Differential and Integral Inequalities: Ordinary Differential Equations v. 1: Theory and Applications: Ordinary Differential Equations v. 1

DIFFERENTIAL AND INTEGRAL INEQUALITIES Theory and Applications Volume I ORDINARY DIFFERENTIAL EQUATIONS

This is Volume 55 in R’IATHEMA’I’ICS I N SCIENCE AND ENGINEERING A series of monographs and textbooks Edited by RICHARD BELLMAN, University of Southern California A complete list of the boolts in this series appears at the end of this volume.

DIFFERENTIAL AND

INTEGRAL INEQUALITIES Theory and Applications Volume I ORDINARY DIFFERENTIAL EQUATIONS

V. LAKSHILIIKANTHAM and S. LEELA {Jnioersity of Rho& Islmiii Kiiqstow, Rliotlc Islaiid

A C A D E RI I C P R E SS

New J’ork and London

1969

COPYRIGHT 0 1969, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London W.l

LIBRARY OF CONGRESS CATALOG CARDNUMBER: 68-8425

PRINTED I N THE UNITED STATES OF AMERICA

Preface

This volume constitutes the first part of a monograph on theory and applications of differential and integral inequalities. 'The entire work, as a whole, is intended to be a research monograph, a guide to the literature, and a textbook for advanced courses. T h e unifying theme of this treatment is a systematic development of the theory and applications of differential inequalities as well as Volterra integral inequalities. T h e main tools for applications are the norm and the Lyapunov functions. Familiarity with real and complex analysis, elements of general topology and functional analysis, and differential and integral equations is assumed. T h e theory of differential inequalities depends on integration of differential inequalities or what may be called the general comparison principle. T h e treatment of this theory is not for its own sake. 'The essential unity is achieved by the wealth of its applications to various qualitative problems of a variety of differential systems. T h e material of the present volume is divided into two sections. T h e first section consisting of four chapters deals with ordinary differential equations while the second section is devoted to Volterra integral equations. T h e remaining portion of the monograph, which will appear as a second volume, is concerned with differential equations with time lag, partial differential equations of first order, parabolic and hyperbolic respectively, differential equations in abstract spaces including nonlinear evolution equations and complex differential equations types. T h e vector notation and vectorial inequalities are used freely throughout the book. Also, because of the several allied fields covered, it becomes convenient to use the same letter with different meanings in different situations. This, however, should not cause confusion, since it is spelled out wherever necessary. T h e notes at the end of each chapter indicate the sources which have V

vi

PREFACE

been consulted and those whose ideas are developed. Some sources which are closely related but not included in the book are also given for guidance. We wish to express our warmest thanks to our colleague Professor C. Corduneanu for reading the manuscript and suggesting improvements. Our thanks are also due to Professors J. Hale, N. Onuchic, and C. Olech for their helpful suggestions. We are immensely pleased that our monograph appears in a series inspired and edited by Professor R. Bellman and we wish to express our gratitude and warmest thanks for his interest in this book.

V. LAKSHMIKANTHAM S. LEELA Kingston, Rhode Island December, I968

Contents

PREFACE

V

ORDINARY DIFFERENTIAL EQUATIONS Chapter 1 . 1.0. Introduction 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10. 1.11.

Chapter 2. 2.0.

2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9. 2.10. 2.11. 2.12. 2.13. 2.14. 2.15 2.16.

3

Existence and Continuation of Solutions Scalar Differential Inequalities Maximal and Minimal Solutions Comparison Theorems Finite Systems of Differential Inequalities Minimax Solutions Further Comparison Theorems Infinite Systems of Differential Inequalities Integral Inequalities Reducible to Differential Inequalities Differential Inequalities in the Sense of Caratheodory Notes

3 7 11 15 21 25 27 31 37 41 44

Introduction Global Existence Uniqueness Convergence of Successive Approximations Chaplygin’s Method Dependence on Initial Conditions and Parameters Variation of Constants Upper and Lower Bounds Componentwise Bounds Asymptotic Equilibrium Asynlptotic Equivalence A Topological Principle Applications of Topological Principle Stability Criteria Asymptotic Behavior Periodic and Almost Periodic Systems Notes

45 45 48 60 64 69 76 79 84 88 91 96 100 102 108 120 129

vi i

...

CONTENTS

Vlll

Chapter 3. 3.0. Introduction 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.1. 3.8. 3.9. 3.10. 3.1 I . 3. I ? . 3.13. 3.14. 3.15. 3.16. 3.11. 3.18. 3.19. 3.20. 3.21.

Basic Comparison Theorems Definitions Stability Asymptotic Stability Stability of Perturbed Systems Convcrse Theorems Stability by the First Approximation Total Stability Integral Stability I,”-S ta hi I i ty Partial Stability Stabilit) of Differential Inequalities Boundcdness and Lagrange Stability Eventual Stability Asymptotic Behavior Relative Stability Stability with Respect to a Manifold .-\lmost Periodic Systems Uniqueness and Estimates Continuous Dependence and the Method of Averaging Notes

Chapter 4 . 4.0. Introduction 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9. 4.10.

Main Comparison Theorem Asymptotic Stability Instability Conditional Stability and Boundedness Converse Theorcms Stability in Tube-like Domain Stability of Asymptotically Self-Inwriant Sets Stability of Conditionally Invariant Sets Existence and Stability of Stationary Points Notes

131 131 135 138 145 155 158 177 186 191 199 205 209 212 222 229 24 1 244 245 254 257 264 267 267 269 273 277 284 293 291 305 308 311

VOLTERRA INTEGRAL EQUATIONS Chapter 5. 5.0. Introduction

Integral Inequaiitics 5.2. Local and Global Existence 5.3. Comparison Theorems 5.4. Approximate Solutions, Bounds, and Uniqueness 5.5. Asymptotic Behavior 5.6. Perturbed Integral Equations 5.7. Admissibility and Asymptotic Behavior 5.8. Integrodifferential Inequalities 5.9. Notes 5.1.

31 5 315 319 322 324 327 333 340 350 354

CONTENTS

ix

Bibliography

355

AUTHORINDEX

385

SUBJECT INDEX

388

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DIFFERENTIAL AND INTEGRAL INEQUALITIES Theory and Applications Volume I

ORDINARY DIFFERENTIAL EQUATIONS

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ORDINARY DIFFERENTIAL EQUATIONS

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Chapter 1

1 .O. Introduction This chapter is an introduction to the theory of differential inequalities and therefore forms a basis of the remaining chapters. After sketching the preliminary existence and continuation of solutions of an initial value problem for ordinary differential equations, we develop fundamental results involving differential inequalities. Basic comparison theorems that form the core of the monograph are treated in detail. While considering the system of differential inequalities (finite or infinite), we find it convenient to utilize the notion minimax solutions, and consequently our treatment rests on this notion. Certain useful integral inequalities that can be reduced to the theory of differential inequalities are also presented. Some results on differential inequalities of Caratheodory's type are also dealt with.

1.1. Existence and continuation of solutions Let R" denote the real n-dimensional, euclidean space of elements u = ( u l ,u2 ,..., un). Sometimes, we shall denote also the (a + 1)-tuple (t, u l , u, ,..., un) as an element, and Rn+I shall denote the space of elements ( t , u l , uz ,..., un) or ( t , u). Let 11 u 11 be any convenient norm. As usual, we shall use R instead of R'. Let E be an open (t, u)-set in I?"+'. We shall mean by C [ E ,R"] the class of continuous mappings from E into R". Iff is a member of this class, one writes f E C[E, R"]. Let us consider a system of first-order differential equations with an initial condition u' = g(t, u ) , u(t0) = u 0 , (1.1.1) where u' = du/dt,u0 = ( u I 0 ,u,,, ,..., unO),and g E C [ E ,R"]. A solution of the initial value problem (1.1.1) is a differentiable function of t such 3

4

CHAPTER

1

that ~ ( t , , = ) ZL, , ( t , u(t))E E, and u'(t) = g(t, u(t)) for a t-interval J containing t,, . This means that u ( t ) has a continuous derivative. From these requirements on the continuous function u(t), it follows that it satisfies the integral equation

I n order to prove the classical Peano's existence theorem, we have to introduce the notion of an equicontinuous family of functions. 1)EFINITION 1.1.1. A family of functions F = { f ( u ) } defined on some ti-set E C R" is said to be equicontinuous if, for every E > 0, there exists a S = S ( E ) , independent of f E F and also u l , u2 E E, such that 11 f ( u J -f(u2)ll -E, whenever / / u1 - u2 I/ < 6. T h e following theorem shows the fundamental property of such a family of functions, the proof of which will be omitted.

THEOREM 1.1.1. (Ascoli-Arzela). Let F = ( f ] be a sequence of functions defined on a compact u-set E C R", which is equicontinuous and equibounded. Then, there exists a subsequence ( f n } , n = 1, 2, ..., which is uniformly convergent on E. THEOREM 1.1.2. (Peano's Existence Theorem). Let g E CIRo , Rn], whcrc R,, is the set [ ( t ,u ) : to t to a, 11 u - u, /I 61; 11 g(t, u)\l M on R,, . Then, the initial value problem (1.1.1) possesses at least one solution u ( t ) on t, t t, 01, where 01 = min(a, 6 / M ) .

< < +

<

<

< < +

Pyoof. 1,et uo(t) be a continuously differentiable function, on [to 5, t,,], S > 0, such that uo(t,) = u, , 11 u,(t) - u,, /I 6, and 11 u i ( t ) 11 M . For 0 < E 6, we define a function u f ( t )= u,(t) on [to- ~ -S, to]and -

% ( t ) = UO

on [t, , t, and

<

<

<

+ 4, where

+ It g(s, u,(s

-

( 1.1.2)

.)) ds

t0

C Y = ~

min(ol, E ) . Observe that uXt) is differentiable

I/ u d t ) - *o II

+


(1.1.3)

on [to 8, to EJ. If 0 1 ~< 01, we can use (1.1.2) to extend uXt) as a continuously differentiable function over [to- 6, to a 2 ] , a2 = min(iu, 2 ~ )such , that (1.1.3) holds. Continuing in this way, uc(t) can be defined over [to 6, to a ] so that it has a continuous derivative and -

+

~

+

1.1.

5

EXISTENCE AND CONTINUATION OF SOLUTIONS

satisfies (1.1.3) on the same interval. Furthermore, I/ u:(t)ll < M , and therefore { u c ( t ) }forms a family of equicontinuous and uniformly bounded functions. An application of Theorem 1.1.1 shows the existence of a sequence {en> such that > > *.. en+O as n--t a,and u(t) = limn+mu,(t) exists uniformly on [to - 6, t,, a]. Since g is uniformly continuous, we obtain that g(t, u,,(t - E , ) ) tends uniformly to g ( t , u(t)) as n + co,and, hence, term-by-term integration of (1.1.2) with E = E , , a1 = 01 yields

+

This proves that u ( t ) is a solution of (1.1.1). T h e following corollary of Peano's Theorem is useful in applications. COROLLARY 1.1 .I. Let E be an open ( t ,u)-set in Rn+l and E, be a and 11 g(t, u)II M on E. compact subset of E. Suppose that g E C[E, Rrb] Then, there exists an a = a(E, E,, , M ) such that, if (to , u,,) E E, , (1.1.1.) has a solution, and every solution exists on [ t o ,t, a]. I n that case, when g is not bounded on E, we can replace the set E by an open subset El having a compact closure in E and containing E,, .

<

+

T h e next theorem deals with the problem of extending the solutions up to the boundary of E.

THEOREM 1.1.3. Let E be an open ( t ,u)-set in Rn+l,and let g E C[ E ,R"]

< < a,.

and u ( t ) be a solution of (1.1.1) on some interval to t u(t) can be extended as a solution to the boundary of E.

Then

Proof. Let E l , E , ,... be open subsets of E such that E = (J E,; the closures El, E, ,... are compact, and En C It then follows from Corollary 1.1.1 that there exists an E , > 0 such that, if ( t o ,uo)E E m , all solutions of (1.1.1) exist on t,, t to E , . Choose n, so large that (a,, , ~(u,,))E Enl . Then, ~ ( tcan ) be extended E,,], and, if (a, E , ~, u(ao E,,)) E En1, over an interval [a,,, a, u(t) can be further extended over [a,, + E , , , a, + 2 4 . This argument can be repeated until we get the extension of u(t) over the interval t,, t a, , where a, = a, N,E,,, N , is an integer 2 1 , such that

< < + +

+

< <

(a1

, u(a1)) 4 En, .

+

+

Choose n, so large that (al , u(al))E Enz . Arguing as before, we arrive at an integer N 2 3 1 such that u(t) can be extended over t, t a2, 0 2 = a, N2E?t2 7 and ( a , 7 u ( 4 ) 4 Enz . Proceeding in this way, we are led to a sequence of integers

+

< <

6

1

CHAPTER

n, < n2 < ... and numbers a, < a, < a2 < . - -such that u(t) has an extension over [to, a), where a = lim,+m a, and that (a,, u(a,)) $ . Thus, the sequence (a,, ,u(uk)) is either unbounded or has a cluster point on the boundary of E. T o show that u(t) tends to the boundary of E as t ---t a, we must show that no limit point of {tk , u(tk))is an interior point of E as t, 4a. Since this follows from the lemma below, the theorem is proved.

Erik

LEMMA 1. I . I . Let g E C[E,R"], where E is an open (t, u)-set in Rn+l.Let u(t) be a solution of (1.1.1) on an interval to t < a, a < 00. Assume that there exists a sequence {t,) such that to tk+ co as k -+ co and uo = 1imk+%u(tk) exists. If g ( t , u) is bounded on the intersection of E and a neighborhood of ( a , uo),then

<

lim u(t) t-a

=

<

( I . 1.4)

uo.

If, in addition, g(a, uo) is defined such that g(t, u ) is continuous at ( a , uo),then u(t) is continuously differentiable on [ t o ,a] and is a solution of (1.1.1) on [ t o ,a].

Proof. Let E

I/ u

uo /I

<

> 0 be sufficiently small. Consider the set I?: O < a - t < E,

<

E. Let M ( e ) be so large that IIg(t, u)ll M ( E ) for ( t , u ) E E n I?. If, for R sufficiently large, 0 << a - t, e / 2 M ( ~and ) 11 u(t,,) - uo // ~ / 2 then , -

<

<

I1 4 t ) - u(t,c)ll for t ,


M(E)(a- tk) G 4 2

a. If this is not true, there is a t, such that t,<< t,

/ / u(tl) - u(t,,) I( = rcl(c)(a - t k )< ~ / 2 It. therefore follows that :c

(1.1.5)

< a,

< & + / / u(tk) uo / / < for t, < t < t , . This implies 11 u'(t)/I < M ( E )for t,c < t < t, . Consequently, 11 u ( t ) - uo 11

-

E

II 4 t l ) - u(t,Jll G M(E)(t, - tk) < M ( 4 ( a - t k ) . This proves (1.1.5), which, in turn, shows that (1.1.4) holds. T h e last part of the lemma follows from the fact that u'(t) = g(t, u(t))+ g ( a ,

COROLLARY I . 1.2. Let g E

=

E

[ ( t ,u ) : t ,

u")

as

t -+ a.

C[E, R"], where

< t < t" + a ( a < a),u E R"].

1.2.

7

SCALAR DIFFERENTIAL INEQUALITIES

Let u(t) be a solution of (1.1.1). T h e n the largest interval of existence of u(t) is either [ t o ,to + a] or [ t o ,a), 6 to + a and /I u(t)l/ -+ og as t46.

<

1.2, Scalar differential inequalities We adopt the following notation for Dini derivatives: D+u(t) = lim sup h-l[u(t h-O+

+ h)

-

u(t)],

D+u(t) = liin inf h-l[u(t + h ) - u ( t ) ] , h-Oi

D-u(t)

= lim

sup h-'[u(t

D-u(t)

= lim

inf h-l[u(t

+ h)

h-*O-

h-0-

+ h)

-

-

u(t)],

u(t)],

+

where u E C[(to, to a), R].When D+u(t) = D+u(t),the right derivative will be denoted by u;(t). Similarly, uL(t) denotes the left derivative. DEFINITION1.2.1. Let E be an open ( t , u)-set in R2 and g E C[E, R]. Consider the scalar differential equation with an initial condition u' = g(t, u ) ,

u(t,) = ug

.

+

Suppose z, E C[[t,, to a ) , R], z,;(t) exists for t E [to, to (t, v(t))E E. If v(t) satisfies the differential inequality a t )

< g(t>a ( t ) ) ,

t

fE [to

9

to

(1.2.1)

+ a),

and

t- a ) ,

it is said to be an under-function with respect to the initial value problem (1.2.1). On the other hand, if q t )

> g(t, +)),

t

6

[to , to

+ a),

v(t) is said to be an over-function.

A fundamental result on scalar differential inequalities is the following:

THEOREM 1.2.1. Let E be an open ( t , u)-set in R2 and g E C[E, R]. Assume that 21, w E C[[t,, to a), R] and ( t , v(t)), ( t , w ( t ) ) E E , t

E

[to, to

+ + u). Suppose further that .(to)

< w(to),

(1.2.2)

8

1

CHAPTER

and, for t E (to, to

+ a ) , the inequalities (1.2.3) (1.2.4) (1.2.5)

Proof.

If assertion (1.2.5) is false, then the set

z

=

[t E [to, to

is nonempty. Defining t , Furthermore,

=

+ a): w ( t ) < v(t)]

inf 2, it is clear from (1.2.2) that to

< t, . (1.2.6)

v(t1) = 4 t l )

and

4t) < 4 t h

t

E

Using (1.2.6) and (1.2.7), we obtain, for small h 74tl

+ h) h

-

u(t1)

>

+ h)

4tl

(1.2.7)

[to , tl).

h

< 0,

-4

tl)

7

which in its turn implies D-v(tl)

2 D-w(t,).

(1.2.8)

T h e inequalities (1.2.3), (1.2.4), and (1.2.8) together with (1.2.6) lead us to the contradiction At1 7 4 t l ) )

>A t 1

7

4tl)).

Hence 2 is empty, and the statement (1.2.5) follows.

REMARK1.2.1. I t is obvious from the proof that the inequalities (1.2.3) and (1.2.4) can also be replaced by D-4t) < g(t, v(t))t D-w(t) >> g ( t , w ( t ) ) ,

respectively. Note that the proof does not demand the validity of the inequalities (1.2.3) and (1.2.4) for all t E (to, to a). T h e following refinement is a consequence of this observation.

+

1.2.

9

SCALAR DIFFERENTIAL INEQUALITIES

THEOREM 1.2.2. Let the assumptions of Theorem 1.2.1 hold, except that the inequalities (1.2.3) and (1.2.4) are satisfied for t E 2, = [t E (to , t, u ) : v(t) = w(t)]. Then (1.2.5) remains valid. I n fact, Theorem 1.2.1 can be subjected to further refinements. T o this end, we require the following simple lemmas. Although we state them for scalar functions, it is easy to see that they are true for vector functions as well. Unless otherwise specified, let S denote an at-most countable u). subset of [ t o ,t,

+

+

+

LEMMA 1.2.1. (Zygmund). Suppose that u E C[[t,, t, a), R] and the inequality Du(t) 0 for t E [to , to u ) - S, D being a fixed Dini derivative. Then, u(t) is nonincreasing in t on [ t o ,to u).

+

<

+

+

LEMMA 1.2.2. Let n, w E C [ [ t , , t, a), R ] , and for some fixed Dini derivative Dv(t) < w ( t ) for t E [ t o , to u ) - S. Then, D-v(t) < w ( t ) for t E [to, to a).

+

Proof.

Define the function m(t) = v ( t ) -

+

st

w(s) ds.

to

It then follows, from the assumption, that Dm(t) = Dv(t)

~

W(t)

< 0,

2E

[ t o ,to

+ a) s. -

Hence, by Lemma 1.2.1, m(t) is nonincreasing in t on [t, , to Consequently, D-m(t) = D-v(t)

-

<

~ ( t ) 0,

t E [ t o , to

+ u).

+ a),

and the lemma is proved. REMARK1.2.2. I n the light of Lemma 1.2.2, it is clear that Theorem 1.2.1 remains true when the inequalities (1.2.3) and (1.2.4) hold for t E [ t o ,t, u ) - S , D being any fixed Dini derivative. It will now be shown that any solution of the initial value problem (1 2.1) can be bracketed between its under- and over-functions.

+

THEOREM 1.2.3.

Let n(t), w ( t ) be under- and over-functions with respect to the initial value problem (1.2. l), respectively, on [to , to + a). If u(t) is any solution of (1.2.1) existing on [t, , t, + u ) such that ~ ( t ,= ) un = w(t,),

(1.2.9)

10

CHAPTER

1

then v(t) < u(t) < w(t),

E

[t”, t,

+ a).

(1.2.10)

Proof. We shall prove the right half of the inequality (1.2.10). Similar reasoning can bc used for the left half. Let w(t) and u(t) be an overfunction and a solution of (1.2.1), respectively. Lct m(t) = w(t) - u(t). Then, m;(f,,) > 0 because of (1.2-9). I t follows that m ( t ) is increasing to the right of to in a sufficiently small interval to t t, E , which implies that

< < +

u(t,

Furthermore,

+

€)

u’(t)

and for t E [t,,

+

< g(t, 4 t ) ) s(t,749)

w’, (t) E,

to

< w(t, 4-€).

+ a). A direct application of Theorem 1.2.1 yields that u(t)

.= w ( t ) ,

t

t

+ a).

[t,, , t ,

This proves the theorem.

COROLLARY 1.2.1. and

Let E be an open ( t , u)-set in R2,g, ,g, ’?l(t, u)

< gz(t, u),

E

C[E,R],

( t , u ) E E.

Let ul(t),u2(t)be any two solutions of u; = g d t , u),

respectively, existing on [ t o ,to udt) ,%(f), f E [ t o 9 to 4.

+

~

4 = &(t, u),

+ a ) such

that ul(t,)

< uz(t,).

Then

COROLLARY I .2.2. Let E be an open ( t , u, v)-set in R3, and g E C [ E ,R ] , and g ( t , u, v) is nondecreasing in v for each fixed t and u. Let u , v t C[[t,, t, 4- a ) , I?] such that u’(t), v’(t) exist, (t, u(t), ~ ’ ( t ) ) , ( t ,~ ( t ~ ) ,’ ( t E) B ) for t E [to , to u). Assume that the inequalities

+

4 t h u’(t)) < 0, hold for t t [ t o ,to t E [ t o , to a).

+

+ u).

A t , U ( t ) , v’(t)) ‘Then, u(t,)

0

< v(t,)

implies u(t) < u(t) for

1.3.

11

MAXIMAL AND MINIMAL SOLUTIONS

1.3. Maximal and minimal solutions T h e notion of maximal and minimal solutions of (1.2.1) will now be introduced.

DEFINITION 1.3.1. Let r ( t ) be a solution of the scalar differential equation (1.2.1) on [ t o ,t, a). Then r ( t ) is said to be a maximal solution of (1.2.1) if, for every solution u(t) of (1.2.1) existing on [ t o ,t, a), the inequality

+

+

u(t)

< r(t),

t E [ t o to 7

+ a)

(1.3.1)

holds. A minimal solution p ( t ) may be defined similarly by reversing the inequality (1.3.1). We shall now consider the existence of maximal and minimal solutions of ( I .2.1) under the hypothesis of Peano’s existence theorem.

C[R, , R], where R, is the rectangle to t t, a, I u - uo 1 6, and I g(t, u)I M on R, . T h e n there exist a maximal solution and a minimal solution of (1.2.1) on [t,,, to a ] , where 01 = min(a, 6/(2M 6)).

THEOREM 1.3.1.

< < +

Let g

E

<

<

+

+

Proof. We shall prove the existence of the maximal solution only, since the case of the minimal solution is very similar. Let 0 < E b/2. Consider the differential equation with an initial condition

<

+

u‘ = g(t, ).

Observing that gdt,

u(t,) =- uo

E,

4 =g(t,4 +

+

(1.3.2)

€.

E

is defined and continuous on R, : t ,

< t < to + a ,

+

<

1u

- (ug

+ €)I

< b/2,

lg, (t, u)I M (b/2) on R,, we deduce from Theorem 1.1.2 that the initial value problem (1.3.2) has a solution u(t, E ) on the interval [ t o ,to a ] , where 01 = min(a, b/(2M 6)). For 0 < E~ < el E , we have R, C R, and

+

<

.(to

7

€2)

+

< to 4 ,

+

U ’ ( t , €2)

< R(t, u ( t ,

U ’ ( t , €1)

> g (t, u(t, €1)) -t €2

€2))

€2, 7

t E [to , t” 4-.I.

12

1

CHAPTER

We can apply Theorem I .2.1 to get t E [to tn

u ( t , el),

~ ( l€ 2, )

9

i a].

Since the family of functions u(t, E ) is equicontinuous and uniformly bounded on [to, t, 011, it follows by Theorem 1.1.1 that there exists a decreasing sequence { E ~ ]such that eTL-+ 0 as n --t m, and the uniform limit r ( t ) = lim u(t, en)

+

n-*m

+

exists on [ t o ,t, a ] . Clearly, r(t,) = Z L ~. T h e uniform continuity of g implies that g(t, u(t, E , ) ) tends uniformly to g(t, r(t)) as n -+ a, and thus term-by-term integration is applicable to u(t, 4 = uo -1-

'n

+ s( R(&

u(s,

to

4 )ds,

which in turn shows that the limit r ( t ) is a solution of (1.2.1) on [f"

, 2"

+

(YI.

We shall now show that r ( t ) is thc desired maximal solution of (1.2.1) on [t,,, to t] satisfying (1.3.1). Let u(t) be any solution of (1.2.1) existing on [z,, , t, 4-n ] . Then,

+

~ ( t ,= ) ug u'(t)

u'(t, 6 )

for t

E

[to, t,,

+

X]

and

E

U(t)

< ug

+

E

== U ( t n ,

df, 4 t ) ) + ', At, u(t, .)I +

, ,

E),

€7

< h/2. By Remark 1.2.1, we obtain that < u(t, €1,

t

E

[to , to

-t

a].

T h e uniqueness of the maximal solution shows that u(t, e ) tends uni0. This proves the theorem. formly to r ( t ) on [to , to 1.1 as E

+

--f

This cxistence theorem, together with the extension Theorem 1.1.3, implies the following:

TIIEOKEM 1.3.2. Let g E C [ E , R ] , where E is an open ( t , u)-set in R2 and ( t o, u,,) E E. Then (1.2.1) has maximal and minimal solutions that can be extended to the boundary of E. T h e lemmas given below are useful in certain later applications.

I~EMMA 1.3.1. Txt the hypothesis of Theorem 1.3.2 hold, and let [t,,, to 4-u ) be the largest interval of existence of the maximal solution

1.3.

13

MAXIMAL AND MINIMAL SOLUTIONS

+

r ( t ) of (1.2.1). Suppose. [ t o ,t l ] is a compact subinterval of [ t o , to u). T h e n there is an E , > 0 such that, for 0 < E -< c0 , the maximal solution r ( t , E ) of Eq. (1.3.2) exists over [ t o ,t l ] , and lim r ( t , E )

=

c-*O

r(t)

uniformly on [ t o ,t l ] .

Proof. Let En be an open bounded set, I?,) C E, and ( t , r ( t ) ) E E for t E [t,,, t l ] . We can choose a b > 0 such that, for t E [to, t l ] , the rectangle R t : [t,t

is included in Eo for E that

+ 61,

Iu

(r(t)

~

+ .)I

< 6,

< b/2. Let 1 g(t, u)I < M on Eo . T h e n it is evident I g(t, u )

+ I < + 6/2 6

<

on R t , for t E [ t o ,tl] and 0 < E b/2. Consider the rectangle RI, . It follows from Theorem 1.3.1 that the maximal solution r ( t , E ) of (1.3.2) exists on [t, , to 711, 7 = min(b, 26/(2M 6)). Note that 7 does not depend upon E . Furthermore, proceeding as in Theorem 1.3.1, we can conclude, in view of the uniqueness of the maximal solution r ( t ) of (1.2. I), that lim r ( t , G) = ~ ( t )

+

+

€-O

uniformly on [ t o ,to

+ 71. This implies that

+ 7,

l i i Y(t"

= Y(t"

6)

+ 7).

< 6/2 such that, for 0 < <

Consequently, there is an

E

Y(t0

+ 7, < r(t0 + 7) + 6)

E

~

we , have

E.

We can now repeat the foregoing argument with respect to the rectangle Rfo+,, E < , to show that there exists an c2 < such that, for E < E$ , the maximal solution f ( t , E ) of 21'

exists on [to

=g(t, u)

+

E,

+ q, to + 2711, and

+ 7)

U(t,

lim f ( t , E )

+ 7,to + 2711. For r ( t ,E )

= f(t,

€1,

E

t

+ 7) +

E

= r(t)

E-0

uniformly on [to r(t, E ) by defining

= r(t0

E

< E $ , we can extend the function [to

+ 7, t o + 271.

14

1

CIIAPTER

It is clear that r ( t , E ) is the maximal solution of (1.3.2) on [ t o ,to and lim v(t, C ) = r ( t )

+ 271,

E-0

+

uniformly on [ t o ,to 271. By induction, it can be shown that there is an c0 = E , such that [ t o ,tl] C [ t o ,t, nq], that the maximal solution r(t, E ) of (1.3.2) exists on [to, to nq] for 0 -: E < E , , and that

+

+

lim r ( t , C ) C-0

= r(t)

+ nq]. T h e lemma is thus proved. LEMMA1.3.2. Let g E C[[t,, to + u] x R, R] and nondecreasing for each t E [to , to + a]. Assume that uniformly on [ t o ,to

g(t, 0)

I g(tj .)I

<M

0,

on

[to

7

to

in u

(1-3.3)

+ a] x R ,

(1.3.4)

and u ( t ) = 0 is the unique solution of u' = R(4 u ) ,

on [to, t ,

(1.3.5)

u(to) = 0

+ a]. Then, the successive approximations u"(t) = M(t - t o ) ,

As,

%+l(t) = Jf

ds

'0

are well defined; 0 and

< untdt) < % ( t )

lim u J t )

n-m

Moreover, for every n u' = g (t, u )

exists on [ t o , t,

L:

0

on

[to to 9

+ a],

uniformly on [ t o ,to

+ a].

(1.3.7)

3 I , the maximal solution m(t) of -I- k ( t , U n - l ( t ) ) ,

%(to)

=

0,

4-a ] , and lim rn(t) = 0

n-.m

(1.3.6)

uniformly on

[ t o , to

> 0,

+ a].

(1.3.8)

<

Proof. An easy induction proves ( I .3.6). Since, by ( 1 .3.4), I uk(t)l M , using Theorem 1.1.1, we can conclude that limn+mu,(t) I- u(t)uniformly

1.4.

15

COMPARISON THEOREMS

on [to , to + a ] .It is clear that u ( t )satisfies u'(t) = g ( t , u(t))and u(t,) By (1.3.5), it follows that u ( t ) = 0, and (1.3.7) is proved. Given E > 0, there is an n 3 a(.) such that

1 k ( t , un-1(t)I <

t

€9

[t" , t o

=

0.

+ a],

because of (1.3.3) and (1.3.7). Now an argument similar to that of Lemma 1.3.1 proves (1.3.8). 1.4. Comparison theorems

An important technique in the theory of differential equations is concerned with estimating a function satisfying a differential inequality by the extremal solutions, of the corresponding differential equation. One of the results that is widely used is the following comparison theorem:

THEOREM 1.4.1. Let E be an open ( t ,u)-set in R2 and g E C [ E ,R]. Suppose that [ t o ,to a ) is the largest interval in which the maximal a ) , R ] , ( t , m ( t ) ) E E for solution ~ ( tof) (1.2.1) exists. Let m E C[(t,, t, t E [to, to a ) , rn(t,) uo , and for a fixed Dini derivative,

+ <

+

+

Dm(t)

t

E

[to , to

+ a)

-

S. Then, m(t)

Pyoof.

< g(t, m ( t ) ) ,

< +),

t

6 [to

, to

(1.4.1)

+ a).

(1.4.2)

From Lemma 1.2.2, it follows that (1.4.1) can be replaced by D-m(t) ,< 'dt,m ( t ) ) ,

t

+

(t" > to

E

(1.4.3)

a).

Let to < T < to + a. By Lemma 1.3.1, the maximal solutions (1.3.2) exist on [to , T ] for all E > 0 sufficiently small, and O E'

m(t) < r(t, E ) ,

t

of

(1.4.4)

~ ( t= ) lim r ( t , c)

uniformly on [ t o ,TI. Using (1.3.2) and Theorem 1.2.1, we derive that

u ( t , c)

(1.4.3)

[ t o , 71.

and

applying (1.4.5)

T h e last inequality, together with (1.4.4), proves the assertion of the theorem.

16

1

CHAPTER

REMARK1.4.1. If the inequality (1.4.1) is reversed and m(t,) 2 u,, , then we have to replace the conclusion (1.4.2) by m(t) 3 p ( t ) , where p ( t ) is the minimal solution of (1.2. I).

Theorem 1.4.1 can also be proved under a weaker hypothesis.

THEORFM 1.4.2.

2

Let m(t),r ( t ) be as in Theorem 1.4.1, and =

[t t [t,,, t,

+ a) :

Y(t)

< m ( t ) < r ( t ) + €01,

for some c0 > 0. If (1.4.1) is satisfied for t E 2 at-most countable subset of 2,then (1.4.2) holds.

-

9,where 3

(1.4.6)

is an

Proof. I t is enough to prove (1.4.5). As before, Lemma 1.2.2 implies that (1.4.3) is satisfied for t E Z. Proceeding as in the proof of Theorem 1.2.1, we arrive at a t, such that m(td

In view of (1.4.4), there exists an r ( t , 6)

=Y(t, en

9

.).

> 0 such that

< y(t) i€0,

t

E

[to

, 71.

RIoreover, we have r ( t ) -'r(t, , E), and hence there results the inequality r ( t ) < r ( t , €)


t

€

[ t o , 71.

It thereforc follows from (1.4.6) that

which implies that t , E 2. Hence, (1.4.3) is satisfied for such a t , , and this is sufficient to establish the desired result. We now give a modification of Theorem 1.2.1. It is evident that the proof of Theorem 1.2.1 breaks down if we do not assume one of the relations (1.2.3) and (1.2.4) to be a strict inequality. This, however, can be relaxed provided g satisfies a further assumption.

THEOIIEM 1.4.3. Let the hypothesis of Theorem 1.2.1 hold except that the inequalities (1.2.3) and (1.2.4) are replaced by

d d t l U(t))>

[ 1.4.7)

D-w(t> 3 d t , w ( t ) )

(1.4.8)

f)-v(t)

1.4.

17

COMPARISON THEOREMS

+

for t E ( t o ,to u). Assume further that, for each t E [to , TI,g satisfies the condition g( t >~

2

1 - )g ( t > ~ 2 )

where G E C[[t,, to of

-G(T

+

-

T E

t , ~1 - uZ),

( t o ,t,

~1

3~

+ u ) and 2

,

(1.4.9)

+ a) x R, R], and r(t) = 0 is the maximal solution G(t,u),

U' =

~ ( t ,= ) 0.

T h e n (1.2.5) holds. Proof. Proceeding as in the proof of Theorem 1.2.1, there exists a t, E (t, , to a) such that 4 t l ) = W(tl>, (1.4.10)

+

and

~ ( t< ) ~ ( t ) , to

+


< ti.

Define vl(t) = n(t, t, t ) and w l ( t ) = w(t1 view of (1.4.10) and (1.4.11), yields that ~

+ to

(1.4.1 1) -

t). This, in (1.4.12)

ui(to) = Wi(tn):

s(t)< W l ( t ) ,

(1.4.13)

t E [ t o > tll.

Setting m ( t ) = w l ( t ) vl(t), the definitions of n, , w1 and the assumptions (1.4.7) and (1.4.8) imply the inequality ~

D-m(t)

=

D-w,(t)

where gl(t, u ) = -g(tl (1.4.9) to arrive at

D-s(t)

~

+ to

D-m(t)

-

< g1(t, W l ( t ) )

- gl(4

s(t)),

t, u). Since (1.4.13) holds, we can use

-< G(t,m(t)),

t E [to

9

tll.

By Theorem 1.4.1, we have m(t)

< r(t),

t

E

[to , tll,

(1.4.14)

where r ( t )is the maximal solution of u' = G(t,u), such that r(t,) = m(t,). From the definition of m ( t ) and (1.4.12) and (1.4.13), we deduce that m ( t ) 3 0, t E [t, , t l ] , and nz(t,) = 0. Then, the inequality (1.4.14) and the assumption r ( t ) = 0 show that v ( t ) = ~ ( t ) , t E [to , ti],

which, however, is contrary to the assumption (1.4.1 1) and the definition of t , . Hence, the set 2 is empty, and the theorem is proved.

18

CIIAPTER

1

T o give another comparison theorem that, in certain situations, is more useful than Theorem 1.4.1, we require the following result:

+

THEOREM 1.4.4. Let E be the product space [to, to a ) x R2 and g E C [ E , R ] . Assume that g is nondecreasing in v for each t and u. Suppose that ~ ( tis) the maximal solution of the differential equation u' = g(t, u, u ) ,

existing on [to , to

u ( f o )= u,

+ a ) , and

t € [to, t,

r ( t ) 3 0,

(1.4.1 5)

2; 0

(1.4.16)

4-u).

Then, the maximal solution r l ( t ) of u(t,) = uo ;3 0,

u' = gl(t, u ) ,

where g l ( t , 21)

= g(t, u, ~ ( t ) )exists ,

r ( t ) = rdt),

on [to, t, t

E

[t" , to

(1.4.17)

+ u ) and

+ a).

(1.4.18)

Proof. By Theorems 1.3.1 and 1.3.2, the maximal solution r l ( t ) of (1 .4.17) exists on an interval [t,,, to $- (Y], (Y a, which can be extended to the boundary of E. This implies that either r l ( t ) is defined over [ t o ,t, a ) or there exists a t , < t, a such that

+

+

and this yields, from Theorem 1.4.1, that

-

as far as r l ( t ) exists. I t follows from (1.4.16), (1.4.19), and (1.4.20) that Yl(t,)

(1.4.21)

-km

as t,. -+ t l - . We shall show that (1.4.21) cannot be true. For this purpose, consider the maximal solution r ( t , E) of u' = g (t, u, u )

+

E,

u(to)= u g

+

6,

ug

2 0,

(1.4.22)

1.4.

19

COMPARISON THEOREMS

+

+

which, by Lemma 1.3.1, exists on [to, t, v ] , v > 0, and t , v < to for sufficiently small E > 0. Moreover, we have from (1.4.22) that

+ a,

and Hence, one gets, from Theorem 1.2.1, the inequality r(t) < r(t,E),

t

-t .I.

[to , t ,

E

(1.4.24)

Since g is nondecreasing in o, (1.4.23) and (1.4.24) lead to

> g d t , r ( t , .I),

r'(4 .)

t

E

[to , t,

+ .I.

) ri(to, €1. But ri(t) = g(t, ri(t), r(t)) gi(t, ri(t)), t E [to > ti), and ~ i ( t o < An application of Theorem 1.2.1 again shows that ri(t> < r ( t , E ) ,

t

E [to

7

( I -4.25)

ti)-

Since r ( t , €) exists on [to , t , + v], v > 0, (1.4.2t) leads us to a contradiction because of (1.4.25), and this proves the existence of r l ( t ) on [to to 9

+ a).

T o prove (1.4.1 S), we now see that (1.4.20) is true for t Furthermore,

E

[t, , t,

+ a).

r;(t> = A t 7 rdt>>= d t t rl(t>?+>)-

From the monotonic character of g in

ZI

and (1.4.20), one gets

G s(4 r d t ) , ri(t))* Theorem 1.4.1 now gives that ri(t)

G ~ ( t ) , t~

[ t o , to

+ a).

This inequality, together with (1.4.20), proves (1.4.18), as is desired.

THEOREM 1.4.5. Let the hypothesis of Theorem 1.4.4 hold; m E C [ [ t o, to a), R ] such that ( t , m ( t ) , o) E E, t E [to , to a ) , and m(t,) < uo . Assume that for a fixed Dini derivative the inequality

+

+

Dm(t>

is satisfied for t E [to , to we have

< g(t, m(t),4

(1.4.26)

+ a ) - 5'. Then, for all < r ( t ) , t E [to , to + a ) ,

m(t)

ZI

<

Y(t),

t

E

[to, t" -4- a).

(1.4.27)

20

CHAPTER

1

+

<

r ( t ) , t E [to, to a). Then, using the monotonicity of g Proof. Let a in z‘, the inequality (1.4.26) reduces to

where gl(t, m ( t ) ) = g(t, m(t), r ( t ) ) . If r l ( t ) is the maximal solution of (1.4.17), ‘Theorem 1.4.4 shows that r l ( t ) exists on [to, to a ) and (1.4.18) is true. Now a straightforward application of Theorem 1.4.1 assures the inequality (1.4.27).

+

Assume that, for each dition

T

(to , to

E

g ( t , ul) - ~ ( t ,u2) 2: -G(T

whcre G E C[[t,,, to of

+ a ) and t E [to, TI, g satisfies the con+ to

-

t a ) x R , R ] , and U’ =

G(t,u),

t , u1

--

uZ),

u1

3 U2

9

r ( t ) - . 0 is the maximal solution ~ ( t ,: ) 0.

+

Then vz(to)< u,, implies m ( t ) < u(t), t E [ t o ,to a ) , where u ( t ) is any ) u,,, existing on [to , to + a). solution of u’ = g ( t , u), ~ ( t , ,= T h e maximal and minimal solutions may be defined to the left of t o , and their existence may be proved using the previous arguments with necessary modifications. A result parallel to Theorem 1.4.1, concerning the minimal solution to the left, is useful in later applications. TTc shall state this as a theorem, omitting its proof.

THEOREM 1.4.6. Let E be an open ( t ,u)-set in R2 and g E C [ E ,R ] . Suppose that m E C[(t(, a, t o ] ,R ] , ( t , m ( t ) )E E for t E (to- a, to], m(t,,) 2 uO, and for any fixed Dini derivative ~

Dm(t)

< g (f, m(t)),

t

t (to

-

a, to).

Then

f 4 t ) ‘z dt), as far as p ( t ) exists to the left of to , p ( t ) being the left minimal solution of (1.2.1).

I .5.

21

FINITE SYSTEMS OF DIFFERENTIAL INEQUALITIES

1.5. Finite systems of differential inequalities Many of the results considered so far for scalar differential inequalities will now be extended, in the sections that follow, to finite systems of differential inequalities. T o avoid repetition, let u s agree on the following: the subscript i ranges over the integers 1, 2, ..., n; let 0 k n ; the subscriptsp and q range over the integers 1, 2 ,..., k and k 4- 1, k 2 ,..., n, respectively. We shall be using vectorial inequalities freely, with the understanding that the same inequalities hold between their corresponding components. We shall consider the differential system with an initial condition, written in the vectorial form

< < +

u'

= g(t, u),

u(tn) = u o ,

whereg E C[E, R"] and E is an open ( t , u)-set in

(1.5.1)

R"1-1.

+

DEFINITION 1.5.1. Let v E C [ [ t ,, to a), R " ] ; ( t , v ( t ) )E E, and v;(t) exists for t E [tn , to a). T h e function v ( t )is said to be a k under ( n k ) over-function with respect to the initial value problem (1.5.1) if

+

~

%J,+(t)< g,(t, 4 t ) ) >

+

74,d t )

> g,(t, 4 t ) )

hold for t E [t,, , to a). If v(t) satisfies the reversed inequalities, it is said to be a k over ( n - k) under-function. These definitions clearly include the definitions of under- and overfunctions as special cases, viz., K = 0 or k = n. We require that the function g(t, u ) should satisfy certain monotonic properties, which are listed below. DEFINITION 1.5.2. T h e function g(t, u ) is said to possess a mixed quasimonotone property if the following conditions hold:

(i) gp(t,u ) is nondecreasing in u j , j = 1, 2,..., k , j f p , and nonincreasing in uQ. (ii) g,(t, u ) is nonincreasing in up and nondecreasing in u j , j = k + 1 , k + 2 ,..., n, j # q . Evidently, the particular cases k = n and k = 0 in the mixed quasimonotone property correspond to quasi-monotone nondecreasing and quasi-monotone nonincreasing properties of the function g(t, u), respectively. Furthermore, g(t, u ) is said to possess mixed monotone property if, in conditions (i) and (ii), j # p , j f q are not demanded.

