) does not adequately reflect our geometric image of the continuous turning of a subspace. §3. The asymptotic methods presented in the basic portion of the section were begun in the work of N. N. Bogoljubov and I. M. Krylov. These methods were developed for various classes of equations by a large group of their students and followers. A complete survey of the corresponding literature is in the monograph [228]. For abstract differential equations in Hilbert space with bounded operators these methods were first applied in a paper of Daleckii and the author [19]. Later Daleckii carried them over to equations in Banach space and equations with unbounded operators [14], [15]. We have written the equations to which the asymptotic methods are applied in the form of equations with a small parameter. Frequently one encounters analogous problems for equations with a large parameter JL. Thus the equation considered in subsection 6 may be written in the form dx/dt = JLA(t)x

+ B(t,JL)x,

where the operator B(t, JL). is given by an expansion .,

B(t,JL) =

.L Bk(t)JL- k• k~O

In subsection 7 we considered the question of asymptotic approximations to a particular solution of the nonhomogeneous equation in the simplest example of such an equation. One may analogously consider the equation edx/dt

=

A(t)x

+ tB(t,e)x + {(t,e),

where B(t, e) and {(t, t) are given by expansions in powers of e. In this form this equation was also studied in the papers cited above. One may consider the question of solution of operator equations more general than those studied in Lemma 3.1. For the equation

with bounded operators Al and A2 under the hypothesis that the polynomial

375

CHAPTER V

does not vanish when ~ and fJ. run through the spectra of the operators Al and A2 in the spaces El and E 2, the solution is given by the formula

X

= _

~ 4'11"

r r R A2(~) YR

Jrl .Jr2

A1 (fJ.)

dAdfJ..

P(A, fJ.)

This formula was used in unpublished papers of M. G. Krein and was found later and independently by Daleckii [13], [15]. Later yet, for the case of equation (3.24), it was again found by M. Rosenblum [88]. In conclusion we note that Daleckii in [123] considered an interesting type of equations containing rapidly oscillating terms:

d X = A (t) x dt

E

+ EL: Bm (t, E) e(im../·)tx. m

Here the operators Bm(t,E) are again given by expansions in powers of E. In order to find a solution with initial data from Yo one applies asymptotic decompositions analogous to those described in subsection 6. The most interesting case is that of intrinsic resonance, when the isolated portions Ao(t) and Al (t) can have points differing by the quantity imw. It is a singular fact that in the presence of resonance, when one seeks a null approximation one succeeds in solving a nonlinear differential equation which in the scalar case is the Ricatti equation. These results are presented in the monograph [3]. Chapter V The first theorem on the convergence of the finite-difference method for the solution of the evolution equation in Banach space was the theorem of Lax (see [65]). In a general way it asserts that in the case of a uniformly correct Cauchy problem for the equation x' = Ax, convergence will follow from the consistency and stability conditions. From the point of view of the applications this theorem is deficient. The approximate equation is considered in the same space as the original one. The consistency condition requires the convergence of the approximating operators to the original one on an everywhere dense set of solutions of the evolution equation without any indications as to the choice of this set. These deficiencies were rectified in

376

REMARKS ON THE LITERATURE

the papers [39] and [40] of L. I. Jakut, in which the approximate equations were treated as equations in factor spaces, which corresponds to the passage from functions given in a region to functions given only at the nodes of a net (see §2.7). In this direction Jakut succeeded in obtaining theorems on the convergence of finite-difference methods for those equations for which the existence of the solution is proved. Still earlier H. F. Trotter in [208], starting also from a critique of Lax' theorem, constructed a theory in which the spaces in which the approximate equations are solved are arbitrary and connected with the original space only by a system of uniformly bounded mappings. The theorem he obtained on the approximation of the corresponding semigroups was presented in [8]. N. N. Gudovic [23] proposed an abstract scheme for finite-difference methods and investigated in detail the connections among stability, convergence and approximation. His results constitute the content of §1. In §2 we are basically presenting the results of Jakut referred to above, reworked and supplemented using the general approach of §1. Both sections of Chapter V were written by the author jointly with N. N. Gudovic. We note further that· certain difference schemes for the solution of the evolution equations were actually investigated by Daleckii [17] in connection with the problem of representation 'of the solutions by means of functional integrals.

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