22

CHAPTER

1

An extension of Theorem 1.2.1 which plays an equally important role is the following: T H E O R E M 1.5.1. Let (i) g E C[E, R"], where E is an open ( t , u)-set in R"-I'; (ii) zi, w E C[[t,, to a), R"], ( t , v(t)), ( t , w(t)) E E for t E [t,,, t,, a ) ;and (iii)g(t, u ) possess a mixed quasi-monotone property. Assume further that

+

+

(1 5 3 ) (1 5 4 ) (1 5 5 ) (1 S.6)

Proof. Define mlJ(t)= wIJ(t) because of (1.5.2), ~

n J t ) and m J t )

m,(t*) 1 0 ,

=

vq(t) - wq(t). Then,

i = 1, 2)...)n.

(1.5.8)

Suppose that the assertion (1.5.7) is not true. Then, the set

z

(J [ t E [to , to 4-a): m,(t) < 01 11

=

2

1

is nonernpty. Let t , = inf 2. By (1.5-8), it is obvious that t, > t o . Since the set Z is closed, t , E Z , and consequently there exists a j such that m,(tl) = 0.

(1 5 9 )

If (1.5.9) is not true, one would have m,(t,) <<0, which implies m,(t) < 0 in a sufficiently small neighborhood to the left of t, . This contradicts the definition of t , , and therefore (1.5.9) is valid. Moreover, (1.5.10) and (1 5 1 1 )

1.5. FINITE Suppose that 1 < j gives

SYSTEMS OF DIFFERENTIAL INEQUALITIES

< k. Th en (1.5.11), &(tl

9

4tlN

< &(tl

23

along with (1.5.3) and (1.5.5), 9

4tl)).

( 1.5.12)

T h e mixed quasi-monotone property of g ( t , u ) in u, in view of (1.5.9) and (1.5.10), yields (1.5.13) &(tl 4 t l ) ) < &(tl w(t1)). 1

1

T h e inequalities (1.5.12) and (1.5.13) lead us to a contradiction. If, on the other hand, k 1 j \< n, arguing as before, we arrive at the contradiction

+ < &(tl

>

4tlN

> &(tl

7

Ntl)),

using the relations (1.5.4), (1.5.6), (1.5.9), (1.5.10), (1.5.11), and the mixed quasi-monotone property of g(t, u ) in u. Hence the set 2 is empty, and (1.5.7) is proved.

COROLLARY I .5.1. Let conditions (i), (ii), and (iii) of Theorem 1.5.1 be satisfied. Assume that, for t E (to, to a ) , the inequalities

+

DWt)

< g(t, W ) ,

D-m(t)

> g(t, 4))

hold. Then, v(to) < w(to)implies v(t) < 4 t h

t

E

[to , t,

+ a).

REMARK1.5.1. Notice that the proof of Theorem 1.5.1 remains unchanged even when the inequalities (1.5.3)-( 1.5.6) are replaced by

< sdt, f4th D-.,(t) 2 g*(t, 4 t ) ) , D--Wp(t) 2 g p ( t ,w ( t ) ) , D-w*(t) < g,(C 4 t ) ) . D-.,(t)

REMARK1.5.2. One can, in Theorem 1.5.1 and the following corollary, use any fixed Dini derivative D in place of D-, the corresponding inequalities being satisfied only for t E [to , to + a) - S. This follows from Lemmas 1.2.1 and 1.2.2. T h e next theorem is an analog of Theorem 1.2.3.

24

CHAPTER

1

THEOREM 1.5.2. I,et v(t),w ( t ) be k under (n - k) over-, k over ( n - k) under-functions, respectively, for t E [t,,, t, + a ) , with respect to the initial value problem (1.5.1). Assume that g ( t , u ) has mixed quasimonotone property. Let u ( t ) be any solution of (1.5.1) existing on [ t o ,to a ) such that

+

v(t,)

Then

for t

E

[to , to

+ a).

(1.5.14)

= U" = W ( t " ) .

% ( t ) < %(t> < zuv(t),

( 1.5.15)

%(t) > 4

(1.5.16)

4 > %(t)

Pmof. If (1.5.15) and (1.5.16) hold for t, < t < fo , fo sufficiently close to t o , then one can deduce the assertion of the theorem by the application of Theorem I .5. I and the subsequent Remark 1.5.1. Indeed, such a f, exists. For, defining 7n,(t)

=

%(t)

~

m,(q

%(t),

% ( t )- % ( t )

and noting thdt m,(t,,) = 0 because of (1.5.14), it is easy to deduce that mi, +(to)> 0, which implies m,(t) is increasing in a sufficiently small neighborhood of t o , say t, t t , . Similar argument with

< <

m;T(t) = Z U D ( t )

~

mZ(t) - U * ( t )

UD(t),

-

w,(t)

< <

shows that nz:(t) increases in t, t t, , t, being sufficiently close to t, . Now, the desired 2, min(t, , t J . T h e proof is therefore complete. ~

COROLLAKY 1.5.2. Let v ( t ) ,w ( t ) be under- and over-functions, respectively, with respect to the initial value problem (1.5.1) for t E [to, to a ) . .Assume that g(t, u ) is quasi-monotone nondecreasing in 21. Let u ( t ) be any solution of (1.5.1) existing on [t, , t, a ) such that

+

+

v(t,)

= U"

== Z"(t,).

Then, v(f)

< u ( t ) < U..(t),

t 6 [ t o , f"

-I a).

COROLLARY 1.5.3. Let (i)f, g E C [ E , R"], where E is an open (t, u)-set in R"I1; (ii) either f or g possess a mixed quasi-monotone property; g,,(t, u ) , f;,(t, ). >g,(t, u), ( 4 u ) E E ; ( 4 4 t h 44 be any (iii) f,,(t,). two solutions of u' = f ( t , u), v' = g(t, v), existing on [ t o ,to a), respectively, such that u0,],<. Z~,,I , u,,* > o,,*. T h e n

+

%(t)

< %(t)'

4 4 > 4%

t

E

[to , to

+ a).

1.6.

25

MINIMAX SOLUTIONS

1.6. Minimax solutions DEFINITION1.6.1. Let r(t) be a solution of the differential system (1.5.1) existing on [ t o , to a) such that, for every solution u(t) of (1.5.1) on [ t o , to a), the inequalities

+

+

4 4

< rzI(t),

% ( t )2 Y J t ) ,

3 rp(t),

uq(t)

or ~ p ( t )

< rq(t),

+4

(1.6.1)

, to t-a)

(1.6.2)

[to , t o

t

E

t

E [to

are satisfied. I n case of ( I .6. I), r(t) is called a k max ( n k) mini-solution of (1.5.1), whereas, in case of (1.6.2), it is said to be a k mini ( n - k) max-solution. I n either case, r(t) is said to be a minimax solution. A k max ( n - k) mini-solution reduces to a maximal solution when k = n and to a minimal solution when k = 0. Similarly, a k mini ( n - k) max-solution coincides with a minimal solution and a maximal solution when k = n and k = 0, respectively. ~

As minimax solutions include both maximal and minimal solutions as special cases, we consider below the existence problem for minimax solutions only.

THEOREM 1.6.1. Let g E CIRo , R"], where Ro : t o

<

< t < to + a,

/I u

- uo II

< 6,

and 11 g(t, u)II M on R, . Assume further that g(t, u) possesses a mixed quasi-monotone property. Then, there exists a k max ( n - k) miniand a k mini ( n - k) max-solution of (1.5.1) on [ t o , to 71, where 7 = min(a, 6/(2M b)).

+

+

Proof.

Let 0

< E < h/2. Consider the initial value problem

+

4 = g,(C

u) E, u:, = g,(t, u ) - E ,

Up(to) =

uo.?,

uq(to)= U o g

+

6;

- E.

1

Observe that g, E C[R,, R"], where g,(t,

R, : [ ( t ,U ) E Rn+': to

and R , C R, . Also,

4 = g(t, 4 i

< t < to +

U,

6,

/I u

- (uo

<)I1 < 6/21,

(1.6.3)

26

1

CHAPTER

It therefore follows from Peano’s Theorem 1.1.2 that the initial value problem (1.6.3) has a solution u(t, c) on the interval [ t o ,t, 71, 7 = min(a, 6/(2M -1 6)). Let 0 -1 e2 < E . Then, we have

+

<

"Ate , € 2 ) < up(to 4,

uq(tn

7

ux4

< g d t , u(t,

€2)

4(f,€2)

‘I

U X f ? €1) > ‘

-t-

€2))

, €2) > .,(to

, €1);

;

€2

& ( t , u(t, 4) - €2 ;

s,(t,

ZGf,€1) < g,(C

44 €1)) u(t, 4)

+

~

;

€2 €2.

An application of Theorem 1.5.1 yields

+

%At,

€2)

< %(t, 4

udt, €2)

> ua(t,€1).

for t E [t, , t , 771. Since the family of functions u ( t , E ) is equicontinuous and uniformly bounded, one can establish that lim u(t, E , )

=

en 4

+

~(t)

71 and that r ( t ) is a solution of (1.5.1) on uniformly on [to , t, [fn > 4, 171. T o show that r ( t ) is a k max ( n - k) mini-solution of (1.5.1) on [t,,, t, 171, we have to prove that (1.6.1) is satisfied. Let u(t) be any solution of (1.5.1) existing on [ t o ,t, -k 171. Then,

+

+

< U J t n , e), uo(tn) > ug(tn uC(t) < R B ( t , 4 t ) ) + E ;

Z4tn)

gq(t,u ( t ) )

uXt)

~

u x t , €1 2 g,(t, u ( t , 6 ) )

4 ( t , .) < g,(4

for

E

< b/2. By Theorem

u(t,

e;

+

E;

-E

1.5.1, it follows that

4 4 < %(t, E ) , % ( t ) > uo(t, €1

for t E [to, t,

+ 71. Consequently, <

u D ( t ) lim u,,(t, 6 ) = r D ( t ) , <-n

for t

E

[to

+ 771.

u,(t)

> lirn u,(t, c) <-to

=

r,(t)

9

c);

1.7.

FURTHER COMPARISON THEOREMS

27

T h e existence of k mini ( n k) max-solution can be established by changing the signs of E in (1.6.3) and proceeding on similar lines. This proves the theorem. ~

This existence theorem for minimax solutions, together with the extension Theorem 1.1.3, implies the following:

THEOREM 1.6.2. Let hypotheses (i) and (iii) of Theorem 1.5.1 hold. Then, if ( t o ,uo)E E, (1.5.1) has minimax solutions that can be extended

to the boundary of E.

COROLLARY 1.6.1. Let the hypothesis of Theorem 1.6.2 hold. Let [to, t,, u ) be the largest interval of existence of k max ( n - k) minisolution r ( t ) of (1.5.1). Suppose [ t o ,t l ] is a compact subinterval of [ t o ,to u). T h e n there exists an co > 0 such that, for 0 < E < to , the k max ( n - k) mini-solution r ( t , E ) of the system ( I -6.3) exists on [to, t l ] , and lim r ( t , C) = ~ ( t )

+ +

e-,O

uniformly on [to, t l ] .

1.7. Further comparison theorems We shall be concerned, in this section, with comparison theorems for finite systems of differential inequalities. These are, naturally, extensions of some of the results in Sect. 1.4. As will be seen, minimax solutions play an essential role.

THEOREM 1.7.1. Assume that the hypotheses

(i) and (iii) of Theorem 1.5.1 hold. Suppose that [ t o ,to a ) is the largest interval in which the k max ( n - k) mini-solution r ( t ) of (1.5.1) exists. Let m E C"to t o a ) , R"],( t , 4 9 ) E E, t E [ t o , to -1- a),

+

?

+

m,(t,) G

*o,n

>

mc(to) 2 *o,c

?

(1.7.1)

and, for a fixed Dini derivative, the inequalities (1.7.2) hold for t E [to, to

+ u)

-

S. Then,

28

CHAPTER

1

Proof. By Lemma 1.2.2, it follows that (1.7.2) is equivalent to the inequalities (1.7.4)

+

+

for t E ( t o ,t, u). Lct T E ( t o ,to a). Then, the existence of k max - k ) mini-solution r ( t , c) of (1.6.3) on [ t o ,71,for all E > 0 sufficiently small, satisfying r ( t ) = lim r ( t , 6) (I -7.5) <-0 (n

uniformly on [to, TI, is a consequence of Corollary 1.6.1. By Thcorern 1.5.1 and the relations (1.6.3) and (1.7.4), we deduce that (1.7.6)

for t E [ t o ,TI. T h e last inequalities, in view of (1.7.5), prove the conclusion (1.7.3).

REMARK1.7.1. If, in Theorem 1.7.1, the inequalities (1.7.1) and ( I .7.2) are reversed, the assertion (1.7.3) becomes % ( t ) ‘2 P,,(t)>

mdt)

where p ( t ) is the k mini ( n requires obvious changes.

-

< PQ(t),

t E It” > to

+ 4,

k ) max-solution of (1.5.1). T h e proof

T h e following corollary of Theorem 1.7.1 is important in later applications,

COROLLARY 1.7.1. Let condition (i) of Theorem 1.5.1 be satisfied. Suppose that g is quasi-monotone nondecreasing in u. Let [ t o ,to a ) be the largest interval of existence of the maximal solution r ( t ) of (1.5.1). Let m E C [ [ t o to a), and, for a , a),R”],( t , m ( t ) )E E, t E [ t o ,to fixed Dini derivative, the inequality

+

-+

+

Dm(t)

< g(t, m(t))

(1.7.7)

holds for t E [to, to -{- a ) - 5’. Then, 4to)

implies m(t)


G uo t

E

[to , t o

(1.7.8)

+ a).

(1.7.9)

1.7.

29

FURTHER COMPARISON THEOREMS

REMARK1.7.2. If, in Corollary 1.7.1, the inequalities ( I .7.7) and ( I .7.8) are reversed, then the conclusion (1.7.9) is to be replaced by m(t) 3 p(t),

t E [to , f ,

-t a),

where p ( t ) is the minimum solution of (1.5.1). This follows from Remark 1.7.1. T h e next theorem is analogous to Theorem 1.4.2 in this general framework.

THEOREM 1.7.2. Let the hypothesis of Theorem 1.7.1 hold, except that the inequalities (1.7.2) are replaced by ( I .7.10)

where

z, = [t E [ t o , to + a ) : d,(t) < 01, d,(t)

d,(t) = r d t ) - m&),

=

m,(t)

-

r*(t),

and Si is an at-most countable subset of Zi , for each i. Then (1.7.3) remains valid. Proof. T h e proof requires minor changes up to (1.7.5) of the proof of Theorem 1.7. I . Now, proceeding to prove (1.7.6) as in Theorem 15 1 , we arrive at a t , and a j such that 1 j n and

< <

4 t l ) =YAtl

, €1.

(1.7.1 1)

Moreover, it is easy to obtain from Theorem 1.5.1 that YAt)

<

> r,(t, €1

y,(t)

€)>

+

(1.7.12)

for t E [ t o ,71, 7 E ( t o, to a), where r ( t , e ) is the k max ( n - k) minisolution of (1.6.3) which exists on [ t o ,.], by Corollary 1.6.1. I t then follows from (1.7.11) and (1.7.12) that r j ( t l )< m j ( t l )

or

r3(tl) > mj(tl)

if

if

K

1

<j < k

+ 1 < j < n.

This implies, from the definitions of Zi and di(t), that t , E Zj . Hence, the j t h inequality in (1.7.10) is satisfied for such a t , . T h e rest of the proof is identical with the proof of Theorem 1.5.1 in order to arrive at (1.7.6), and this is sufficient to draw the conclusion (1.7.3).

30

CHAPTER

1

THEOREM 1.7.3. Let the hypothesis of Theorem 1.5.1 hold, except that the inequalities D-.*(t) D-wD(t)

+

3 g,(t,

( I .5.4*)

.(t)),

> g,(t, w ( t ) )

(1.5.5*)

are satisfied for t E (to, to a ) , instead of (1.5.4) and (1.5.5). Assume further, for each T E ( t , , t, a), t E [ t o ,71, and for each i, that g satisfies the condition gi(t,U)

-

+

g,(t, U) 3 -G(T

+ to

ui 3 z& , u j = zZi (i # j ) , where G E C[[to, to is the maximal solution of U'

=

G(t,u),

-

t , ui - Ci),

(1.7.13)

+ a ) x R, R], and r(t) = 0

~(t,= ) 0.

Then (1.5.7) holds. Proof. Following the proof of Theorem 1.5.1, we arrive at a t, E ( t o, to u ) and a j ( 1 j TZ) satisfying

+

< <

m,(t1)

=

( 1.7.1 5 ) < t < t, . C ( t ) = v(t, + to t ) , G ( t ) = ~ ( t+, t, t),

rnj(t) 2 0,

<

(I .7.14)

0, to

Let 1 < j k. Define g(t, u) = -g(t, to t, u), and d ( t ) = Gi(t) Then, using (1.5.3) and (1.5.5*), we obtain

+

~

-

~

Ci(t) for t E [to , t J .

This inequality implies, along with (1.7.16) and the assumption (1.7.13), the scalar differential inequality D-d(t)

< G(t,d ( t ) ) ,

t E [to , tll.

Since d(t,) = 0 because of (1.7.14), arguing as in Theorem 1.4.3, one deduces that vj(t)

1~

j ( t ) ,

t E [to tll, 7

1.8.

31

INFINITE SYSTEMS OF DIFFERENTIAL INEQUALITIES

which contradicts the assumption (1.5.2) and the definition of t, E (to , to a). A repetition of the argument to the case Jz 1<j n yields a similar contradiction. This proves that the set 2 is empty, and the desired result (1.5.7) follows.

+

+

<

+

COROLLARY 1.7.2. Let E be the product space [ t o ,to a) x R2n and g E C[E, R"].Assume that g is quasi-monotone nondecreasing in u for each (t, v) and monotone nondecreasing in 21 for each ( t , u). Suppose that r ( t ) is the maximal solution of the differential system u' = g(t, u, u),

u(to)= uo 3 0

+

existing on [to , to a ) , and that r ( t ) 3 0, t maximal solution r l ( t ) of

[to, to

+ u). Then,

the

u(to) = uo 3 0

u' = gl(t, u),

exists on [ t o ,to

E

+ a ) , where

gdt, u ) = g(t, u, r ( t ) )

and r ( t ) = rl(t), t E [to, to

+ a).

COROLLARY 1.7.3. Let the assumptions of Corollary 1.7.1 hold; m E C [ [ t o to , a), R"] such that ( t , m(t), u ) E E, t E [tu , to a), and m(to) uo . Assume that for a fixed Dini derivative the inequality

<

+

+

Dm(t)

+

< g(t, m ( t ) ,v)

is satisfied for t E [to , to a ) -- S. Then, for all 21 we have m(t) r ( t ) , t E [ t o ,to a).

<

+

< r ( t ) , t E [to, t, + a),

1.8. Infinite systems of differential inequalities

A classical result of Perron is that the maximal solution of a scalrr differential equation can be obtained as the least upper bound of the family of functions m(t) that satisfy the inequality

with the same initial condition m(to) = uo . Similar arguments hold for infinite systems of differential inequalities, provided that the maximal solution of a single equation is known. We shall first formulate an abstract

32

CHAPTER

1

version of this method and then apply it to show the existence of minimax solutions for an infinite system of differential equations and also obtain inequalities. Let El ,Fl be two partially ordered sets with the partial ordering We use the same symbol of order relation, namely, for both the sets. Assume that the following conditions hold:

<.

<,

x,y,Z

E

El,

x

x,yeEl,

y

x
f,y, 2 eF1, f


f

<j ,

€F1 implies

f


y < x

9

< f.

< .%

< z;

(1.8.1)

x=y;

(1.8.2)

< 2;

(1.8.3)

imply x imply

imply f

(1.8.4)

Corresponding to the sets E , and Fl , let us consider two partially ordered sets I&, F2 , with the dual order relation, denoted by the symbol 3, satisfying conditions (1.8.1 *)-(1.8.4*) analogous to (1.8.1)-(1.8.4). We shall use u , v, w and U,5,W for elements belonging to E, and F, , respectively. Let the operators P I , P, be defined on E l , E , , taking values in 17, , F2 , respectively. Furthermore, let the functions Q1 ,Q2 be defined on El x El x E, , El x E, x E , , taking values in Fl ,F 2 , respectively. Consider the simultaneous equations (1.8.5)

By a solution of (1.8.5), we shall mean an ordered pair (x,u), x E El , E, such that x, u satisfy Eqs. (1.8.5) simultaneously. X solution Y = ( E , 71) of (1.8.5) is said to be a minimax solution, if for every solution (x, 21) of (1.8.5) the relations

ZL E

x
U > T

are satisfied. T h e functions 0,, Q, are said to possess a mixed quasi-monotone property whenever the following conditions hold: y1 ,y 2 E El , y1 y , imply that

<

and

Q1(x, y1 , 21) QZ(Y1

u1 , uz t E ,

7

-sQ1(x,y z , u ) ,

x € El ,

*>a) ;Q2(Y,, u, a),

u E B, ;

(1.8.6)

u, 21 E E, ;

(1 23.7)

x,Y E E, ;

(1.8.8)

, u1 3 u, imply that L)l(X?

Y , u1)

<

Y , u2),

Ql@,

1.8.

INFINITE SYSTEhlS OF DIFFERENTIAL INEQUALITIES

33

We now define the sets Ul

= [X E

El : P1(x)< Q1(x,X, u),u E E,];

( 1.8.10 )

(1 3 . 1 1) E, : P,(u) 2 Q,(x, U , u),x E El]. T h e following theorem is concerned with the existence of the minimax solution of (1.8.5).

U,

= [U E

THEOREM 1.8.1. Let P, , P, ,Q, ,Q, be as defined previously. Suppose that Q1 , Q, have the mixed quasi-monotone property. Assume further that there exist two functions z,,x, defined on El, E, such that zl(E,) C E l , z,(E,) C E, , satisfying the follwing conditions: ( I .8.12)

and (1.8.1 3)

<

imply that x zl(y) , u >, z2(v).Let the sets U , , U , defined in (1.8.10) and (1.8.11) be nonempty. Then,

Zl(E1)c U, ,

Zz(E2)

c u, -

(1.8.14)

Moreover, the existence of (sup U , , inf U,) implies the existence of (sup zl(U,), inf z2(U,)), and vice versa. Also, sup U, = sup zl(U,), inf U , = inf z,( U,), and r = (sup U , , inf U,) is the minimax solution of ( I .8.5).

Proof.

Let x E E, , u E E,

. Then,

from (1.8.12), ( 1.8.15)

34

CHAPTER

1

which, in view of the definitions of the sets U , , U , , imply that Z1(X) t

u,,

Zz(U) E

u, .

This proves the assertion (1.8.14). We shall show that z1 , are increasing functions. For, let y1 y 2 , zj >, 21, , where y1 ,y, E El and 71, , v2 E E, . Using again (1.8.6) and (1.8.9) in (1.8.12), we get

<

Pl(Zl(Y1))

G Ql(Zl(Y1)I Yz >

PZ(%(.l))

2 Q,(Yl

7

0 2

9

4

7

Z2(.1)),

and this, because of (1.8.13), proves that

d Z1(3’2)>

ZI(Y1)

Suppose now that that

2 .2(.21.

Z Z ( 4

8 = sup U , , 7 = inf U , ; we Zl(S)

< 6,

have, by (1.8.14), (1.8.17)

, 4 1 7 2)17.

<

On the other hand, since x 6,u 3 7 for any x E U , , u E U , , the mixed quasi-monotone property of 0, , Q 2 , together with the definitions of U , , U, , yields Pl(.) < V l ( X , S, T ) , P.,(,) 2 I ) z ( t , 7, U).

I t then follows from (1.8.13) that x f

Zl(S),

u >, 47).

Consequently, f = z,(<), 7 = ~ ~ ( from 7 ) (1.8.17). I t is evident from (1.8.12) that Y = ( f , 7) is a solution of (1 A.5). Let y y * , 71 3 u* for y E zl( U,), v E z2(U J . Then

<

Zdt) f

y*,

Zz(T)

3 .*-

T h e monotonic nature of z1 , z2 yields

zl(x) < ~ ~ ( 6 )for all y E z1(U1),

y

=--

v

= z&)

I t therefore implies that

> z2(r])

8

for all

sup xl( U,), 7

:

uE

=

z2(U2).

inf x,( U,). T h e fact that

.=(5 , 7) is- the minimax solution of (1.8.5) follows from the definitions of U , , r J z .

1.8.

35

INFINITE SYSTEMS O F DIFFERENTIAL INEQUALITIES

If ( = sup zl(U,), 7 = inf z,( U,), using similar arguments, one can prove that ( = sup U , , 7 = inf U , , respectively, and that Y = 7) is the minimax solution of (1.8.5). This completes the proof of the theorem.

(e,

Consider the system u; = f ; ( t , u1 , u2 ,... u, ; v1 , v, )...)v,), )

Ui(t") =

uio,

1
v;

= &(t, u1 , u2 ,... u, )

; v1 , v,

)...,us),

(1.8.18)

V j ( t 0 ) == Zljo,

1

<.I'< q,

whcre p , q are arbitrary (may be infinite). T h e existence of minimax solutions for finite systems of differential equations follows when p , q are both finite. T h e minimax solution for infinite systems is covered by the other choices of p and q (either p or q or both may be infinite). I n case p , q are both infinite, the functions f i and g, in (1.8.18) are to be interpreted as f i (t, u1 , u, ,...; v 1 , v, ,...) and g j ( t , u1 , u2 ,...; vl , v 2 ,...), respectively. Let the functions fi and gj satisfy the following assumptions: and (i) f i ( t , u1 , u2 ,..., ur, ; v1 , v2 ,..., a), gj(t, u1 , u2 ,..., up ; vl , v2 ,..., v,) are defined for [to, to a] and arbitrary u1 , u, ,..., u, and v 1 , v 2 ,..., vq , (ii) There exists constants M iand Nisuch that

+

Ifdt, u1

I g j ( t , ui

9

u2

up

>.--3

; v1 ,

uz )-..,up ; ~1

fort~[t,,t,+a],l
9

,**-,

~2

j...,

< Mi uQ)I< Nj %)I

9

<j
(iii) T h e functions fi and gj are continuous in the sense that, if t'Y-+t,uia+ui,uje+vj(l < i < p , 1 < j < q ) , t h e n fi(ta, &(t", u l a ,

%Iy,...,

u2a

-

upa;vile, v,", ...,%*) -+fi(t,

,...,up";vl", COZ"

,... a*") )

u1

,ug ,..., u, ; 81 ,w 2 ,...,v,),

gj(t, u1 , u2

,..., up; v1 , v2 )...)v,).

(iv) T h e functions f i and g . ossess mixed quasi-monotone property, p i.e., for each i = 1, 2 ,...,p a n d j = 1, 2,..., q, (al) f i ( t , u1 , u, ,..., up ; v 1 , v, ,..., v,) is monotonic nondecreasing in u l , uz ,..., u i p 1 ,ui+l,..., up and monotonic nonincreasing in 'u1 , 'u2 ,...,vuq ; J.

36

CHAPTER

1

,..., ull ; vl , v, ,..., v,) is monotonic nonincreasing ZL, , , til ,..., ti1, and monoconic nondecreasing in vl , v, ,...,v. 1-1 , V ] 11 ,..., vq . (al) g,(t,

in

11,

Xov we have the following:

TIIEOREM 1.8.2. Assume that the functions f i and gj satisfy conditions (i)-(iv). Then there exists a minimax solution ( u t ( t ) ,vT(t))of (1.8.18) on [t,,, f , -1- a ] . Furthermore, if mi(t), nj(t)are continuous functions defined on [t,,, t,, a ] ,satisfying the inequalities

+

nz,(to)

< uin,

nj(to)

>

7170,

ml(t), ~ a z ( t ) , - *m*D,( t ) ; nl(t), nz(t),.-*> nq(t)),

D ; m j ( t )s < f ; ( f ,

D+/L,(t) 1.3g,(t, q ( t ) ,m d t ) , . . . , m&); fil(t), nz(t),..., @)),

and nlL(t)

Pmg. [t,, , I , ,

< &t),

t E [to, to

n,(t) 5 v3*(t),

+ a].

Let (+i(t)),{ $ j ( t ) }be two sequences of continuous functions on

+ a] such that, for t E [ t o , to + a ] ,

Denote

+,(t)

<

$j(t)

::vjo :

ZLj"

+ Mi(t l- N,(t

-

to),

i

=

1, 2,...)p ,

-

to),

j

=

1, 2 ,..., q.

i+i(t)S = (+l(t), + d t ) Y > 4 f l ( t )= ) YY {$j(f))

($l(t),

1

%z(t),..*,

% q ( t ) ) =--V *

I,et E l , E, stand for systems of sequences y = {qh,(t)},ZI = {$j(t)}.If $#p(t)], y, = {$:."(t)} are two sequences such that, if y 1 ,y z E E, , then yl y 2 implies +;"(t) $ y ) ( t ) on [to, to a] and for each i. Similarly for z', , u2 E E, , v 1 3 v, means that $ j l ) ( t )3 $iz)(t) on [to , f , -1- a ] and for each j . Let F , , P, stand for the systems of sequences of continuous functions on [t,,, t , + a] which take values in the real extended line, the order relations in F , , F2 being the same as those in El , E, , respectively. It is easy to verify that conditions (1.8.1)-( I .8.4) and (1.8.1 *)-( 1.8.4") are satisfied. Let us now define the operators P I ,P, and the functions Q1 , 0,as follows:

3'1

~

<

<

+

{wL(q;, P2,(71) = P ' % j ( t ) > > i 01(*~, Y , a) = . f L ( t , bl(t),.*.i (bi-l(t), ~ i ( t )4i+l(t)>..*,+g(t); PdY)

=

O~(JJ> 71, u) = gj(t,+l(t),

+D(t);

$l(t),$At),***, 4q(t)),

%l(t>,.*., %f-l(t),'At), 4j+l(t),***,4Qb(t)).

1.9.

INTEGRAL INEQUALITIES REDUCIBLE T O DIFFERENTIAL INEQUALITIES

37

Clearly the functions PI , Q2 satisfy the mixed quasi-monotone property. For any pair of sequences {$i(t)}, {t,bj(t)},define f i ( t , Y)

=

.fdt,95i(t),.--~$ ’ i - ~ ( ~ yi) ,

$l(t),

$~+i(~),..-,

Zi(t,P ) = gj(t, A(t),...,+,tt); $l(t)>..*,$j-l(t)>

~j

1

&dt),..*,d ’ ~ ( ~ ) ) ;

d’i+l(t),...,d’n(t)>.

Let, for each i, ri(t) be the maximal solution of Y’ =

h(t,Y),

Yi(t0) == uio,

and, for eachj, pj(t) be the minimal solution of P’ = Ej(G P ) ,

Pj(t0)

=

vjo.

Since the functions f i and gj satisfy (ii), the existence of r i ( t ) and pj(t) on [to , to u ] is ensured. Let us now define the functions z, , z2 by

+

%({$At)))= {fj(t)>.

~ , ( ~ g l i ( l ) )= ) {Yi(t)}>

From Theorem 1.4.1, it follows that the functions z, , z2 satisfy (1.8.13). Moreover, the sets U , , U , are nonempty, since uio - Mi(t - to)E U , and zjjO + Ni(t - t o )E U , , because of (ii). Furthermore,

I Yk(t)l

< Mi

9

I P;(t)l

< Nj .

Therefore, the family of functions {ri(t)},{ p j ( t ) } are equicontinuous and uniformly bounded. This proves that sup z1( Ul)=: {sup Y i ( t ) S , inf z,( U,)

+

= {inf p j ( t ) }

are continuous on [t,, t, u ] . T h e assertion of Theorem 1.8.2 now follows from Theorem 1.8.1.

1.9, Integral inequalities reducible to differential inequalities We shall consider, in this section, only those integral inequalities that are reducible to differential inequalities. We begin with one of the simplest and most useful integral inequalities.

THEOREM 1.9.1.

+

Let m, v E C[[t,, t, a), I?+], where R , denotes the nonnegative real line. Suppose further that, for some nonnegative constant C , we have (1.9.1)

38

CHAPTER

1

Then m(t)

< C exp j t

v(s) ds,

t

[to, to

E

t,l

Proof.

If C

+ u).

(1-9.2)

> 0, it follows from (1.9.1) that

which, by integration, yields C

+ J:vv(s)m(s) dsI

~

log C

<

fo

u(s) ds.

This incquality, together with ( I .9. l ) , gives (1.9.2). If C 0, then (1.9.1) holds for every constant C, > 0, and therefore the previous argument gives (1.9.2) with C = C, . Letting C, + 0 implies m ( t ) 7 0. This proves the theorem.

COROLLARY 1.9.1. 1,et m, u E C[[t,, to and satisfy the inequality m(t)

< n(t) -1-

It

+ a ) , R,], n E C [ [ t ,, to + a), R],

v(s)m(s)ds,

to

t

E

[ t o , to

+ u).

Then we have

If, in addition, the derivative n'(t) exists for t

E

[to, to + a ) , then

A generalization of Theorem 1.9.1 is the following analog of Theorem 1.4.1 which, however, requires the monotony of g with respect to u.

THEOREM I .9.2. Let E be an open ( t , u)-set in R2 and g E C[E,R]. Suppose that g(t, u ) is monotonic nondecreasing in u for each t. Let m E C"t0 to a), RI, (4 4 4 ) E E, t f [to , t o a), m(t,) %J , and 9

+

+

<

1.9.

INTEGRAL INEQUALITIES REDUCIBLE TO DIFFERENTIAL INEQUALITIES

Then m(t)

< r(t),

t

6 [to

+ 4,

, t"

39

(1.9.4)

where r ( t ) is the maximal solution of (1.2.1)existing on J.

Proof.

Define

so that (1.9.5)

and Since g is monotonic in u, using (1.9.5), we obtain the differential inequality v'(t)

< g ( t , v(t)),

t

E [to

, to

+ a).

From an application of Theorem 1.4.1, we deduce that v(t)

< r(t),

t E [to ,to

+ a).

T h e assertion (1.9.4) is now immediate because of (1.9.5).

REMARK1.9.1. Notice that one could prove Theorem 1.9.1 using an argument similar to that of Theorem 1.9.2, although we have given the classical proof.

+

COROLLARY 1.9.2. Let m, v E C[[t,, to a ) , R,], g E C[R+, R,], g(u) monotone in u and g(0) = 0. Assume that m(t>

< m, + 1't o v(s)g(m(s)>ds,

t

E

[ t o ,to

+ a).

Then m(t)

< w-l[zo(m,>

+ Ft

to

74s) dsl,

to

< t < t, ,

where

N1.

=

jud7/g(7), UO

uo

> 0;

+

w-'(u) is the inverse function of w(u), and (to, t l ) C [to, to a ) such 1 that w(mo)+ Jt,v(s) ds is in the domain of definition of w-'(u).

40

CIIAPTEK

1

COKOLL~ARY 1.9.3. Let m , p E C [ [ t , , to + a), R] and m(t)

< m ( b ) + J tt o [ K m ( s )+ p(s)l ds,

K > 0.

Then m(t) ' ::m(to)exp[K(t- to)]

+ j'p ( s ) e x p [ K ( t

-

s)] ds,

to

t tz [to , t o

+ a).

COKOI,I.ARY 1.9.4. Let the assumptions of Theorem 1.9.2 be satisfied except that the integral inequality (1.9.3) be replaced by

where

TZ E

C[[t,,, t,

+ a ) , R]. Then (1.9.4) takes the form

4 4 ,< 4 t ) + Y ( t ) ,

t

E:

[to , to

+ a),

where r ( t ) is the maximal solution of u'

= g(t, n(t)

+ u),

u(2,) = 0

existing on [to , t , 4-a ) , It is casy to extend Theorem 1.9.2 to finite systems of integral inequalities. Actually, we prove such a result in a more general form.

THEOREM 1.9.3. Let assumption (i) of Theorem 1.5.1 hold, and suppose that g(t, u ) has the mixed monotone property in u. Let m E C[[k,,, to a ) , R"],( t ,m ( t) )E E , t E [to, to a ) , and the inequalities

+

+

m,(to) G

% (t)<: ;

hold for t E [to, to

for t t [ t o , to (1.5.1).

U0.D

%(to)

+ a). Then % ( t ) e .,(f>,

+ a), where r ( t ) is

%(to)

+ Jt g&,

> U0.Q

3

4 s ) ) ds

to

m,(t) b .dt>

(1.9.6)

the K max (n - k) mini-solution of

1 .lo.

Proof.

41

DIFFERENTIAL INEQUALITIES IN THE SENSE OF CARATHEODORY

Let the vector function v(t) be equal to

so that mP(t)

and

< %(t), v'(4

%(t)

24

(1.9.7)

2 )

= g(t, v ( t ) ) .

T h e mixed monotonic character of g in u shows, in view of the inequalities (1.9.7), that % (t)

< g,(t, 4 t ) ) l

VXt)

>, gdt, n ( t ) ) ,

t E [ t o t" 9

+ 4.

Theorem 1.7.1 is now applicable, and we get %(t)

<

YP(t),

% ( t ) 3 r*(t),

t

E

[to , t o

+

a).

T h e inequalities (1.9.6) result from (1.9.7), and the theorem is proved.

I .lo. Differential inequalities in the sense of C a r a t h e o d o r y Let the function g(t, u ) be defined on an open ( t , u)-set E C R2,taking values in R. g(t, u ) is said to satisfy the Caratheodory condition if (i) g(t, u) is continuous in u for each fixed t and Lebesgue measurable in t for each fixed u ; and (ii) M ( t ) is a summable function on [to , to + u] and I g(t, ).I < M ( t ) , (4 u) E E. By a solution u(t) of the differential equation with an initial condition u'

= g(t, u),

.(to) = uo ,

(1.10.1)

we mean an absolutely continuous function u(t) satisfying (1 .lo. 1) almost everywhere on [ t o ,to u]. By the classical theorem of Caratheodory, there exists a solution of (1.10.1)under the foregoing conditions. Moreover, existence of maximal and minimal solutions and the problem of extension of solutions can be shown in just a similar way as before. T h e following theorem on differential inequalities of Caratheodory type is of interest.

+

42

CHAPTER

1

THEOREM 1.10.1 Let (i) the function g(t, u ) be defined on an open ( t , u)-set E C Rz, taking values in R and satisfying the Caratheodory's condition; and (ii) r ( t ) bc the maximal solution of (1.10.1) existing on [ t o , t,) -1- u ] . Assume that rn E C[[t,, t , -1- a],R] and is of bounded variation on [ t , , t, u ] such that its singular part is a nonincreasing function. Suppose further that

+

d ( t ): . ;

almost everywhere on [ t o ,to

+ a ] . Then 4t")

implies m(t)

Proof.

(1.10.2)

f ( t , m(t))

< r(t),

<

(1.10.3)

Ul)

t € [t" , to

+ u].

(1.10.4)

Define the function f ( 4 ).

=

<

if m ( t ) u, if u 6 m ( t ) ,

dt,U)

lg(t, m ( t ) )

which satisfies the Caratheodory's condition. 1,et r l ( t )denote the maximal solution of u(t,) =: U" . ( I . 10.5) u' : f ( t , u ) , We claim that ~

m(t)

<:

t

r,(t),

€

[ t o , to

+ u].

( I . 10.6)

If this were not true, let, without any loss of generality, (tl , t z ) be an open interval such that

(1.10.7)

4 t l ) = Yl(t1)

and " ( t , -1- 12) >; rl(t,

11 > 0 sufficiently small. For t ( 1.10.5) the inequality m'(t)

--

rXt)

E

+- h),

( I . 10.8)

( t l , t z ) , we obtain from (1.10.2) and

< s(t,4 4 ) - f ( t ,

Vl(t)).

( 1 .10.9)

T h e definition of f ( t , u), together with (1.10.7), (1. IO.8), and (1.10.9), gives "'(t) - r ; ( q 0, which in its turn implies

<

( 1.10.10)

1.10.

DIFFERENTIAL INEQUALITIES I N T H E SENSE OF CARATHEODORY

43

Since m(t) is of bounded variation, we have m(tl

+ h)

m(h)

=

+

tl+h

m’(s) ds

tl

+ B(tl + h),

(1.10.11)

where P ( t ) is the singular part of m(t). The relations (1.10.7), (1.10.10), and (1.10.11) lead to m(t,

-t-

h)

P(tl

~

+ h) <

Yl(t,

+ h).

As P(tl) = 0 and P ( t ) is nonincreasing by assumption, -P(tl and consequently

+ h) <

m(t,

Yl(t1

+ h) 3 0,

+ h),

which contradicts (1.10.8). This proves (1.10.6). T h e definition of f now shows, because of (1.10.6), that r l ( t ) is a solution of (1.10. I), and therefore Yl(t>

e r(t>,

t E [ t o , to

+ .I

This completes the proof.

THEOREM 1.10.2. Let assumptions (i) and (ii) of Theorem 1.10.1 be satisfied. Assume that m ( t ) is absolutely continuous for t E [to, to + a] and satisfies (1.10.2) almost everywhere on [t,, , to a]. Then (1.10.3) implies (1.10.4). If m(t) is absolutely continuous on [ t o ,to a ] , the singular part P ( t ) of m ( t ) is identically zero. This remark shows that Theorem 1.10.2 is a consequence of Theorem 1.10.1.

+

+

THEOREM 1.10.3. Let assumption (i) of Theorem 1.10.1 hold. Assume that g ( t , u ) 2 0; m E C [ [ t o, to a ] , R] and satisfies, for small h > 0,

+

Then (1.10.3) implies (1.10.4). Proof. It follows from the inequality (1.10.12) that m(t) is absolutely continuous over any interval in [ t o ,to a]. Consequently, m’(t) exists almost everywhere on [ t o ,to a]. Moreover, (1.10.12) implies that the derivative m’(t) satisfies the relation

+

I m’(t)l almost everywhere on [ t o ,to from Theorem 1.10.2.

+

< g(t, m(tN

+ a]. T h e assertion (1.10.4) now follows

44

CHAPTER

1

1.11, Notes

Theorems 1.1.2 and 1. I .3 are taken from Hartman [5]. Theorem 1.1.2 is due to Peano [2], and the proof uses a device of Tonelli [l]. For the type o f results on differential inequalities in Sect. 1.2, see Babkin [l], Cafcrio [ I , 21 and Chaplygin [ 11. AIaxim‘il and minimal solutions are considered by Peano [I]. Also see I, 0. Theorem 1.4.5 is new. ‘I’he results of Sects. 1.5, 1.6, and 1.7 have been adopted from the work o f Rurton and Whyburn [l], who introduced the notion of minimax solutions. See also Kamke [2] and Wazewski [3, 4, 61 for results on e\tremal solutions for systems and corresponding theorems on ditferential inequalities. Section 1.8 contains the work of Lakshmikantham and Leela [4]. For allied results, see Mlak and Olech [l] and Mlak [2]. I o r ‘I‘heorem 1.9.1, see Bellman [3], Giuliano [I], Gronwall [l], and Peano [I]. Corollary 1.9.2 is due to Bihari [I]. Also, see Lakshmikantham [9] and Langcnhop [l]. ‘I’heorem 1.9.2 is due to Viswanatham [2, 41. See also Haiada [ l ] and Cafiero [3]. For a generalization of this result to systems, see 1,akshmikantham [2] and Opial [l]. Theorem 1.9.3 is new. Theorem 1.10.1 is taken from Olech and Opial [l]. ‘l’he result of Theorem 1.10.3 is due to Lakshmikantham [l]. See also Olech and Opial [I]. For Caratheodory’s existence theorem, see McShane [ 11. N e m proofs of ‘l’heorems 1.1.2, 1.3.1, and 1.4.1 are given by Corduncanu [ 171. For differential inequalities with the limiting initial conditions, see hIamedov [I]. For collateral reading on differential and integral inequalities, see Szarski [l] and Walter [3].

Chapter 2

2.0. Introduction T h e most important techniques in the theory of differential equations involve a systematic use of the theory of differential inequalities, or what may be termed as the integration of differential inequalities. We present, in this chapter, a number of varied results depending essentially on this approach. Many of our theorems and their proofs will involve estimates with respect to some convenient norm. T h e choice of a norm as a medium for our arguments is a natural one, although it is seldom the best choice. A better candidate is, of course, the so-called Lyapunov function, which is more flexible. An approach based on Lyapunov functions will be postponed to later chapters.

2.1. Global existence We shall use the Tychonoff's fixed point theorem for locally convex linear spaces to prove the global existence of solutions of the differential system x' = f ( t , x),

x(t,) = xo ,

t,

> 0,

(2.1 .I)

where f~ C [ J x R", R"], J being the half-line [0, 03). Let us state the fixed point theorem of Tychonoff in the following form.

THEOREM 2.1.1. Let B be a complete, locally convex, linear space and B, a closed convex subset of~B. Let the mapping T : B + B be continuous and T(B,) C B, . If T(B,) is compact, then T has a fixed point in B,. T h e main result of this section runs as follows. 45

46

2

CHAPTER

THEOREM 2.1.2.

L c t f E C[J x R”, R”], and, for ( t , x)

llf(t! .x)ll

E

< g(t7 I1 x It),

J x Rn, (2.1.2)

where g E C [ J x R , , R , ] and the function g(t, u ) is monotonic nondecreasing in u for each t E J . Assume that, for every u, > 0, the scalar differential equation u’ = R ( t , u ) ,

u(t,)

ug

,

t,

5 ,

0

(2.1.3)

has a solution u ( t ) = u(t, t o , u,,) existing for t 3 t, . Then, for every x,,E R7’such that 11 xo I/ u, , there exists a solution x ( t ) = x ( t , t, , x,) of (2.1.1) for t 2 t, , satisfying

<

/I x(t)lI

<~ ( t ) ,

t

3 to.

Pyoof. T o apply Theorem 2.1.1, lct us consider the real vector space B of all continuous functions from [ t o ,m) into Rn, the topology on B being that induced by the family of pseudo-norms (p,(x)}:=,, , where for x E B,

II 4t)ll.

pn(.x) = t,”yP,

A fundamental system of neighborhoods is then given by { IjrL}:=i=l ,

where

V,

[x E B : pn(x)

~:

< 11.

Under this topology, B becomes a complete, locally convex, linear space. Let us now define a subset B,, of B as follows: B,

=

[x t B : /I x(t)/l

< u(t),t 3 f,],

(2.1.4)

where u(t) is a solution of (2.1.3) existing for t 3 t, . I t is clear that, in the topology of B, the set B, is closed, convex, and bounded. Consider the integral operator defined by T ( x ) ( t ) = Y,,

+

1:”

f ( ~x(s)) , ds,

(2.1.5)

whose fixed point corresponds to a solution of the system (2.1.1). Evidently, the operator T is compact in the topology of B , and therefore T(B,) is compact in view of the boundedness of B, . T o prove the theorem, it remains to be shown that T(B,) C B, . T o this end, we observe that for any x E B, , (2.1.6)

2.1.

47

GLOBAL EXISTENCE

because of (2.1.5) and (2.1.2). Using the monotonic character of g(t, ZL) in u , the definition of B,, , and the fact that u ( t ) is a solution of (2.1.3) such that 11 x,,I/ u, , it follows from (2.1.6) that

<

II W(t)ll G 49. This implies T(B,,)C B, and completes the proof. COROLLARY 2.1.1.

Let f e C[f x R", Rn]and

Ilf(t, 41 ,< YtMll II)

(2.1.7)

for ( t , x) E J x Rn, where h(t) 3 0 is continuous for t E J; g(u) 3 0 is continuous for u 3 0, g(0) = 0, g(u) > 0, zc > 0, and g(u) is nondecreasing in u. If

1"

(2.1.8)

du/g(u) = a3

un

for u,, > 0, then, for every x,,E Rn, there exists a solution of (2.1.1) for t 3 to . Proof. T h e result follows from Theorem 2.1.2, if we show that the differential equation (2.1.3) has a solution existing for t 3 to . Clearly, the equation u' = A(t)g(u),

U ( t " ) = ug

>0

(2.1.9)

may be solved. For, if we write

it is easily seen that the function G(u) is strictly increasing in u, and so the inverse function exists. I n view of the assumptions concerning g, the domain of the inverse function is [0, a), and therefore the solution u(t) of (2.1.9) is defined for t 3 to .

REMARK2.1.1. It is possible to obtain, from Theorem 2.1.2, some qualitative information of solutions of (2.1. l), knowing the behavior of solutions of the related scalar differential equation (2.1.3). T h e result that follows illustrates this fact.

THEOREM 2.1.3. x'

Consider the system = A(t)x

+f(t,

XI,

x(t,)

= xo ,

to >, 0,

(2. I. 10)

48

CHAPTER

2

where A ( t )is a continuous, n x n matrix for t E J and f E C [ J x R", R"]. Let U ( t )denote the principal matrix solution of the linear system x'

=

A(t)x

where g E C [ J x R, , A!,] g(t, u) is monotone nondecreasing in u for each fixed t , and the scalar differential equation u' = -0u

-1- kg(t,u ) ,

has a solution u ( t ) existing for t

u(t,) = uo

>0

3 to such that

lim u ( t ) = 0. 1-

riLi

Then, there exists a solution x ( t ) of (2.1.10) defined for t

to such that

lim x ( t ) = 0.

t

-0,

2.2, Uniqueness A simple criterion that implies the uniqueness of solutions is the following Perron's condition.

THFORFM 2.2.1. Assume that (i) the function g(t, u)is continuous and nonnegative for t,, t to a, 0 u 2b, and, for every t , , to . t , . t, a, u(t) = 0 is the only differentiable function t , , which satisfies t on to

+

<

< < + u'

for to


< t,

= g ( t , u),

< <

(2.2.1)

u(to) = 0

; (ii) f E C[R, , R"],where

R, : t"

< t < to + a,

11 x - XG 11

< b,

2.2.

49

UNIQUENESS

4,, IIf(t, 4 -.f(t,-Y)lI < g(t, /I x --Y 11).

and for (4 x), ( t ,Y ) E

(2.2.2)

Then, the differential system x' = f ( t , x),

has at most one solution on to

x(t,)

(2.2.3)

= x,

< t < t, + a.

Proof. Suppose that there are two solutions xl(t) and x 2 ( t )of the system (2.2.3) on t, t t, a. Define m ( t ) = 11 x,(t) - x2(t)11. Then,

< < +

DWt)

< I1 4 ( 4 - "G(t)ll = Ilf(t, xdt)) - f ( C < g(t, W ) ,

X7Lt))ll

using (2.2.2). Also, m(t,) = 0. For any t , such that to < t , we obtain from Theorem 1.4.1 the inequality m(t)

< r(t),

t,

< to + a,

< t < t, ,

where r ( t ) is the maximal solution of (2.2.1). T h e assumption (i) now assures that mft) = 0 on to t < t , , proving the theorem.

<

COROLLARY 2.2.1. T h e function g(t, u) = Ku, K > 0, is admissible in Theorem 2.2.1. It is an easy exercise to verify that g(t, u ) satisfies assumption (i) of Theorem 2.2.1. I n this case, the condition (2.2.2) just reduces to the well-known Lipschitz condition. Although Corollary 2.2.1 is a direct consequence of Theorem 2.2.1, we give below a proof that is instructive. Proof of Corollary 2.2.1. Let m ( t ) be the same function defined previously, and let m ( t ) = n(t)eLt, where L > K is a constant. It is enough to show that n ( t ) = 0 on t, t to a. Suppose, on the contrary, that max n(t) > 0, t o < t 4t o f a

< < +

and that the maximum occurs at t = u. We have, at On the other hand, using (2.2.2), we obtain n'(o)eLu

+ Ln(a)eLu= m'(a) < llf(% x d 4 ) - f ( % xz(4)ll < Kn(u)eLu.

u, n'(o) =

0.

50

CHAPTER

< L , that n‘(n) < 0, contradicting

This implies, because of the choice K n’(o) 0. Thus, n(t) 0 on t, t ~

2

< < to + a.

~

T h e next result is known as liamke’s uniqueness theorem, which is, evidently, more general than that of Perron and is sufficient for many practical cases, since it includes as special cases many known criteria.

THFOKEM 2.2.2.

Assume that (i) the function g(t, u)is continuous and nonnegative for t, < t t, -1- a, 0 u 2b, and, for every t, , t, 2, to a, u ( t ) .~ 0 is the only function differentiable on t, t < t , and continuous on t,, t -.t , , for which e .

-_

<

+

< <

<

L ,

u’(4 =

Af,u ( t ) ) ,

u(t,)

u&)

=

=

f,

< t < t, ,

(2.2.4) (2.2.5)

0;

(ii) the hypothesis (ii) of Theorem 2.2.1 is satisfied except that the condition (2.2.2) holds for (t,x), ( t ,y ) E R, , t # 1, . Then, the conclusion of Theorem 2.2.1 is valid. We shall first prove the following: T H F O R ~ h ~2.2.3. I J x t the function g(t, u ) verify hypothesis (i) of Theorem 2.2.2. Assume that the function ~ , ( tu, ) is continuous and nonnegative for t, t t, a, 0 u 2h, gl(t, 0) = 0, and

< < + ~ i ( t u, )

< <

< s(t,u ) ,

+

Then, for every t , , t, < t , t, a, u ( t ) function on t,, t it, , which satisfies

<

for t ,


i ,

u’

---

gl(t, u),

(2.2.6)

t f to. ~

0 is the only differentiable (2.2.7)

u(t,J = 0

. t, .

Proof. Let us show that the maximal solution r ( t )of (2.2.7) is identically zero. Suppose, on the contrary, that there exists a n, t , CT -< t, a, such that ~ ( o> ) 0. Because of the inequality (2.2.6), we have : 4

v’(t)

.< g ( t , r ( t ) ) ,

If p ( t ) is the minimal solution of

t,


< u.

+

2.2.

51

UNIQUENESS

an application of Theorem 1.4.6 shows that P(t)

< +),

(2.2.8)

as far as p ( t ) exists to the left of u. T h e solution p ( t ) can be continued to t = t o . If p(T) = 0, for some T, to < T i u, we can effect the continuation by defining p ( t ) = 0 for to < t < T . Otherwise, (2.2.8) ensures the possibility of continuation. Since r(t,) = 0, liml_,to+ p ( t ) = 0, and we define p(to) = 0. Furthermore, since gl(t, u ) is continuous at ( t o ,0) and gl(to , 0) = 0, r;(to) exists and is equal to zero. This, because of (2.2.8), implies that p;(to) exists and p;(t,) = 0. But we have assumed that g(t, u ) satisfies hypothesis (i) of Theorem 2.2.2. Xence, p ( t ) =- 0. This contradicts the fact that p(o) = r(u) > 0. Therefore, r ( t ) = 0, and the proof is complete.

COROLLARY 2.2.2. T h e function g(t, u ) = A(t)u, where A ( t ) 3 0 is to a, satisfies the requirements of Theorem continuous on t, < t 2.2.3, provided that

< +

+ A(t)]e-”(t) > 0,

lim+sup[l

t-to

where p(t) =

Proof.

J

t0

A ( 4 ds,

(2.2.9)

t f t” .

(2.2.10)

Consider the differential equation (2.2.1 1 )

u’ = A(t)u.

+

T h e solutions u ( t ) 0 of (2.2.11) are nonvanishing constant multiples of the function e - p ‘ l ’ , p ( t ) being given by (2.2.10). T h e derivative of this function is A ( t ) e - P ( [ ) . Since A ( t ) 3 0, it follows from assumption (2.2.9) that every solution u(t) 0 of (2.2.11) violates at least one of the two limiting conditions (2.2.5). Hence, the function g(t, u) = A(t)u satisfies hypothesis (i) of Theorem 2.2.2.

+

COROLLARY 2.2.3. Let the assumptions of Theorem 2.2.3 hold except that the function gl(t, u ) is continuous on to < t t, a, 0 u 26. Then, for every t , , to < t, < t, a, u(t) 5-0 is the only function differentiable on to < t < t, and continuous on to t < t, , for which u;(t,) exists,

< + <

+

u’(t) = gdt, 4

and (2.2.5) holds.

th

to


9

< <

52

CHAPTER

2

+

Pmof. \Ye have to show that all the solutions u(t) 0 are such that they violate the limiting properties (2.2.5). Assuming the contrary and proceeding as in the proof of Theorem 2.2.3, it is easy to prove the stated result. Pmof of Theorem 2.2.2.

Define the function (2.2.12)

+

<

<

for to t < to a, 0 < zi 2b. Since f ( t , x) is continuous on I-?, , g,(t, 7[) is continuous on t, < t t, a, 0 u 26. It is clear that condition (2.2.2) holds for the function gf(t, u ) because of (2.2.12). 31oreol er, gr(t, ).

<

< +

< <

< g(C ).

< <

for t,, . t to + a, 0 zi 2h. Theorem 2.2.3 is now applicable with gl(t, 7 1 ) = g,(t, zi), and therefore g,(t, u ) satisfies the assumptions of Theorem 2.2.1. This establishes Theorem 2.2.2. COROLLARY 2.2.4. T h e conclusion of Thcorem 2.2.2 holds if condition (2.2.2) is replaced by

0, then g(t, u ) = X(t)+(u) is admissible in COROLLARY 2.2.5. If t, Theorem 2.2.2 provided that A ( t ) 3 0 is continuous for 0 < t a; +(u) is continuous for zi 2 0 and +(O) = 0, + (u ) > 0 for u > 0; and

<

:

T h e following is yet another criterion of uniqueness cf solutions which generalizes the earlier ones. T h e statement of the results involves the existence of two controlling functions.

TIIFOR~--LI 2.2.4. Alssuniethat (i) the functions A ( t ) ,B(t)are continuous and nonnegatixe on t,, < t < to a such that A(t,) = B(t,) = 0, K ( t ) ,0, t > , and

+

f,

limAL4(t)/B(t) = 0;

t *fo

(2.2.13)

2.2.

53

UNIQUENESS

(ii) the functions gl(t, u), g 2 ( t ,u) are continuous and nonnegative for t, < t t, + a, 0 u 2b; (iii) all the solutions u(t) of

<

< <

(iv) the only solution v(t) of (2.2.15)

D' = '?At, ).

on t,

< t < to + a such that (2.2.16)

lim+v(t)/B(t) = 0

t-to

is the trivial solution; (v) f E C[R, , R"],and, for ( t ,x), ( t ,y ) E R,, t f to, (2.2.17) Then, the differential system has at most one solution on to

< t < t, + a.

Before proceeding to the proof of Theorem 2.2.4, it is convenient to prove the following:

THEOREM 2.2.5. Let the functions A(t), B(t), gl(t, u), and g z ( t ,u ) fulfill hypotheses (i), (ii), (iii), and (iv) of Theorem 2.2.4. Suppose that the function g ( t , u ) is continuous and nonnegative for t, t to a, 0 u 2b, g(t, 0) e 0, and

< < +

< <

(2.2.18) Then, u ( t ) = 0 is the only differentiable function on t, which satisfies for to

< t < to + a.

u'

= g(t, u ) ,

U(t") =

0

< t < to + a (2.2.19)

Proof. We shall show that the maximal solution r ( t ) of (2.2.19) is identically zero. Assuming, on the contrary, that there exists a 0 such that Y(U) > 0 and proceeding as in the proof of Theorem 2.2.3, making use of relations (2.2.18) and (2.2.19), we obtain PAt)

< r(t),

(2.2.20)

54

CHAPTER

2

as far as p 2 ( t ) exists to the left of u, where p 2 ( t ) is the minimal solution of (2.2.15) such that pL(~) = r(u). As before, we can continue p2(t) up to t, by defining pz(to) : 0. Since p2(t) 0, we have

+

t-tO+ lim

# 0,

pz(t)P(t)

which, in view of (2.2.20), implies that lim+v(t)/B(t) # 0.

t-tto

This, together with assumption (2.2.13), shows that there exists a t, such that r(t1)

>4tJ.

(2.2.21)

Let p l ( t ) be the minimal solution of (2.2.14) such that pl(tl) = r ( t l ) . Then it can be shown, arguing similarly, that p l ( t ) can be continued up to t, , pl(t,) = 0, and

0


t"


(2.2.22)

Since, by hypothesis (iii), all solutions u(t) with u(t,) = 0 of (2.2.14) must obey u ( t ) A(t),to t t, a, we must have

< < -+

<

Pl(4

<4 t h

to

< t < t , + a.

This is absurd because of (2.2.21) and the fact that pl(tl) Hence, pl(to) > 0, which implies, in view of (2.2.22), that 0
contradicting the assumption r(t,)

=

=

r(tl).

< +")> 0.

Pmof qf Theorem 2.2.4. Consider the function g(t, u ) gf(t, u), where gf(t, u ) is the function defined by (2.2.12). By combining the respective -~

arguments in the proofs of Theorems 2.2.2 and 2.2.5, it is easy to show that gf(t, u ) verifies Perron's uniqueness conditions of Theorem 2.2.1, which is sufficient to establish the uniqueness of solutions. REMARK2.2.1. Whenever f ( t , x) is assumed to be continuous on R, , it follows from the foregoing considerations that the uniqueness conditions of Theorems 2.2.2 and 2.2.4 can be reduced to that of Perron's condition. If the pair of functions g,(t, u), g 2 ( t ,u ) satisfies the hypotheses of Theorem 2.2.4, we can also show that there exists a function g(t, u )

2.2.

55

UNIQUENESS

that fulfills the uniqueness criteria of Kamke as given in Theorem 2.2.2. This is the content of the following:

THEOREM 2.2.6. Let the functions A(t), B(t), gl(t, u), and gz(t,u ) satisfy hypotheses (i), (ii), (iii), and (iv) of Theorem 2.2.4. Then, there exists a function g(t, u ) verifying assumption (i) of Theorem 2.2.2.

Proof. Define the function g(t, u ) by g(t,

4 = min[g,(t,

u), gdt,

41.

(2.2.23)

Then g satisfies (2.2.18). T o prove the stated result, it is enough to show that no nontrivial solution of (2.2.4) fulfills the limiting conditions (2.2.5). I n fact, the assumption that there exists a differentiable function u(t) satisfying the differential equation (2.2.4) and the conditions (2.2.5) for which u(a) > 0, to < CT < to a, leads, following the proof of Theorem 2.2.5, to the contradiction that u(t,) > 0.

+

COROLLARY 2.2.6. T h e functions gl(t, u ) = K1ua, gz(t,u ) = K,(u/t) are admissible in Theorem 2.2.4, if 0 < 01 < I , K2(1 - a ) < 1, with A(t) = K,(l - ~ ~ ) t l / ( l and - ~ ) , B(t) = t K 2 . We shall now show that, if certain conditions of Theorem 2.2.2 are violated, Eq. (2.2.3) has nonunique solution. We prove this for the case n = 1 and t, = 0.

<

THEOREM 2.2.7. Let g(t, u ) be continuous on 0 < t a, 0 < u < b, Suppose that, for each t , , g ( t , 0) = 0, and g(t, u ) > 0 for u > 0. 0 < t , < a, u(t) 0 is a differentiable function on 0 < t < t , , and continuous on 0 < t < t, for which uL(0) exists,

+

0 < t < t, ,

u’ = g ( t , a),

and u(0) = u i ( 0 ) = 0.

Let f E C I R o ,R], where R, : 0 ( t ,Y ) E R, t # 0,

< t < a, 1 x I < b,

and, for ( t , x),

7

I f ( t , x) -f(t,Y)I

>g(t,

I x -Y I ) .

(2.2.24)

Then, the scalar differential equation x’ = f ( t , x),

has at least two solutions on 0

x(0)

< t < a.

=0

(2.2.25)

56

CHAPTER

2

Pmf. Let us first suppose that f ( t , 0) = 0, so that, putting y we obtain the inequality

=

0,

lf(t,41 > d t l I x I), because of condition (2.2.24). Sincef(t, x) is continuous and g(t, u) > 0 for u > 0, it follows that eitherf(t, x) < 0 or f ( t , x) > 0, for x .f 0. This implies that either (2.2.26)

or (2.2.27)

By hypothesis, there exists a u, 0 < u < a, such that u(u) > 0. Let y ( t ) be the minimal solution of x’ = f ( t , x), y(a) = u(u). Then, using an argument similar to that in the proof of Theorem 2.2.3 and the inequality (2.2.26), it can be shown that y ( t ) u ( t ) to the left of (T, as far as y ( t ) exists. Moreover, y ( t ) can be continued u p to t = 0 and

<

0

< y ( t ) < u(t),

0


< u.

Since u(0) z u;(O) = 0, we obtain from the foregoing relation y(0) = y’,(O) = 0. This proves that the differential equation (2.2.25) has a solution y ( t ) , not identically zero. On the other hand, since f ( t , 0) 0, (2.2.25) admits the identically zero solution. Hence, we have two different solutions for Eq. (2.2.25). Corresponding to the case (2.2.27), we can employ a similar reasoning to arrive at the same conclusion. We shall now remove the restriction f ( t , 0) = 0. Let x o ( t ) be a t < a. Using the transformation solution of (2.2.25), existing on 0 - x x,,(t), we get

<

v

-

-

2’ =

x’

-

=f ( t ,

x

xi(t)

= f ( t , x) - f ( t , x,(t))

+ x,(t)

- f ( t , xo(t))

(2.2.28)

= F ( t , 2).

Evidently, F ( t , 0) L 0. I t follows, therefore, that x = 0 is one solution of (2.2.28) through (0, 0). But the previous considerations show that (2.2.28) has a solution z ( t ) different from the identically zero solution. This implies that x ( t ) = x ( t ) x,,(t) is not identically equal to x,(t), and the theorem is proved. We have assumed the continuity of f ( t , x) in the uniqueness results

+

2.2.

57

UNIQUENESS

that are discussed so far. We shall consider, in what follows, the differential system x' =f(t,

x),

(2.2.29)

x(0) = 0,

< <

<

t a, 11 x 11 b. A solution of where f ( t , x) is defined on R, : 0 (2.2.29) in the classical sense will mean a function x(t) that is continuous on 0 t a, differentiable on 0 < t < a, and that satisfies (2.2.29) for 0 < t < a. Suppose that x(t), y ( t ) are two solutions of (2.2.29), existing on 0 t a ; then, the requirement

< <

< <

which is satisfied when f ( t , x) is continuous at (0, 0), is a necessary condition for the uniqueness of solutions. This can be generalized by the following: (2.2.30)

<

a, and where the function B(t) is continuous, positive on 0 < t B(O+)= 0. That condition (2.2.30) is not sufficient for uniqueness is seen by the following:

LEMMA 2.2.1. Suppose that the function B(t) is continuous and a such that B(O+)= 0. Then there exists an infinity positive on 0 < t of functionsf(t, x) such that (2.2.29) has more than one solution satisfying the condition (2.2.30).

<

Proof. t

0

We first construct a function A(t)having a nonnegative derivative

< < aand

lim A(t)/B(t)= 0.

t-0+

We proceed as follows. Divide the interval [0, a] into subintervals I, such that z1= (42, a), z, = (a/4,a/2)).... Suppose that 6 , such that

=

inf, B(t). Find a positive linear function I,, on Il L,(a)

=

b, ,

L,(a/2) ,< &(a).

58

CHAPTER

2

Then find L, on I , such that L,(a/2)

= b,/2

< &,(u/2) L 2 ( 4 4 ) e 4L2(u/2). L,(a/2)

if 6,/2

< L,(a/2),

if b,/2

> L,(u/2),

We continue this process and then connect the linear functions near the points a/2% by suitable functions having nonnegative derivatives (for example, by arcs of parabolas). This modification gives us the function A(t) with the required properties. Having constructed the function A(t), it is easy to define f (t, x) by f ( t , x) = x"A'(t)

(0 < a

< 1).

Then, x(t) = 0 and x ( t ) = (1 - c ~ ) l / ( l [- ~A)(t)]/(lpm) are solutions of (2.2.29). It is clear that any two solutions satisfy the condition (2.2.30), and, hence we have the proof. It is also easy to prove the following fact.

LEMMA 2.2.2. Suppose that f ( t , x) is defined on R, and is continuous at (0, 0). T h e n there exists a function B(t) on 0 t 1 such that (2.2.30) is satisfied.

< <

Proof.

Let x(t), y ( t ) be two solutions of (2.2.29) on 0

< t < a. Define

m(t> = /I x(t> - Y(t)Il.

Observe that m(0) = 0, and, because of the assumed continuity of x) at (0, 0), we have lim m ( t ) / t = 0.

f(t,

t-.o+

Setting B(t) = sups

tQ

m(s)/s, it is easily verified that lim m ( t ) / B ( t )= 0. -o+

I

+

This is possible if m(s) 0 in some neighborhood of the origin; otherwise, the existence of B(t) is trivial. We notice that the continuity requirement off(t, x) at (0,O) is stronger than the condition (2.2.30). T o see this, definef(t, x) as follows:

I

1, f ( t , x) = x / t ,

0,

x > t, 0 < x < t, x < 0.

2.2.

59

UNIQUENESS

< <

T h e solutions of (2.2.29) are then given by x(t) == Izt, where 0 Iz 1. Take B(t) = t 1 / 2 . Clearly, the relation (2.2.30) is satisfied even though f ( t , x) is not continuous a t (0,O). These considerations lead to

THEOREM 2.2.8. Suppose that x(t), y ( t )are any two solutions of (2.2.29) a satisfying (2.2.30), where B(t) is positive, continuous on 0 < t with B(O+)= 0. Let the functiong(t, u ) 3 0 be continuous on 0 < t a, 0 u b, and the only solution u(t)of

< <

< <

on 0

u’ = g(t, u )

< t < a such that

lim u ( t ) / B ( t )= 0

t-O+

is the trivial solution. Assume further that the functionf(t, x) is defined on Ro and satisfies I I f ( t 7

< g(t7 I1 x - Y II)

4 - f ( t ,Y)II

for ( t , x), (t,y ) E Ro , t # 0. Then there exists at most one solution of (2.2.29) on 0 t a.

< <

Proof. Define m(t) = (1 x(t) - y ( t ) (I, where x ( t ) , y ( t ) are any two solutions of (2.2.29) existing on 0 t a. Then m(0) = 0, and

< <

o.44 G llf(t, 4 t ) ) -f(t,y(t))lI

< g(t, m(t>).

If we suppose that, for some u, 0 as in Theorem 2.2.3, that p(t)

< u < a,

<4 t h

t

<

m(u) > 0, we can show,

0,

as far as p ( t ) exists, where p ( t ) is the minimal solution of u’ = g(t, u ) ,

u(u) = m(u).

Furthermore, as before, p ( t ) can be continued up to t = 0 and p(t) m(t), 0 t 0. Then, because of the assumed condition (2.2.30), we have

0

<

<

< <

0

< lim p(t)/B(t)< lim m ( t ) / B ( t )= 0, t-O+

t-O+

which, by hypothesis, implies that p ( t ) = 0. This contradicts p(u) m(u) > 0, and hence m(t) = 0, 0 t a. T h e proof is complete.

< <

=

60

CHAPTER

2

2.3. Convergence of successive approximations The answer to the question of whether or not a solution of the system (2.2.3) can always be obtained as a limit of the sequence or a subsequence of the successive approximations is negative. It is not difficult to construct an example such that the solution of (2.2.3) is unique, although no subsequence of the successive approximations converges to that unique solution. It turns out, however, that, with an additional restriction of monotony ofg(t, u ) in u in Theorems 2.2.1,2.2.2, and 2.2.4, convergence of successive approximations to the unique solution follows. Suppose that g ( t , u ) of Theorem 2.2.2 is monotone nondecreasing in zi for fixed t, in addition to the hypotheses of the theorem. Then, defining

< < +

< <

instead of (2.2.12) for t, t to a, 0 u 26, we note that g,(t, u ) is monotone nondecreasing in u for each t. Thus, it follows that

and, by Theorem 2.2.3, g,(t, u) satisfies the hypotheses of Theorem 2.2.1. Similarly, if we assume that gl(t, u ) and g2(t, u ) of Theorem 2.2.4 are monotone nondecreasing in u for each t, then Theorem 2.2.5 shows that g,(t, u ) = g ( t , u ) also verifies the assumptions of Theorem 2.2.1, in view of the fact that

It is therefore enough to prove the convergence of successive approximations for Theorem 2.2.1 with an additional restriction of monotony on g ( t , u).

THEOREM 2.3.1. Let the assumptions of Theorem 2.2.1 hold. Suppose M on further that g ( t , u ) is nondecreasing in u for each t, l l f ( t , x) 11 R, , and = min(a, b / M ) . Then, the successive approximations defined by

<

(Y

(2.3.1)

< <

exist on t, t t, -k 01 as continuous functions and converge uniformly on this interval to the solution x ( t ) of (2.2.3).

2.3.

61

CONVERGENCE OF SUCCESSIVE APPROXIMATIONS

< < to +

Proof. Suppose that xk(t)is defined and continuous on to t and satisfies 11 xk(t) xo 11 b for R = 0, 1, 2 ,..., n. Write -

<

xn+1(t) = xo

+ sIo

f(S, x n ( 4

01

ds.

Then, since f ( t , x,(t)) is defined and continuous on to the same holds for x n t l ( t ) . Moreover, it is also clear that

< t < to +

01,

< Ma < 6. Thus, by induction, the successive approximations are defined and continuous on to t to 01 and

< < + II xn+i(t)

-

xn

I1 < 6,

=

0, 1,2,....

We shall now define the successive approximations for Eq. (2.2.1) as follows: udt) = M(t

-

to),

un+i(t) = fOg(s, un(s)) ds,

tn

< t < to +

01.

Then,

(2.3.2) (L.3.3)

the inequalities (2.2.2), (2.3.3), and the monotonic character of g ( t , u ) in u give

Thus, by induction, the inequality (2.3.4)

62

CHAPTER

2

(2.3.5)

because of the monotonicity of g ( t , u ) in u.An application of Theorem 1.4.1 yields that

II % ( t )

-

Xm(t)ll

< yn(t),

to

< t < to +

01,

where r,(t) is the maximal solution of Y'

=d t , Y )

-i- 2g(t, un-l(t>),

%&(to)

=

0

for each n. Since the conditions of Lemma 1.3.2 are satisfied r,(t) -+ 0 uniformly on to t to 01, as n + CO. This implies that xn(t) converges uniformly to x ( t ) on to t to 01 as n + CO. By Theorem 2.2.1, the solution of (2.2.3) being unique, this x ( t ) is the unique solution of (2.2.3).

< < +

< < +

Another proof of Theorem 2.3.1. It can be easily shown that the sequence of approximations (2.3.1) is uniformly bounded and equicontinuous on to t to 4-N, and therefore there exists uniformly convergent subsequences. Suppose that xn(t) - xnPl(t) + 0 as n + 00; then (2.3.1) implies that the limit of any such subsequence is the unique solution x(t) of (2.2.3). It then follows that a selection of a subsequence is unnecessary and that the full sequence x,(t), xl(t), x2(t),... converges uniformly to x(t).

< <

2.3.

63

CONVERGENCE OF SUCCESSIVE APPROXIMATIONS

Thus, to prove Theorem 2.3.1, it is sufficient to show that m(t) = 0, where m ( t ) = lim sup (1 x n ( t ) - xn-l(t)ll. n-w

(2.3.6)

We shall first show that m ( t ) is continuous for t E [t, , t, Ilf(t, x) 11 M on R, , we see that

<

II 4 t l ) - xn-l(tl)ll

< ll xn(tz) xn-l(tz)ll + 2 M I tl < m(tz) + 2MI tl t , I + -

-

for large n, if

E

-

+ a ] . Since

tz

I

E

> 0. Hence, we have m(t1)

< m(t2) + 2MI tl

As t, , t, can be interchanged and

E

I 4tl) -4tz)I

-

t,

I

+

E.

> 0 is arbitrary, we obtain

< 2 M I tl

t, I 9

~

which proves the continuity of m ( t ) . T h e assumption (2.2.2), together with the relation (2.3. l), yields

< J t g(s, II xn(s) xn-l(s)II) ds. For a fixed t in the interval ( t o , t, + a ] , there is a sequence of integers II xn+dt> - xn(t)ll

n, < n2 < and that

0 . .

-

t0

such that 11 ~ ~ + ~ xn(t) ( t )11 tends to m(t) as n m*(s)

exists uniformly on to

=

lim

n = n,-m

/I xn(s)

-

=

nk 4 co,

X~-~(S)II

< s < to + a. Thus, (2.3.7)

Since g is assumed to be monotone nondecreasing in u and m*(s) we obtain from (2.3.7) the inequality

< m(s),

<

By Theorem 1.9.2, m(t) r(t), where r ( t ) is the maximal solution of (2.2.1). As Eq. (2.2.1) is assumed to possess only identically zero solution, it follows that r(t) = 0, which in turn shows that m(t) = 0 on [ t o ,to a ] . This completes the proof.

+

64

2

CHAPTER

2+4. Chaplygin’s method

We are interested in establishing a method of approximation of the solution of a given differential equation by means of solutions of an associated linear equation and in estimating the difference between them. This is precisely what the Chaplygin’s method accomplishes. For convenience, we shall first consider the case of scalar differential equation.

THEOREM 2.4.1. Let f E C I R o ,R ] , where R, is the rectangle to t 0 in R, . Let to + 01 the functions uo = uo(t),vo = vo(t)be differentiable for to t such that (t, uo(t)),( t , vo(t))E R, and

+

<

<

< <

< f ( t , *o(t)), vat> > f ( t ,vo(t)>7 *Xt)

*,(to) = xo

7

(2.4.1)

vo(t0) = xo

-

(2.4.2)

Then, there exists a Chaplygin sequence {un(t),vn(t)) such that *At) < * n + d t )

< x ( t ) < vn+1(t) < vn(t), %(to)

= xo =

t

E

[ t o I to

+ 4,

%(to),

where x ( t ) is the unique solution of x‘ = f ( t , x),

(2.4.3)

x ( t o ) = xo

+

existing on [to , to a ] . Also, u,(t) and v,(t) tend uniformly to x(t) on [to , to a] as n -+ GO. If, in addition, for a suitable constant ,f3,

+

0 < vo(t) - *,(t)

then

I %(t)

-

v,(t)l

< 2is/22n,

< is, t

E

[to , to

+ .I.

(2.4.4)

Proof. T h e functions uo(t),vo(t), and x(t) satisfy the assumptions of Theorem 1.2.3, and therefore we have %(t)

< x ( t ) < .o(t),

We now define the functions

t

E

( t o , to

+ .I

2.4. Observe that, for t =

65

CHAPLYGIN’S METHOD

to,

fAt0

9

, x; uo

x; uo 7 no) = f&o

Y

.o).

Let ul(t), vl(t)be the solutions of the linear differential equations 4 ( t ) =f1(t, u d t ) ; uo , %I,

%(to) = xo

>

(2.4.7)

q t ) =fdt, s(t); uo %),

Vl(t0) =

xo

1

(2.4.8)

9

which exist on [to, to offi in (2.4.5) result

+ a ] . From the inequality (2.4.1) and the definition uLxt)

< f(4 uo(t)) = fdt, uo(t);*o

>

no),

which, because of Theorem 1.2.1 and the following remark, yields uo(t)

< %l(t),

t

E

(to 9 to

+

. I ] .

(2.4.9)

A similar reasoning with (2.4.2) and (2.4.6) shows that

We shall next show that the functions ul(t), vl(t) also satisfy the differential inequalities (2.4.1) and (2.4.2), respectively. Since fz(t, x) is strictly increasing in x, using (2.4.5), (2.4.7), and the mean value theorem, it is easy to deduce that

On the other hand, 4 ( t ) < f ( t , uo(t)) = f2@,

uo(t>;uo >

Vo),

and consequently, we have, applying Theorem 1.2.1 again,

Furthermore, it is readily seen that (2.4.13)

66

2

CHAPTER

and .f(t>s ( t ) ) = f(tJuo(t)) + f z ( t ,

uo(t)"dt)

+ i f z z ( t , O[vi(t)

-

uo(t)l udt)

- uo(t)12,

< E < ni(t).

(2.4.14)

T h e relations (2.4.8), (2.4.1 I), (2.4.12), (2.4.13), and (2.4.14), together with the repeated applications of mean value theorem and the assumption M t , t) > 0, imply v;(t) = fi(f, v1(t>; uo vo) 1

>f ( t ,s ( t ) ) ,

+ 4.

t E (to to 1

(2.4.15)

Since the functions ul(t), vl(t), and x(t) verify the assumptions of Theorem I .2.3, we obtain %(t) < x ( t ) < s(t), t

E

( t o to 7

+

011,

which, in view of the inequalities (2.4.9) and (2.4.10), gives uo(t) < ul(t) < x ( t ) < vl(t)

< vo(t),

t

E

( t o ,to

+ a].

(2.4.16)

T h e foregoing considerations define a transformation T that assigns to a given couple of functions (uo(t),vo(t))a new couple (ul(t),vl(t))satisfying the same inequalities (2.4.1) and (2.4.2), respectively, such that (2.4.16) holds. This implies that (u1 7 v1) =

mu0

9

.")I.

It therefore follows that we can apply the transformation T to the couple ( u l ,ul) to get (u2, v2). A repeated application of the transformation T provides a well-defined Chaplygin's sequence (%+l

> %+l) =

W

n

>

741

of functions satisfying the following relations:

6) (ii) (iii)

uXt)

4 ( t ) > f ( t , v,(t)), u,(t)

(3 4 + * ( t ) (v)

< .f(t,un(t)),

%(to)

=

xo ;

vn(to) = xn ;

< ~ , + ~ (< t ) x ( t ) < ~ , + ~ (< t )v,(t), = f d t 7

t

E

( t o , to

+ a];

%+At); u,(t>, vn(t));

4 + 1 ( t ) = f d t , 7-5n+1(t);

% ( t ) >vn(t))-

It is clear from (iii) that the sequences {urn},{urn}are monotonic and uniformly bounded on [ t o ,to a ] . Furthermore, they are equicontin-

+

2.4.

67

CHAPLYGIN'S METHOD

uous, in view of the fact that, for each fixed n, u, , v, are solutions of linear equations. Hence, an application of Theorem 1.1.1 proves the uniform convergence of un(t),vn(t)to x ( t ) as n --f 00. Let (2.4.17)

and

<

<

Assume that 0 v,(t) - u,(t) (2Haek@)--l= p. Clearly, (2.4.4) holds for n = 0. Suppose it is true for a certain fixed n, i.e., (2.4.19)

From the definition of ~ % + ~ (~t % ) ,+ ~ (and t ) , the mean value theorem, it follows that

where u,(t)

where u,(t)

we obtain, from (2.4.17), (2.4.18), and (2.4.20),

Furthermore, we also have and

I4

-

I un+1(t)

un(t)I -

< I vTdt>

- Un(t)l

4 1 1 G I vnw - un(t)l-

These estimates, together with (2.4.19) and (2.4.21), lead to the differential inequality 22/32 D+lvn+1(t) - un+1(t)l

< KI %+l(t>

- un+1(t)l

+ H22"+1?

68

2

CHAPTER

which, in view of Theorem 1.4.1, yields

Since

S eK(lPs)ds < meKa, we get 1

to

Thus, by induction, the relation (2.4.4) is true for all n, and consequently we have, by (iii),

I x ( t ) - U,(t)i

and

G 2/3/2'"

1 x ( t ) - v,(t)l ,< 2/3/22". This completes the proof. Let us now consider the differential system x' = f ( t , x),

x(t,) = x,

.

(2.4.22)

I n this case, we shall be able to demonstrate only the lower Chaplygin's sequence {un},under some additional restrictions.

THEOREM 2.4.2.

Let f E C[R,, Rn],where R, is the set,

< t G to + a ,

R, :

<

I1 x

-

xo II

< b.

Let l / f ( t ,x) (1 M on R , . We suppose that f ( t , x) is quasi-monotone nondecreasing in x, for each t E [to , t, + a], and that af ( t , x ) / a x exists and is continuous on R,. Let u,(t) be continuously differentiable on [to , a ] , where 01 = min(a, b / M ) ,( t ,u,(t)) E R,,and ui(t) < f ( t , u,(t)), ",(to) = x, . Furthermore, let

+

f ( t , x) +f,(t,

x)

. ( y - x) < f ( t , y )

if

x


(2-4-23)

Then, there exists a Chaplygin sequence {un(t))such that un(to)= x, , %(t)

< Un+l(t) < 4 t h

t E

[ t o , to

+ 4,

where x(t) is the solution of (2.4.22) existing on [to , to uniformly on [ t o , t,

+ .I].

lim u,(t)

n-m

= x(t)

+ a ] , and

2.5.

69

DEPENDENCE ON INITIAL CONDITIONS AND PARAMETERS

Proof. We notice, first of all, that afi/axj >, 0 for i # j . This follows from the quasi-monotonicity of f ( t , x). Moreover, applying Corollary 1.5.1, we obtain uo(t)

< X(t),

t 6 [to ,to

+ .I.

Corresponding to the linear equation (2.4.7),we have now to consider the linear system defined by

=

(2.4.24)

Y(t0) = xo *

uo(t)),

Observe that f ( t , y; u,(t)) possesses the quasi-monotone property in y because afi/axj2 0, i # j . Hence it follows by Corollary 1.5.1 that uo(t)

< Ul(t),

t E [to t o

+ 4,

where ul(t) is the solution of (2.4.24). The assumption (2.4.23) implies that u;(t> = f ( t ,u l w ; u o w


Thus, we have uo(q

< x@),

< ul(4 < 4 t h

t E [to , t o

+ 4.

t E [to 7 t o

+ 4.

As in Theorem 2.4.1, we can define the transformation verifying u d t ) = T[uo(t)l-

T h e rest of the argument is but a repetition of the proof of Theorem 2.4.1 with appropriate changes. This establishes the method of Chaplygin for systems.

2.5. Dependence on initial conditions and parameters We shall consider the problem of continuity and differentiability of solutions x ( t , to , xo) of the differential system x' = f ( t , x),

(2.5.1)

70

CHAPTER

2

with an initial condition (2.5.2) with respect to the initial values (to , xo).

LEMMA 2.5.1.

Let f~ C [ J x R", Rn],and let

Assume that ~ * ( tto, , 0) is the maximal solution of U' =

G ( t ,u),

through ( t o ,0). Let x(t, t o , xo) be any solution of (2.5.1) and (2.5.2). Then,

ll x ( t , t o ,xo) - g o II e Y * ( t ,

THEOREM 2.5.1.

to I O),

t >, t o -

L e t f E C [ J x R", R"],and, for ( t , x), ( t ,y ) E J x R",

where g t C [ J x R, , R,]. Assume that u(t) = 0 is the unique solution of the differential equation U' = g ( t , u )

(2.5.4)

such that @(to) = 0. Then, if the solutions u(t, t o , uo) of (2.5.4) through every point ( t o ,uo) are continuous with respect to initial conditions

2.5.

DEPENDENCE ON INITIAL CONDITIONS AND PARAMETERS

71

( t o ,uo), the solutions x(t, t o ,xo) of (2.5.1) and (2.5.2) are unique and continuous with respect to the initial values ( t o ,xo).

Proof. Since the uniqueness of solutions follows from Theorem 2.2.1, we have to prove the continuity part only. T o that end, let x(t, t o ,xo), y ( t , to ,y o ) be the solutions of (2.5.1) through (to, xo), (to,yo), respectively. Defining m ( t ) = I1 x ( t , to 7 xo) - Y ( t , to Yo)ll,

the condition (2.5.3) implies the inequality

e

D f f 4 t > g(t, 4 t h

and, by Theorem 1.4.1, we obtain where r(t, to , 11 xo - y o 11) is the maximal solution of (2.5.4) such that u(to)= 11 xo - yo 11. Since the solutions u(t, to , uo) of (2.5.4) are assumed to be continuous with respect to the initial values, it follows that lim r ( t , to , I/ xo

J(o-yo

-

Yo I l l

= r ( t , to

7

01,

and, by hypothesis, r ( t , t o ,0) = 0. This, in view of the definition of m ( t ) , yields that lim x ( t , to , xo)

xo-Yo

= r(t,t o

,Yo),

which shows the continuity of x ( t , t o ,xo) with respect to x,, . We shall next prove the continuity with respect to initial time t o . If x(t, t o , xo), y ( t , t, ,xo), t , > t o , are the solutions of (2.5.1) through ( t o ,xo), ( t l , xo), respectively, then, as before, we obtain the inequality D+m(t)

where Also,

< gft, W ) ) ,

m(t> = /I x ( t , to > xo) - Y ( t , t , 4 t l ) = ll X ( t ,

?

>

to xo) - xo It. ?

Hence, by Lemma 2.5.1, m(t1)

and, consequently, m(t)

< r*(t,

< r"(t),

x0)Il.

7

to 9 01,

t

> t, ,

12

CHAPTER

2

where f(t) =

q t , tl , Y * ( t ,

7

to ,O))

is the maximal solution of (2.5.4) through (tl , r * ( t l , t o , 0)). Since = 0, we have

r * ( t , , t o , 0)

lim f ( t , t, , ~ * ( t,,t o ,0 ) ) = f ( t , t o ,0 ) ,

tl-fo

and, by hypothesis, T ( t , t o , 0) is identically zero, thus proving the continuity of x ( t , t o , xu) with respect to to .

COROLLARY 2.5.1. T h e function g ( t , u ) in Theorem 2.5.1.

= Lu,

L

> 0,

is admissible

THEOREM 2.5.2. Let f E C[E, Rn],where E is an open ( t , x, p)-set in R"+"L+l,and for p = po , let xo(t) = x(t, to , xo , p,,) be a solution of x' = f ( t , x,Po), existing for t

4 t O ) = xo

7

(2.5.5)

3 to . Assume further that lim f ( t ,x , Y ) = f ( t ,x, Po),

w-wo

(2.5.6)

uniformly in ( t , x), and, for ( t , x1 , p), ( t , x2,p ) E E, I l f ( t 7

x1

9

Y) -f(t,

x2

, P)Il

e At, /I

x1 - xzll),

(2.5.7)

whereg E C[/ x R, , R,]. Suppose that u(t) = 0 is the unique solution of (2.5.4) such that u(tn) = 0. Then, given E > 0, there exists a S ( E ) > 0 such that, for every p, 11 p p,, I/ < S ( E ) , the differential system ~

x' = f ( t , x,P),

x(tn) = xo

(2.5.8)

admits a unique solution x ( t ) = x(t, t o ,x,, , p ) satisfying

I1 ~

( t ) xo(t)ll ~

<

t

€9

> to

Proof. T h e uniqueness of solutions is obvious from Theorem 2.2.1. From the assumption that u(t) = 0 is the only solution of (2.5.4), it

+

follows, by Lemma 1.3.1, that, given any compact interval [to , to U ] contained in J and any E > 0, there exists a positive number 7 = V(E) such that the maximal solution r(t, to , 0, 7) of 24'

= g(t, 24)

+7

2.5.

exists on to

DEPENDENCE ON INITIAL CONDITIONS AND PARAMETERS

< t < to + a and satisfies r ( t , to , 079)

< 6,

t

E

[to , to

+ 4.

Furthermore, because of the condition (2.5.6), given 7 a 6 = 6 ( q ) > 0 such that

Ilf ( t , X, P ) - f ( t ,

3 9

PJI

13

> 0, there exists

< 17

provided

I1 P Now, let

E

> 0 be

- Po

II < 8.

given, and define

Nt)= I1

- XO(~)lL

where x(t), x,(t) are the solutions of (2.5.8) and (2.5.5), respectively. Then, using the assumption (2.5.7), we get D+m(t)

< g ( t , 4 t ) ) + llf ( t , xo(%

From this, it turns out that, whenever D+m(t)

11 p

P ) - f(t, xo(t), P0)Il.

- po 11

< 6,

< g(t, 4 t ) ) + ?-

By Theorem 1.4.1, we have

< r(t,t o , O,d,

t

3 to,

II x ( t ) - x,(t)Il < e ,

t

2 to,

m(t)

and hence provided that

/I P

- Po

II < 8.

Clearly, 6 depends on E since q does. T h e proof is complete.

LEMMA 2.5.2. Let f E C [ J x D,R"],where D is an open, convex set in R",and let af /ax exist and be continuous. Then,

74

CHAPTER

2

the convexity of D implies that F ( s ) is defined. Hence, (2.5.9)

SinceF(1) = f ( t , x2) andF(0) (2.5.9) from 0 to I .

=f

( t , xl),the result follows by integrating

T H E O R E M 2.5.3. Assunic that f E C [ J x Rn,Rn] and possesses continuous partial derivatives af /& on J x Rn. Let the solution xO(t)= x(t, t o , x,,)of (2.5.1) exist for t 3 t o , and let

Then

exists and is the solution of Y'

=

H ( t , t o , x0)y

(2.5.10)

such that O(to, to , xo) is the unit matrix; (ii)

ax(t, f, , XI)) at,

exists, is the solution of (2.5.10), and satisfies the relation

First wc shall prove conclusion (i). Let h be a scalar and e, = (eTcl, ..., ek?() be the vector such that ekj = 0 if j f k and ekk = 1. Then, for small h, let

Proof.

x(t, h )

=

x(t, t o ,

x0

+ e&),

which is defined on J , and lim x(t, h ) = xo(t) h-0

uniformly on J . Since

2.5.

DEPENDENCE ON INITIAL CONDITIONS AND PARAMETERS

75

applying Lemma 2.5.2 with x2 = x(t, h), x1 = x,(t), we have

If we write

the existence of ax(t, t o ,xo)/2xo is equivalent to the existence of the limit of x h ( t ) as h 3 0, since ~ ( t, ,h ) = xo ekh, xh(t,) = ek . Thus, xh(t) is the solution of the initial value problem

+

y‘ == H ( t , ‘ 0 ,

9

h)y,

Y(tO) = ek

(2.5.12)

7

where

As x(t, h) + x,(t) as h + 0, by the continuity of lim H(t, to , xo , h) h-0

=

af /ax,

it follows that

H(t, t o ,xo)

uniformly on J . Considering (2.5.12) as a family of initial value problems depending on a parameter h, where H(t, to , xo , h) is continuous for t E J , h being small and y arbitrary, and observing that the solutions of (2.5.12) are unique, it is clear that the general solution of (2.5.12) is a continuous function of h. In particular, limb+,, xl,(t) = x ( t ) exists and is the solution of (2.5.10) on J. This implies that ax(t, t o ,xo)/axo exists and is the solution of (2.5.10). T o prove (ii), define

Since (2.5.1) has unique solutions, we have x ( t , to

+ h, xo) = x(t, to , “(to , to + h,

and therefore hGhh(t)= x(t, tn >

-

.To)),

to , tn + h, xo)) - x ( t , to , xo).

Because ax(t, t o ,xo)/axoexists and is continuous and .(to

, to

+ h, xo)

.(to , to , xo) = xo

as

h

-

(2.5.13)

0,

76

CHAPTER

2

it follows from (2.5.13) that hqt)

xO],

=

0. By the mean value theorem, there exists a 0 ash such that --f

where 0

as h

--f

=

0, ,k

=

(2.5.14) 1 , 2,...,n

< B < 1. Notice that, for each k,

0. Thus, (2.5.14) shows that

0, which implies that ax(t, to , xo)/ato = limh+oi r l ( t )exists and satisfies (2.5.1 1). This completes the proof.

as h

--f

2.6. Variation of constants Let us prove some elementary facts about linear differential systems,

x'

=

(2.6.1)

A(t)x,

where A ( t ) is a continuous n x n matrix on J. Let U ( t ) be the n x n matrix whose columns are the n-vector solutions x ( t ) , x(t), being so chosen to satisfy the initial condition U(t,) = unit matrix. Since each column of U ( t )is a solution of (2.6. l ) , it is clear that U satisfies the matrix difierential equation U'

=

U(to)= unit matrix.

d(t)U,

THEOREM 2.6.1.

(2.6.2)

Let A ( t )be a continuous n x n matrix on J. T h e n the fundamental solution U ( t )of (2.6.2) is nonsingular on J , More precisely, det U ( t )

exp

1 :

/ t t0

where tr A ( t ) =

CF=laii(t).

tr A ( s ) ds,

t E J,

2.6. Proof.

VARIATION OF CONSTANTS

77

T h e proof depends on the following two facts:

(i) d(det U ( t ))/d t= sum of the determinants formed by replacing the elements of one row of det U ( t )by their derivatives. (ii)

T h e columns of U ( t )are the solutions of (2.6.1).

Simplifying the determinants obtained in (i) by the use of (ii), we get d dt

- det U ( t ) = tr A(t)det U ( t ) .

T h e result follows, since U(to)= unit matrix.

THEOREM 2.6.2.

Let y ( t ) be a solution of Y’

=

A(t)Y + F ( t , Y ) ,

(2.6.3)

where F E C[J x Rn,R”],such that y(to)= y o . If U ( t ) is the matrix solution of (2.6.2), then y ( t ) satisfies the integral equation

This, because of (2.6.2), yields z’(t) = U-’(t)F(t,Y ( t ) ) ,

whence z ( t ) = Yo

+

Jt

U-l(s)F(s,y(s))ds.

t0

Multiplying this equation by U ( t )gives (2.6.4). COROLLARY 2.6.1. Let A(t) be a continuous n x n matrix on J such that every solution x ( t ) of (2.6.1) is bounded for t 3 to . Let U ( t )be the fundamental matrix of (2.6.1). Then, U-l(t) is bounded if and only if

is bounded from below.

78

2

CHAPTER

We shall now consider the nonlinear differential system (2.5.1). T h e following theorem gives an analog of variation of parameters formula for the solutions y(t, to , x,,)of

Y' = . f ( t , y ) +F(t,Y).

(2.6.5)

THEOREM 2.6.3.

Let ~ , F CE[ J x RrL,R"], and let af /ax exist and be continuous on J x R". If x ( t , t o , x,,) is the solution of (2.5.1) and (2.5.2) existing for t >, t o , any solution y ( t , t o , xo) of (2.6.5), with ~ ( t , ,= ) xo , satisfies the integral equation Y ( t , to .o> 7

=

x ( t , to 7 %>

+ It @ ( t ,

s,

to

Y ( S , to , x,,))F(s,Y ( S , to , x,,)) ds

(2.6.6)

for t 2 t,, , where @(t,t o , x,,) = ax(t, to , xo)/axo. Proof.

Write y ( t ) = y(t, t o , x,,). Then, dx(t, s , y ( s ) ) ds

-

ax(t,s,y(s))

+

q t ,S?Y(S))

aY

as

= @ ( t ?s,Y N ) [ Y ' ( S )

.Y" (2.6.7)

YG))l,

using Theorem 2.5.3. Noting that x(t, t , y ( t , to , x,,)) = y(t, t,, , x,,) and y'(s) - f ( s , y(s)) = F(s, y(s)), by integrating (2.6.7) from to to t, the desired result (2.6.6) follows.

THEOREM 2.6.4. Let f E C [ J x Rn, R"], and aflax exist and be continuous on J x R". Assume that x ( t , t o , x,,) and x(t, t,, ,y o ) are the solutions of (2.5.1) through (to , x,,) and (to ,yo), respectively, existing for t 3 t o , such that x,,, y o belong to a convex subset of R". Then, for t >, t o , x ( t , t" , Y") - x ( t , to xo) = 7

[I;

@(G to , xo

+

S(Y0 - XI)))

4.

(Yo - xo).

(2.6.8)

Proof. Since xo,y o belong to a convex subset of R", x ( t , to,x,, is defined for 0 s I . Thus,

< <

and hence the integration from 0 to 1 yields (2.6.8).

+ s( yo

-

x,,))

2.7.

79

UPPER AND LOWER BOUNDS

2.7. Upper and lower bounds Consider the differential system

and the differential inequality

where f , fl E C [J x R", R"], and 6 E C [J , R,]. DEFINITION 2.7.1.

By a &approximate solution of

on [to, co), we mean a function y ( t ) such that y E C [J , R"],y ' ( t ) exists on J - S , S being an at-most countable subset of J , and satisfies (2.7.2) on J - S.

THEOREM 2.7.1. that

Let g E C [ J x R, , R,] and u, a, 8 E C [ j ,R,] such (2.7.4)

for t

> t o . Let D

=

[x,y

E

Rn : 1) x - y 11

=

u(t)

and

Ij x - y jl

= v(t),t

> to].

Assume that f,fl E C [J x Rn,R"], and

for t > to and x, y E Q. If x ( t ) , y ( t ) be a solution and a &approximate solution of (2.7.1) and (2.7.3), respectively, on [ t o , co) such that

(2.7.6)

Proof. If (2.7.6) is not true, the set

80

CHAPTER

2

is nonempty. Arguing as in the proof of Theorem 1.2.1, we arrive at a t , > to such that either or In either case, it follows from the definition of 9 that, at t = t, , x(tl), y ( t l )E $2, and therefore, defining m ( t) = I/ x ( t ) - y ( t ) 11, we get

I m;(tl)l

< I/ x’(t1) -Y‘(tl)ll G

Ilf(t1

x(t1))

-fAh ?Y(tl))ll

+ I/ Y’(t1) -fdG

?Y(tl))ll.

This, together with (2.7.2) and (2.7.5), implies

A repetition of the rest of the proof of Theorem 1.2.1, with appropriate changes, proves (2.7.6).

THEOREM 2.7.2.

Let the assumptions of Theorem 2.7.1 hold except that (2.7.4) and (2.7.5) are replaced, respectively, by (2.7.7)

for t > to , x, y E Rtl. Suppose further that, for each t E [to , 71,g satisfies the condition

1 g(t7 ~

1 - )g ( f ,

uZ)I

< G(7 + t o

-

t , ~1 - uz),

7E

~1

[to , 00) and

2~

2 , (2.7.9)

where G E C [ J x R, , R,] and r ( t ) = 0 is the maximal solution of U’ =

G(t,u),

~ ( t , )= 0.

Then, the inequality (2.7.6) remains valid.

(2.7.10)

2.7.

81

UPPER AND LOWER BOUNDS

Proof. By a repeated application of Theorem 1.4.3, we can prove (2.7.6). For this purpose, it is enough to see that g 6 and -(g 6) satisfy the condition (1.4.9), in view of (2.7.9). Also, (2.7.8) implies that

+

+

for t

> to.

+ S(t)l < m;(t) < g(t7 4 t ) ) + s(t)>

-[g(t, 4 t ) )

THEOREM 2.7.3.

Let g E C [ J x R, , R,], 6 E C [ J ,R,], and r ( t ) , p ( t ) be the maximal and the minimal solutions of

respectively, existing on [to, 00). Let Q = [ x7 Y and

where

c1

E

R" :P ( t ) - €2 II x - Y

r(t)

<

< II x

- y I1 < p(t> II < ~ ( t ) el , t 2 to],

+

, c2 > 0. Assume that, for t > to , x, y

E 9,

IIf(t! x) - f d ~ > Y ) l l G g(t, II x - Y 11).

(2.7.12)

If x(t), y ( t ) be a solution and a &approximate solution of (2.7.1) and (2.7.3), respectively, on [ t o , 00) such that

< II -2^o- Y o II < uo ,

no

then p(t)

G II 4 t ) - Y(t)ll

< r(t),

t

(2.7.13)

2 to.

Proof. Let T E [to, 03). By Lemma 1.3.1, the maximal and the minimal solutions r ( t , E ) and p ( t , c) of u' = g(4 u )

v'

=

+ s(t) +

-[g(t, v)

exist for sufficiently small E

4tO)

€7

+ 8 ( t ) + €1,

= UO

v(to)= vo

+

~

E, E

> 0, and r ( t ) = lim r(t, E ) , 6-0

p(t)

=

lim p ( t , t-0

E)

uniformly on [ t o ,TI. In view of this, there exist r ( t , €1 < r ( t ) p(t9

c)

+

€1

> p ( t > - €2

,

, c2 > 0 such that

82

CHAPTER

for t for d

2

E

[t,, , TI. Furthermore, an application of Theorem 1.2.1 yields that,

E

[f,

, 71,

r ( t ) < r(4 € ) r

f(t)

At,

4.

It now follows that, f o r t E [ t o ,TI, (2.7.14)

T o prove (2.7.13), it is enough to show that

Assuming the contrary and arguing as in Theorem 2.7.1, we get either

or These relations show, because of the inequalities (2.7.14), that, in either case, x ( t l ) , y(tJ E 8. By following the rest of the standard argument, it is easy to prove (2.7.15). This completes the proof.

REMARK2.7.1. Evidently, Theorem 2.7.3 holds when the condition (2.7.12) is satisfied for all x,y E Rn instead of 8. Similar comment is valid for Theorem 2.7.1 also. T h e bounds obtained in the foregoing theorems are on a general setup. They include a number of special cases. For instance, if 6(t) I= 0, we get the estimates of the difference of solutions of (2.7.1) and (2.7.3), respectively; whereas if, in addition, f ( t , x) = fl(t, x), the same results yield the growth conditions between any two solutions of the system (2.7.1). On the other hand, if fl(t, x) f ( t , x), error estimates between a solution and a 6-approximate solution of the system (2.7.3) are obtained. Furthermore, if S ( t ) E 0 and fi(t, x) = 0, these results provide the upper and lower bounds of solutions of the system (2.7.1). ~

For future use, the following well-known result is stated as

COROLLARY 2.7.1.

Let f~ C [ J x IIf(f,

).

-

f ( f >Y)II

R",R"],and, for t 3 @,

< L(f>llx

~

Y

I/

1

x, y E R",

2.7. where L

E

83

UPPER AND LOWER BOUNDS

C [ J ,R,]. Then, for t >, to ,

where x ( t ) , y ( t ) are any two solutions of the system (2.7.1), through ( t o xo), ( t o Yo), respectively. I n the foregoing results, the upper bounds obtained are increasing functions of t, since the assumptions demand that g(t, u) 3 0 and 8 ( t ) > 0, and therefore give very little information about the growth of solutions for large time. We give below a different set of assumptions that yield sharper bounds because the function g(t, u ) need not be restricted to be positive. 9

7

THEOREM 2.7.4. Let g E C [ J x R, , R],6 E C [ J ,R,], and r ( t ) be the maximal solution of u' = g(t, u )

+qt),

u(t,) = u g ,

existing on [ t o , 00). Assume that, for t

II X - Y

+ h[f(t,

X) -fi(t,

Y>III

for all sufficiently small h

< /I x

-Y

II

E

J , x, y

+ hg(t,I/ x

E

Rn,

-

y 11)

+ O(h),

(2.7.17)

> 0. Then, II xo - Yo II < uo

implies

I/

-Y(t)ll

< r(t)l

t 3 to

1

(2.7.18)

x(t), y ( t ) being a solution and a &approximate solution of (2.7.1) and (2.7.3), respectively, existing on [ t o , 00). Proof.

Consider the function

44 We have, for small h

> 0,

=

II x(t>-Y(t>ll.

84

CHAPTER

where r ( h ) / h+ 0 as h (2.7.17), that

2

It therefore follows, using (2.7..2) and

+ 0.

D+m(t)

< g(t, m(t))+ q t ) ,

which, by Theorem 1.4. I , yields the estimate (2.7.1 8). 2.8. Componentwise bounds

Instead of the differential inequality (2.7.2), we shall be considering a system of differential inequalities given by

I Y’

- f A t , Y)l

< s(4,

(2.8.1)

where 6 E C [ ] ,R,”]. Here and in what follows, we mean by j x 1 a vector whose components are I x1 1, I x2 1, ..., I x, I for any x E R”. Note that 8(t) is a scalar function in (2.7.2), whereas it is a vector in (2.8.1). In this case, the &approximate solution of (2.7.3) must satisfy (2.8.1) in place of (2.7.2).

THEOREM 2.8.1. Let g E C [ ] x R,”, R,”] and possess the quasimonotone nondecreasing property. Let u,v, S E C [ J , R,”] such that, for t > t , , (2.8.2)

Suppose that f,fl E C [ ] x R”, RTL], and, for t

> t, , x,y E Qi, (2.8.3)

If x ( t ) , y ( t ) be a solution and a 8-approximate solution of (2.7.1) and (2.7.3), respectively, on [ t o ,a)such that

(2.8.4)

2.8.

85

COMPONENTWISE BOUNDS

Proof. T h e proof runs parallel to that of Theorem 2.7.1. However, in this situation, the assumption that the set n

2=

(J [t 3 to : q ( t ) < 1 Xi(t) -yi(t)i

< Ui(t)j

i=l

is nonempty leads to the existence of an index j , 1 t , > to such that either

< j < n, and

a

or which shows that x ( t l ) ,y ( t l )E Qj . Consequently, as in Theorem 2.7.1, it is easy to show, using (2.8.3), that - M t l9

+ %(tl)l G 4 A t d < &(tl

"1))

9

m(t1))

+ Utl).

Making use of the quasi-monotone property of g(t, u)and the arguments of Theorem I .5. I , we can prove (2.8.4). The next theorem is analogous to Theorem 2.7.2 for componentwise bounds, the proof of which can be deduced from Theorem 1.7.3, with an observation similar to that of Theorem 2.7.2.

THEOREM 2.8.2. Assume that, in place of (2.8.2) and (2.8.3), we have

for t > t o , x,y E R",other assumptions being the same as in Theorem 2.8.1. Moreover, let, for each T E [to, a),t E [to, T] and for each i - 1 , 2,..., n,

I gdt, u ) u,

>, ii, ,

4

< G(7 + to u, = U, ,

-

t , u,

-

4,

i #j,

where G E C[J x R, , R,], and r ( t ) = 0 is the maximal solution of (2.7.10). Then, the assertion of Theorem 2.8.1 remains true.

86

CHAPTER

2

THEOREM 2.8.3. Let g E C [ J x Rn+,R,"] and possess the quasimonotone nondecreasing property. Assume that r ( t ) ,p(t) are the maximal and the minimal solutions of

and, for each t

> tn , x,y I f i ( t ,x)

E Qi

-fi.i(t?Y)I

Then an

implies P(t)

,

< Ri(G I x -Y

I).

< I xn -yo I < un

< I x ( t ) -Y(t)l < r ( t ) ,

t

3 to

7

x ( t ) , y ( t ) being a solution and a &approximate solution of (2.7.1) and

(2.7.3), respectively, existing on [t, , a). T h e proof of this theorem can be constructed by following the respective arguments of Theorems 2.7.3, 2.7.1, and 1.5.1 with necessary modifications.

THEOREM 2.8.4. I,et S E C [ J ,Rtn], g E C [ / x R+n, R"], and g possess the mixed quasi-monotone property. Suppose that f,fiE C [J x Rn, Rn], a n d , f o r e a c h t > , t , , , p - 1 , 2,..., k , q = k + l , k + 2 ,..., n,

and x ( t ) , y ( t ) are any solution and a &approximate solution of (2.7.1) and (2.7.3), respectively, on [ t o , a). (i) If r ( t ) is the k max(n u'

-

= g(t, ).

k) mini-solution of

+qt),

u(t,)

=

u,

2.8.

then, for t

87

COMPONENTWISE BOUNDS

> t o ,we have (2.8.5)

whenever

for t > t o ,then (2.8.6) implies (2.8.5) provided that, for each T E [ t o ,GO), t E [to,TI, and for each i = 1, 2,...,n,

+

g i ( t , u ) -gg,(t, c)

3 -G(T to - t , ui - $), ui 3 zii , ui = z i i , i #j,

where G E C [ J x R, , R,], (2.7.10).

and r ( t ) = 0 is the maximal solution of

THEOREM 2.8.5. Let g E C [ J x R+",R,"] and possess the quasimonotone nondecreasing property. Let r ( t ) be the maximal solution of u' = g(t, 4

+qt),

existing on [to, co),where S E C [ J ,R,"].

I x -Y

+ h [ f ( t ,x) -fdt,r)ll

4td

=

uo

9

Assume that, for t

< I x -Y

I

E

J, x,y

E

+ k ( t , I x -Y I) + O(h)

R",

88

CHAPTER

for all sufficiently small h

2

> 0. Then,

~ ( t )y,( t ) being a solution and a 6 approximate solution of (2.7.1) and

(2.7.3), respectively, existing on [to, a).

2.9. Asymptotic equilibrium

We shall continue to consider the differential system (2.7.1). DEFINITION 2.9.1. We shall say that the differential system (2.7.1) has asymptotic equilibrium if every solution of the system (2.7.1) tends to a finite limit vector f as t 03 and to every constant vector .$ there is a solution x ( t ) of (2.7.1) on to t < co such that limt+mx(t) = 4. T h e following theorem gives sufficient conditions for the system (2.7.1) to have asymptotic equilibrium: ---f

THEOREM 2.9.1.

<

Let f~ C [ J x R",R"] and

where g E C [ J x R ~ ,,R,] and monotone nondecreasing in u for each t E J . Assume that all solutions u ( t ) of u'

= g ( t , u),

u(t,)

= uo

>0

(2.9.2)

are bounded on [ t o , 03). Then the system (2.7.1) has asymptotic equilibrium. Proof. Let x ( t ) be any solution of (2.7.1). Then, it is easy to deduce from Theorem 2.7.1 that

where r ( t ) is the maximal solution of (2.9.2) such that [I xo 11 = uo . Since, by assumption, every solution of (2.9.2) is bounded on [to , a), it follows from (2.9.3) that every solution x ( t ) of (2.7.1) is bounded on [to a). ?

2.9.

Furthermore, for any t

89

ASYMPTOTIC EQUILIBRIUM

> t , > t o ,we have

II 44

- x(t1)ll

<

<

st

llf(S,

tl

f1&,

e J;,.(s>

+))I1

ds

I1 x(s)ll)

ds

y(s))

ds

(2.9.4)

= r(t) - Y(tl),

using (2.9.1) and (2.9.3) and the monotonic character of g(t, u) in u. Since g is nonnegative, every solution u(t) of (2.9.2) is nondecreasing in t , and hence the boundedness of all solutions of (2.9.2) shows that the maximal solution ~ ( ttends ) to a limit as t + co. This implies that, given an E > 0, we can choose a t, > 0 sufficiently large so that 0 < u(t)

-

r(tl) < 6

for all

t

> t, .

It then follows, because of (2.9.4), that

// x(t)

-

for all

x ( t , ) / /< E

which proves that

t

> t, ,

lim x ( t ) = 5.

(2.9.5)

t-w

T o prove that the system (2.7.1) has asymptotic equilibrium, it remains to be shown that, for every constant vector 5 such that 11 5 1) uo , there exists a solution x ( t ) of (2.7.1) on [ t o ,00) such that (2.9.5) holds. For this purpose, let x,(t) be a solution of (2.7.1) such that

<

xn(t,

+ n) = 5

(n = 1 , 2, ...).

(2.9.6)

+

If r,(t) is the maximal solution of (2.9.2) with Y n ( t , n) = 1) 411, because of the nondecreasing character of every solution of (2.9.2), it follows that We claim that

/I 5 II yn(t)

<

Yn(&

+ < r(t, + n).

< r(t),

It this were not true, let, for some

t D

2 to

+ n.

> to + n,

> r(.).

(2.9.7)

90

CHAPTER

2

Then, by taking the larger of r , ( t ) and r ( t ) , we can construct a solution of 5 11) whose value at is greater than that of the maximal solution r ( t ) ,which is absurd. Hence, (2.9.7) is true. As before, for any t, > t,, and t > t, ,

(2.9.2) through ( t o ,11

(5

I1 , y j L ( t )

xn(h)Il ,<

~

<

Since 11 x j L ( t 11) r?,(t), t tonicity of g, yields that

-I

.1

tl

g(s, I/ .T;~(.~)II) ds.

3 to + n, (2.9.7), together with the mono-

I1 '%(t)

4tl)ll

~

<
Y(tl),

~

which assures that ~ , ~ tends ( t ) to a limit vector x,,(GO) as t + GO uniformly in n. Therefore, Ilf(t, x,,(t))/ / is bounded on every bounded t-interval uniformly in n. Since x X t ) = f ( t !xn(t)),

the family (.x?,(t)f is equicontinuous on every bounded t-interval. Thus, there is a subsequence { x j l k ( t ) )that converges uniformly on every bounded t-interval as k + co. As limt+mx,,(t) + x,(co) for each n, this subsequence may be chosen such that the corresponding sequence of limits {x,,,(co)} also converges as k + co. Now, {xTlb(t)}converges uniformly on [t,,, GO) to a continuous limit function x ( t ) as k + co. Evidently, x ( t ) is a solution of (2.7.1), and therefore lim x ( t ) = ~ ( c o ) .

t-.u

Also, as k -+

GO, SUP

to'

t<m

II x,,(t)

-

%,(.n)

- x(t)ll

-

07

4.n),

and these two facts, together with (2.9.6), imply that x(c0) = (. Now x ( t ) is the desired solution, and this completes the proof of the theorem.

COROLLARY 2.9.1.

If the function g ( t , u ) in (2.9.1) is of the form

' d t ,u> = X(t)&u),

2 0 is continuous for t~ J and +(u) 3 0 is continuous for 3 0, +(0) = 0, +(u) > 0, u > 0, and montonic nondecreasing in u,

where A ( t ) u

and if

(2.9.8)

the conclusion of Theorem 2.9.1 holds.

2.10.

91

ASYMPTOTIC EQUIVALENCE

Theorem 2.9.1 has a corollary for the case that (2.7.1) is replaced by x'

(2.9.9)

A(+ + F ( t , x),

=

where A(t) is a continuous n x n matrix and F Let X ( t ) be a fundamental matrix for

E

C [ J x R", R"].

X ( t o ) = unit matrix,

x' = A(t)x,

(2.9.10)

so that the transformation

x

reduces (2.9.9) to y'

=

X(t)y

(2.9.1 1)

= x-1(t )F(t, X ( t ) Y ) .

Thus, an application of Theorem 2.9.1 to (2.9.11) gives

COROLLARY 2.9.2. Let A(t) be a continuous matrix for t E J and X ( t ) be a fundamental matrix for (2.9.10). Let F E C[J x R", R"], and, for ( t ,Y ) E J x R", /I X-Yt)F(t, X(t)Y)lI < h(t)llY 11, (2.9.12) where h(t) >, 0 is continuous for t

E

J , and

s z X ( s ) ds

< a.

Furthermore, let x ( t ) be a solution of (2.9.9) on some t-interval to the right of to . T h e n x ( t ) exists for all t 2 to , lim X-l(t)x(t) = [,

(2.9.13)

t-m

and, conversely, given a constant vector f , there is a solution x ( t ) of (2.9.9) satisfying (2.9.13). An interesting special case in which the hypotheses of Corollary 2.9.1 are satisfied is that of the linear homogeneous system (2.9.10), where

1; I/

4 s ) l l ds

< a-

I t is enough to take h(t) = 11 A(t)11 and $(u) = u.

2.10, Asymptotic equivalence Suppose we are given the following two differential systems:

x'

=f1(t,

x),

Y'

=f i ( t , Y ) ,

x(h)

=

xo

Y(to) = Yo

1

(2.10.1 ) (2.10.2)

92

CHAPTER

where fi, f 2 valence.

E

2

C [ J x R”, R”]. We shall first define asymptotic equi-

DEFINITION 2.10.1. T h e differential systems (2.10.1) and (2.10.2) are said to be asymptotically equivalent if, for every solution y ( t ) of (2.10.2) [a(t) of (2.lO.l)], there is a solution x ( t ) of (2.10.1) [ y ( t ) of (2.10.2)] such that as

x(t) -y(t)+O

THEOREM 2.10.1

2)

co.

>g(t,u)

for t 3 to such that u ( t ) + 0 as t Suppose further that

+ h(fi(t,

+

Let zc(t) be a positive solution of u’

II X -Y

t

4

< I/ x -Y

-fZ(t,Y))Il

I1

GO,

where g E C [ J x R, , R].

+ hg(t, II x - y 11) + O(h)

(2.10.3)

for all sufficiently small h, t 3 to , and I] x - y 11 = u(t). Then, the systems (2.10.1) and (2.10.2) are asymptotically equivalent. If, in addition, one of the systems has asymptotic equilibrium, then the other system also has asymptotic equilibrium.

Proof. Let us first suppose that y ( t ) is a solution of (2.10.2) defined for t >, t,, . Let x ( t ) be a solution of (2.10.1), defined on some t-interval

to the right oft,, such that

/I 4 t o )

- Y(4J)ll

< .(to>.

Clearly, such a solution exists. Define Then as far as x ( t ) exists. If this assertion is false, let t , be the greatest lower bound of numbers t > t o , for which m ( t ) u(t) does not hold. Since m ( t ) and u(t) are continuous functions, we have, at t = t, ,

<

4 t l ) = 4tl)

and m(t,

+

12)

> U(t,

+ h),

h

> 0.

This implies the inequality (2.10.4)

2.10.

93

ASYMPTOTIC EQUIVALENCE

I n view of the condition (2.10.3), one also gets, at t

=

t, ,

D+m(t,) G d t l 7 m(tl)),

which is a contradiction to (2.10.4). Hence,

/I x(t) -y(t)ll

< u(t)

is true as far as x(t) exists. Now, using Corollary 1 .I .2, it follows that x(t) exists for all t 3 to , since y ( t ) and u ( t ) are assumed to exist for t 3 to . Moreover, as limL+m u(t) = 0, lim I/ x(t) -y(t)ll

t--.m

=

(2.10.5)

0.

On the other hand, if x ( t ) is a solution of (2.10.1) existing on [ t o , a), arguing as before, we can conclude that there exists a solution y ( t ) of (2.10.2) on [ t o , KI) such that (2.10.5) is satisfied. It therefore follows that the systems (2.10.1) and (2.10.2) are asymptotically equivalent. If one of the systems has asymptotic equilibrium, the asymptotic equilibrium of the other system is a consequence of (2.10.5). T h e proof is complete. T h e next theorem gives sufficient conditions for the asymptotic equivalence of the systems (2.9.9) and (2.9.10).

THEOREM 2.10.2. Let A(t) be a continuous matrix for t E J and F E C [J x Rn,R"]. Suppose that

I1F(t>41 < Yt)ll x It. where A ( t )

3 0 is continuous for

t

E

J , such that

s I h ( s ) ds

< 03.

Assume that all the solutions of (2.9.10) are bounded as t lim inf t-m

J:o

(2.10.6)

tr A(s)ds

>

+ CQ

-03.

and (2.10.7)

Then, the systems (2.9.9) and (2.9.10) are asymptotically equivalent. Proof.

Let Y ( t )be a fundamental matrix of (2.9.10). Setting Y(t),(t) = r(t),

94

2

CHAPTER

it is easy to verify that x ( t ) is a solution of (2.9.9) if and only if v ( t ) satisfies a' = Y-'(t)F(t, Y(t),). (2.10.8) Using (2.10.6), (2.10.7), and the assumption that all the solutions of (2.9.10) are bounded, we get

II W f ) F ( t ,y(wll < II W ~ ) l l IY(t)llll l 21 Ilqq

< KII

?J

llA(9,

where K is some constant. Hence, Corollary 2.9.1 implies that (2.10.8) has asymptotic equilibrium. Now, any solution y(t) of (2.9.10) can be written as Y ( t ) = Y(t)S,

4 being a constant column vector. Therefore, 44

-y(t)

=

Y(t)[,(t>- [I,

and the desired result follows, since Y ( t )is bounded on ( t o , a). T h e asymptotic equivalence of the systems (2.10.1) and (2.10.2) can also be considered on the basis of the variation of parameters formula for nonlinear systems developed in Theorem 2.6.3.

THEOREM 2.10.3.

Assume that (i)fi

,.f2

E

C [ J x Rn,Rn],

exists and is continuous on J x R";(ii) dj1(t, t, , x,)(Q2(t, t o ,yo))is the fundamental matrix solution of the variational system

2.10.

ASYMPTOTIC EQUIVALENCE

95

Then, there exists a solution x ( t ) of (2.10.1) [ y ( t ) of (2.10.2)] on [to , a) satisfying the relation lim x ( t ) - y ( t ) = 0. (2.10.10) i-m Pyoof. Let y ( t ) = y ( t , to , y o ) be a given solution of (2.10.2) existing on [ t o , a).Define a function x ( t ) by the relation x(t) =Y(t)

+\

W

' t

@l(hs,y(s))[f,(s,y(s)) -fl(s>Y(S))lds.

(2.10-11)

Since the integral converges by assumption (iii), it follows that x ( t ) is well defined, and, consequently, (2.10.10) is satisfied. It therefore suffices to prove that x(t) is a solution of (2.10.1). For this purpose, we observe, as in Theorem 2.6.3, that

Here use is made of the relation (2.5.11) and the fact that

T h e relations (2.10.1 1) and (2.10.12) yield

Moreover, we have

96

CHAPTER

2

1,et us differentiate (2.10.1 l), recalling that y ( t ) ,Q l ( t , t, , x,) are the solutions of (2.10.2) and (2.10.9), respectively, and using (2.10.12) to obtain

This reduces to, in view of (2.10.14),

T h e relation (2.10.13) implies that x ( t ) is a solution of (2.10.1) with

On the other hand, if x ( t ) is a solution of (2.10.1) existing on [ t o , a), we can show exactly in a similar way that there exists a solution y ( t ) of (2.10.2) on [ t o , m) such that (2.10.10) holds. Thus the theorem is established.

2.1 1 . A topological principle

This topological principle is concerned with the differential system Y'

= f ( t , x),

.(to)

= X"

,

to 2 0,

(2.1 I. 1)

where f E C [ E ,RT1],E being an open ( t , x)-set in Rn+l. Let E, be an open subset of E , r?E, the boundary, and l?, the closure of E, .

DEFINITION 2.1 1.1. A point ( t o , x,) E E n aE, is said to be an egress point of E,) with respect to the system (2.11.1) if, for every solution x ( t )

2.1 1.

97

A TOPOLOGICAL PRINCIPLE

< <

of (2.1 l.l), there is an E > 0 such that ( t , x ( t ) ) E Eo for to E t to . An egress point ( t o ,xo) of Eo is called a strict egress point of E, if to E , for a small E > 0. ( t , x ( t ) ) E, for to < t T h e set of all points of egress (strict egress) is denoted by S(S*). It is clear that S* C S. ~

< +

DEFINITION 2.1 I .2. If A C B are any two sets of a topological space and T : B 4 A is a continuous mapping from B onto A such that W( p ) = p for every p E A , then T is said to be a retraction of B onto A. When there exists a retraction of B onto A , A is called a retract of B. T h e following examples would sufflce to illustrate the concept of retraction.

<

Example 1. Let B = [x E R" : /I x 11 cx] and A = [x E R" : /I x 11 = a ] . T h e n A is not a retract of B. For, if there exists a retraction w : B + A , then there exists a continuous map of 3 into itself, x 4 -n(x), without fixed points. This contradicts the Brouwer's fixed point theorem. Example 2.

Let C = [ ( x , u ) E Rnfm: 11 x

(1

[(x, uo) E Rnim: /I x 11

B

=

A

=B

nC

=

=

01,

<

01,

u

arbitrary],

uo fixed],

[(x, uo) E Rn+" : 11 x I/

=

01,

u,fixed].

From example 1 , it can be seen that A is not a retract of B, whereas it is a retract of C, because we can choose a retraction ~ ( xu, ) = (x,u,).

THEOREM 2.1 1 . 1 . Let f E C [ E , R"], where E is an open ( t , %)-set in Rn+l. Assume that, through every point of E , there passes a unique

solution of the system (2.11.1) and that the solutions depend continuously on initial values. Let Eo be an open subset of E . Suppose that all egress points of E, are strict egress points, i.e., S = S*. Let Z be a nonempty subset of Eo u S such that 2 n S is a retract of S , but is not a retract of Z. Then, there exists at least one point (to , xo) E Z n Eo such that the solution arc ( t , x ( t ) ) of (2.1 1.1) remains in E , on its maximal interval of existence to the right of t o . Proof. Suppose that the conclusion of the theorem is not true. Then, for every ( t o ,xo) E Z - S, there exists a t , = t , ( t o , x,), t , > to such that the solution x ( t ) = x(t, to , xo)of (2.1 1.1) exists on [to,t , ] , ( t , x ( t ) )E E, for [to , t,) and ( t , , x ( t , ) ) E S.

98

CHAPTER

Define a map T , : Z (I)

~ , ) ( ,t x,) ,

-

(ii) ~ , ) ( t,,xt,)

--f

2

S such that

( t , , x(tl))

( t o , x,)

if if

( t o ,x,)

, x,)

(f,

E

EZ

~

S;

Z n S.

We shall show that no is continuous because of the assumption S = S* and the continuous dependence of the solutions on the initial values. Let ( t o , x,))E Z n En and ( t * , x*) be sufficiently near to ( t o ,xo). T h e n the solution x ( t , t*, x*) exists on [t*, t , €1 for some small E > 0 and

+

(i)

( t ,s ( t , t * , x*))

E

E, ,

[ t * , t,

(ii) ( t , r(t, I*, x*)) $ E,, ,

t

=

-

t,

€3;

4-e .

This implies that, at t t t>(t*,x*), ( t t ,x(tf, t", x*)) E S and E , which shows that (tf, x(tF, t*, x*)) is a continuous j tt t, I function of ( t %, x*). T h i s proves that T , is continuous at ( t o ,xo). A similar reasoning holds if (to , x,) E 2 n S. 1,et T be a retraction of S onto 2 n S. I t then follows that the composite map m r , : 2 + 2 n S is a retraction. This contradicts the assumption that X n S is not a retract of 2. T h e theorem is proved. -

T o give an idea of the interplay of the conditions in Theorern 2.1 1.1, let u s consider (2. I l . l ) , where E = J x R,E, = [(t,x) : t E J , 1 x I < 61. T h e boundary aE0 consists of the half-lines x : fb. T h e assumption that f ( t , 6) > 0 , f ( t , -6) 0 guarantees that S = S" = aE0. T h e set Z can be chosen as 2 [ ( t ,x) : t t, , 1 x I < b] and 2 n S as the set of two points (to , i b ) . T h e n 2 n S is a retract of S but not of 2. Theorem 2.1 1 . I now shows that there exists at least one point (to, xo), 1 yo 1 6, such that a solution of (2.1 1.1) exists and satisfies I x ( t ) I < b for t 2 t o . Given a differential system, the choice of the set E,, , for which Theorem 2.11.1 can be applied successfully, may be rather difficult. However, in some cases, it is possible to overcome this difficulty. 1,et ZL E C [ E ,K"] and x ( t ) be a solution of (2.1 1.1). T h e function u ( t , x) is said to possess a trajectory derivative u'(t, x) at the point (t,), xo) along the solution x ( t ) of (2.1 1. l ) if u(t, x ( t ) ) has a derivative at t t, , in which case

-

-

-

:

.'(to

>

=

Luff, x(t))l;pt".

If ~ ( tx), is continuously differentiable with respect to ( t , x), the trajectory derivative d ( t , x) exists and is equal to

2.1 1.

99

A TOPOLOGICAL PRINCIPLE

where the centered dot denotes the usual scalar product of vectors. T h e following theorem provides a suitable choice of E, and the set of egress points in terms of certain functions.

THEOREM 2.11.2. Let f E C [ E ,R"], u E C [ E ,R p ] , and u E C [ E ,R*].Let (i) E,

=

[ ( t ,x) : u j ( t ,x)

< P , 1 < fz < 41; L, = [ ( t ,x) : u,(t, x) = 0 and u3(t,x) < 0, %(t, x! < 0, 1 < j < P , 1 < < sl; M , = [ ( t ,x) : v,(t, x) = 0 and u,(t, x) < 0, %(t, x) < 0, 1 < j < P , 1 < < 41. %(t, ).

(ii)

(iii)

and

<0

< 0, 1 < j

Assume further that the trajectory derivatives ui(t, x), v;(t, x) exist on

L, , M8 and satisfy

u x t , ).

> 0,

( t ,). E L , ,

4 ( t ,).

< 0,

( t ,).

E MI3

(2.11.2) (2.1 1.3)

,

respectively, along all solutions through ( t , x). Then 3)

a

,=l

B=1

UL,-uM,

S=S*=

Proof. We shall first show that S n M , is empty. For, if ( t o ,x,) and x ( t ) is a solution of (2.11. l ) , from (2.11.3) we have vB(t,x ( t ) )

> 0,

for small

[to- E , to)

E

E

M, ,

> 0,

which shows that ( t , x ( t ) )4 E, because of (i). This means that ( t o ,xo) is not a point of egress. Since

it follows that a

21

s*csc(a~,,nE)-( J M , C U L , a=l

a-1

On the other hand, if

u n

( t o > xo) E

m=l

u a

L,

-

B=1

MB

7

a

( J M ~ ~ . (2.11.4)

8-1

100

CHAPTER

2

<

then, from (ii) and (iii), we get uj(to, xo) 0 and vk(t0, x,,) < 0, j I ,..., p , k I ,..., q. T h e assumption (2.11.2) yields that there is an E :> 0 such that 2

~

if ( t o, xo) $ L j ; and + ( t , x(t))

< 0,

[ t o - E , t,

+ €1

for all k.

Hence, ( t o ,xo) E S*, and

4-1

n=l

This, together with (2.11.4), establishes the theorem. 2.12. Applications of topological principle

Considcr the two differential systems x' = f d t , 4,

(2.12.1)

Y' = f i ( t , Y ) ,

(2.12.2)

where fi ,f.L E C [ J x R",R"]. Let g E C [ J x R+ , R] and u(t) be a positivc solution of the differential inequality u'

< g(t, u).

(2.12.3)

Givcn that y ( t ) is a solution of (2.12.2) for t there is a solution x ( t ) of (2.12.1) such that

II x ( t ) - Y(t)ll < 4 t h

t

3 t o , we shall show that 3 to .

(2.12.4)

THI~OREM 2.12.1. Let fi ,f2 E C [ ] x Rn,R"] and g E C [ J x R+ , R]. Assume that the systems (2.12.1) and (2.12.2) possess unique solutions through every point and that the solutions depend continuously on the

2.12.

APPLICATIONS OF TOPOLOGICAL PRINCIPLE

101

initial values. Let y ( t ) be a solution of (2.12.2) and u ( t ) a positive solution of (2.12.3), for t >, t, . Suppose further that

+ h ( f l ( t , 4 -f2(t,Y(t)))ll 2 I/ x Y(t)ll + hg(t, II x Y(t)ll)

II x

-Y(t),

-

(2.12.5)

-

for all sufficiently small h and 11 x - y ( t ) /I = u(t). Then, if T > t, is given, there exists a solution x(t) of (2.12.1) defined for t >, T satisfying (2.12.4) for t 3 T . Proof. We wish to apply Theorems 2.11.1 and 2.11.2 to deduce the result. Defining =

[ ( t ,4 : I/ x -y(t)ll

u(t,

=

II x

u(t, x)

=

to - t ,

4

-y(t)ll

-

< 4% t > t o ] ,

4th

it follows that

< 0, v(t,x ) < 01,

Eo

=

[ ( t ,x ) : u ( t , x )

L

=

[ ( t ,x ) : u ( t , x) = 0,v(t,x )

M

=

[ ( t ,x ) : u(t, X)

< 01,

< 0,~ ( t , = 01. X)

If ~ ( t= ) /I x ( t ) - y ( t ) 11, where x ( t ) is a solution of (2.12.1) such that, for some t = t , >, t, , a ( t l ) = u(tl), the condition (2.12.5) yields a(t1

+ h)

= =

I1 “ ( t ,

+ h)

+ h)ll

-At1

II 4 t l ) + hfl(t1

2 4 t l ) + hg(t1 >

+4 4 4 G ) ) + +), 7

x(t1))

- Y ( t d - hf2(tl

where el(h)/h,e,(h)/h, and e(h)/hall tend to zero as h the inequality m’(t1)

3d t l * 4 t l ) )

=d t l

,4 t l ) ) .

Using the inequalities (2.12.3) and (2.12.6), we obtain u’(t, , XI)

= a’(tl) - U ’ ( t l )

3 g(t1 > 4 t l ) ) 0,

(tl

3

-

u’(t1)

x1) E L ,

where x1 = x(tl). Moreover, v’(t, , xl)

=

-1,

(tl , X1) E M.

7

Y(fl>) - 62(h)ll

+ 0.

This implies (2.12.6)

102

CHAPTER

2

Thus, in view of Theorem 2.1 1.2, we have

,c = s*= L - M. Let

T

> t,, be given. Since AS

=

[ ( t ,x) : t

> t o ,[ ( x -y(t)ll

defining [ ( t ,-y) : t

7,

one sees that Z nS

-

[ ( t ,x) : t

=

11 %.

= T,

- y(t)l/

=

u(t)],

< U(T)],

/ / x - y ( ~ ) l= l .(.)I.

Observing that Z is a closed ball in Rn and 2 n S is the boundary of the ball 2 in R", it is clear that 2 n S is not a retract of 2.However, the mapping

T

:S

S n 2 given by

~ ( tx), x*

-

( t * , x*),

34.1

-~

+

with

t*

= T,

("X - Y ( w 4 T ) / u ( t ) l ,

is a retraction. Consequently, we conclude from Theorem 2.1 1.1 that there exists at least one point ( T , xo) E Z - S such that the solution arc ( t , x ( t , T , xJ) of (2.1 2.1) remains in Eo on its maximal interval of existence. Sincc z r ( t ) and y ( t ) cxist for all t T 3 t o , it follows that the maximal intcrval of existence of x ( t ) is [ T , a). Hence (2.12.4) holds, and the proof is complete.

REMARK 2.12.1. If, for every solution y ( t ) of (2.12.2) [ x ( t )of (2.12.1)], there exists a t,, and g(t, a ) such that (2.12.3) is satisfied by some positive function u ( t ) that tends to zero as t -+ GO and (2.12.5) is satisfied for 11 A y ( t )11 - u(t) (11 ~ ( t ) y 11 u ( t ) ) , then the systems (2.12.1) and (2.12.2) are asymptotically equivalent. :

2.13. Stability criteria Wc consider the differential system x' - f ( t , x),

x(to) = xo ,

where f E C [ / x S , , R"],S, being the set So

--

[x E R" : I/ x / j

< p].

(2.13.1)

2.13.

103

STABILITY CRITERIA

Assume that f ( t , 0) = 0, so that (2.13.1) admits the trivial solution. Let x ( t ) = x(t, t , , x,,) be a solution of (2.13.1) through ( t , , x,,).

DEFINITION 2.13.1. T h e trivial solution of (2,13.1) is said to be (i) stable if, for every E > 0 and to E J , there exists a 6 > 0 such that Ij x, 11 < 6 implies /I x ( t ) 11 < E , t to; (ii) asymptotically stable if it is stable and if there exists a 6, > 0 such that 11 x, 11 < 6, implies x ( t ) + 0 as t -+ co.

>

THEOREM 2.13.1. Let g E C [ J x R , , R] and g(t, 0) the functionf(t, x) satisfies

II x 4-W t , .)I1

< II x I1 -1- Mt,II x Ill-t O(h)

for ( t ,x) E J x S, and for all sufficiently small h or asymptotic stability of the trivial solution of u’

= 0. Assume

= g(t, u),

that

(2.13.2)

> 0. Then, the stability (2.13.3)

u(t,) = u,,

implies likewise the stability or asymptotic stability of the trivial solution of the system (2.13.1). Pyoof. Let the solution u = 0 of (2.13.3) be stable. Then, given 0 << E < p, and to E J , there exists a 6 > 0 with the propcrty that

u,,< 6 implies u(t, t , , u,,) < E , t 3 t o . It is easy to claim that, with these E and 6, the trivial solution of (2.13.1) is stable. If this were false, there would exist a solution x(t) of (2.13.1) and a t , > to such that

I1 x(t,)ll For t

E

=

/I x(t)ll

E,

<

t,

E,

< t < t,

[ t o ,tl], using the condition (2.13.2), it follows that D+m(t>

< s(t,m(t>),

(2.13.4)

where m(t) = /I x ( t ) 11, and hence, by Theorem 1.4.1, choosing 11 x, we obtain

/I 4t)ll

< ~ ( tto,

7

II xn II),

11 = u 0 , (2.13.5)

t E [to

where r(t, t, , 11 x, 11) is the maximal solution of (2.13.3). At t we therefore arrive at the following contradiction: t

=

I/ X(tl)ll

thus justifying our claim.

< f(t,

>

to

7

II Xo lo <

€7

=

t,

,

104

CHAPTER

2

Suppose that the solution u - 0 is asymptotically stable. Since this implies, by definition, stability of u = 0, the stability of the trivial solution of (2.13.1) is a consequence of the foregoing argument. This means that the inequality (2.13.4) holds for all t 3 to , and hence (2.13.5) is valid for t >, t,, . It is now clear, by hypothesis, that, if /I xo I( < S o , lim->, x ( t ) = 0. 'The proof is complete. Let us demonstrate the significance and practicability of the assumptions of Theorem 2.13.1 by an example. Letf(t, x) = Ax,where A is an n x n constant matrix. Since

I1 A! it follows that

+ hAx Ii < I/ I + hA /Ill x II,

'Thus, defining the logarithmic norm p ( A ) == lim

[IlZ -1

h-O+

where e(h)/h-+ 0 as h

-j

h

I1 - 11

(2.13.6)

0. T h e function g(t, u ) is therefore given by g(t, u ) = p(A)u.

Clearly, g(t, 0)

E

0, and the general solution of (2.13.3) is u(t, t , uo) = uo = p w w

-

4~1.

<

Thus, the trivial solution u = 0 of (2.13.3) is stable if p ( A ) 0 and asymptotically stable if p ( A ) < 0. Hence, the stability or asymptotic stability of the trivial solution of (2.13.1) follows from Theorem 2.13.1. From the definition (2.13.6), it is easily seen that p ( u A ) = O(p(A),

O(

3 0,

< II A 4'2 + B) < 44 + P ( m I P(A)I

(2.13.7) (2.13.8)

/I7

(2.13.9)

and, from (2.13.8) and (2.13.9), IPV) -

< /I '4

-

B

ii.

(2.13.10)

2.13.

105

STABILITY CRITERIA

T h e value of p ( A ) depends on the particular norm used for vectors and matrices. For example, if /I x 11 represents the Euclidean norm, p ( A ) is the largest eigenvalue of & [ A A*], A* being the transpose of A, whereas the corresponding matrix norm /I A I/ is the square root of the largest eigenvalue of A*A. On the other hand, if I] x jl = I xi I, and 11 A I/ = sup,, IT=, I aili 1, then

+

r

n

1

We further remark that every eigenvalue of A has real part less than or equal to p ( A ) . For, if h is an eigenvalue of A , and x a corresponding eigenvector of norm 1, then il(I+ h A ) x 11

~

=

jl x 11

< II Z + hA /I

On the other hand, ll(I+hA)x 11

~

<

+ hX 1

11 x 11

11

-

-

1

-

h Re X

1

hp(A)

for

h + Of.

for

h + O+.

Therefore, Re h p(A). Let us now take f ( t , x) = A(t)x, where A(t) is a continuous n x n matrix on J. I n this case, g(t, u) = p [ A ( t ) ] u . We observe that p [ A ( t ) ] is continuous on J , by virtue of the inequality (2.13.10) and the continuity of A(t).T h e general solution of (2.13.3) is of the form u(t, to uo) 7

=

uo exp

and, hence, the trivial solution u lim sup t-m

and is asymptotically stable if lim sup t -rm

Jlo

]lo

=

[Jt

to

P[A(S)I ds],

0 of (2.13.3) is stable if

p[A(s)]ds

< 00

p[A(s)]ds =

03.

Therefore, the corresponding stability properties of the trivial solution of (2.13.1) follow from Theorem 2.13.1.

THEOREM 2.13.2. Assume that (i) f E C[J x S o ,R"],f ( t , 0) E 0, and fz(t, x) exists and is continuous on J x So;(ii) y [ f z ( t ,O)] satisfies 1

t

lim sup -t-m t - t o J t op[fx(s, O)] ds

=

01

< 0.

Then, the trivial solution of (2.13.1)is asymptotically stable.

(2.13.11)

106

2

(:HAPTER

Proof. Since f ( t ,0) = 0, given such that

E

> 0, it

f ( t , x) = J;.(t, 0)x

is possible to find a S ( E ) > 0 (2.13.12)

4-F ( t , x),

wherc

~ l F ( t , x ) l ~ cllx~l if 11 x / j < 6 (2.13.1 3) uniformly in t. Let E '> 0 and t, E be given. By the condition (2.13.1 l ) , it follows that we have, for large t > t,, , =<

i;g, d . f A S , 0)l and, if

E

is small enough, lim cxp[c(t

t .a

~

Cis

< 3(t

=

t"),

~

J" p[.f,(s, 011 ds] =:

to ) -1-

0.

(2.1 3.1 4)

f0

Thus, K

01

max e x p [ c ( t

t0

-

t,'Y>

exists, and we choose K

to)

+ 1' p[fs(.s, o)] is] '0

2 1 and 6, such that KSl

< 6(c).

(2.13.1 5 )

11 x(,11 -- 6, , we claim that jl x ( t ) -:'6, t 3 t o , where ,x(t, t , , x,,) is any solution of (2.13.1). If this were false, there

Then, if

x(t)

~

exists a t ,

> t, with the property that 11 'Y(t,)lJ

=

6,

11 x(t)ll

< 6,

t"

< t < t, .

(2.13.16)

Defining nr(t) = / / x ( t ) 11, we observe that

I! , x ( t ) I- h f ( t , s(t)lI < 11 I

+ 4fz(fv0)Iiil 4t)JI + hll F ( f ,x(t))ll,

for t E [t, , t,], because of (2.13.2), and hence ?at)

.< p[f&

O)l.z(t)

+ 11 F ( t , .2.(t))ll,

which, in view of thc relations (2.13.16) and (2.13.13), yields m; ( t )

,< [ P [ f n . ( f ,0)l

I - EIm(t),

t

E [to

t

tll.

Theorem 1.4.1 then implies, choosing u ~= , m(t,), that, for t m(t)

.< m(to) exp [ ~ (--t to)-1-

it '0

p [ f z ( s , O)] ds],

(2.1 3.17) E

[to , t J ,

(2.13.18)

2.13.

107

STABILITY CRITERIA

and we are led to an absurdity:

because of relations (2.13.15), (2.13.16), and the fact that /I xo 1) < 6,. This proves that, whenever 11 xo ( 1 <’ 6,, we have 11 x ( t ) 11 < 6, t 3 t o , and therefore the inequality (2.13.17) is true for all t 3 to . Consequently, (2.13.18) holds for all t 3 t, . It now follows from (2.13.14) and (2.13.18) that Iim{+, x ( t ) = 0, if E is small enough, which establishes the stated result.

THEOREM 2.13.3. Assume that the hypothesis of Theorem 2.13.1 holds except that the inequality (2.13.2) is replaced by x . f ( t ,x)

< I1 x llR(t, II x II),

( t ,).

E

J x

AT,.

(2.13.19)

Then, the conclusion of Theorem 2.13.1 remains valid. Proof,

We proceed as in Theorem 2.13.1 to get

I/ 4tl)lI

= E,

II 4t)lI

e

Now, using (2.13.19) and setting m ( t ) the inequality m(t)m’(t)

e

E,

=

to

< t < t,

11 x ( t ) (1, we obtain, for t E [ t o ,t l ] , (2.13.20)

g(t, m w .

Choose 11 xo 11 = uo . We wish to prove the relation (2.13.5). For this purpose, it is enough to show that

where u(t, E ) is a solution of

being sufficiently small positive quantity. Assuming the contrary and following the proof of Theorem 1.2.1, we arrive at a t, , to < t, t, such that E

<

m(t2) = U ( t , ,

c),

m’(t,)

2 u’(f,,

€>.

(2.1 3.22)

108

2

CHAPTER

Ry assumption on g(t, u), we see that ~ ( t , c,) > 0, and therefore m(t,) > 0. Thus, the relations (2.13.20), (2.13.21), and (2.13.22) lead to the contradiction g(t2 , u(t2 , .))

proving m(t)

+ eg(t2 , m(t2)), E

u(t, E ) , t E [to , t J , which implies (2.13.5) because of the

,<

fact that lim,,,, u(t, c ) = r ( t , to , 11 xo 11) uniformly on [to, tl]. T h e rest of the proof is the same as in Theorem 2.13.1.

2.14. Asymptotic behavior

We shall present here several results on the asymptotic behavior of solutions of differential systems.

THEOREM 2.14.1. small h > 0,

I1 .t'

~

Assume that F

<

Y I- h(F(t7x) - F(~,Y))II /I x

E

-

C [ J x Rn, R"], and, for sufficiently Y I/

+ hg(f, I/ x -yII) + O(h),

(2-14.1)

where g E C [ J s R, , R]. If every solution u ( t ) of u' = g ( t , u )

+ II F ( t , 0)lL

(2.14.2)

44,) = *o > 0

tends to zero as t + CO, then every solution x(t) of x'

Proof.

= F ( t , x),

Let x(t) be any solution of (2.14.3) such that 11 xo 11 m ( t ) = I1

Th en , for small h

---f

~

<

ZI,

. Define

4t)N

> 0, we have

whcrc eCh)/lz 0 as h 0, we obtain

y

(2.14.3)

%(to)= xo

-

0. Now, using the condition (214.1) with

2.14.

109

ASYMPTOTIC BEHAVIOR

and this gives, by Theorem 1.4.1, the estimate

< r(t),

/I 4t)lI

t

3 t"

(2.14.4)

f

where r ( t ) is the maximal solution of (2.14.2). T h e conclusion follows from the hypothesis and the relation (2.14.4).

THEOREM 2.14.2.

Assume that F E C [ J x R", R"] and (aF/ax)(t, x) exists and is continuous on J x R". Let x . [ H ( t ,x)

+ f f * ( t ,41. < 211 x Ilg(t, I1 x II),

(2.14.5)

where H * is the transpose of H , which is given by H ( t , x)

=

J1

n

(aF/ax)(t,xs) ds,

and g E C [ J x R, , R]. Then every solution x ( t ) of (2.14.3) tends to zero as t -+ 00, if every solution u ( t ) of (2.14.2) tends to zero as t + co. If, in particular, F ( t , 0) = 0, then, the trivial solution of (2.14.3) is asymptotically stable whenever the null solution of (2.14.2) is asymptotically stable.

Proof.

If x ( t ) is any solution of (2.14.3), write m"t)

=

11 X ( t ) l l 2 .

Then 2m(t)m'(t) = 2x(t) - F ( t ,x(t)).

Since F ( t , x) - F ( t , 0) =

J

aF( t ,xs) ~-

ax

0

ds

. X,

using (2.14.5), we get the inequality 2m(t)m'(t) G 211 4t)llllF(t,0)Il

+ 211 x(t)llg(t, 11 4t)Ih

which implies, arguing as in Theorem 2.13.2, that

/I 4t)ll

< r(t),

t

2 to,

r ( t ) being the maximal solution of (2.14.2) such that stated result is clear from this estimate.

THEOREM 2.14.3.

11 x,,/I

< uo . T h e

Let U ( t )be the fundamental matrix solution of x'

=

A(t)x,

(2.14.6)

110

CHAPTER

2

A ( t ) being a continuous n x n matrix and U(t,) F E C[/ x Rn, R r t ]F, ( t , 0) == 0, t E J , and

I1 U-'(t)F(t, r4t)Y)ll

=

unit matrix. Let

< g(4 I/ Y ll),

(2.14.7)

where g E C [ J x R , , K,]. Assume that the solutions u(t) = u(t, t o ,u,,) of

u' = g ( t , u ) ,

(2.14.8)

u(t,) = U"

are bounded for t 2 t o . Then, the stability properties of the linear differential system (2.14.6) imply the corresponding stability properties of the null solution of x'

Proof.

=

- l ( t ) x -tq t , x ) ,

x(t,)

= X"

.

(2.14.9)

T h e linear transformation x =

reduces (2.14.9) into

U(t)y

WW(t, U ( t ) Y ) .

y'

(2.14.10)

<

Let y ( t ) be any solution of (2.14.10) such that y(tJ - x,,and 11 x, 11 uo . Then, if m(t) - /I y ( t ) 11, it is easy to obtain, in view of (2.14.7), the differential incquality D+m(t)

' .<

g ( t , m(t)),

and hence, by Theorem 1.4. I , we arrive at the inequality lIY(t)ll

< r(t)9

t

to

(2.14.11)

?

r ( t ) being thc maximal solution of (2.14.8). If x ( t ) is any solution of (2.14.9), we deduce, from the relation (2.14.1 I ) and the transformation x L'(t)y, that II x(t)ll < Y ( t ) l l q t ) l l , t 2 t" . (2.14.12) ~~

Since all the solutions of (2.14.8) are assumed to be bounded, it follows from (2.14.12) that the stability properties of the null solution of (2.14.9) are implied by the corrcsponding stability propertics of (2.14.6).

THEOREM 2.14.4. Assume that the fundamental matrix solution U ( t ) of the linear system (2.14.6) vcrifies

11

U(t)ji

<M

11 U-l(s)lI

< M,

t,

< s < t.

(2.14.13)

2.14.

111

ASYMPTOTIC BEHAVIOR

(2.14.14)

(2.14.1 5 )

Then, the stability properties of the null solution of (2.14.9) depend on the corresponding stability properties of the linear system (2.14.6). Proof. By Theorem 2.6.2, any solution x ( t ) of (2.14.9) satisfies the integral equation ~ ( t=) U(t)xo -t

SI

U(t)U-l(s)F(s,x(s)) ds.

(2.14.16)

T h e estimate (2.14.14) reduces (2.14.16) to the inequality

It x(t)ll

e II u(t)ll[llxo II +

Jt

I1 U-Ys)lIYs)lI

x(s)ll dsl,

t0

which, by writing m(t) = 11 x(t)li/ll U ( t )11 and using (2.14.13), shows that

for some constant K > 0. T h e conclusion is now immediate.

THEOREM 2.14.5. Let the assumptions of Theorem 2.14.4 hold except that the inequalities (2.14.13) are replaced by

11

u(t)il< M ,

11 u(t)U-l(s)II < N ,

to

< s < t.

(2.14.17)

Then, all solutions x ( t ) of (2.14.9) exist for t 3 to and verify the estimate

/I 4t)Il

< KIIx,

11,

t 3 to

7

(2.14.18)

for some K > 0. If, in addition, y ( t ) is the solution of the linear system (2.14.6) with y(to) = xo such that limt+wy ( t ) = 0, then 1imL.,%x ( t ) = 0.

I12

CHAPTER

2

Pmof. T h e integral equation (2.14.16) gives, using the conditions (2.14.17) and (2.14.14),

I/ ?c(t)Il /<

+ J^'

I1

Mll.~"

tu

which, by Theorem 1.9.1, leads to

/I 4t)ll

t 3 to ,

MA(s)lI x(s)ll ds,

< Mll Xo I1 exp

[I:w 4

< A-11 .L(,/I,

t

U

25 to ,

> 0, proving (2.14.18). If y ( t ) is the solution of (2.14.6) such that Iimt+=y ( t ) = 0, given E > 0, there exists a T ( E )such that 11 y(t)li < E for all t >, T ( E ) .Hence, for t 22 T(c), we h a w , successively, using (2.14.17) and (2.14.14), for some constant K

s(t) =y ( t )

<

/ / .v(t)lj

+

E

Jt

U(t)U-l(s)F(s,x(s)) ds,

'0

exp

MA(s) ds

J^:,,

< RE,

for some constant k > 0, which is independent of that lirnl ,I, x ( t ) = 0.

E

and T . This proves

THEOKIJM 2.14.6. Assume that (i) A is an n x n constant matrix and the characteristic roots of A have negative real parts; (ii) F E C[J x R", Rn], and, given any E > 0, there exist 6(t), T ( E )> 0 such that

I/ F ( t , .)I1

< €11

X

I/

provided 11 x 11 < 6 ( ~ )and t 3 T ( E ) (iii) ; G E C [ J x Rn,R"] and there OL and t E J , exists an .I >- 0 such that, if 11 x /I i

I1 G(t,xlll where y

E

< 74%

C [ J ,R,] and tfl

p(t) =

j t

y(s) cis -+

o

as

t

+

co.

2.14.

113

ASYMPTOTIC BEHAVIOR

Then, there exist To 3 0, 6 > 0 such that, for every to 2 To and I/ xo 11 < 6, any solution x ( t ) = x(t, to , xo) of the differential system X' =

A X + F ( t , X) + G(t,x),

satisfies

lim x ( t )

t t m

=

x(to) = xo

(2.14.19)

0.

If, in particular, (2.14.19) possesses trivial solution, then the trivial solution is asymptotically stable.

Proof. o

By assumption (i), it follows that there exist constants K

> 0 such that

11 eAt Ij

< KcUt,

t

0.

2 1 and (2.14.20)

Choose E so that 0 < E < min(a/K, a). Because of assumption (ii), we can choose T ( E )3 1 and 6 ( ~ ) E . Let To 3 T ( E )be so large that t To implies that

<

1:

exp[-(o

-

&)(t

- s)]y(s) ds

< S(e)/2K = 6, .

(2.14.21)

We shall prove below that such a choice is possible. Observe that

= Jt

Also, for p

~ ( udu )

t0

> 0,

whence e-Bt

J' eBsy(s) ds < e-Bt 1

Applying L'Hospital's rule on

for t

> to > 1.

I14

CHAPTER

2

it can be easily verified that

j t essy(s)ds

lim

t-m

for all /3

=0

1

> 0. T h e validity of (2.14.21) is now clear.

Let to 2 To and 11 x,,// x ( t ) = exp[A(t

-

-26, .

to)]x,

Then, as long as 11 x(t)/(< 6(~), we have

+

/:o

eA(t-s)[F(s,x(s)

+ G(s,x(s))]

ds,

from which, using assumptions (ii), (iii), and the estimate (2.14.20), it follows that

II x(t)lleuf

+

< Kal exp(ot,)

By Corollary 1.9.1, we obtain /I x(t)/jeot < kT6, exp(ut,,) exp[Kc(t

~

so that

I/ x(t)ll

< KS, e x p - ( o

-

Kc)(t - t,,)]

i:

to)]

[ell

x(s)ll

+

1'

+ y(s)]KeuSds.

Keusy(s)exp[Kc(t - s)] ds,

to

+ K It

exp[-(a

-

Kc)(t

~

(2.14.22)

which, using (2.14.21), yields

11 x(t)]l < KSl <<

s)] y(s) ds,

to

KS,

+ K J: exp[-(o

4-$a(€)

=

Kc)(t - s)] y(s) ds

~

S(r).

Thus, the inequality (2.14.23)

II 4t)ll <

holds on [to , a), which implies that (2.14.22) is true for t 3 to . Hence, lim x ( t )

t-m

= 0.

If x = 0 is the solution of (2.14.19), it is clearly asymptotically stable. For, we have immediately from (2.14.23) that (( x(t)\( < E whenever // x,,/ [ < a(<), since we have chosen S ( E ) E . T h e proof is complete.

<

Notice that p ( t ) = J, y(s) ds + 0 as t as t --t n3 or J" y(s) ds < co . 0 This observation suggests the following: 1+1

-+

co holds if either y ( t )

---f

0

2.14. COROLLARY 2.14.1. system

115

ASYMPTOTIC BEHAVIOR

T h e conclusion of Theorem 2.14.6 is valid for the

x' = Ax

+ F ( t , + Gl(t, + G2(t, 2)

x)

2)

if conditions (i) and (ii) of Theorem 2.14.6 hold and, for small 11 x 11, G,(t, x) + 0 as t + m uniformly in x and 11 G,(t, x) I/ h(t)l/x 11, where X E C[J,R,],and 1 : X(s) ds < co.

<

THEOREM 2.14.7.

Assume that f E C [ / x S,, , R"],f ( t , 0) exists and is continuous on J x S , , and, for t E J ,

IIfAt, 4 -fdt)

0)li

< K /I x I/.

= 0, f,(t,

x)

(2.14.24)

Suppose that lim sup ( t t-m

-

j t p [ f z ( s ,O)] ds

t,)-l

= uo

to

< 0.

(2.14.25)

Let y(t) be the solution of the variational system Y'

where x(t)

=

=fz(t, x ( t ) ) y ,

for t

(2.14.26)

x(t,) =

X"

(2.14.27)

3 to . Then, we have the estimate

3 to and lim,,,y(t)

Proof.

= xn

x(t, to , xo) is the solution of x' = f ( t , x),

existing for t

to>

=

0.

Consider the function m ( t ) = IIy(t)ll. Observe that

where e(h)/h-+ 0 as h -+ 0. Furthermore, from the definition of p ( A ) given by (2.13.6), the inequality (2.13.10), and the assumption (2.14.24), we obtain

~ [ f d 4t))l t, Hence, it follows that

< &it7

0)l + Kli 4t)ll.

(2.14.29)

116

CHAPTER

2

-4s thc hypotheses of Theorem 2.13.2 are satisfied, the trivial solution of (2.14.27) is asymptotically stable, and therefore, if 11 x, I/ is small t 2 t, . Consequently, choosing 11 x, I/ sufficiently enough, 11 x(t)ll small, we have, from (2.14.30),

v,

which, by Theorem 1.4. I , leads to the estimate (2.14.28). Moreover, by condition (2.14.25), it results that, if 7 is small,

This, together with (2.14.28), implies that liml+my ( t ) = 0.

REMARK2.14.1. T h e condition (2.14.25) implies that the solutions yo(t)of the variational system

have the property that liml+myo(t)= 0. For, setting m(t) = IIy,(t)ll,

we obtain

D+m(t)

< P [ f Z ( t , O)lm(t),

and hence, by Theorem 1.4.1,

T h e assumption (2.14.25) assures that liml->va yo(t) = 0. Thus, in essence, Theorem 2.14.7 guarantees the asymptotic behavior of the solutions of (2.14.26), whenever there exists a similar behavior for the solutions of (2.14.31). From these considerations, we infer the following lemma, which is interesting in itself. T,EMMA 2.14.1. Let A(t) be a continuous n x n matrix on J . If x(t) is the solution of s' = A ( t ) x ,

we have

x(t,) = .xo ,

2.14.

COROLLARY 2.14.2.

Under the assumptions of Theorem 2.14.7, if ,t

1

then u

117

ASYMPTOTIC BEHAVIOR

(2.14.32)

< 0.

Proof. Since the trivial solution of (2.14.27) is asymptotically stable by Theorem 2.13.2, choosing (1 xo 11 small, we can make 11 x(t)ll < 7, t 3 t o , and hence we have 0

<

00

+ K.I

because of (2.14.29). It therefore follows that, by choosing 7 sufficiently small, u can be made less than zero whenever go < 0.

THEOREM 2.14.8. Assume that f E: C[J x R", R"],and f Jt , x) exists and is continuous on J x R". Suppose further that x ( t ) = x ( t , t o ,xo) is the solution of (2.14.27) with the property lim x ( t )

t-m

= 0.

(2.14.33)

If defined by the relation (2.14.32) is less than zero, then every solution x(t, to ,yo) of (2.14.27) such that xo ,yo belong to a convex subset of Rn satisfies lim x ( t , to ,yo) = 0. (2.14.34) t-m (5

Let x ( t , to , xo), x ( t , to , y o ) be the solutions of (2.14.27) such that xo , y o belong to a convex subset of Rn.Then, by Theorem 2.6.4, we get Proof.

and hence, for t 3 t o ,

II x ( t , t o ro)ll

< I! x ( t , to

> x0)ll

+ I/ yo

-

xo ll exp

[j t I-L[f&, t,

x(4)lds],

by virtue of Lemma 2.14.1. T h e relation (2.14.32) implies that

for sufficiently large t, and therefore the assumption u 0.

This, together with (2.14.33), assures (2.14.34).

< 0 yields

118

CHAPTER

2

COROLLARY 2.14.3. In addition to the hypotheses of Theorem 2.14.8, if we assume that f ( t , 0) = 0, then the trivial solution of (2.14.27) is asymptotically stable.

THEOREM 2.14.9. Let f E C [ J x R",R"], f ( t , 0) = 0, and f 2 ( t , x) exist and be continuous on J x R". Let G E C [ J x R", R"],and, given any E > 0, there exists a 6 = S ( E ) > 0 such that

II G(t,Y)ll

<4 Y

I19

<

provided IIy 11 a(€). Assume that the trivial solution of (2.14.27) is asymptotically stable and that u defined by (2.14.32) is negative. Then, the trivial solution of the perturbed system

is asymptotically stable. Proof. Let x ( t ) = x ( t , to , y o ) ,y ( t ) = y ( t , to , yo) be the solutions of (2.14.27) and (2.14.35), respectively. Then, by Theorem 2.6.3,

Moreover, by Theorem 2.6.4, we infer that

Let

E

> 0 be given. By assumption on G ( t , y ) ,it follows that

2.14.

119

ASYMPTOTIC BEHAVIOR

provided IIy(t)ll < a(€). Theorem 1.9.1 readily gives the estimate

for t 3 to , which implies that 11 y(t)l/ remains less than S ( E ) if 1) yo )I is small enough, because the condition u < 0 shows that, for sufficiently small E > 0,

Thus, (2.14.36) is valid for t 3 t, , and the asymptotic stability of the trivial solution of (2.14.35) follows.

THEOREM 2.14.10. Assume that (i) f E C [ J x R",R"],f ( t , 0)= 0, and f z ( t , x) is continuous on J x Rn;(ii) p [ f z ( t , O)] -u, u > 0, t E J ; (iii) G E C [ J x Rn,R"],G(t,0) = 0, and there exists an a > 0 such that, if 11 x 11 < a, t E J , 11 G(t,x)II y(t), where y E C [ J ,R,] and

<

<

p(t> =

1

t+l

t

y(s) ds -+

o

as

t

-j

co.

Then, the trivial solution of (2.14.35) is asymptotically stable. Proof.

Let E

> 0 be given such that 0 < E < min(u, a ) . Choose To 3 1

so large that, for t

s:

T o ,we have

exp[-(cr

-

c)(t

- s)]y(s)

ds

< YE) - = 6, , 2

(2.14.37)

<

E. This choice is possible, as shown in Theorem 2.14.6. where a(€) It is easy to show that, whenever 11 xg I] < a,, /I x(t)ll < S ( E ) , t >, to . For, otherwise, there would exist a t, > to 3 'Tosuch that

II 4tl)ll Define m ( t ) inequality

=

= 6(E),

ll 4t)ll

< 6(c),

t

E

[to >

4.

11 x ( t ) 11. Then, for t E [ t o ,t l ] , we obtain the differential

m;(t>

< P [ f Z ( t ? O)lm(t) + /I F(t7 4t))ll + It G(t,4t))ll < -(a 4 4 t ) + y(t). -

Here we have used assumptions (ii) and (iii) of the theorem, in addition to the relations (2.13.12) and (2.13.13) and the argument employed in Theorem 2.13.2.

120

CHAPTER

2

An application of Theorem 1.4. I gives

/I 4t)ll

< /I xo I/ exP[-(u

+ for t

E

[t, , t,]. At t S(c)

=

Jl0

exP[-(a

-

.)(t

-

441

- E)(t -

s)1 A s )

(2.14.38)

t , , there arises an absurdity

< 6, -k j:'

< 6,

+ 6,

exp[-(o =

-

c)(tl

-

s)] y(s) ds

S(E),

because of (2.14.37). This proves that, if 11 x,,11 < a,, Ij x(t)ll < a(€), t 2 t, , which, in its turn, implies the inequality (2.14.38) for all t 3 to . Since S(E) E , the stated result follows, as in Theorem 2.14.6.

<

+

COROLLARY 2.14.4. T h e function f ( t , x) = Ax F ( t , x), where A is an n x n constant matrix such that p ( A ) -0 and F ( t , x) satisfies assumption (ii) of Theorem 2.14.6, is admissible in Theorem 2.14.10.

<

2.15. Periodic and almost periodic systems We shall be concerned in this section with the existence of periodic and almost periodic solutions of differential equations. Let us first state the following:

LEMMA 2.15.1. Let E be an inductively ordered set, and let T be a transformation from E into E such that, for any x E E, we have T ( x ) x. Then, there exists at least one point x E E satisfying T ( x ) = x. As an application of this lemma, we prove an existence theorem for periodic solutions. THEOREM 2.15.1. Assume that (i) g E C [ J x R., , R],g(t, u) is nondecrcasing in u for each t E J , periodic in t with a period W , and the differential equation (2.1 5.1) u' = At, u) admits a periodic solution of period w ; (ii) f E C [ J x R",R"],f ( t , x) is periodic in t with a period w , and, for t E J , x,y E R", and sufficiently small h > 0,

2.15.

121

PERIODIC AND ALMOST PERIODIC SYSTEMS

where fi E C [ J x R",R"];(iii) the functions f,fi , and g are smooth enough to assure the existence and uniqueness of solutions, and the system Y' = S d h Y ) (2.1 5.3) has a bounded nondecreasing solution. Then, the differential system x'

=f

admits a periodic solution of period

(2.15.4)

( t , x)

w.

Proof. Let y ( t ) be the bounded monotonic solution of (2.15.3) such that y(to)= y o , to >, 0. Suppose that u(t) is the periodic solution of (2.15.1) of period w . It is possible to choose to and uo such that u(to) = u g >, 0,

>, 0,

u(t) - ug

t

>, t o .

(2.15.5)

Define m ( t ) = 11 x ( t ) - y(t)ll, where x(t) is the solution of (2.15.4) with the property x(to) = y o . Clearly, m(to) = 0. We have, using the condition (2.15.2), the differential inequality D+m(t>

< g ( 4 m(t)>.

Consider the solutions u(t, c) of u' = g(t, u )

for sufficiently small

E

+

u ( t o ,E )

E,

=

uo

+

E

> 0. Setting

because of the nondecreasing nature of g(t, u ) we get

According to Theorem 1.2.1, we infer that m(t) < p(t, E),

t

But limp(t, E ) <-0

= lim E-0

[u(t, E )

-

(uo

2 to .

+

E)]

= u(t)

-

uo

uniformly in t, and therefore (2.15.6)

122

CHAPTER

2

+

Define the point x(t,)by p , . Take the point ~ ( t , w ) on the solution x ( t ) and denote it by p , . Let T be the transformation that takes any point p, t o p , by the foregoing process. Since the functionf(t, x) is periodic in t with a period w , any solution passing through the fixed point, under the preceding transformation, is clearly a periodic solution. Hence, it is enough to show the existence of a fixed point, under the transformation T defined previously. Since u ( t ) is periodic and u(t,) = u, , we have u(t,,

+ nu)

~

U" = 0,

n

= 0,

I , 2 ,....

It therefore follows from (2.15.6) that X(t,

+- nu)

= y(t,

+ nu),

n

= 0,

1 , 2,....

By assumption, y ( t ) is a bounded monotonic solution of (2.15.3). Thus, nw) form a bounded, monotonic, and denumerably infinite set. If their upper bound is also included, the set becomes inductively ordered. Hence, the application of Lemma 2.15. I yields a fixed point, and the theorem is proved.

+

it is evident that the points %(to

COROLLARY 2.15.1.

If, in addition to the hypotheses of Theorem 2.15.1,

fi(t, y ) is periodic in t with a period w , the assertion of Theorem 2.15.1 remains valid. I n particular, fi(t, y ) = 0 is admissible. We remark that the monotony of g(t, u ) can be dispensed with if the

periodic solution u(t) of (2.15.1) has the property that, for some t, 0, u(t,,) = 0 and u ( t ) 3 0 for t 3 0. Another set of sufficient conditions for the existence of a periodic solution is given by

THEOREM 2.15.2. tial equation

Assume that (i) g E C [ J x u' = g ( t , u ) ,

u(0) = uo

R, , R], and the differen20

(2.15.7)

has unique solution u(t, 0, uo) such that lim u ( t , 0, u,) t-m

=

0;

(ii) f E C [ J x R",R"],and f (t,x) is periodic in t with a period w and is smooth enough to guarantee the existence and uniqueness of solutions of the system (2.15.4), and, for t E J , x, y E R?&,

I1 x

~

Y -1- h [ f ( t ,X) - f ( t ,

.Y)III < I/ x - Y I1 + hg(t, II x - y 11)

+ O(h)

(2.15.8)

2.15.

123

PERIODIC AND ALMOST PERIODIC SYSTEMS

for sufficiently small h > 0; (iii) the system (2.15.4) has a bounded solution existing on [0, a). Then, the system (2.15.4) admits a periodic solution of period w . Proof. Let x,(t) = xo(t,0, xo) be the bounded solution of (2.15.4) defined for t >, 0. We shall show that, under the assumptions of the theorem, we have lim xo(t W ) - x o ( t ) = 0. (2.15.9) f-m

+

Let x ( t ) = xo(t

+ w)

-

x o ( t ) . Then,

z'(t>= f ( t

+

W,

=f ( t , x d t )

xo(t

+

W))

- f ( t , xo(t))

+ z(t)>- f ( C

xo(t>),

because of the periodicity of f ( t , x) in t. Hence, setting m ( t ) = I/ x(t)ll, we observe, for small h > 0, that mft

+ h) < II .o(t + -f(C

W)

- xo(t)

xo(t))lll

+ II

+ h [ f ( t ,x , ( t ) + z ( t ) )

+)I19

where c(h)/h-+ 0 as h -+ 0. Using condition (2.15.8), it is easy to get the inequality D+m(t)

< g(t, 49,

which implies, by Theorem 1.4.1, choosing uo = 11 x0(w) - xo 11,

II xo(t

+

W)

%(t)ll < 4 4 0,II X O ( W )

~

-

xo Ill,

t

2 0,

u(t, 0, uo) being the unique solution of (2.15.7). Consequently, in view of assumption (i), the relation (2.15.9) is valid. Since xo(t) is bounded, the sequence {xn} = {xo(nw)) is bounded. Hence, a subsequence {xnJ can be extracted which converges to a point x*. It follows from (2.15.9) that, for any n, lim xo[(n

and thus,

n-m

+l)~]

-

xo(nw) = 0 ,

lim xn,+* = x*.

(2.15.10)

k+m

+

Observing that the functions defined by x(t, 0, x,) = xo(t nw) are the solutions of (2.15.4) through (0, xn), where x, = xo(nw),in view of the periodicity off(t, x) in t , we see that the fact lim xnr

k-m

= x*

124

2

CHAPTER

implies Iim .v(t, 0 , xn,)

k

= x ( t , 0, x * ) ,

t-

and hence, from (2.15.10), we obtain lim x ( t , 0 , x,,,,)

0 x*).

= x(t,

k+n

However, because of the uniqueness of solutions,

Since

lim x ( t , O

k-m

=

.lcn,+,)

=

we have X(t,

0 , x*)

=

lim x ( t

k-*m

x(t

x(t

-1

+

w,

w , 0,

xnk)

0 , x*),

+ w , 0, x*).

This means that the solution that satisfies the initial condition x(0) is periodic with period W. T h e proof is complete.

=

x*

An analogous result holds for almost periodic systems. We shall start with the following

DEFINITION 2.15.1. A functionfe C [(- GO, co) x Rn,R"]is said to be almost periodic in t uniformly with respect to x E S, for any compact subset S C R", if, given any 7 > 0, it is possible to find a l(7) such that, in any interval of length 1(7), there is a T such that the inequality llf(t

is satisfied for t

E

+

). - f ( t , x)ll

7,

< 17

(- co, a), x E S.

THEOREM 2.15.3. L e t f E C[(--00, co) x R", R"],andf(t, x) be almost periodic in t uniformly with respect to x E S, for any compact set S C R", and be smooth enough to ensure the existence and uniqueness of solutions of (2.15.4). Furthermore, suppose that, for sufficiently small h > 0,

II x - y I W t , 4

-f(t,y)lll

< II x - y

11[1

-

ah]

+ O(h),

t

2 0, (2.15.1 1)

2.15.

125

PERIODIC AND ALMOST PERIODIC SYSTEMS

where 01 > 0, and that the almost periodic system (2.15.4) admits a bounded solution x ( t , t o , x,) with a uniform bound B. Then, (2.15.4) admits an almost periodic solution that is uniformly asymptotically stable.

Proof. Let x(t) be the bounded solution of (2.15.4), defined on [ t o ,co) so that I/ x(t)ll B, t 3 t o , where B does not depend on t o . Let rk be a sequence of numbers such that rk -+ co as k -+ co and

<

f(t f

Tk

Y

x, - f ( t , ).

-

uniformly for t E (- co, 00) and x E S, any compact set in R". For any p, let k, = k,(p) be the smallest value of k for which r k , p 3 t o . We have

+

/I x ( t

+

Tk)l/

< B,

t

2 B,

k 3 k,(B).

+

We shall now show that the sequence of functions { x ( t rJr)), k 3 kO(,3) converges to a continuous bounded function w ( t ) defined on [p, m), with a bound independent of /3, and that the convergence is uniform on all compact subsets of [p, CO). Since the boundedness of x ( t ) is uniform with respect to t o , it is sufficient to prove that the sequence ( x ( t rX.)>, k 2 k,(& forms a Cauchy sequence on any compact subset of [/ICO). , Let U be any compact subset of [p, a),and let E > 0 be given. Choose an integer n, = no(€,p) 3 k, so large that, for k, >, n o ,

+

(2.15.12)

m(t)

Setting t ,

=

t

+ r6,, t, = t + rk,, we

have

D+m(t)

where 6 is an E@-translation to 6 3 0, that is,

+

number of f ( t , x) for x E S such that (2.15.13)

126

CHAPTER

2

where x E S, any compact set in R”, and t E (- co, GO). I n view of the condition (2.5.1 l ) , we deduce

which, because of relations (2.15.12) and (2.15.13), yields D%Z(t)

< -am(t) + ( € 4 2 ) ,

t

2 to.

By Theorem 1.4.1, we get

This proves the existence of a function w ( t ) defined on bounded by B. Since /3 is arbitrary, w ( t ) is defined

[p, co) and

This proves the existence of a function w ( t ) defined on [/3, a)and bounded by B. Since /3 is arbitrary, w ( t ) is defined on (- co, co), and we have as K co, v(t 7 k )- w ( t ) + 0

+

---f

uniformly on all compact subsets of (- co, a). Next we show that w ( t ) is differentiable and that it satisfies (2.15.4). Observe that lim, ,, x’(t 4-T / \ ) exists uniformly on all compact subsets of (- co, GO), and, consequently,

1

=

lim lim

=

lim -I [w(t

h-0

~t-o

k-.-o

h

h ~

[x(t

+

+ h)

T~

-

-1h )

~(t)],

2.15.

127

PERIODIC AND ALMOST PERIODIC SYSTEMS

which proves that w'(t) exists. Also, w'(t) = lim x'(t k-m

= =

+

T

~

)

+ lim [ f ( t + x(t + lim f ( t f

Tk

k+m

x(t

7

Tk))

Tic,

k-r,

+ lim [ f ( t + k-m

Tk

Y

Tk))

-f(t

+

Tk

9

w(t))]

w(t))]

=At, w(t)), proving that w ( t ) is the solution of (2.15.4). It remains to be shown that w ( t ) is almost periodic. For this purpose, we need to show that, for any E > 0 and any r for which

Ilf(t + T > x )- f ( t , 411 < E

uniformly on (-

00,

co) and all x E S , we have

II w(t

+

7) - w(t)/l

< EL

uniformly for t E (- co, co), where L is some constant independent of E and 7. Suppose y is such that

< E/~BoI.

e-"Y

Let z ( t ) = w(t

+

T)

- w(t), so

that

+

w(t

z'(t) = f ( t

7,

+

T))

- f ( 4 w(t)).

Let 6 be an <-translation number off(t, x) for R?&,such that t - y 8 3 0, that is,

+

llf(t

+ 8, x) - f ( t , .)I1

< E,

If m ( t ) = I/ x(t)ll, we get, for small h m(t

+ h) < /I w(t + ).

-

w(t)

(2.15.14)

S , any compact set in

xE

x E s,

t

(-a, a).

> 0,

+ h [ f ( t+ 8, w(t +

7))

-f(t

+ 8, w(t))lll

+ hllf(t + 8 , 4 t + - f ( 4 w(t + .)I1 + hllf(t + 6 , 4 t ) ) - f ( C 4t))Il -t hllf(t + r , w ( t + .)) - f ( t , w(t + .)I1 + II +)lL TI)

which, as before, yields the differential inequality D+m(t)

< -OIm(t) + 3t,

128

2

CHAPTER

whence, by Theorem .4.1, we have

I/ 4

t

+

T)

3E +; [I

- e-Ov].

T h c boundedness of w ( t ) and the relation (2.15.14) show that

uniformly in t , where L = 4/a. T h e uniform asymptotic stability of w ( t ) can be easily verified. T h i s coinpletes the proof

COROLLARY 2.15.2. Assume that (i) A ( t ) is a continuous n x n matrix on (- m, m), almost periodic in t , and p [ A ( t ) ]<:

a!

-01,

> 0,

t

2 0;

(ii)fE C [ ( - co, a), R7q,a n d f ( t ) is almost periodic in t and is bounded. Then, the system x’ A ( t ) x - l - f ( t ) (2.15.1 5 ) admits an almost periodic solution that is uniformly asymptotically stable. Proof. It is easy to check that the assumptions of Theorem 2.15.3 are satisfied except the existence of a bounded solution. Hence, it only needs to be verified that (2.15.15) has a bounded solution x(t, t o , xo) with a uniform bound 13 for t 3 t , . I n fact, under the assumptions, it turns out that all the solutions x (t , t , , xo), to E (- m, a), xo E R” are uniformly bounded. Let -m

SUP ,(.

a

llf(t)ll

=

B,

1

and let x ( t ) x ( t , t,, , xo) be any solution of (2.15.15) such that t,, E (- m, co) and /I xo / / R,/a. Then, it can be shown that

<

:

11 ~ ( t ) l< l 5B1/2m = B, If this were not true, there would exist a t ,

I1 r(t,)ll

=

B,

II 4t)Il

< B,

t

2 to.

> to such that t

E

[to

7

tll.

(2.15.16)

2.16.

129

NOTES

Let I9 be a BJ3-translation number of A ( t ) x + f ( t ) for x E S , a compact set in R”,such that t, I9 >, 0, that is,

+

+

where ~ ( h ) /+ h 0 as h --+ 0. Observing that t fl 2 0, assumption (i), together with the relations (2.15.16) and (2.15.17), yields

Hence, by Theorem 1.4.1,

which which leads leads to to an an absurdity, absurdity, using using the the fact fact that that I/xJ I/xJ (2.15.16); (2.15.16);

<
This (2.15.15) and and T h i s proves proves the the uniform uniform boundedness boundedness ofof solutions solutions ofof (2.15.15) establishes establishesthe thecorollary. corollary. 2.16, Notes

T h e results of Sect. 2.1 are due to Stokes [I]. For a more general global existence theorem using Tychonoff’s fixed point theorem, see Corduneanu [I]. Theorem 2.2.1 is given by 0. Perron, (see Kamke (I]). T h e proof of Corollary 2.2.1 is new. T h e general uniqueness theorem 2.2.2 is due to Kamke [l]. T h e proof given in the text is based on that of Olech [4]. For Theorem 2.2.3, see Lakshmikantham [12] and Olech [4]. Corollary 2.2.2 is a result of Wintner [l5]. Corollaries 2.2.4 and 2.2.5 are Nagumo’s and Osgood’s uniqueness criteria, respectively. Theorem 2.2.4 is due to Brauer [2]. Theorem 2.2.5 and the proof of Theorem 2.2.4 are taken from Walter [2]. Corollary 2.2.6 is a result of Krasnosel’skii and Krein [2].

130

CHAPTER

2

See also Brauer [l], Kooi [I], and Luxemburg [l]. Beginning from nonuniqueness Theorem 2.2.7, the remaining results of Sect. 2.2 are due to Lakshmikantham [5, 121. T h e proof of Theorem 2.3.1 is taken from Wazewski [9], whereas the second proof of Theorem 2.3.1 is from Hartman [5]. See Olech and Plis [l] for the proof that the monotony assumption in Theorem 2.3.1 cannot be dropped in general. See also Bihari [I], Cafiero [2, 41, Coddington and Levinson [l], Diaz and Walter [I], Dieudonne [l], LaSalle [l], and Viswanatham [I]. The results of Sect. 2.4 are taken from Chaplygin [l] and Lusin [I]. Lemma 2.5.1 and Theorem 2.5.1 are new: the idea is taken from Turowicz [l]. Theorem 2.5.2 is adopted from Antosiewicz [7]. T h e rest of the results of Sect. 2.5 are taken from Hartman [6]. Theorems 2.6.3 and 2.6.4 are due to Alekseev [I]. Most of the results of Sects. 2.7 and 2.8 are adopted from Lakshmikantham [7] and Walter [3]. See also Bihari [2], Brauer [5], and Langenhop [I]. Section 2.9 contains the work of Brauer [ll]. Theorem 2.10.1 is from 1,akshmikantham and Onuchic [I]. Theorems 2.10.2 and 2.10.3 are adopted from Brauer [4, 121. See also Cesari [I] and Wintner [4, 5, 7-91. Theorems 2.11.1 and 2.11.2 are due to Wazewski [2] and are very useful in the study of differential equations. T h e proofs are taken from Hartman [5]. Theorem 2.12.1 is due to Lakshmikantham and Onuchic [l]. For further results and references on the application of Wazewski’s topological principle, see Cesari [l] and Hartman [5]. Theorem 2.1 3.1 is adopted from Lakshmikantham [7], whereas Theorem 2.13.2 is due to Brauer [lo]. For Theorem 2.13.3, see Conti and Sansone [l]. Theorems 2.14.1 and 2.14.3 are adopted from Lakshmikantham [7]. For Theorem 2.14.2, see Krasovskii [4] and Zubov [I]. See Cesari [I] for Theorems 2.14.4 and 2.14.5. Theorem 2.14.6 is due to Strauss and Yorke [I]. Theorems 2.14.7, 2.14.8, and 2.14.9 are taken from Brauer [lo]. See also Cesari [l], Coppel [l], Hartman [5], and Markus and Yamabe [I]. Theorem 2.15.1 is due to Lakshmikantham [7]. Theorem 2.15.2 is new. T h e proof uses a result from Halanay [2]. Theorem 2.15.3 is new. See also Cartwright [l], Deysach and Sell [l], Massera [2], Miller [3], Seifert [5, 61, and Sell [4].

Chapter 3

3.0. Introduction As is well known, Lyapunov's second method has its origin in three simple theorems that form the core of what he himself called his second method for dealing with questions of stability. It is widely recognized, today, as an indispensable tool not only in the theory of stability but also in studying many other qualitative properties of solutions of differential equations. T h e main characteristic of this method is the introduction of a function, namely the Lyapunov function, which defines a generalized distance from the origin of the motion space. As a result, the concept of Lyapunov function, together with the theory of differential inequalities, furnishes a very general comparison principle under much less restrictive assumptions. We present, in this chapter, a variety of qualitative problems bringing out the real significance of the comparison technique.

3.1. Basic comparison theorems Consider the differential system x' = f ( t , x),

where f~ C [ J x R", R"]. For function

x(t0) = x0

,

to 6

1,

(3.1.1)

V E C [ J x R",R,], we define the

1 D+V(t,x) = lim sup - [V(t ti-.^+ h

+ h, x + hf(t, x))

-

V(t,x)]

(3.1.2)

for ( t , x) E J x Rn. Occasionally, we write D+V(t, to denote that the definition of D+V(t,x) is with respect to the system (3.1.1). We can now formulate the basic comparison theorems. 131

132

CHAPTER

3

THEOREM 3.1.1. Let V E C [ J x RE,R,] and V ( t ,x) be locally 1,ipschitzian in x. Assume that the function D+V(t,x) of (3.1.2) satisfies Z l V ( t ,x)

< g (t, V(t,x)),

( t , x)

E

J x Rn,

(3.1.3)

where g E C [ J x R, , R]. Let r ( t ) = r ( t , to , uo) be the maximal solution of the scalar differential equation u' = g ( t , u),

existing to the right of to . If x(t) existing for t 3 to such that

u(to) = uo =

2 0,

(3.1.4)

x ( t , to , xo) is any solution of (3.1.1)

(3.1.5)

then (3.1.6)

Proof. Let x(t) be any solution of (3.1.1) defined for t 3 to such that (3.1.5) holds. Define m(t) = q t , X ( t ) ) ,

so that m ( t J

< u o . For

sufficiently small h

> 0,

we have

where ~ ( h ) /+ h 0 as h + 0. Since V ( t ,x) is locally Lipschitzian in x, we get, using (3.1.3), the inequality L> ' m ( t ) < s(t,m ( t ) ) ,

t E J.

Applying Theorem 1.4.1, we obtain the desired result (3.1.6). REMARK 3.1.1.

Let S, = [x E Rn : 11 x

I/ < p ] ,

and assume that the condition (3.1.3) holds for ( t , x) E J x S, . If x ( t ) is any solution of (3.1.1) such that 11 xo 11 < p, then (3.1.5) implies (3.1.6) as far as x ( t ) remains in S, to the right of to . COROLLARY 3.1. I . zero, i.e.,

If the function g(t, u ) in Theorem 3.1.1 is identically W V ( t ,x)

< 0,

( t , X) E J x

Rn,

3.1.

BASIC COMPARISON THEOREMS

133

then the function V ( t ,x ( t ) ) is nonincreasing in t , and V ( t ,X(t))

< V(to

9

t

Xo),

3to.

COROLLARY 3.1.2. Assume that the hypotheses of Theorem 3.1.1 hold except that the condition (3.1.3) is to be satisfied only for ( t , x) E J x 52, where SZ = [X E R" : ~ ( t< ) V ( t ,X) < ~ ( t+ ) co , t 3 to], c0 being some positive number. Then, the conclusion of Theorem 3.1.1 is true.

Proof. We choose uo = V ( t , , xo) and proceed as in the proof of Theorem 3.1.1 to obtain Z being the set

D+m(t)

< g(t, m ( t ) ) ,

t

E

z,

z= [t E J : r ( t ) < m ( t ) < Y ( t ) + 61. Theorem 1.4.2 now assures the stated result. Sometimes, the following variants of Theorem 3.1.1 are more useful in applications.

THEOREM 3.1.2. Assume that the hypotheses of Theorem 3.1.1 hold except that the inequality (3.1.3) is replaced by A ( t P + V t X) ,

+ v, X)D+A(t)< g(t, V ( t ,x)A(t)),

(3.1.7)

for ( t , x) E J x R", where the function A(t) > 0 is continuous for t E J , and 1 D+A(t) = lim sup - [A(t h++ h

Then U t o xo)A(to)

imp1ies q t , X(t))A(t)

Proof.

+ h)-

A(t)].

< uo

< r(t),

t

(3.1.8)

2 to -

(3.1.9)

Defining VAt, X)

=

V ( t ,X)A(t),

it is easy to show that Vl(t,x) satisfies the assumptions of Theorem 3.1.1.

134

For, if h

CHAPTER

3

> 0 is sufficiently small,

and therefore, using the assumption (3.1.7), we get

T h e estimate (3.1.9) follows immediately from Theorem 3.1.1.

THEOREM 3.1.3. Let the hypotheses of Theorem 3.1.1 hold except that, instead of the inequality (3.1.3), we now assume

o+v(t, x) + 9(ll

.X

Ill < R(& v(t,x)),

( t , x) E I x I?”,

(3.1.10)

where +(u) 3 0 is continuous for u 3 0, $(O) = 0, and $(u) is strictly increasing in u. Suppose further that g(t, u ) is nondecreasing in u for each t E J . T he n (3.1.5) implies that

Proof.

Consider the function

<

m(t). T h e monotonic character of g(t, u ) in u, together with the condition (3.1.lo), now yields so that V ( t ,x ( t ) )

D+ 4 t )

< R(t, m ( t ) ) ,

and the assertion (3.1.11) follows from Theorem 1.4.1.

DEFINITION 3.1.1. T h e function V ( t ,x) is said to be mildly unbounded if, for every T > 0, V ( t ,x) + as 11 x 11 co uniformly for t E [0, TI. T h e mild unboundedness of V ( t ,x) guarantees that, whenever V ( t ,x ( t ) ) is finite, 11 x(t)ll is also finite. T h e assumption that the solutions x(t) of (3.1.1) exist for all t 3 t o , therefore, becomes superfluous, if V ( t ,x) is further assumed to be mildly unbounded in the foregoing theorem. From this observation stems the following global existence theorem. ---f

3.2.

135

DEFINITIONS

THEOREM 3.1.4. Let V EC [ J x Rn, R,], V(t,x) be mildly unbounded and locally Lipschitzian in x. Suppose that g E C [ J x R, , R] and r ( t ) is the maximal solution of (3.1.4) defined for t 3 t, . Assume that (3.1.3) holds. Then, every solution x(t) of (3.1.1) exists in the future, i.e., for all t 3 to , and (3.1.5) implies (3.1.6). Proof. Suppose that the assertion that every solution x(t) of (3.1.1) exists for all t >, t, is false. Then, by Corollary 1.1.2, there exists a t, > t, such that x(t) cannot be extended to the closed interval t, t t , , which implies that there cannot exist an increasing sequence {t,} + t,- such that 11 x(tn)1l is bounded. This, in its turn, yields that 11 x(tn)lI + 00 as t, + t,- . On the basis of Theorem 3.1.1, it follows that (3.1.5) implies (3.1.6) for to t t, . By the assumption that V ( t ,x) is mildly unbounded, the fact that r ( t ) exists for all t >, t, , and (3.1.6), there arises a contradiction as t, + t,- . Hence, the global existence of solutions x(t) of (3.1.1) is proved, which, in turn, assures the estimate (3.1.6) for t 3 t, whenever (3.1.5) holds. T h e proof is complete.

< <

< <

3.2. Definitions Let x ( t , t o , xo) be any solution of the differential system x' = f ( t ,XI,

.(to)

=

,

to

2 0,

(3.2.1)

where f E C[J x S,, , Rn],S, being the set

s,= [x E Rn : 11 x 11 < p].

(3.2.2)

Assume that f ( t , 0) = 0, t E J , so that x = 0 is a (trivial) solution of (3.2.1) through ( t o ,0). We now list a few definitions concerning the stability of the trivial solution.

DEFINITION 3.2.1.

T h e trivial solution x

=

0 of (3.2.1) is

(S,) equistable if, for each E > 0, t, E J , there exists a positive function 6 = 8(t,, E ) that is continuous in t, for each E such that the inequality IlxnIl

implies

/I x ( t , to .")I1 9

<8

< E,

t

3 to ;

(S,) uniformly stable if the 6 in (S,) is independent of t, ;

136

CHAPTER

3

(S,) quasi-equi asymptotically stable if, for each E > 0, to E J , there exist positive numbers IS,, = &,(to) and T = T ( t o ,C ) such that, for t t,, -I T and 11 x ~/I, -: 8, ,

>

I/ x ( t , t,,

7

x0)Il

< E;

(S,) quasi uniformly asymptotically stable if the numbers 6, and T in (S,) are independent of to ; (S,) eqzti-asymptotically stable if (S,) and (S,) hold simultaneously; (S,) un;formly asymptotically stable if (S,) and (S,) hold together; ( S , ) quasi-equi asymptotically stable if, for each E > 0, 01 > 0, and to E J , there exists a positive number T = T(t, , E , a ) such that // xo (1 01 implies

<

I/ x ( t , t o , xo)ll < E,

t

2 to

+ T;

(S,) quasi uniformly asumptotically stable if the T in (S,) is independent of t" ; (S,) completely stable if (S,) holds and (S,) is verified for all 01, O < Y co; (Slo) uniformly completely stable if (S,) holds and (S,) is verified for all ct, 0 < (Y -< m; unstable if ( S , ) fails to hold. (S'J REMARK3.2.1. Sometimes the notion of quasi-asymptotic stability may be relaxed somewhat as in (S,) and (S8).Clearly the E, 01 given in the preceding definitions must be less than p of (3.2.2), and therefore thc concepts (S,)-(S,) are of local nature. If, on the other hand, p = co, so that S , R",the corresponding concepts of stability would be of global character. These considerations lead to (S,) and (Sl,,).We note further that the definitions (S,)and (S,) may hold even whenf(t, 0) + 0. I n other words, the assumption about the existence of the trivial solution is not necessary. I n characterizing Lyapunov functions, it is convenient to introduce certain classes of monotone functions. DEFINITION 3.2.2. A function + ( r ) is said to belong to the class .X if 4 E C[[O,p), K,],#(O) = 0, and +(r) is strictly monotone increasing in Y . I~EFINITIO 3.2.3. N 4 function ~ ( tis) said to belong to the class 9 if C [ ] ,R,],u(t) is monotone decreasing in t , and ~ ( t+ ) 0 as t -+ co.

uE

3.2.

137

DEFINITIONS

DEFINITION 3.2.4. A function + ( t , r ) is said to belong to the class %A? if 4 E C [ J x [0, p), R+], 4 E S” for each t E J , and 4 is monotone increasing in t for each Y > 0 and +(t, Y ) -+ 00 as t oc) for each r > 0. -j

DEFINITION 3.2.5. A function V ( t ,x) with V ( t , 0) = 0 is said to be positive deJinite (negative deJinite) if there exists a function 4 ( r ) E X such that the relation

DEFINITION 3.2.6. A function V ( t , x) with V ( t , 0) = 0 is said to be strongly positive dejinite if there exists a function + ( t , r ) E XA? such that

q t , ). 3 d ( t , I1 x ll),

( t , x) E

J

x

s, .

DEFINITION 3.2.7. A function V ( t , x) 3 0 is said to be decrescent if a function + ( Y ) E S” exists such that V(t,x) <+(I1

x

(44

II),

E

J x s,.

T o use the second method of Lyapunov, which attempts to make statements about the stability properties directly by using suitable functions, we need to study the scalar differential equation u’

= g(t, u),

u(to) = uo

2 0,

tn

0,

(3.2.3)

where g E C [ J x R, , R].We suppose that g ( t , 0) 3 0 so that u = 0 is a solution of (3.2.3) through ( t o ,0). Furthermore, this assumption also implies that the solutions u ( t ) = u ( t , t o , uo) of (3.2.3) are nonnegative for t 2 to so as to assure that g ( t , u ( t ) ) is defined. Corresponding to the stability definitions ( Sl)-(&), we designate by (Sf)-(S$)the concepts concerning the stability of the solution u = 0 of (3.2.3). DEFINITION 3.2.8. The trivial solution u = 0 of (3.2.3) is said to be if, for each E > 0, to E J , there exists a positive function 6 = 6 ( t o , E) that is continuous in to for each E such that

(Sp) equistable

u(t, to uo)

<

provided uo

€7

< s.

t

3 tn,

138

3

CHAPTER

T h e definitions (S$)-(S$) may be formulated similarly. Notice as before that, for the concepts (ST) and (23:) to hold, the assumption g ( t , 0) F- 0 is not necessary.

3.3, Stability We begin with the following stability criteria, recalling the definition of the function D+V(t, x) given in (3.1.2).

THEOREM 3.3.1. Assume that there exist functions V(t,x) and g ( t , u) satisfying the following conditions:

(i) g E C [ J x R, , R] and g ( t , 0) = 0. (ii) V E C [ J x S o ,R,],V ( t ,0) E 0, and V(t,x) is positive definite and locally Lipschitzian in x. (iii)

for ( t , x)

E

J x So ,

< g(t, q t , x)).

D+V(t,).

Then, the equistability of the trivial solution of (3.2.3) implies the equistability of the trivial solution of the system (3.2.1).

Proof.

By assumption, a function b(r) of class 3? exists such that V ( t ,x) 3 b(ll x \I),

( t , x) E

J x

So.

(3.3.1)

Let 0 < E < p and toE J be given. Since the solution u = 0 is equistable, given b ( ~ > ) 0, to E J , there exists a positive function S = S(t, , E ) that is continuous in to for each E , such that uo 8 implies

<

u ( t , t o , uo)

< b(E),

t

3 to.

(3.3.2)

Choose uo = V ( t o xo). , Since V ( t ,x) is continuous and V(t,0) E 0, it is possible to find a positive function 8, = S,(t, , E ) that is continuous in to for each E , satisfying the inequalities

/I xo /I

<

simultaneously. We claim that, if

I1 x(t7 t o

9

V t o %>

61,

xo)ll

9

(1 xo 11 <

< a,,

€3

t

<6

2 to

(3.3.3)

*

Suppose that this is not true. Then, there would exist a solution x ( t ) x ( t , to , xJ with /I xo I/ 8, and a t , > to such that

<

/I x(t,)ll

= E?

I1 4411 < 6 ,

t

E

[to

f

tll,

=

3.3.

139

STABILITY

(3.3.4)

because of (3.3.1). This means that (1 x(t)ll < p for t E [to , t l ] ,and hence the choice uo = V ( t o xo) , and condition (iii) give, as a consequence of Theorem 3.1.1, the estimate v(t,4 t ) )

< Y(t> to

9

uo),

t

E [to

, tJ,

(3.3.5)

where r ( t , t o , uo) is the maximal solution of (3.2.3). T h e relations (3.3.2), (3.3.4), and (3.3.5) lead to the contradiction b(c)

< V(tl

>

4tl))

<

Y(t1

, t o , uo) < b ( 4 ,

proving (&). T h e proof of the theorem is complete.

COROLLARY 3.3.1. T h e function g ( t , u ) = h(t) +(u), where A ( t ) is continuous on J , +(u) 0 is continuous for u 2 0, +(u) > 0 for u > 0, is admissible in Theorem 3.3.1 provided

>

(3.3.6)

for some uo > 0, every to 2 0 and to

< t < co.

Proof. All that we have to verify is that the solution u is equistable. Define

=

0 of (3.2.3)

s,"

otherwise, J(u) = J" ds/+(s), for sufficiently small E > 0. If ds/+(s) = R co, then J ( u ) s'; a monotone function mapping the interval [0, co) homeomorphically onto the interval [0, R). T h e solution u(t, t o , uo) of (3.2.3) is given by

<

(3.3.7) as long as

or

1 40

CHAPTER

3

Thus, from (3.3.7) and the fact that J is a homeomorphism, it is easy to see that u = 0 of (3.2.3) is equistable when (3.3.6) holds. This proves the corollary. 3.3.2. Under the assumptions of Theorem 3.3.1, the uniform stability of the solution u = 0 of (3.2.3) also implies the cquistability of the trivial solution of (3.2.1). 1'IIEOREM

T h e proof follows from the proof of Theorem 3.3.1. In this case, although 6 is independent of t o , the relation (3.3.3) shows that 6, is not independent of t o . Consequently, one gets only the equistability of the trivial solution of (3.2.1).

I'Toof.

COROLLARY 3.3.2.

Assume that there exists a function V ( t ,x) verifying

the following conditions:

(i) V E C [ J x S, , R,], V ( t ,0) = 0, V ( t ,x) is positive definite and locally Iipschitzian in x. 0, ( t , x) E J x So. (ii) D-I V ( t ,x)

<

Then, the trivial solution of (3.2.1) is equistable. It is important to note that, when (ii) holds, the scalar differential equation (3.2.3) reduces to u' =

and as a result Theorem 3.3.2.

0

u(t,) = u,

7

(S2) is

3 0,

to

2 0,

satisfied. Thus, Corollary 3.3.2 follows from

THEOREM 3.3.3. I n addition to the hypotheses of Theorem 3.3.1, assume that V ( t ,x) is decrescent. Then, the equistability of the null solution of (3.2.3) assures the equistability of the solution x = 0 of (3.2.1). Proof. Since V ( t ,x) is decrescent, there exists a function u(r) E .X such that ( 4 4 E 1 x s,. v t , x) < a(ll x II), We follow the proof of Theorem 3.3.1 except that we choose uo = x,) 11). Hy assumption, (SF) holds, and therefore 6 = 8(t,, E ) depends on t,, . As U ( T ) E X , the existence of a positive function 6, = Sl(to , c) satisfying the inequalities

I1 x" II

< 6,

9

4 xo It)

<

(3.3.8)

simultaneously is clear. T h e rest of the proof is very much the same.

3.3.

141

STABILITY

COROLLARY 3.3.3. T h e function g ( t , u ) tinuous on J and / I A ( s ) ds

=

h(t)u, where h(t) is con-

< 03

for every to 3 0, is admissible in Theorem 3.3.3.

THEOREM 3.3.4. Let the hypotheses of Theorem 3.3.1 hold. Assume further that V ( t ,x) is decrescent. Then, the uniform stability of the solution u = 0 of (3.2.3) guarantees the uniform stability of the trivial solution of (3.2. I). Proof. Following the proof of Theorem 3.3.3, it is easy to see that 6, does not depend on t o . For, by assymption of the uniform stability of the null solution of (3.2.3), 6 is independent of t o ,and (3.3.8) shows that 6, is also independent of t o . COROLLARY 3.3.4. T h e function g ( t , u ) = h(t) +(u) defined in Corollary 3.3.1 is admissible in Theorem 3.3.4 if (3.3.6) holds and if either J.", ds/$(s) < 00 or (i) l.",ds/+(s) = cc and (ii) h(s) ds is bounded above for every t , to t < co uniformly in t o .

l;o

<

COROLLARY 3.3.5. Assume that there exists a function V ( t ,x) fulfilling the following assumptions: (i) V EC [ J x S, , R,], V ( t ,x) is positive definite and decrescent, and V ( t ,x) is locally Lipschitzian in x. (ii)

D+V(t,x)

< 0, ( t ,x) E J

x S, .

Then, the trivial solution of (3.2.1) is uniformly stable. T h e definition of uniform stability of the solution x = 0 given in (S,) can also be formulated by means of a monotone function, as can be seen by the following

THEOREM 3.3.5. T h e trivial solution of (3.2.1) is uniformly stable if and only if there exists a function u(r) E % verifying the estimate

II x(t, to for /I xo II

< P.

I

x0)Il

< 4 xo It),

t

2 to

(3.3.9)

Proof. T h e sufficiency of the condition is immediately clear. T o prove the necessity, consider, for a given E > 0, the least upper bound of all positive functions a(€), and designate it by 8 = 8 ( ~ ) .Then 11 xo I/ 8

<

142

3

CHAPTER

<

implies 11 x ( t , t o , xo)ll E for t >, t o , and, if 6, > 8, there exists at least one xo such that, for /I xo I/ < a,, I( x ( t , t o , xo)lI exceeds the value E at some time t. Clearly, the function S(E) is positive for E > 0; it is nondecreasing and tends to zero as E + 0; and it may be discontinuous. We now choose a continuous, monotonically increasing function S * ( E ) satisfying S*(E) 8(e). Then, the inverse function a(r) = 6*-l(r) satisfies (3.3.9). T h e proof is complete.

<

We shall now prove a result that gives sufficient conditions for unstability of the solution x = 0 of (3.2.1).

THEOREM 3.3.6. Let there exist functions V(t,x) and g ( t , u ) satisfying the following properties:

e,

V ( t ,x) = 0 (i) V E C[G, R,], V ( t ,x) is locally Lipschitzian in x on for all ( t , x) E - G, and V ( t ,x) is positive and bounded on G, where G C J x Sp is some open set such that G has at least one boundary point ( T ,0), T > 0.

e

(ii) g E C[J x R, , R+],and D+V(t,x) >> g ( t , V ( t ,x))

2 0,

( t ,x) E G .

(iii) For to > T , the solutions u(t, t o , uo) of (3.2.3), for arbitrarily small uo > 0, are either unbounded or indeterminate, for t 2 t o . Then the trivial solution of (3.2.1) is unstable. There exists a point ( t o ,x,,)E G, xo f 0, in the vicinity of ( T , 0). Let x(t) = x(t, t o ,x,,) be any solution of (3.2.1). Then, the Lipschitzian nature of V(t,x) and condition (ii) yield Proof.

V ( t ,x ( t ) ) 2 V ( t , , xg)

=

uo

> 0,

(3.3.10)

for all t >, 0, for which ( t , x ( t ) ) E G. Since V ( t ,x) = 0 for all (t,x) E - G, it follows from (3.3.10) that (t, x(t)) E G for t 2 t o . Moreover, we also have

e

which, in view of Remark 1.4.1, implies that V t , 4 t ) ) 2 d t , to

7

U"),

t

2 to ,

(3.3.11 )

where p ( t , t o , uo) is the minimal solution of (3.2.3). Since V ( t ,x) is bounded by assumption, the estimate (3.3.1 1) leads to an absurdity,

3.3.

143

STABILITY

if we assume the trivial solution of (3.2.1) is stable. This proves the theorem.

THEOREM 3.3.7. L e t f E C[(-co, co) x S o ,R"] andf(t, x) be periodic in t with a period w . Then, under the hypotheses of Theorem 3.3.1, the trivial solution of (3.2.1) is equistable for to E (- 00, co). Proof. Let 0 < E < p and to E (- co, co) be given. I t is possible to choose an integer k such that to kw 2 0. Since the solution u = 0 of (3.2.3) is equistable, given b ( ~ > ) 0, t, kw >, 0, there exists a positive function 6 = 6 ( t o , c) such that uo 6 implies

+

u(t

u(t

+

+ kw, to + kw, uo) < We),

<

t

+ kw, to + kw, uo) being any solution of u' = g(t

+ kw, u ) ,

u(to

+ kw)

3to,

(3.3.12)

uo .

(3.3.13)

=

+ kw, xo) and obtain 6 , = S,(t, II xo I1 < 8, , w o + kw, xo) < 8

We choose uo = V(to

, c) satisfying

simultaneously, as in the proof of Theorem 3.3.1. With this 6, , the equistability of the trivial solution of (3.2.1) follows. Supposing this were false, there would exist a t , > to such that

II 4tl)ll

=

<

/I +Ill

E?

t

€9

E

(3.3.14)

[to Y t11

for some solution x ( t ) of (3.2.1) such that 11 xo 11 function m(t) = V ( t kw, , x ( t ) ) ,

< 6,.

Consider the

+

and, for small h m(t

+ h)

-

> 0,

m(t)

< Jwf(t,

-f(t

+ kw, 49 II

+LI1+)I1 + [V(t + kw + h, 4 t ) + hf(t + V ( t + kw, x(t))l,

kw, 4 t ) ) )

-

where L is the Lipschitz constant and r(h)/h + 0 as h + 0. T h e periodicity of f ( t , x) and condition (ii) of Theorem 3.3.1 give the inequality

o w ) < g(t

+ kw, m(t)),

t

6

[to 7

tll,

which implies the estimate, by Theorem 1.4.1,

v(t + kw, x ( t ) ) < r(t + kw, to + kw, uo),

t E [to ti], 9

(3-3.15)

144

r(t

3

CHAPTER

+ kw, t , 4-kw, uo) being the maximal solution of (3.3.13). b ( ~< ) V(tl + kw, < ~ ( tI , k w , t o + kw, uo) < b(c),

Thus,

~ ( t l ) )

using relations (3.3.1), (3.3.12), (3.3.14), and (3.3.15). This contradiction proves the assertion of the theorem.

If, in particular, the function f ( t , x) of system (3.2.1) is known to be periodic in t or autonomous and is smooth enough to assure uniqueness of solutions, it is possible to infer more information about the stability of the null solution of (3.2.1). T o this effect, we have

THEOREM 3.3.8. IJet f E C[(-co, GO) x S , , R 7 L ] , f (xt), be periodic in t with a period w, and the system (3.2.1) admit unique solutions. Then, the stability of the trivial solution of (3.2.1) is necessarily uniform. Proof. By the periodicity of f ( t , x) in t , it follows that, if x(t, t o , xo) is a solution of (3.2.1), then x(t -1 w , t , , x,) is also a solution. Furthermore, the uniqueness of solutions shows that, for any interger k, ~ ( $t k w , t , & kw, x,) = ~ ( tt , , x,). For each fixed to , t , E (- co, a), let

Since S ( t , , E) is continuous in to for each E , if we let 6, it IS clear that 8, > 0. For D 8 < 6 , , we define

=

inf,,< to+

S(t,),

4 s ) = 020 sup ll "(t" + 0,to %)ll. 9

Notice that c ( 6 ) is a monotone increasing function of 6, and hence, if S ( E ) is the inverse function of €(a), we havc

I/ x ( t , t o

1

~o)lI

< E,

for every t, E [O, w], provided /I x, /I we can choose an integer k such that kw

: t, < ( k + I)w

<

t

2 to

< S(E).

or

-(k

7

Let t , be arbitrary. T h e n

+ l ) w < to < -kw.

< t, + ( k + 1)w < w .

<

Hence, either 0 to - kw w or 0 Consequently, if 11 xo /I 6 ( ~ ) either ,

/I ~ or for t

/I

G X ( ~ ,

tn

9

( f t ,o

xo)lI

<

7

xo)ll

I1 x ( t

11 x ( t

-

kw, t o

-

kw, xo)ll

<E

+ ( k II)w, + (k + l)w, xo)ll <

2 t o , and the uniformity of the stability is evident.

E,

3.4.

145

ASYMPTOTIC STABILITY

3.4. Asymptotic stability We present, in this section, a number of results concerning the asymptotic stability of the solution x = 0 of the system (3.2.1).

THEOREM 3.4. I . Let there exist functions V(t,x) and g(t, u ) fulfilling the following assumptions: (i) g E C [J x R+ , R] and g(t, 0) = 0. (ii) V E C [ J x S o ,R,], V ( t ,0) = 0, and V ( t ,x) is positive definite and locally Lipschitzian in x. (iii) D+V(t, x)

< g(t, V(t,x)),

(t,x) E J x S,,.

Then, the equi-asymptotic stability of the solution u = 0 of (3.2.3) assures the equi-asymptotic stability of the trivial solution of (3.2.1). Proof. Let (S:) hold. Then, by definition, (SF) and (S:) are satisfied, and therefore only quasi-equi asymptotic stability needs to be proved, since Theorem 3.3.1 guarantees (Sl). T o prove (S3),let E > 0, to E J be given. It then follows from (S?) that, given b ( ~ > ) 0, t, E J , there exist positive numbers 6, = S,(to) and T = T ( t o ,E ) such that u(t, t o , uo)


t

2 t"

+ T,

(3.4.1)

<

whenever u, 6,. Choose u, = V ( t o x ,,). Since V ( t ,x) is continuous and V ( t ,0) = 0, we can find, as in the proof of Theorem 3.3.1, a positive number 8, = 8,(t,) satisfying the inequalities

I1 Xo I/

< 8,

V(to > xo)

Y

< 8,

(3.4.2)

together. We have also, in view of (SJ, the inequality (3.3.5) holding for t 3 t o . Suppose, now, that there is a sequence {tk),t, 3 to + T and t, + co as k + co such that I/ x(t, , t o ,xo)lI 3 E , where x(t, t o ,x,) is some solution of (3.2.1) starting in I/ x, /I 8,. This leads to the contradiction

<

b(c)

< V(t,

, 4 t k >

to

> Xo))

< Y(tk , to

7

uo)

< Kc),

because of (3.3.1), (3.3.5), and (3.4.1). Thus, (S,) is proved, which implies the equi-asymptotic stability of the solution x = 0 of (3.2.1). REMARK 3.4.1. It is possible that 8, obtained in (3.4.2) may be such that 8, > 8,. where S$ = S ( t , , p), in which case the inequality (3.3.5)

146

CHAPTER

3

need not hold for all t 3 t, whenever 11 xo 11 < 8, . T o avoid this situation, it is necessary to redefine 8, = min[8, , &,3in the foregoing proof. COROLLARY 3.4.1. T h e function g(t, u ) = A ( t ) $(u), where h(t) is continuous on J , +(u) >, 0 is continuous for u >, 0, +(a) > 0 for u > 0, is admissible in Theorem 3.4.1 provided that there exists a T , to T co, verifying the relation

< <

THEOREM 3.4.2. Under the assumptions of Theorem 3.4.1, the uniform asymptotic stability of the trivial solution of (3.2.3) also implies the equi-asymptotic stability of the solution x = 0 of (3.2.1). Proof. T h e proof runs parallel to the proof of Theorem 3.4.1. First of all, following the proof of Theorem 3.3.2, we note that 6, is not independent of to although 6 is. Similarly, 8, occuring in (3.4.2) is also not independent of t, , even though 6, is. COROLLARY 3.4.2. T h e conclusion of Theorem 3.4.2 remains true if the function g(t, a ) : --+(a), where #(a)E .K.

Proof. We shall show that (S:) holds. Consider the scalar differential equation (3.2.3), which now takes the form u'

+(a),

.(to) = ug

3 0,

t,

2 0,

t

2 to,

whose solutions are easily seen to be u(t, t o , 110)

where

=

I--"J(.,)

-

(t

-

~IJl,

(3.4.3)

and J - l is the inverse function of J . Given any E > 0, we first observe that (S,) holds with S ( E ) = E . Furthermore, we can also conclude from (3.4.3) that ( S z ) is satisfied with 6, = p and T ( E )= J ( p ) - I(€),for any E p. Thus, it follows that the solution u = 0 of (3.2.3) is uniformly asymptotically stable.

.:

3.4.3. Let the hypotheses of Theorem 3.4.1 hold. Suppose further that V ( t ,x) is decrescent. Then, if the solution u = 0 of (3.2.3) is equi-asymptotically stable, the trivial solution of (3.2.1) is equiasymptotically stable.

THEOREM

3.4.

147

ASYMPTOTIC STABILITY

Proof. T h e equistability of the solution x = 0 is immediate from Theorem 3.3.3. T o prove ( S 3 ) ,we follow the proof of Theorem 3.4.1 and choose u,, = u(II x, 11). Then, instead of (3.4.2), we have the inequalities (3.4.4) II xn II < s o ~ ( xo I I 11) < so holding simultaneously. T h e rest of the proof runs almost similar. T h e fact that 8, and T are not independent of to shows that (S,) holds. This and the proof is complete. proves (S5), COROLLARY 3.4.3. tinuous on J and

T h e function g ( t , u ) ! ; A @ ) ds

for every to

---f

= A(t)u,

where

A(t)

is con-

--Go

> 0, is admissible in Theorem 3.4.3.

THEOREM 3.4.4. Let the hypotheses of Theorem 3.4.1 hold, and let V ( t ,x) be decrescent. Then, uniform asymptotic stability of the solution u = 0 of (3.2.3) guarantees likewise the uniform asymptotic stability of the null solution of (3.2.1). Proof. Since uniform stability of the solution x = 0 follows from Theorem 3.3.4, it remains to be shown that (S,) holds. T o do this, we follow the proof of Theorem 3.4.3 and observe that, in view of (3.4.4), 8, is independent of t,. Th at the number T ( E )depends only on E follows from the condition (5'2). Hence, (S,) is satisfied, which, in its turn, proves (S6).

THEOREM 3.4.5. Assume that there exist functions V(t,x) and g(t, u ) obeying the following conditions:

(i) g E C [ J x R, ,R] and g(t, 0) = 0, t E J . (ii) V E C [ J x S,, , R,], V ( t ,0) = 0, t E J , and V ( t ,x) is strongly positive definite and locally Lipschitzian in x.

<

D+V(t, x) g(t, V(t,x)), ( t , x) E J x So . Then, if the solution u = 0 of (3.2.3) is equistable, the trivial solution of (3.2.1) is equi-asymptotically stable. (iii)

Proof. By assumption (ii), V(t,x) is strongly positive definite, which implies that there exists a function b(t, u ) E %Zsuch that

q t , x) 3 b(t, II x II),

( t , x) E

J x

s,.

(3.4.5)

148

Define b,(u)

CHAPTER

=

3

b(0, u). Then

p be given and t , E J . Since (ST)holds, given bl(q) > 0, Let 0 7 to E J , there exists a positive function 6 = 6(t, , q), which is continuous in t , for each 7 such that u,, 6 implies I

<

Choosing uo = V ( t , , xu), we can find a positive function 5, as in the proof of Theorem 3.3.1, such that the inequalities

=

& ( t o , q),

hold together. Furthermore, by Theorem 3.3.1, we see that the solution = 0 of (3.2.1) is equistable, using (3.4.6). Let q be fixed, and let 5, denote the number S,(t, , 7 ) . T o prove (&), let 0 < E < q, to E J be given, and let 11 xu 11 6, . Since b(t, u ) E XZ,there exists a T = T(to,6) satisfying the relation

x

<

+

If (tr.) is a sequence such that t , >, to T , t , -+ m as k + 00, such that I( x(t,,. , t o , xo)ll >, E whenever 11 x, 11 < a,, it would follow from (3.4.5), (3.4.7), (3.4.9), and Theorem 3.1.1 that

This is a contradiction, since b(t, , c) + co as t, + co. Thus, (S,) and (S,) hold simultaneously, and the theorem is proved.

THEOREM 3.4.6. Under the assumptions of Theorem 3.4.5, the uniform stability of the solution u = 0 of (3.2.3) also implies the equi-asymptotic stability of the solution x = 0 of (3.2.1). Pmof. By assumption, 6 is independent of to in the foregoing proof. However, 8, 1 6, is not independent of to because of (3.4.8). Moreover, 7' also depends on t o , in view of (3.4.9), and thus (S,) holds.

COROLLARY 3.4.4. Theorem 3.4.6.

The

function

g(t, u)

E

0

is

admissible

in

3.4.

149

ASYMPTOTIC STABILITY

THEOREM 3.4.7. Assume that there exist functions V ( t ,x), g(t, u),and A(t) satisfying the following properties: A(t) > 0 is continuous for t

(i)

E

J and A ( t ) -+ co as t

---f

co.

(ii) g E C [ J x R, , R ] and g(t, 0 ) = 0, t E J. (iii) V EC [ J x S, , R,], V ( t ,0 ) = 0, t E J , and V ( t ,x) is positive definite and locally Lipschitzian in x.

<

A(t)D+V(t,x)+ V ( t ,x ) D + A ( t ) g(t,V(t, x ) A ( t ) ) ,( t , x) E J x S,. Then, if the null solution of (3.2.3) is equistable, the solution x = 0 of (3.2. I ) is equi-asymptotically stable. (iv)

Proof. Let 0 < 7 < p, to E J be given, and let u = min,,,A(t). By assumption (i), u > 0. Define 7” = ob(7), where b(u) E .X is the same function as in (3.3.1), obtained because of the positive definiteness of V ( t ,x). Assume that (Sf) holds. Then, given 7* > 0, to E J , there exists a 6 = 6(t, , 7 ) that is continuous in t, for each 7 such that 4 4 t o uo) < ?I*,

t

3

<

(3.4.1 1)

2 to,

if ua 6. Choose ua = A(t,) V(t,, x,). Arguing, as in Theorem 3.3.1, we conclude the existence of a positive function 8 = 8(t, , 7 ) such that

/I xnI/ < 8,

~ ( t o ) ~ ( txu> o,

<8

(3.4.12)

hold jointly. With this 8, condition (8,)holds. For otherwise we are led to a contradiction, as in the proof of Theorem 3.3.1,

4 ~ < A(ti)v(ti ) ,~

( f 3, to

xo))

<

ti

t o , uo)

< rt*,

because of the definition of 7” = ob(7) and the application of Theorem 3.1.2. For a fixed 7 < p, designate by So = 6,(to) the number 8(to, 7). Let now 0 T * ,

t

2 to + T .

(3.4.13)

T h e choice u,, = A(t,) &‘(to, x,,), the positive definiteness of V(t,x), and Theorem 3.1.2 give A(t)b(II x ( t , t n , xo)II)

for t 11 x, [I

(3.4.14)

where x ( t , to , xo) is any solution of (3.2.1) such that 6,. If there exists a sequence { t k } ,t,b 2 to + T , t, -+ co as

2 t,, i

< A ( t ) V ( t ,x ( t , fo , xg)) < ~ ( t o, , uo)

150

k

4

j / x,, / /

3

CHAPTER

I

such that 11 x ( t , &t,, , , xo)il 3 E for some solution satisfying 8, , we obtain from (3.4.1 1) and (3.4.14) the inequality

< rl*,

.4(t,c)b(t)

which contradicts (3.4.13). This proves that (S,) holds, and consequently the solution x = 0 is equi-asymptotically stable. T h e theorem is proved.

THEOREM 3.4.8.

Under the hypotheses of Theorem 3.4.7, the uniform stability of the solution ZL 0 of (3.2.3) also implies the equi-asymptotic stability of the trivial solution of (3.2.1). ~

Proof. As in the proof of Theorem 3.4.6, it is easy to see that 6, and T depend on t o , and consequently (8,) holds.

THEOREM 3.4.9. Assume that there exists a function V ( t ,x) enjoying the following properties: and V ( t ,x) is positive definite, decrescent, (i) I/ E C [J x S, , R,], and locally Lipschitzian in x. (ii)

D+V(t,x)

< -+(I]

x li), ( t , x) E J

x S,, , where 4 E X".

Then, the trivial solution of (3.2.1) is uniformly asymptotically stable.

Proof. Let 0 . t p, t,, E J be given. Since V ( t ,x) is positive definite and decrescent, there exist functions a, h E 2"such that ~

h(ll ?L'Il)

< L7(t,).

,< .(I1

2

ll),

s,.

( t ,). fzJ x

(3.4.15)

By Corollary 3.3.3, it follows that (S,) holds. Hence, condition (ii) and Theorem 3.1.3 with g(t, u ) I 0 give the inequality

for t

t,. Designate by 6, the number Q),

and choose T ( E )=

<

u(S,,)/+(S(c)), mhcre S ( E ) corresponds t o E in (&). Suppose that )I x,, 11 6, and that we would have / / x ( t , t,, , x,,)ll S(t) for t,, t t,, T(E).

Then, for t E [t,,, t,,

+ 7 ' ( ~ ) ] ,we get, from (3.4.16),

1 I t , 4 t ) ) -2 q t , , , .")

s 44,)

which, for t 0

=

t,,

~

+ T ( t ) , reduces to

< b(qc))

.L. LTt,,

0.

~

4(S(,))(t

.b(S(,))(t

~

~

< < +

t")

t,,),

+ T ( c ) ,$4, + 7'(€)))i.(So)

-

4(8(E))T(E)

151

3 . 4 . ASYMPTOTIC STABILITY

This contradiction proves that there exists a t , E [ t o ,to that 11 x(t, , t o , xo)\\< 6(e). Thus, in any case, we have

+ T ( E ) ]such

<

whenever (1 xo I( S o , proving the uniform asymptotic stability of the solution x = 0 of (3.2.1). Notice that condition (ii), together with (3.4.15), yields that D+V(t,x )

< -+[u-"V(t, .))I

ESE

-+*(V(t, x)),

where $* E .X, and therefore the conclusion of Theorem 3.4.9 follows right away from Corollary 3.4.2. However, the proof given previously is of interest in other situations.

THEOREM 3.4.10. Assume that there exist functions V ( t ,x), g(t, u) satisfying the following properties: (i) V E C [ J x S, , R,], and V ( t ,x) is positive definite, decrescent, and locally Lipschitzian in x. (ii) g E C [ J x R, , R ] , and, for every pair of numbers 01, p such that 0 < 01 p < p, there exist 8 = 8(01,p) 3 0, K = k(01,p) > 0 satisfying

<

g(t,u)

< -A,

01

< u < p,

t

.> 0.

(iii) D+V(t,x) d g(t, V ( t ,x)), ( t , x) E J x S,, . (iv) For any function X E C [ J ,R,],

Then, the trivial solution of (3.2.1) is uniformly asymptotically stable. Proof. Let x ( t ) = x ( t , t o ,xo) be any solution of (3.2.1). Defining m(t) = I/ x(t)ll, we obtain, from (3.4.17), m;(t)

< I1 x'(t)ll < Ilf(4 x(t))ll < Y t ) m ( t ) .

By Theorem 1.4.1, it follows that

152

3

C'IIAPTER

as far as (1 x(t)(\ p. Consider thc interval [ t o ,T] for some p. Then, since which 11 .z(t)l/ <

Ji0

h(s) cls

w e obtain

T

>, to for

-< j'h(s) ds, 0

I1 4f)ll< I1 X" II e N T ,

[to , TI,

t

(3.4.18)

where N N ( T )= supo A(t). Since V ( t ,,x) is positive definite and ctescresccnt, (3.4.15) holds. Let 0 < c c p , toE J be given. Choose 8, S , ( c ) such that 481) < 4 6 ) . -

-

It is clcar that s1 E. Let 8 : B[a(S,), A(€)], tZ = k[a(S,), b ( ~ ) ] and , 8 ( ~= ) 8,ecN0. Wc choosc I/ xo 11 6 so that (3.4.18) assures that 11 x(t)ll < 6 , , to t B, which implies that

< <

s .

.(I1 4f)ll)< a(S,),

C'(t, t ( t ) )

t"

< t < 8.

We now claim that "(t, x(t)) < b(€),

t

2 8.

If this wcrc not truc, thcrc exist t , , t, such that t,

> t, >, 8, satisfying

qf, , x(tl)) = ~(s,), Tt2

and n(S,)

Hence, at t

--

1

r(t2))

-

b(c),

< " ( t , x(f)) < h ( € ) ,

t

E

[tl , t,].

(3.4.19)

t , , there results I ) ' K(tl , x(tl)) > 0.

(3.4.20)

On the other hand, as t , 2 8 and (3.4.19) holds, we obtain, from conditions (ii) and (iii) and the fact that V ( t ,x) is locally 1,ipschitzian in x, the inequality I ) ' l'(f,

>

dtl))

< A t , , Vtl :< -k < 0 ,

7

x(t1)))

which contradicts (3.4.20), thus proving that V ( t ,x ( t ) ) < A(€) for t 3 8. It therefore follows that, if (1 xl, (1 -'8, V ( t ,x ( t ) ) << b(a), t >, t o , and

3.4.

153

ASYMPTOTIC STABILITY

consequently, in view of (3.4.15), the uniform stability of the trivial solution of (3.2.1) is proved. Let us denote by So the number 6 ( p ) obtained by setting E = p, and let 0 < E < p . Let 6 = S ( E ) be the same function as before. Assume 11 x,,11 < 8,. Choose

where

T o prove uniform asymptotic stability, it is enough to show that there exists a t, E [to , to TI satisfying /I x(t,)ll < S ( E ) ; since it would then follow that 11 x(t)ll < c, t 3 to T in any case. Suppose there is no such t , ; then,

+

+

a(€) G I1 4t)ll G P ,

t

E

[t" t o 9

+ TI,

which, because of (3.4.15), shows that

Using assumptions (i) and (iii), we get

This absurdity proves that 11 x(t)iI < S ( E ) < E , t 3 t, and the theorem is completely proved.

11 x,,11 < 8,,

-t T , whenever

We have seen in Theorem 3.3.5 that the uniform stability of the trivial solution of (3.2.1) can be formulated by means of monotone functions. We may likewise state the following theorem with respect to uniform asymptotic stability.

154

3

CHAPTER

THEOREM 3.4.1 1. T h e trivial solution of (3.2.1) is uniformly asymptotically stable if and only if there exist functions a E X , (r E 9such that

I/ . ~ ( tto, , xo)ll for

1) xo I/

< 41xn I l ) ~ ( f

-

t

to),

2 to >

(3.4.21)

-, p.

Pyoof. If (3.4.21) is satisfied, it is easy to verify that (S,) holds, and hence sufficiency of the condition is evident. Assume now that the trivial solution of (3.2.1) is uniformly asymptotically stable, so that (S,) holds. Let { e n } be a positive, monotonic sequence, converging to zero as n + GO. Let T , ( E )= inf T ( C ? ~ + ~for ) en+,

Then,

I/ X" I/ < 6 0 ,

t

I1 .x(t, to , xojll < [E,,+~,

en],

TI(€,),

E

E,

.

> t" iTI(€)

assures that

Define T*(e) linear in T,(E,, , l ) . Note that

< <

<

T*(E,+,)

6.

T l ( ~ 1 1 +and 2 ) , T*(r,)

:

lim ?'](en)

n-r

=

03.

T h e equality T1(~,,+l) = may occur on finite parts of the sequence which can be eliminated. T h e function T*(e) is continuous, monotone decreasing, and lim6-,oT * ( E )= CO. Moreover, t 3 t,, T*(e) implies t

Hence, if

entl

< < E

5

to i- T*(%)= t"

E,,

, we have

+ Tl(%I+l).

+

This shows that the function T(e) occurring in (S,) can be chosen continuous and monotonic. Let q ( T ) be the inverse function of T ( E ) . Then, it follows that

3.5.

155

STABILITY OF PERTURBED SYSTEMS

whenever I/ x, 11 < 6, , which proves that uniform asymptotic stability is equivalent to (3.4.21).

3.5. Stability of perturbed systems We shall consider the perturbed system

where j ; R

E

C [J x S, , Rn].

THEOREM 3.5.1.

Let there exist a function V ( t ,x) satisfying the following conditions:

(i) V E C[J x S, , R,], V(t,x) possesses continuous partial derivatives with respect to t and the components of x, and

for ( t , x) E J x S o . V(t,x) B [I x (la, A , B being positive constants. (iii) A 11 x 114 (iv) w E C [ J x R, , R,], w ( t , 0) = 0, w ( t , u) is monotone nondecreasing in u for each t, and

<

<

11 R(t, x)ll

< w ( t , II x 1 ").

Then, the stability properties of the trivial solution of (3.2.3), with U"

L + All w(t, ~

Urn),

imply the corresponding stability properties of the solution x

(3.5.1). Proof.

Consider the function U ( t ,x)

=

(3.5.2) =

0 of

156

CIIAPTER

3

In view of thc assumptions of the theorem, if we define

we obtain, after some computations, that

< U ( t ,x) < Bl'Q/I x 11,

I/ 2 /I

U ' ( t ,). ,< 'dt, V t , x)),

.>

( t ,x) €

J x so,

(t,

J

'R

>

where ~ ( tu ,) is the same function given by (3.5.2). T h e conclusion of the theorem is then a direct conseqeunce of Theorems 3.3.3, 3.3.4, 3.4.3, and 3.4.4. COROLLARY 3.5.1. Let the assumptions of Theorem 3.5.1 hold with 2, m = 1, and w ( t , u ) = X(t)u, where h(t) 3 0 is continuous on J such that N =

Then, the solution x

=

0 of (3.5.1) is asymptotically stable.

THEOREM 3.5.2. Let there exist functions V(t,x), g l ( t , u), and w(t, u ) fulfilling the following conditions: (i) V t C [ J x SR, R ,], V ( t ,x) is Lipschitzian in x for a function k ( t ) 3 0 continuous on J , and

411x II)

< L'(t, x) < 411x II),

t , x) E J

x s, ,

(3.5.3)

where a, h E f . (ii) g,

E

C [ J x R, , R],g,(t,0)

= 0, and

n + l 7 ( t 4? ( 3 2.1) < gl(t? f7(k x)),

( t ,x) E J x

s,

1

(iii) w E C [ J x R, , R,], w(t, 0) = 0, w ( t , u ) is nondecreasing in u for each t, and I / R(t,.y)II

< w ( t , II x II)?

( t ,).

E

Then, the stability properties of the solution u

At, ).

= Rl(t>).

J x s,. =

0 of (3.2.3) with

4- k ( t ) w ( t ,W U ) ) ,

(3.5.4)

where b ' ( u ) IS the inverse function of b(u), imply the same kind of stability properties of the trivial solution of (3.5.1).

3.5.

157

STABILITY OF PERTURBED SYSTEMS

Proof. Let us define the function D+V(t,x) with respect to the perturbed differential system (3.5.1) as follows:

using assumptions (ii) and (iii). This, together with (3.5.3) and the monotonicity of w ( t , u ) in u, leads to the differential inequality D'

v(t,

X)(:5.5.1)

< g(t, V ( t ,x)),

( t ,).

E

J

x s, ,

where g(t, u ) is given by (3.5.4). Now, it only remains to apply Theorems 3.3.3, 3.3.4, 3.4.3, and 3.4.4 to get the desired result. COROLLARY 3.5.2. T h e functions b(u) = u,gl(t, u ) = -au, a > 0, and w(t, u ) = X(t)u, X ( t ) 3 0 being continuous on J and satisfying

are admissible in Theorem 3.5.2, to guarantee the uniform asymptotic stability of the solution x = 0 of (3.5.1) provided k is the Lipschitz constant for V ( t ,x). COROLLARY 3.5.3. T h e functions w ( t , u ) = h ( t ) $ ( u ) , gl(t, u ) = 0, where X E C [ J ,I?,], 4 E X , are admissible in Theorem 3.5.2 to yield the uniform stability of the trivial solution of (3.5.1)) provided that k is the 1,ipschitz constant for V(t,x) and :J- X(s) ds < GO. COROLLARY 3.5.4. T h e functions w(t, u ) = h(t)$(u), gl(t, u ) = -C(u), where h E C [ J ,I?+], 4,C E .Y, are admissible in Theorem 3.5.2 to assure that the trivial solution of (3.5.1) is uniformly asymptotically stable, provided that Fz is the Lipschitz constant for V ( t ,x), k 4 ( k 1 ( u ) )< ,C(u), for some 01 > 0, and limt+m[-t 01 h(s) ds] = -a for all to 3 0.

+ J-7'

158

CIIAPTER

3

3.6. Converse theorems This section will be devoted to a variety of results concerning the construction of Lyapunov functions. Let us first define the notion of generalizcd exponential asymptotic stability. DEFINITION 3.6.1. T h c trivial solution of (3.2.1) is said to be (,Sll) generalized exponentially asymptotically stable if

where K ( t ) > 0 is continuous for t E J , p E X' for t E J , and p ( t ) + GO as t + co. I n particular, if K ( t ) = K > 0 , p ( t ) = at, a: > 0. We have the exponential asymptotic stability of the trivial solution of (3.2.1)

THEOREM 3.6.1. Assume that the solution x = 0 of (3.2.1) is generalized exponentially asymptotically stable and thatf(t, x) is linear in x. Suppose further that p'(t) exists and is continuous on J . T h e n there exists a function V ( t ,x) satisfying the following properties: (i) V E C [ J x S o , R,], and V ( t ,x) is Lipschitzian in x for the function K(t).

<

<

V ( t , x) K ( t )/I x 11, ( t ,x) E J x S, . (ii) 11 ?L^ 11 D + V ( t , x) -p'(t) V ( t ,x), ( t ,x) E J x S, . (iii) Proof.

<

Define

Then, from (3.6. l ) , it follows that (ii) is satisfied. Let s, y E So . Then,

3.6,

159

CONVERSE THEOREMS

To arrive at this estimate, we have used the facts thatf(t, x) is linear in x and that the solutions x(t, t o , x,,) obey the inequality (3.6.1). We shall now prove that V ( t ,x) is continuous. Let 6 >, 0; then

+

Since V ( t ,x) is Lipschitzian in x and x( t 6, t , x) is continuous in 6, the first two terms on the right-hand side of the preceding inequality are small when /I x - x* I( and S are small. Let us consider the third term. Observe that x(t

+ 6 + u, t + 6, x ( t + 5,t , x))

= x(t

+6 +

t , x).

0,

Hence we have

we notice that a(S) is nondecreasing and tends to a(0) as S + 0, since I\ x(t u, t , x)ii exp[ p ( t D ) - p(t)] is a bounded continuous function for all u 2 0. Thus,

+

+

implies that the third term tends to zero as 6 the continuity of V ( t ,x).

+ 0.

T h u s we have verified

160

CHAPTER

3

Furthermore, using the uniqueness of solutions and the definition (3.6.2), 1

D+T-(t,x ( t ) ) = lim sup - [I'(t -1- h, x ( t ih, t , .)) h-0F h =

lim sup h.0'

=

lim sup A-0'

1 ~

12

1

h

~

[sup / / x ( t -t- h 030

sup

[I/ x(t

U ' h

-4-

u, t

11

x {..p(p(t)

<

-

V ( t ,x)]

+ h, x ( t + h, t , %))I1

+ u, t , x)ii e x p ( p ( t -1-

< lim sup 1 [sup 1 1 .z.(t + u, t , )I./ rO &.O+

~

u)

c s p ( p ( t -t u)

~

~

p(t

+ h))

p(t))

0

-

p ( t -1- h ) )

~

I}]

p ' ( t ) l ' ( t , ,x).

Since, for small h > 0, r-(t 7-h, 2

+ hf(t, x))

~-

+ h, t , x) s - hf(t, .)/I r q t 4-h, s(t + h, t , x)) V ( t ,x),

< K ( t )I/ .x(t

l-(t,Y)

-1-

~

-

it easily follows that fl' l'(t, x)

' \

~

p'(t)V(t,x),

proving (iii). T h e theorem is completely proved.

A similar result is true even when f ( t , x) is nonlinear in x provided it is assumed to satisfy a Lipschitz condition in x for a constant L. T h e next theorem substantiates this remark.

THEORE~I 3.6.2.

1,et the trivial solution of (3.2.1) be generalized cxponcntially asymptotically stable and f ( t , x) satisfy a Lipschitz condition in x for a constant 1, = L ( p ) > 0. Assume that p ' ( t ) exists and is continuous for t E J . Suppose further that K ( t ) is bounded and, for some q, 0 ,. q 1, there cxists a number T > 0 such that <.

K ( t )c x p [ - q ( p ( t I T ) - P ( t ) ) l

< 1,

t

E

1.

(3.6.3)

3.6.

161

CONVERSE THEOREMS

Then, there exists a function V(t,x) fulfilling the following conditions:

(i)

V E C [ ] x SDo, R,], 0 < p0

I U t , 4 - V(t,Y)l (ii) (iii) Proof.

< eLT0 sup

.o-’T

< p, and V ( t ,x) obeys

exp[(l

~

mJ(t

+

0) -

P(t))lll x - Y 11.

< v(t,x) < K ( t ) I! x ll, (t, x) E J x So0. P V ( t , x) < -(1 - q)p’(t) V ( t ,x), ( t , x) E J x !I x II

SDo .

Let q, T be given so that the relation (3.6.3) holds. Define V ( t ,x )

=

sup /I x(t --I- (I, t , x)ll exp[( 1 - p){p(t uao

+

0) -

p(t))]. (3.6.4)

Since K ( t ) is assumed to be bounded and p ( t ) E X , the function V(t,x) is defined for ( t ,x) E J x SD,, where p0 = p / M , M = supLGJ K(t). T o prove that V(t,x) satisfies the Lipschitz condition, we first observe that

and, since (3.6.3) holds, we have

Accordingly, by Corollary 2.7.1, we get

T h e relations (ii) and (iii) can be verified as in the proof of Theorem 3.6.1. T h e proof is complete.

COROLLARY 3.6.1. If the trivial solution x = 0 of (3.2.1) is exponentially asymptotically stable, that is, p ( t ) = at, CY > 0, K ( t ) = K > 0, then T = log Klqct, and M = KL+(-q)a/qa, where n/l is the Lipschitz constant for V ( t ,x), are admissible in Theorem 3.6.2. It is possible to prove the previous theorems, under milder assumptions, in a different way.

162

3

CHAPTER

THEOREM 3.6.3.

Assume that

(i) f E C [ J x S, , R”],f(t,0) = 0, andf(t, x) satisfies

< L ( t ) I/ x

IIf(t, x ) -f(t,Y)II

-y

11,

( t , x ) , ( t ,y ) E

1X

L(t) 0 being continuous on J ; (ii) There exists a p E X’ for t E J , p ( t ) 4 co as t exists, and

11 x ( t , 0, xo)ll

< K II xo II exp[-P(t)l,

t

S,

, (3.6.5)

-+00,

2 t o , K > 0,

p’(t)

(306.6)

where x(t, 0, xo) is the solution of (3.2.1) through (0, xo). Then, there exists a function V ( t ,x) enjoying the following properties: (1) V E C [ J x S o ,R,], and V ( t ,x) is Lipschitzian in x for a continuous function K ( t ) > 0.

<

(2) I1 x I1 G V(t,4 K ( t ) I1 x 11, ( t , x) E J x s, . ( 3 ) Di-V(t, x) -p’(t) V ( t ,x), ( t , x) E J x S , .

<

Pmof. Let us denote x = x(t, 0, xo) so that xo = x(0, t , x), because of the uniqueness of solutions of (3.2.1), which is assured by condition (i). We now define V ( t ,x) = Ke-fi(t)11 x(0, t , %)I\. (3.6.7) I t is then evident that V E C [ J x S, , R,], V ( t ,0) of (3.6.6), we have II x II < q t , x).

= 0,

and, because

Since the solutions of (3.2.1) are unique, it follows that V(t

+ h, x ( t + h, t , x)) = Ke-p(t+h)jl x(0, t + h, x ( t + h, t , .)I1 -

Ke-P(t+h’I/ 40,t , x)IL

and hence we obtain

=

+(t)V(t, x).

If x(t, 0, xo), x(t, 0, yo) are the two solutions of (3.2.1) through (0, xo), (0, yo), respectively, the condition (3.6.5) yields

3.6.

I63

CONVERSE THEOREMS

by virtue of Corollary 2.7.1. Letting x = x(t, 0, xo), y get, from the preceding estimate, the inequality

=

x ( t, 0, yo), we

Consequently,

= K ( t )/I

* - Y /I)

+

if we define K ( t ) = K exp[-p(t) l:L(s) ds]. This proves that V ( t ,x) satisfies a Lipschitz condition in x for a function K ( t ) > 0. T h e upper estimate in (2) follows from the Lipschitz condition by setting y = 0 and observing that V ( t ,0) = 0. As in the proof of Theorem 3.6.1, one can deduce (3) from (3.6.8). T h e theorem is proved.

3.6.2. If in Theorem 3.6.3, the functions L(t) and p ( t ) COROLLARY are such that J:L(s) ds p(t), and V ( t ,x) satisfies the Lipschitz condition in x for a constant K > 0. We have already seen that the concepts of stability and asymptotic stability can be defined by means of simple inequalities involving certain monotone functions. We give below some converse theorems in terms of differential inequalities. As will be seen, the approach depends upon the differentiable properties of solutions with respect to the initial values and yields, in a unified way, a method of constructing Lyapunov functions.

<

THEOREM 3.6.4. Suppose that (i) the function j~ C[J x So,R"],f(t,0) exists and is continuous for ( t , x) E J x So; (ii)

= 0,

PI,p2 E Z such that < I1 x ( t , 0 , < PA1 xo ID,

and af(t, %)/ax

there exist functions

A(11xo 11)

X0)ll

tE

I,

(3.6.9)

where x(t, 0, xo) is the solution of (3.2.1) through (0, xo); (iG) the function g E C [ J x R, , R],g(t, 0) 3 0, and ag(t, a ) / & exists and is continuous for (t, u ) E J x R, ;

164

CHAPTER

3

(iv) the solution ~ ( t0,, uo) of (3.2.3) fulfills the estimate

< 44 0 , U,,) < Y 2 ( U o ) ,

Yl(U,)

t

E

J,

(3.6.10)

where y 1 , y 2 E .T. Then, there exists a function V ( t ,x) with the following properties: ( 1 ) V E C [J x S, , K,], and V ( t ,x) possesses continuous partial derivatives with respect to t and the components of x.

(2) h(lI x 11) (3)

< V ( t ,x) < 411x II), =g(t,

( t , x) E

J

q t , x)), ( t , x) E

x S, , a, b E z-.

J x so.

Proof. I n view of hypothesis (i), the existence and uniqueness of solutions of (3.2.1), as well as their continuous dependence on the initial values, is assured. Also, the solutions x ( t , t, , x,), ( t o ,x,,)E J x S, are differentiable functions with respect t o the initial values, and x = 0 is the trivial solution. Furthermore, on the basis of Theorem 2.5.3, we have and

where @(t,to , x,,) = Bx(t, t o , xo)/axois the fundamental matrix solution of the variational system

such that @(t,,, t, , xo) is the unit matrix. Similar conclusions hold for the solutions of (3.2.3) as (iii) is satisfied. Let now x ( t , 0, xJ, u(t, 0, u,) be the solutions of (3.2.1), (3.2.3) through (0, xo), (0, uo), satisfying the inequalities (3.6.9) and (3.6.10), respectively. Denote x(t, 0, x,,) by x so that x,,= x(0, t , x). T h i s is clear by virtue of the uniqueness of solutions of (3.2.1). Choose any continuous function p(x) possessing continuous partial derivatives ap(x)/ax for x E S, such that rnl(I1

II)

< d x ) < %(/I

x II),

Define the function V ( t ,).

=

u[t, 0 , &(O,

a1

7

t , .))I.

a2 E

x-0

(3.6.12)

3.6.

165

CONVERSE THEOREMS

Because of the continuity of x(0, t , x), p(x), and u(t, 0, u,) with respect to their arguments and the fact that u = 0 is the trivial solution of (3.2.3),

it follows that V E C [ J x Y, ,R,]. Since the functions x(0, t , x), u(t, 0, u,) are differentiable with respect to the initial values and ap(x)/ax is assumed to exist, we see that

ax(o, t , at,

+

Thus, (1) and (3) hold. Since x = x ( t , 0, x,) and x, (3.6.9) that

P;l(l

x

II)

ax(o, t , x) . f ( t , x) ax,

=

= 0.

x(0, t , x), it follows from the inequality

< /I 407 t , 4 < P;l(l

x ll),

(3.6.13)

where Pr', / 3 ~ lare inverse functions of PI , p2 and hence belong to X . Using the inequalities (3.6.10), (3.6.12), and (3.6.13) successively, the definition of V ( t ,x) gives J q t ,x)

=

u[t,0, p(x(0, t , 411

2 Yl(cL(X(0,t , 4)) 2 Y d 4 1 40,t , x)II)) 2 Y d ~ l ( P i l(11 x 11)) = b(11 x 11) and

166

3

CHAPTER

Obviously, the functions a, b E 37,since the functions y1 , 0 1 ~,pz', y z , a 2 , and P;' all belong to class ,X. This proves (2), and the proof is complete. COROLLARY 3.6.3. Assume that f E C[J x S, ,R"],f ( t , 0) = 0, and af(t, x)/ax exists and is continuous for ( t , x) E J x S,. Suppose also that, for any solution x(t, 0, x,,) through the point (0, x,,) of (3.2.1),

Then, there exists a function V ( t ,x) such that: (1)

V E C [ J x S, , R,], V ( t ,0)

0, and V ( t ,x) possesses con-

=

tinuous partial derivatives with respect to t and the components of x;

(2) b(ll x 11) (3)

< V(t,x),(t,). V ( t ),.

E

x So 6 E ,x;

J

1

a v t , ).

= ____

at

= 0,

(b,

+

av(t7 4 . f ( t , ). a x-

x) E J x S " .

T h e following variant of Theorem 3.6.4 is of interest in some situations.

THEOREM 3.6.5.

Let

(i) f~ C [ J x S, , R"],f ( t , 0) Ilf(t!

=

-f(t,y)ll

0, andf(t, x) satisfies

< Ldt) /I x

-

Y II

for ( t , x), ( t , y ) E J x S, , where L,(t) 3 0 is continuous on J ; (ii) there exists a function p2 E .X such that

II x(t, 0, %)I1

< P,(ll

xo

10,

t

2 0,

x(t, 0, xo) being the solution of (3.2.1); (iii) g E C [ J x K, , R],g ( t , 0) 1 0, and g(t, u ) verifies

I At, 4 - g(f7 v)l

< L,(t) I

zL - 7l

I

for t E J , u, ZI >, 0, L,(t) 2 0 being continuous on J ; (iv) the solution u(t, 0, u,,) of (3.2.3) fulfill the estimate n(.o)

< 4 t >0, a,),

t

2 0,

Y1 €37.

3.6.

167

CONVERSE THEOREMS

Then, there exists a function V(t,x) with the following properties: (I) V E C [ j x S, , R,], and V ( t ,x) is positive definite and satisfies a Lipschitz condition for a continuous function K ( t ) 3 0.

<

D + W , x) g(t, V ( t ,XI), ( t ,4 E J x So. Proof. Since the uniqueness of solutions, as well as their continuous dependence on initial values, is guaranteed by (i) on the basis of Corollary 2.5.1, if we let x = x (t, 0, xJ, it follows that xo = x(0, t , x) as previously. We define

(2)

V ( t ,4 = u(t, 0, !I 4 0 , t , x)ll),

where u(t, 0, uo) is the solution of (3.2.3) through (0, u,,). Note that, by assumption (iii), u = 0 is the trivial solution of (3.2.3) and that the solutions u(t, t o , u,,) are unique. Furthermore, we infer from uniqueness of solutions of (3.2.1) that V(t

+ k , x ( t + k , t , x))

=

=

and therefore

+ k , 0, /I 40, t + k , x ( t + h, t , x))ll> u(t + k , 0, /I 40, t , u(t

1 D+V(t,x ( t ) ) = lim sup - [ V ( t &+Of k

X)II)>

+ k , x ( t + h, t , x))

-

V ( t ,x)]

= u’(t,0 , !I 40,t , .)!I) = g(t,

V t , XI).

According to Corollary 2.7.1, if we let x = x(t, 0, x,,), y condition (i) implies, as in Theorem 3.6.3, the inequality

Moreover, assumption (iii) also implies that

Thus, for ( t , x), ( t , y ) E J x S o ,we have

(3.6.14) =

x(t, 0, yo),

168

CHAPTER

3

+

if K ( t ) = exp[Ji (L,(s) L2(s))ds]. This, together with (3.6.14), enables us to deduce, as in Theorem 3.6.1, that

where P(T) = /322c22(0)T. Moreover, using the lower estimate of (3.6.20), it follows that

COROLLARY 3.6.4. Theorem 3.6.5.

The

function

g(t, u) = 0

is

admissible

in

T h e next theorem deals with the converse problem for asymptotic stability.

THEOREM 3.6.6.

Let assumptions (i) and (iv) of Theorem 3.6.4 hold. Suppose that the solution x ( t , 0, xo) of (3.2.1) satisfies

< P,(I

II x ( t , 0, .T")ll

t

X" Il)o(t),

where p2 E X , c E 9'. Assume that there exist functions y such that y'(u)

3K

b

0,

and r(u")qt)

t

E

.X, 6 E 9

K, > 0,

8 ( t ) 3 klU(t),

< u ( t , 0, uo),

(3.6.15)

2 0,

3 0.

(3.6.16)

Then, there exists a function V(t, x) satisfying V ( t ,0 ) 0, (1) V E C [ J x s, , R,], V ( t ,x) is positive definite and possesses continuous partial derivatives with respect to t and the components of x; :

(2)

V'(4

x) = g(t,

V t , x)),

(4 x) E

1 x so.

Pmof. Let x ( t , 0, x,,), u ( t , 0, uo) be the solutions of (3.2.1), (3.2.3) obeying the estimates (3.6.15), (3.6.16), respectively. Choose any

a

continuous function p(x) having continuous partial derivatives ap(x)/ax for x E S,, such that y(0) = 0 and

Bz(ll x II)

Defining

V(4 ).


(3.6.17)

W

= u [ t , 0, P ( X ( 0 , t ,

4)1,

it can be readily shown as in Theorem 3.6.4 that V E C [ J x So,R,] and satisfies (2). Moreover, V(t,0) = 0 follows from x(0, t , 0) = 0, R > 0, y(0) = 0, and u(t, 0,O) = 0. From the assumption y’(u) there results Y(u1uz) ku1uz. (3.6.18)

z

Furthermore, by virtue of the fact that x the inequality (3.6.15) yields

=

x(t, 0, xo) and xo = x(0, t, x),

(3.6.19)

where P(T) = /322c22(0)T. Moreover, using the lower estimate of (3.6.20),

which implies, on account of the assumption 8 ( t ) is positive definite. T h e proof is complete.

2 Rla(t), that V(t,x)

COROLLARY 3.6.5. If u E 2 is a differentiable function for t E J , then the function g ( t , u ) = [a’(t)/u(t)]uis a candidate in Theorem 3.6.6. REMARK 3.6.1. Notice that, in Theorems 3.6.5 and 3.6.6, we have not assumed that the trivial solution of (3.2.3) is stable and asymptotically stable, respectively, since we do not need, in the proof, such

170

3

CHAPTER

a specific assumption. However, these hypotheses are required to prove

direct theorems. Nevertheless, the lower estimates of the solutions ~ ( t0,, u0) of (3.2.3) are compatible with the corresponding stability requirements. It can be seen from the proof of Theorem 3.6.4 that the lower estimate on x ( t , 0, xo) and the upper estimate on u(t, 0, uo) are useful only to prove the dccrescent nature of V ( t ,x). Observe also that we need only the stability information of solutions starting at to = 0, and this is a definite advantage. Undcr the rather general assumptions of Theorem3.6.6, it is not possible to show that. V ( t ,x) is decrescent. This can, however, be achieved in the following: 'rHE O R EM 3.6.7. Let assumptions (i) and (iv) of Theorem 3.6.4 hold. Suppose that, in place of (3.6.15), we have

Plll %J Y 4 t ) < I1 x ( t , 0, X")ll

< All

where R, p1 , p2 > 0 are constants, and of (3.2.3) allow the estimate 4u"qt)

where A, , A,

p > 0,

G

E

Xo

t >, 0,

Ilao(t),

2. Let the solution u(t, 0, uo)

< 4 t , 0, uo) < huo8(t),

t

z 0,

> 0 are constants, and 6 E 9 such that,

for some constant

syt) = U B ( t ) .

Then, there exists a function V ( t ,x) that is decrescent and that obeys (I), (2) of Theorem 3.6.6.

Proof.

By choosing a continuous function p ( x ) so as to satisfy

kill X /I0 < ("(x)

< k,lI

.^c

k , I R,

I T 3

I

B

0,

and following the proof of Theorem 3.6.6 with necessary modifications, we can easily construct the proof of this theorem.

THEOREM 3.6.8. Let assumption (i) of Theorem 3.6.4 hold, and let there exist functions u l , u2 E 9 such that

PlII xo / / ~ i (-t to)

< /I

4)

~ ( t l

~o)ll

< Bzll

Xo

IluAt

-

to),

t

> to

(3.6.20)

p,, p2 > 0 being constants and x ( t , t o ,XJ being the solution of (32.1). Then, there exists a function V ( t ,x) satisfying the following with

properties:

3.6.

171

CONVERSE THEOREMS

(1) V E C [ J x S, , R,], and V ( t ,x) is positive definite, decrescent, and possesses continuous partial derivatives with respect to t and the components of x. -eV(t, x), N > 0, ( t , X) E J x S o . (2) V'(t,X)

<

Proof.

Define the function, for some fixed T

> 0 that we choose later,

Because of assumption (i), one can argue, as before, to show that V E C [ J x S, , R,] and V ( t ,x) is continuously differentiable. Furthermore, from the upper estimate of (3.6.20), we have

and hence

(3.6.21)

where P(T) = /322c22(0)T. Moreover, using the lower estimate of (3.6.20), it follows that

Thus, we have shown that (1) is verified. To prove (2), observe that

O n the other hand, using relation (3.6.1 l), we see that

Consequently,

172

CHAPTER

3

We now fix T by choosing it so large that

This is possible, since upE 2.Evidently, from this choice results the inequality V ' ( t ,x ) < - g 11 x 112, which, in view of (3.6.21), leads to

_-=

setting

&i=

~

d ( t ,x ) ,

1/[2P(T)]. T h e theorem is proved.

COROLLARY 3.6.6. condition

Instead of the lower estimate in (3.6.20), the llf(t,

1)I.

< -%)I1

ZL'

I/,

( t ,x ) E

I

x

s,

is admissible in Theorem 3.6.8.

TIIEOREM 3.6.9. Let the trivial solution of the system (3.2.1) be uniformly asymptotically stable. Suppose that

Ilf(t, XI)

-f(t,

%>ll

< L(t)/l

x1

for ( t ,q),( t , xg)E J x S, , where L(t)

~

xz

I1

0 is continuous on J , and

Then, there exists a function V ( t ,x) with the following properties:

, V ( t ,x) is positive definite, decrescent, (1) V E C [ J x S o ,R ! ] and and satisfies 1

v(t,.TI)

Y t , %)I

< MI1x1

-

% II

for ( t , xi), ( t , EJ x . D + V ( t , x ) C ( V ( t , x ) ) , ( t , x )J~ x S,,CE%. (2) C y J

<

AJls(s,)

Pyoof. Let us choose a function G(r) such that G(0) G ( Y )> 0, G"(r)> 0, and lct cx > 1. Since G (T)=

J: du

G"(v)dv 0

=

0, G'(0)

=

0,

3.6.

173

CONVERSE THEOREMS

and

we have, setting u

w/a,

=

( 1) = - i: dw [w'm G"(v)dv

G-

0

<1 [dw f"G"(v)dv 0 1 0

1

= - G(y). 01

0

(3.6.22)

Define

(3.6.23)

If ~(8) is the inverse function of

a(€),

I1 x(t + 0

and therefore

G(ll x ( t

+

U>

Consequently, observing that ( 1

we have

t,4 11 < 41* 11))

1

t , X)ll)

< G(4l * 11))-

+ ma)/( 1 + a) < a, it follows that

v(t*x) < m G ( 4 x 11)).

Since u

> T ( E )implies 11 x ( t + u,t , *)I1 I1 x (t

if

u

2 T(lj x li/a). Thus, G(ll x(t

which, in turn, leads to G(I/4 t

+

D?

t >.)lo

+

D7

t,

< E,

we get

x>ll < II x llb

+ u>t?x)II) < G(ll x IIbh 1+ I +o< olG *I( 01u

< G(ll x 11)

< V(t,21,

because of relations (3.6.22) and (3.6.23). This shows that

1 74

3

CHAPTER

T h e continuity of the function V ( t ,x) implies that there exists a such that v(t,4 = G(llx(t 0 1 t , "411)1 a(J1.

+

If we let x = x(t, to , xJ, x* of (3.2.1) shows that V(t

Denote cr* 1

1

+h

+ au*

+

+ h, x * )

u*

=

-

-

(5.

(1 (1

1

x(t

=

+

7

+ h, t, x), the uniqueness of solutions

+ + u*, t + h, x*) I )

G(/I~ ( t h

Then

+ au*)(l +

+ U*)(l +

0) -

-

1

(a

(1

-

1 +a,* 1 +u*

+ u* + au + auu* - ah + h

0)

+ au

u1

+ o*)(l + ).

1-

1)h

1 +o

It therefore follows that

< V ( t ,x) [I

-

(a

(1

+ u*)(l -

I)h +am)

1'

using (3.6.23), it is easy to obtain D+V(t,x ( t ) )

=

1 lim sup - [ V ( t h-Q+ h

+ h, x ( t + h, t , x))

-

V ( t ,x)]

(3.6.24)

3.6.

because of the fact that limb+*+ I( x* 11 function. We have seen previously that

where

11 x2 /I < S(p),

175

CONVERSE THEOREMS

=

(1 x (1 and that T ( E is) a decreasing

be such that

so that the solutions x ( t , to , xl), x ( t , to , x2) remain in

Let

s,.

If r2 3 rl , we have G(r,) 3 G(r& and hence

On the other hand, if r2

(3.6.25) Since f(t, x) satisfies the Lipschitz condition, by Corollary 2.7.1, we obtain

Furthermore, choosing

the relation (3.6.25) leads to

using the monotonic decreasing character of T ( E ) It . follows from these relations that

176

CHAPTER

3

and thus that

,,

V ( t ,xl) - aAl/ XI

~

x2 11.

These considerations show that, in all cases, V ( t ,xZ)

-

V ( t ,xl)

2

-

o~i4/lx1 - x2 11.

By interchanging the roles of x l ,x2 , we obtain V ( t ,XI) - V ( t ,x2) 3 --olR/l XI - x2 I/,

and therefore there results (3.6.27)

provided xl,x2 f 0. If xg = 0, (3.6.26) yields 0

< V ( t ,21) < 4 1x1 11,

and hence (3.6.27) is true even when x2 = 0. If x1 = x2 = 0, the relation (3.6.27) is trivially satisfied. As previously, it is now easy to obtain (2) from relations (3.6.24) and (3.6.27) and the descrescent character of V ( t , x). Finally, it remains to prove that we can choose G(r) satisfying the required conditions. For this purpose, we may take G(r) == A

J: exp [-KT(!E1)]

dr.

One can easily verify that G(0)

=

0,

G’((Y)= A exp[--KT(S(r)/a)

> 0,

G’(0) = 0,

0, T(0) = a,G’(r) is monotone increasing, and thus since 6(0) G”(r)exists almost everywhere and is positive. T h e proof is complete. Although we have used Theorem 3.6.9 only t o consider stability properties of perturbed systems, we give below a result that makes such a treatment easier.

THEOREM 3.6.10.

Under the assumptions of Theorem 3.6.9, there exists a function w(t, x) satisfying (1) and D+w(t, x) -w(t, x).

<

3.7.

STABILITY BY THE FIRST APPROXIMATION

177

Proof. By Theorem 3.6.9, there exists a function V ( t ,x) such that (1)

and (2) hold. Without loss of generality, we may assume that C(u) is differentiable, C’(0) = 0, and J ds/C(s) = GO. If C(u) does not have 0 these properties, we can choose such a function C,(u) satisfying C,(u) C(u). Consider the function

<

A(.>

=

exp

[J,

ds

for

o < u < p.

It is clear from the properties of C(u)that X(0) = 0 and X’(u) exists and is continuous on 0 u < p, so that I X’(u)i < K. We now definc the desired function w(t, x) by

<

w ( t , x)

= h(V(t,x )).

It is easy to check that this function verifies the required properties. T h e proof is therefore complete.

3.7. Stability by the first approximation Let x,(t) be a solution of (3.2.1). Set y = x -- x,(t), and obtain the equation y’ = x’ - x’( 0 t ) = f ( C 4 - f ( t , xo(t)) =f -

( 4Y

+ xo(t))- f ( 4

ax

xo(t))

+ O(l] y 1 ).

A natural question is whether we can legitimately neglect the terms of the form O(1l y 11). I n the theory of stability by the first approximation, this procedure is justified. T h e following theorem is to that effect.

THEOREM 3.7.1. Suppose that the trivial solution x = 0 of (3.2.1) is exponentially asymptotically stable and f ( t , x) in (3.2.1) is linear in x. Assume further that the function R(t, x) in (3.5.1) satisfies the relation

<

/I R ( t ?)1.I

< Clt x I/

(3.7.1)

p, C being a sufficiently small constant. Then, the trivial for 11 x /I solution x = 0 of (3.5.1) is exponentially asymptotically stable.

Proof. By Theorem 3.6.1, there exists a function V ( t ,x) having the following properties: (i) V EC[/ x S o ,R,], and V ( t ,x) is Lipschitzian in x for a constant K > 0.

178

CHAPTER

3

D+I/(t, X ) ( d . S . l ) G D 'l'(t> 4 b . 2 . 1 )

+ KIIR(t,

(3.7.2)

X)IL

using the fact that V ( t ,x) is Lipschitzian with a constant K . Define m(t) I.'(t, x ( t ) ) ,where ~ ( tis) any solution of (3.5.1) such that 11 xo 11 < &/K. Because of (ii), whenever /Ix,I/ < i p / K , we have m(to) < &p. We claim that m ( t ) < p for t >, t o . If this is not true, there exist numbers t , and t, such that ~

and Thus, D+m(t,) >, 0. On the other hand, since m ( t ) we have, by (ii), that

< P,

II 4t)Il

t"

< p for to < t < t , ,

< t < tl .

Hence, using condition (iii), together with (3.7.1) and (3.7.2), leads to

DWt,)

< -am(t,) + KCIIx(t,)ll = m(t,)[-a

<

+ KC],

because /I x I/ V ( t ,x). Since C is sufficiently small, there exists a > 0 such that C ( N - y ) / K . This implies that D+m(t,) < 0, and this contradiction provcs that m ( t ) < p for t 3 t o . Consequently, (1 x(t)i/ -1p for t >, t o . Thus, whenever 11 xo (1 < & p / K , we have y

<

D+q t , x ( t ) ) < -rV(t, x(t)), and therefore, by Theorem 1.4.1, V ( t ,. x ( t ) )

< V(to

?

xu) exp[-y(t

-

tdl,

t >, t,

*

It is easy to obtain from this inequality a further inequality

/I x ( t , t" > XdIl

< P expl--r(t

~

to)],

t

2 to

9

which proves the stated result. T h e proof of the theorem is complete. T h e linearity o f f ( t ,x) in x can be dropped, iff(t, x) satisfies a Lipschitz condition in x for a constant L = L ( p ) > 0. T h e next theorem is therefore a generalization of Theorem 3.7.1.

3.7.

179

STABILITY BY THE FIRST APPROXIMATION

THEOREM 3.7.2. Assume that the trivial solution x = 0 of (3.2.1) is exponentially asymptotically stable and f ( t , x) satisfies a Lipschitz condition for a constant L = L ( p ) > 0. Let

where N = N ( p ) is sufficiently small. Then, the trivial solution of (3.5.1) is exponentially asymptotically stable. Proof. By Theorem 3.6.2, there exists a function V ( t ,x) with the following properties:

(i) V E C [ J x S, , R,], and V(t,x) satisfies a Lipschitz condition with M = K [ L + P @ ~ 0l / < ~ ~q , < 1. (ii) (iii)

< K I1 x 11, < -qolV(t, x).

I1 x I/ < V t , 4 D+V(t,x)

so.

( t ,4 E J x

Now, following the proof of Theorem 3.7.1, one can prove the stated result. Here we have to choose y > 0 such that N (q y ) / M .

<

~

THEOREM 3.7.3. Assume that the trivial solution of (3.2.1) is exponentially asymptotically stable and that f ( t , x) is linear in x. Suppose further that (i) F E C [ J x S, , R"],and, given any such that IIF(t,x)ll

<

Ell

x I/,

/I x II

E

> 0, there exist a(€), T ( E )

< 6(€),

t

2 q€);

(ii) R E C [ J x S, , Rn], R(t, 0) = 0, and there exists an q such that, if I/ x 11 q,

<

where y

E

>0

C [ J ,R,] and

jy

y(s) ds + 0

as

t

+

a.

Then, there exists a To 3 0 such that, for to 3 To , the trivial solution of x' = f ( t , ).

is asymptotically stable.

+ F ( t , x) + R ( t ,x)

(3.7.3)

180

3

CHAPTER

Pyoof. Since the solution x = 0 of (3.2.1) is assumed to be exponentially asymptotically stable, there exists, by Theorem 3.6.1, a function V ( t ,X) satisfying (a)

V E C [ j x S,, , K,],and V ( t ,x) is Lipschitzian in x for a constant K > 0;

<

<

(b) !I x I! V ( t ,). K /I x /I, ( t , x) E J x so; (c) D+V(t,~ ) ( 3 . 2 . 1 ) < - w V ( t , x), 01 > 0, ( t , X) E J x S o . Let E be given such that 01 ' large that, for t 2 T , , we have A-

<

j': exp[-(a

-

< min(cli/K, 7). Choose T, 2 1 so

E

K€)(t- s)]y(s) ds

< @(E)

= 8,,

(3.7.4)

where 6 ( ~ ) E . As shown in Theorem 2.14.6, this choice is possible. If 11 x Ij . p, it is easy to obtain

+ K[llF(t,x)ll + II &t,

< 1)' v(t>

D LI.'(t,.Y)(a.7.3)

X)C3.2.1)

4111.

(3.7.5)

Consider the function m ( t ) = V ( t ,x ( t ) ) ,x ( t ) = x(t, t,, , x,,) being any solution of (3.7.3). We maintain that, whenever 11 x,,11 < 6,/K, we have I/ x(t)ll a(<), t 3 t,, . If this were false, there would exist a t , > t, >, T, such that

II 4 t J

= 8(€),

II 4t)ll

< S(E),

t

[to

6

ti].

In view of conditions (i) and (ii) and (3.7.5), there results the differential inequality D+m(t)

< -(a

-

+

K€)W(t) K y ( t ) ,

t

<

E

[ t o ,ti].

<

Here, wc have used that /I x(t)ij V ( t ,x ( t ) ) and D+V(t,x ) ( ~ . ~ . ~ ) - ; y V ( t , x). According to Theorem 1.4.1, we can deduce m(t)

< m(to)e x p - ( a

-

K € ) ( t- t")]

3.7.

STABILITY BY THE FIRST APPROXIMATION

using relation (b). At t

=

t , , we shall then have a contradiction

+ 6,

I1 X(tl)ll < 6,

=

S(E)

181

=

S(6)-

>

Thus, 1) xo I/ < S,/K implies 11 x(t)ll < a(€), t to 2 To . Consequently, t o , and the asymptotic stability of the trivial (3.7.6) is valid for all t solution of (3.7.3) follows, as in Theorem 2.14.4.

>

THEOREM 3.7.4. Assume thatfE C [ J x S, , R n ] , f ( t 0) , s 0, af(t, x)/ax exists and is continuous for ( t , x) E J x S o , and that the trivial solution of the variational system x’ = f z ( t ,

(3.7.7)

0)x

is exponentially asymptotically stable. Suppose that assumption (ii) of Theorem 3.7.3 holds. Then, there exists a To >, 0 such that, for to >, T o , the trivial solution of (3.5. I ) is asymptotically stable. Since f ( t , 0)

Proof. have

= 0 and

af(t, x)/&

f ( 4 ).

f d t , 0)x

=

exists and is continuous, we

+ F ( t , x),

where F ( t , x) satisfies assumption (i) of Theorem 3.7.3. Hence, the differential system (3.5.1) takes the form

.’

= f z ( t , 0)x

+ q t , x) + R ( t , x).

It is therefore clear that the stated result follows by Theorem 3.7.3.

THEOREM 3.7.5. Let us suppose that the trivial solution of (3.2.1) is generalized exponentially asymptotically stable and that f ( t , x) is linear in x. Suppose further that the perturbation R(t, x) verifies the estimate

II R(t, .)I1 where w

u for t

< w ( t ,I1 x Ill,

(t,

4E J

x

s,,

C [ J x R, , R,], w(t, 0) = 0, and w(t, u ) is nondecreasing in J. Then, the stability or asymptotic stability of the trivial

E

E

solution of

u‘

=

-p’(t)u

+ K ( t ) w ( t ,u),

u(to) = uo >, 0,

implies the equistability or equi-asymptotic stability of the trivial solution of the perturbed system (3.5.1). Proof. On the basis of Theorem 3.6.1, there exists a function V ( t ,x) fulfilling the following conditions:

182

CHAPTER

3

(i) V E C [ J x S o ,R,], and V ( t ,x) is Lipschitzian in x for a functiorl K ( t ) 3 0. (ii)

< v(t,x) < K ( t )ll x IN,

( t , x) 6 J x so. ni v(t,x)(3.2.1) G -P'(t) v(t,x), ( t , x) E J x

I/ x II

(iii) Thus, whenever

.

/ / x Ij < p, it can be readily verified that

D t V t , X)(3.6.1) G D+V(t,4 ( 3 . 2 . 1 )

which yields a further inequality I)'

s p

I.'(t, X)(a.s.1)

< -p'(t)

I.'(t,

+ K(t)l/w t , x)l'

4 + K(t)w(t,q t , 4)

because of (ii), (iii), and the monotonic character of w(t, u)in u. We can now apply Theorems 3.3.1, 3.3.2, 3.4.1, and 3.4.2 t o obtain the desired result.

THEOREM 3.7.6. Let f E C [ S p ,R"], f(0)= 0, f(m) = olmf(x), m > 1, and the trivial solution of x' = f ( x ) (3.7.8) be asymptotically stable. Then, the trivial solution of the system

(3.7.9)

is exponentially asymptotically stable.

Proof.

Let y(s, s o , x,,) be a solution of (3.7.9) and

s ( t ) being the inverse function of t(s). Set x( t) = y(s(t),so , xo). Then,

3.7.

STABILITY BY THE FIRST APPROXIMATION

183

Furthermore, x(to) = x o , where to = t(so). Since the solution x = 0 of (3.7.8) is assumed to be asymptotically stable, the functionf(x) being autonomous, we have

/I 4t)ll

< 4 xo Il)o(t

-

t

to),

3 to >

where a E X , (T E 9. It therefore follows that

/I Y ( S , so

x0)ll

< 4 xo Il)u[t(s)

- t(s0)l

< 411xo Il)aCP(s - so)] =

4 xo I l ) 4

-

(3.7.10)

so),

where o1 E 9, using the fact that y(u, so , xo) is bounded, and so

From the evaluation (3.7.1 O), the uniform asymptotic stability of the solution y = 0 of (3.7.9) is evident. Clearly, F( y ) is homogeneous in y of first degree. Hence, because of uniqueness of solutions, it results that Y ( &10 , axe)

=

U Y ( % so xo). 9

Moreover, using (3.7.10), we derive that

which implies that a*(.) is linear in u. One can now conclude, on the basis of Corollary 3.6.6 and the facts that a*(.) is linear in u and F ( y ) is homogeneous in y of first degree, that the trivial solution of (3.7.9) is exponentially asymptotically stable.

THEOREM 3.7.7.

Let f E C [ S ,, R"],f ( 0 ) = 0, d)Lf(x) = f ( a x ) , rn > 1, and the trivial solution of (3.7.8) be asymptotically stable. Assume that R E C [ J x S, , R"] and

II Nt,.)I1

< CII x 1Irn,

(1, x)

E

1 x s, ,

(3.7.1 1)

184

CHAPTER

3

C being a sufficiently small constant. Then, the trivial solution of the system x'

=f(.)

+ R(t,).

(3.7.12)

is uniformly asymptotically stable.

Pyoof.

Let x ( t , t o , xo) be a solution of (3.7.12). Define

and let t ( s ) be the inverse function of s ( t ) . Setting y ( t ) = x(t(s),to , xo), it is easy to check that

verifying that y(s ) satisfies the system +ids

=F(Y)

+ R*(s,y),

T h e conditions of Theorem 3.7.1 being fulfilled, it follows that

/I Y(S)ll G k'll X" I/ exp[--or(s whence

-

dl

/I 4 t ( s ) , to , xo)ll < KII2, /I exp[--n(s

(S"

= s(f,)),

-

dl,

and therefore

Since the solution y(s) is defined for all s 2 so , limt+ms ( t ) = 00. This shows that the integral in (3.7.13) is divergent, proving the exponential asymptotic stability of the trivial solution of (3.7.12). T h e theorem is proved.

3.7.

STABILITY BY THE FIRST APPROXIMATION

I85

T h e next theorem is of less general character, which may prove effective in certain concrete cases. T h e importance of the theorem, however, is that a judicious selection of V ( t ,x), reflecting more closely particular properties of the given system, frequently leads to much more precise results rather than yielding to the temptation of choosing V(t,x) as simple as possible, such as V(t,x) = // x /I.

THEOREM 3.7.8. Let the following assymptions hold: (i) There exists a continuously differentiable matrix G(t), which is self-adjoint and positive, that is, the Hermitian form (Gx, x) is positive definite, and A,, A, > 0 are the smallest and the largest eigenvalues of G(t ). (ii) T h e function q E C [ j ,R] is the largest eigenvalue of the matrix G-I(t)Q(t), where

ec4

+ G(t)A(t)+ A*(t)G(t),

=dG(t)

dt

A(t) being a continuous matrix on J and A*(t) its transpose. (iii)

R E C [ J x S,, , R"],and

< P(t)ll x /la,

/I R ( t , .)I1

0
< 1,

where p E C [ J ,R,]. Then, the stability properties of the trivial solution of u' = q(t)u

+ 2P(t)(X,hr*)'/"u"+"'/2

(3.7.14)

imply the corresponding stability properties of the trivial solution of x'

I n particular, if

N

=

= A(t)x

+ R ( t , x).

1 and

then the trivial solution of (3.7.15) is asymptotically stable.

Proof.

Let us consider the Lyapunov function defined by V ( t ,X)

=

(G(t)x,x).

(3.7.15)

186

CHAPTER

3

We then have

=

(G'x,X) -1 (GAx, X)

=

(G'x,X)

+ (GAx,

= ( Q ( t ) x ,X)

X)

+ (Gx,A X )4-(GR, + (Gx, fz) + (A*Gx, + (GR, + (Gx,R ) X)

X)

X)

+ (GR,x) $- (Gx, R).

I n view of the definition of Q(t), we also obtain G ~ l ( t ) Q (= t ) G-l

and hence (OX,

).

dG dt

-

+ A + GplA*G,

< q(t)(Gx,4 = q(t)V(t,

X).

Furthermore, it is easy to infer that (Gx,R ) -1(GR,X)

Since

x)

< ~[(G..c, x)(GR,R)]"'.

e (Gx,x) < A,(%

x),

it follows, on account of (iii), that (GR, f z f

< h,(R R ) <

4 ! P 2 ( t ) ( X , X)"

= h2h;"p(t)hlyx,x)"

< hzX;"P2(t"(t, 41". Taking into account all of these inequalities, we arrive at V ( t ,x)

< q(t)V(t,x) + 2/3(t)(hZX~")"2[V('(t,

X)](l+")'Z.

We can now use Theorems3.3.3, 3.3.4, 3.4.3, and 3.4.4 to get the stated result. If 01 = 1 and (3.7.16) holds, it is clear that the trivial solution o f (3.7.14) is asymptotically stable, and consequently the asymptotic stability of the trivial solution of (3.7.15) follows.

3.8. Total stability T h e theorems in this section emphasize the importance of uniform asymptotic stability in that, if the trivial solution is uniformly asymp-

3.8.

187

TOTAL STABILITY

totically stable, it also has certain stability properties under different classes of perturbations.

DEFINITION 3.8.1. T h e trivial solution x = 0 of (3.2.1) is said to be T,-totally stable (stable with respect to permanent perturbations) if, for every E > 0, to E J , there exist two positive numbers 6, = 6 , ( ~ )and 6, = a,(€) such that for every solution x(t, t o , xo) of the perturbed differential equation (3.5.1) the inequality

II xtt,

to

1

holds, provided that and

X0)ll

<

t 3 t"

/I Xoll < 8,

< 8,

I/ R ( t , .)I1

for

/I x I1 <

THEOREM 3.8.1. If the trivial solution x asymptotically stable and

=

€7

t

E

J.

(3.8.1)

0 of (3.2.1) is uniformly

llf(tl4 - f ( t , r)ll< Jwllx - Y II for ( t , x), ( t ,y ) E J x S, and

1

Sf+lL(s)ds

then the trivial solution x

=

I<

KI 7.J

I,

0 of (3.2.1) is also T,-totally stable.

Proof. According to Theorem 3.6.9, it follows from the uniform

asymptotic stability of the trivial solution of (3.2.1) that there exists a function V ( t ,x) satisfying the following conditions:

<

< <

(a) b(ll x II) V ( t ,4 4 x ll), a, b E 37. (b) I V(t,x) - V ( t ,y)I M II x - Y II, ( t , x), ( t ,y ) E J x Ss(ao),where 6 ( ~ and ) 6, appear in (S6). (c) D+V(t,x) Let 0

< -C[V(t, x)], c E T .

< E < 6(6,) be given. Choose 6, b(E)

> 481)-

=

a,(€)

such that (3.8.2)

Define m(t) = V(t, x(t)), where x(t) = x(t, t o ,xo) is any solution of (3.5.1) such that 11 xo I( < 6, . This, because of (a), implies m(to) < ~(6,). We claim that m(t) < b(€), t 2 to. (3.8.3)

188

3

CHAPTER

I f this is false, there exist two numbers t, m ( b ) = @l),

> t , > to such that

m(t1) = b(c)

and m ( t ) 3 a(S,),

t,

< t < t, .

Thus, we would have D+m(t,)

(3.8.4)

2 0.

On the other hand, observe that, for to

< t < t , , we have

(3.8.5)

'The fact that leads to

11 x(t2)/1< c , together with relations (3.8.1) and (3.8.5), D+m(t,)

< -C[u(S,)]

+ MS, = 0,

which contradicts (3.8.4) and proves (3.8.3). From this follows that the trivial solution x = 0 of (3.2.1) is T,-totally stable. T h e theorem is proved.

A variant of the notion of total stability with respect to perturbations may be defined if, instead of (3.8.1), we only require that the perturbations be bounded in the mean.

DEFINITION 3.8.2.

T h e trivial solution x = 0 of (3.2.1) is said to be T,-totally stable (stable under permanent perturbations bounded in the mean) if, for every E > 0, to E J , and T > 0, there exist two positive numbers 6, = S,(E) and 6, = S2(c) such that, for every solution x ( t , to, xo) of the perturbed differential system (3.5.1), the inequality

II ~

( tt n, , xn)ll

<

€9

t >, to

3.8.

189

TOTAL STABILITY

THEOREM 3.8.2. Under the assumptions of Theorem 3.8.1, the trivial solution x = 0 of (3.2.1) is T,-totally stable. Proof. We proceed as in the proof of Theorem3.8.1 and choose 6, = a,(€) by relation (3.8.2). Let 11 x,, 11 < 8, and m ( t ) = V ( t ,x ( t) ) , where x(t) = x(t, to, xo) is any solution of (3.5.1). As before, m(to) < a(&), and the claim (3.8.3) is true. If it is not the case, there exists a t, > to such that m(t) b(a) for to t t, , which implies that

<

< <

II x(t)lI Denote t,

-

<

< 8(8n),

tn

< t < t,

to = T , and choose

8, = S,(E) < b ( E )

-

J-1[J(481))l/M>

(3.8.8)

where

and J-l is the inverse function of J . From relations (b) and (c), we obtain, for t E [to , t,],

< - C [ V ( t , 4t))l + MilR(t,4t)ll.

D-'-V(t,49)

If we now define z ( t ) = V ( t ,x ( t ) ) - 4 t ) ,

where v(t) = M

jt II W s , X ( 4 l l ds,

we obtain the inequality D+z(t)

to

< --C[z(t)l,

using the monotonic character of C(u)and the fact that z ( t ) >, V(t,x ( t ) ) , which implies, in view of Theorem 1.4.1, that z(t)

< J-"J(V(to .")I T

~

(t

- to)],

t

E

[ t o ? tll.

I90

CHAPTER

3

Note that the maximal solution of u' = -C(u), u(t,) = V(t, , x,) is just the right-hand side of the foregoing inequality. Thus, it follows that

<

E From this, we derive a further inequality, using the {acts that 11 x(t)ll for to t t, T , V ( t , , x,) < a(8,) and relations (3.8.6), (3.8.7), and (3.8.8),

< < +

b(E)

< V(t" + T ,

X(t,

+ T ) ) < J-"J[a(8,)1

-

77

+ MS, <

WE).

This contradiction assures that m(t) < b ( ~ )t , >, t o , which, in its turn, implies T,-total stability of the solution x = 0 of (3.2.1). This completes the proof. I n the case of certain perturbations that approach zero as t -+ co, uniform asymptotic stability implies total asymptotic stability defined below.

DEFINITION 3.8.3. T h e trivial solution x = 0 of (3.2.1) is said to be totally uniformly asymptoticaZ~stable if, for solutions x(t, t o , x,) of the perturbed differential system (3.5. l), the definition (S,) holds, provided that R(t, 0) = 0 and

/I R(t, .)I1 uniformly for

< 4t),

0E

9,

(3.8.9)

11 x 11 < p.

THEOREM 3.8.3. Under the assumptions of Theorem 3.8.1, the trivial solution x = 0 of (3.2.1) is totally uniformly asymptotically stable, provided R(t, x) also satisfies a Lipschitz condition in x. Proof.

Consider the same function V ( t , x) as in Theorem 3.8.1. If = p, , we would have

/I x I/ < S(6,)

11' V t , X)(3.5.1)

< - W ( t , 4 1 + MllR(t,)1.I < -C[V(t, 4 1 +-Mo(t) = d t , V t ,4).

< < <

Let 0 < N /3 -< po be given, and let K(a, p) = $C(a). Since u E 2, there exists a d(a, p) 3 0 such that o(t) i C ( a ) / M ,t > d(a, p). Thus, if N u p, t 3 @(a,p), we have

<

R(t, ).

=

-C(.)

+ Mu(t)

< -C(Ol) + +C(,) :=

--K(or,B).

3.9.

191

INTEGRAL STABILITY

The conditions of Theorem 3.4.10 being verified, it is easy to see that Theorem 3.8.3 is proved.

3.9. Integral stability We shall continue to study the system (3.2.1) and its perturbed system (3.5.1). However, for the purposes of this section, it becomes necessary to assume that p = GO so that functions f , R occurring therein are such that f,R E C [ J x Rn, R"].

DEFINITION 3.9.1.

T h e trivial solution x

=

0 of (3.2.1) is said to be

( I l ) Equi-integrally stable if, for every 01 3 0 and to E J , there exists a positive function /3 = / 3 ( t o ,n), which is continuous in to for each 01 and /3 E Z for each to , such that, for every solution x ( t , t o , xo) of the perturbed differential system (3.5. l), the inequality

It x ( t , t o > x0)ll < 8, holds, provided that and, for every T

> 0,

It xo I/

t 3 to

>

<%

(I,) Unqormly-integrally stable if the /3 in (I,) is independent of t o . (I,) Equi-asymptotically integrally stable if ( I l ) holds and, for every E > 0, 01 3 0, and to E J , there exist positive numbers T = T ( t o ,a , c) and y = y ( t , , 01, E ) such that, for every solution of the system (3.5.1), the inequality t 3 to T, I/ x ( t , t o , x0)Il < 6 ,

+

holds, provided that IIxoll

and

<

(I4) Uniformly-asymptotically integrally stable if the T and y in (I3) are independent of to and (I2)holds.

192

CHAPTER

3

I n addition to the scalar differential equation (3.2.3), let us consider the following perturbed equation: u' = g(f,a)

where g

E

C [J x R, , R] and

DEFINITION 3.9.2.

+ (b(t),

u(t,)

=

(3.9.1)

uo ,

4E C [J , R+].

T h e null solution u

=

0 of (3.2.3) is said to be

(IF) Equi-integrally stable if, for every 0 1 ~3 0, to E J , there exists a positive function p1 = & ( t o , 0 1 ~ ) that is continuous in t, for each m1 and p1E .X for each t, such that, whichever be the function 4 E C [J , R,] with /;;T(b(5)

<9

ds

for every T > 0, every solution u(t, t o ,u,) of the perturbed scalar equation (3.9.1) satisfies the inequality u(f, t o , ug)

provided that

< 81 ,

%J

t

2 to,

< %.

T h e definitions (lz)-(Ic) may be formulated similarly. T I i E O R E h l 3.9.1. Assume that there exist functions V ( t ,x) and g(t, u ) satisfying the following properties:

(i) g E C[/ x R, , R],g(t, 0) = 0. (ii) V E C [J x R",R,], V ( t ,x) is Lipschitzian in x for a constant M > 0, and there exists a function b E A'" such that b(u) + co as u + CO, and ( t , x ) E J x R". b(ll x 11) < q t , x),

<

(iii) D 'v(t,~ ) ( & . 2 . i ) g ( t , v(t,x)), ( t , x) E x Rn. Then, the equi-integral stability of the null solution u = 0 of (3.2.3) implies the equi-integral stability of the trivial solution x = 0 of (3.2.1). Proof. Let n: 3 0 and t, E J be given, and let I/ xo 11 01. Since V ( t ,x) is Lipschitzian in x for a constant M > 0, we have

J

<

I

qt,

-

V f Y)l ,

< MI1x

<

~

Y

II,

(3.9.2)

from which it follows that V ( t , , x,,) Mm = 0 1 ~. Let x ( t ) = x(t, t o , xo) be any solution of (3.5.1). Then, condition (iii), together with (3.9.2), yields, as far as x ( t ) exists to the right of t, , b"(t,

X)(3.6.1)

< g(t, q t , .I)

+ MI1R ( t , .)I.

3.9.

193

INTEGRAL STABILITY

Define h(t) = M 11 R(t, x(t))ll, and choose u,, of Theorem 1.4.1 shows that

=

V(to, x,,). An application

on the common interval of existence of x( t) and r(t, t o , u,,), where r(t, to , u,,) is the maximal solution of u’ = g(t, u )

+ h(t),

.(to)

= u,,

.

Assume now that (If)holds. Then, given a1 3 0 and to E J , there exists a 8, = P,(t, , al), which is continuous in to for each a, and PI E X for each t,, , such that, for every solution u(t, to , u,,) of (3.9.1), the inequality u(t,to

holds, whenever u,,

>

4 < B1,

t

2 to

7

< a1 and, for every T > 0, /l:T4(s)

ds

< al.

Since assumption (ii) holds, it is possible to choose a satisfying the relation b(B) b B1 7

p

=

&to, a) (3.9.4)

where p1 is the function occurring in (IF). Evidently, /3 is continuous in t,, for each a and p E X for each to . We claim that, with this p, definition (I,) holds. If this is not true, there would exist a t , > to such that (3.9.5)

194

3

CHAPTER

We extend $ ( t ) continuously for all t 3 to such that

("a(.)

ds

< a1 .

to

> t , , satisfying the inequality

T o do this, it is enough to take t,

t,

-

t,

<

2(%

-

1

j)N

ds)

+ +(td

7

to put +(t2) = 0, and to take +(t) linear on [ t l ,t z ] and +(t)= 0 for

t 3 t,. Let r * ( t , t,, , u,J be the maximal solution of the perturbed differential equation (3.9.1) with +(t) chosen as before. Because of (IF), it would (yl and follow from u,,

<

for every T

> 0, that r*(t, t o ,

U")

< PI ,

t

3 t" .

But, on [to, tl], we have r * ( t , t o , U")

=

r ( 4 t"

7

U"),

since h(t) and +(t)are identical on this interval. Hence, r(tl , t,, , u,,) < p1 . Thus, we get, from relations (3.9.5), (3.9.3), (3.9.4), and assumption (ii), the following absurdity: b(P)

< V(t1 4h))< Y(t, , t o >

7

u,,)

< PI d b(B).

This proves the integral stability of the trivial solution of (3.2.1).

COROLLARY 3.9.1. If the function g(t, u ) = h(t)u enjoys the property that h E C [ J ,K] and ds < a,

ps)

then it is a candidate in Theorem 3.9.1.

THEOREM 3.9.2. Under the assumptions of Theorem 3.9.1, the uniform integral stability of the solution ZL = 0 of (3.2.3) assures the uniform integral stability of the solution x = 0 of (3.2.1). Proof. T h e proof is very much the same except to observe from (3.9.4) that /3 is independent of t, since PI does not depend on to by assumption.

3.9.

COROLLARY 3.9.2. Theorem 3.9.2.

The

195

INTEGRAL STABILITY

function

g(t, 21)

= 0 is admissible in

THEOREM 3.9.3. Let the assumptions of Theorem 3.9.1 hold. Assume that the trivial solution of (3.2.3) is equi-asymptotically integrally stable. Then, the null solution of (3.2.1) is likewise equi-asymptotically integrally stable. Proof. On the basis of Theorem 3.9.1, the trivial solution of (3.2.1) is equi-integrally stable. Let E > 0, 01 3 0, and t, E J be given, and let jl x,, 11 LY. As in Theorem 3.9.1, we define aI = Ma. Let ,8 = P(to , 01) be the same function obtained by relation (3.9.4), for which (I,) holds. Since (I?) holds, it follows that, given b ( c ) > 0, a , 3 0, and to E J , there exists a pair of numbers y, = yl(to , a l , E) and T = T ( t o ,I Y ~, E) such that, whichever be the function 4 E C [ J ,R+]with

<

J"x(4 ds < Y1 ,

(3.9.6)

every solution u(t, t o , u,,) of the perturbed scalar differential equation (3.9.1) satisfies (3.9.7) u(t, t o , Ug) < b(c), t 2 to T, whenever u,, so that

<

+

N,

. We now choose a positive number MY

y = y(t,

, 01,

E)

(3.9.8)

Y1

and maintain that, with the positive numbers T and y so defined, ( I 3 ) is satisfied. For otherwise, let {tli} be a sequence such that t, >, to T, t, co as k -+ 00. Suppose that there is a solution x(t) = x ( t, t o , xo) of the system (3.5.1) such that 11 x, 11 and 11 x(t,, t o ,x,,)lI >, E. As in the proof of Theorem 3.9.1, condition (iii), in view of the fact that V ( t ,x) is Lipschitzian, gives

+

<

--f

D+V(t,x(tN

(Y.

< At, V(t7 4 9 ) + MI1R ( t ,x(t))ll.

(3.9.9)

If we now define +(t)= M 11 R(t, x(t))lj, we have to

to

Jz

< MY

SUP I/ R(s, 4 ds

1ia11

0

= y1,

JT

using (3.9.8) and the fact that sup,lxl,48 I/ R(s, x)II ds < y . This implies that, for solutions u(t, t,, , uo) of (3.9.1), (3.9.7) is true, because

196

CHAPTER

3

of (3.9.6). Moreover, by (3.9.9) and the definition of $(t),it follows from Theorem I .4.1 that V ( t ,x ( t ) ) < r ( t , to %),

t

9

3 t" ,

(3.9.10)

where r ( t , t o ,u,,) is the maximal solution of (3.9.1). Hence, relations (3.9.10) and (3.9.7) and assumption (ii) of Theorem 3.9.1 lead us to the contract ict ion b(c)

< c-(t,,

x(t,))

< Y(t,

, t o , U") < Y e ) ,

which proves the equi-asymptotic integral stability of the solution x 0 of (3.2.1), and the theorem is complete. 1

THEOREM 3.9.4. Let the assumptions of Theorem 3.9.1 hold. Assume that the null solution of (3.2.3) is uniformly asymptotically integrally stable. Then the solution x = 0 of (3.2.1) is likewise uniformly asymptotically integrally stable. We note that, in this case, the positive numbers T and y1 are independent of to , and therefore (3.9.8) implies that y is also independent of to . T h e rest of the proof is just the same as that of Theorem 3.9.3. Pyoof.

If thc function g(t, u ) is assumed to be nonincreasing in u for each t E J , we can obtain integral stability notions from the stability notions of the trivial solution u : 0 of (3.2.3). T o this end, we shall prove the following result.

3.9.5. Assume that the hypotheses of Theorem 3.9.1 hold. Let the function g(t, u ) be nonincreasing in u for each t E J . Then, uniform asymptotic stability of the null solution u = 0 of (3.2.3) implies uniform asymptotic integral stability of the trivial solution of (3.2.1). THEOREM

Proof. We first prove the uniform integral stability of the solution x 0 of (3.2.1). By Theorem 3.3.5, uniform stability of the null solution of (3.2.3) implies the existence of a function /I1 E :X such that 7

u(t, t" , uo)

< PI(%),

t

2 to.

<

(3.9.1 1)

Let now 1 >, 0 and to E J be given, and let I( xo (1 CL. Then we have, from (3.9.2), V ( t o ,xu) Ma. Let x ( t ) = x( t, t o ,xo) be any solution a , and let of (3.5.1) with I/ xo 11

<

<

(3.9.12)

3.9. INTEGRAL

197

STABILITY

from which it follows, because of the monotonic nonincreasing character of g(t, u ) in u and the fact that m(t) 3 V ( t ,x(t)),that

< g(t, m(t)). By Theorem 1.4.1, we then have, as far as x ( t ) exists to the right of t o , m(t)

< r ( t , to

7

(3.9.13)

UO),

where r ( t , t o ,uo) is the maximal solution of (3.2.3) with uo = m(to). Let p be so chosen that b(P)

2 P11Ma)

+ Ma.

(3.9.14)

This choice is clearly possible in view of the fact that b(u)4 co as u + a, It is evident that p = p(a) and that p E X . We claim that, with this 8, the trivial solution x = 0 of (3.2.1) is uniformly integrally stable, whenever 11 xo 11 < a and, for every T > 0,

,ZPB/I R(s3 411d5 <

J;T

Assuming that this claim is false, there exists a t ,

I1 X ( t ,

7

to , xo)ll = P

and

/I x(t9 to %)I1 9

(3.9.15)

01.

< P,

> to such that t E [to , tll.

We are then led to the absurdity, because of relations (3.9.12), (3.9.14), (3.9.1 5), and assumption (ii), b(P) G V t l Y 4 t l ) )

thus proving ( I 2 ) .

< r(t,

1

< b(13),

to

9

+

+(tl)

198

CHAPTER

3

T o prove ( I 4 ) ,we have, by uniform asymptotic stability of the solution 0 of (3.2.3) and Theorem 3.4.1 1, the inequality

u =

< Bi(uo)u(t

u(t, to > un)

-

to),

t

2 to,

(3.9.16)

where & E -X and u E 9. If we are now given E > 0, 01 3 0, and t, E J , we make the following choice: /I x,,11 01, M y < b(c), and y = min(y, a ) . Since, for any solution x ( t ) = x ( t , t o , xo) of (3.5.1), (3.9.13) is true, whenever uo = V ( t o xu), , relations (3.9.12) and (3.9.16), together with assumption (ii), given the inequality

<

< Pl(Mab(t

~

to)

Since u E 9, there exists a T

and hence, for t >, t,,

<

(Y

T(a,c) such that

b(ll 4t)ll)

/I x ( t , to

11 x,,I/

=

+ T , we would have

which implies that provided

+ MY.

9

xo)lI

< 6,

< 4€)? t

2 to -t T ,

and (3.9.15) is satisfied. T h e theorem is proved.

COROLLARY 3.9.3. Assume that there exists a function V ( t ,x) satisfying the following properties: (i) V EC [ J x R",R,], V ( t ,0) = 0, t E J , and V ( t ,x) is positive definite and 1,ipschitzian in x for a constant M > 0.

<

(ii) D+V(t,~ ) ( a . ~ . ~ )-C(ll x /I), ( t , x) E J x R", where the function C E z. Then, the trivial solution of (3.2.1) is uniformly asymptotically integrally stable.

<

Proof. By relation (3.9.2), we have V ( t ,x) MI1 x 11. This fact, together with condition (ii), is sufficient to arrive at the differential inequality D+V(t,X h 3 . 2 . 1 ) < g(t, q t , 41,

3.10.

199

LP-STABIIJTY

whereg(t, V )-= - C(V(t,x ) / M ) .It is now immediate, by Corollary 3.4.2, that the null solution of (3.2.3) is uniformly asymptotically stable. Moreover, g(t, u) defined previously has all the properties required by Theorem 3.9.5. Hence, the corollary is a consequence of Theorem 3.9.5.

3.10. LI’-stability DEFIXITIOK 3.10.1.

T h e trivial solution x

7

0 of (3.2.1) is said to be

(I,,) egui-l,r’ stable if (S,) holds and there exists a such that the inequality jl x,,11 6, implies

<

a,,

-

6,(t,)

>0

(3.10.1) (l,J uniform-l2’ stable if (S,)holds, the 6, in (IJ1)is independent of to , and the integral (3.10.1) converges uniformly in to .

T h e following example shows that L l ’ stability need not necessarily imply asymptotic stability, and vice versa. Thus, they are different concepts. Consider the linear equation (3.10.2)

whose general solution

I f g(t) is a continuously differentiable, bounded Z,l function on J which does not tend to zero as t + m, then the trivial solution of (3.10.2) is 1,1 stable but not asymptotically stable. On the other hand, if g(t) [log(t f 2)] I , then the trivial solution of (3.10.2) is asymptotically stable but not I,” stable for a n y p > 0. Analogous to the notions ( L , ) and (L2),we require the L1stability notions for the null solution of the scalar differential equation (3.2.3).

I)EFISITION 3.10.2. The null solution u - 0 of (3.2.3) is ( L f ) eqzci-L1 stable if (ST) holds and there exists a 8, - - So(t,,) such that u0 6, implies

<

J;

U(S7

to

>

Uo)

&

< 03.

T h e definition of uniform LLstability is clear.

(3.10.3)

200

CHAPTER

3

THEOREM 3.10.1. Assume that there exist functions V ( t ,x) and g(t, u ) satisfying the following properties: (i) g E C [ J x R, , R] and g(t, 0) = 0, t E J. (ii) V E C [ ] x S,, , R,], V ( t ,0 ) = 0, t E J , Lipschitzian in x, and

< V ( t ,x),

V ( t ,x) is locally

so. (iii) D+V(t, x) < g(t, V(t,x)), ( t , ,x)E J x S,, . '411 x 11')

(3.10.4)

( t ,x) t J x

Then, the equi-l1 stability of the null solution of (3.2.3) implies the equi-D stability of the null solution x = 0 of (3.2.1).

Proof. Assume that the null solution of (3.2.3) is equi-l1 stable. Then, it is equistable and there exists a 8" = &,(to)such that u,, < 8, implies

(3.10.3). By Theorem 3.3.1, the equistability of the null solution of (3.2.1) follows, and therefore, to prove (Ll), it remains to show that So assures (3.10.1). Since there exists a 6, = S,(t,) such that I/ x,,11 V ( t ,x) is continuous and V(t,0) = 0, t E J , there exists a positive number S, = S,(t,) satisfying the inequalities

<

I/ xo /I

< so

9

v(to7 xo)

<

$0

together. As in the proof of Theorem 3.1.1, using condition (iii) and Theorem 3.1.1, we get, by choosing u,, = V(t,,, xo), the inequality F ( t , x ( t , to , X"))

< r ( 4 to , 4,

t

2 t" ,

<

where x ( t , to , so) is any solution of (3.2.1) such that (1 xo (1 8, and r ( t , t o , so)is the maximal solution of (3.2.3). From this, the desired rcsult (3.10.1) follows, using assumptions (3.10.4) and (3.10.3). T h e proof is thus complete.

THEOREM 3.10.2. Assume that there exists a function V ( t ,x) with the following properties: (i) V t C [ J x So, R , ] , V ( t ,0 ) = 0, t E J , and V ( t ,x) is positive definite and locally Lipschitzian in x. (ii) D+V(t, x) < -C j / x 1/1), ( t , x) E J x So, C > 0. 0 of (3.2.1) is equi-L'' stable. Then the null solution x :

Proof. Ry Corollary 3.3.2, it follows that (5,) holds. Let to E J , and let So S,(t,) > 0 be such that, if 11 xo I/ < So, then ( t ,x(t, to,x,,)) E J x S,, for t 3 t o . This is possible because (S,) is valid. Define m(t) =

v(t,x ( t , to , xu)) + c

1'I/ to

x ( s , to

3

X~)IIP ds.

3.10.

20 1

LP-STABILITY

Then, condition (ii), in view of Theorem 3.1.3, gives the inequality m(t)

< m(to),

2 to,

t

from which there further results ..W

J

to

/I x(s, t” > x0)Il” ds

It is clear that the null solution x

=

<

1

C

V ( t , > xo).

0 is equi-Lp stable.

THEOREM 3.10.3. Let the assumptions of Theorem 3.10.1 Suppose further that V ( t ,4

< 411xll),

( t ,4 E

f

x

s, ,

a

hold.

C3Y.

Then, the uniform-L1 stability of the solution u = 0 of (3.2.3) implies the uniform-LP stability of the solution x = 0 of (3.2.1). Proof. By the assumption, the null solution of (3.2.3) is uniformly independent of to such that (3.10.3) holds stable, and there exists a uniformly in to whenever u, 8,. Consequently, the uniform stability of the solution x = 0 of (3.2.1) follows from Theorem 3.3.4. T o prove that (L,) holds, we follow the proof of Theorem 3.10.1 and we choose uo = u(\\ xo I]), thereby deducing So = u - ~ @ , ) . I t is evident that 6, is independent of to and the integral (3.10.1) is uniformly convergent in to . This proves the theorem.

so

<

COROLLARY 3.10.1. Assume that there exists a function V(t,x) verifying the following conditions: (i)

V E C [ J X S, , R,], V ( t ,x) is locally Lipschitzian, and, for ( t , x) E J X SD 3

All x lI*

< V ( t ,x) < BIJx Il*l,

A , B > 0.

(3.10.5)

<

-C \I x IlP, C > 0, ( t ,x) E J x S o . (ii) D+V(t,x) Then, the null solution of (3.2.1) is uniform-LP stable. Proof.

Assumption (ii), in virtue of (3.10.5), reduces to

D V ( t ,4 < g(t, v t , XI), where g ( t , u) = -Cu/B, and hence it is easy to check that the solution u = 0 of (3.2.1) is uniform-L1 stable. Now, the assertion of the corollary is a consequence of Theorem 3.10.3.

202

CHAPTER

3

Although L?-'stability and asymptotic stability are different concepts, under certain conditions Lr' stability implies asymptotic stability, as shown in the following:

THEOREM 3.10.4. Let the trivial solution of (3.2.1) be let there exist a constant M > 0 such that

Then, the trivial solution x

=

L p

stable, and

0 of (3.2.1) is asymptotically stable.

Proof. Assume that there is a solution x(t, t o , xo) of (3.2.1) such that I/ xo I/ &,(to) and Iimi+%x ( t , t o , xo) # 0. Then, there exists an E > 0 and a sequence {t,;},t,; + co as k co such that

<

---f

11 x ( t , ,

t , , x,,)Il

2 2~

for all k.

By assumption,

I1 ~ ' ( t o, > xo)ll

=- Ilf(t,

and hence there is a number X

II x ( t , to

9

.vo)ll

x ( t , t o , xo))II

<M,

t

3 to ,

> 0 satisfying

3 c,

t,

< t < t, +A

for each k. This contradicts (3.10.1), and the theorem is proved. We shall next consider some converse theorems for L

THEOREM 3.10.5. (i) (ii)

p

stability.

Let us assume that

the function f E C [ J x S,, , R"],f(t,0) = 0, and f J t , x) exists and is continuous for ( t , x) E J x S, ; the solution x(t, 0, xn) of (3.2.1) satisfies

< Bll X" I),

I1 x ( t , 0 , .xo)ll

t

3 0 , p > 0,

(3.10.6)

s,"

(iii)

and / / x(s, 0, xo)\/i'ds < co; the function g E C [ ] x R, , R ] ,g(t, 0) = 0, and g,(t, u) exists and is continuous for ( t , u ) E J x R, ;

(iv) the solution u(t, 0, uo) of (3.2.3) verifies the estimate

< -1o

00

yuo

U(S,

0 , u") ds,

y

> 0.

(3.10.7)

3.10.

203

L~-STABILITY

Then, there exists a function V ( t ,x) with the following properties: (1) V E C [J x S, , R,], V ( t ,0) = 0, V ( t ,x) is continuously differentiable, and All x jjp

< J q t ,x),

( t , x)

E

J x

s,, A > 0;

Proof. By assumptions (i) and (iii), the continuity and differentiability with respect to the initial values of the solutions x(t, t o ,xo), u(t, t, , uo)

of (3.2.1), (3.2.3), respectively, follows as in Theorem 3.6.4. Moreover, denoting x ( t , 0, xo) by x, we see that xo = x(0, t, x), because of uniqueness of solutions. We choose a continuous function p(x) such that p(0) = 0, ap(x)/ax exists and is continuous, and x 112,

< p(x),

01

> 0,

x E s,.

(3.10.8)

We then define the function *W

It is clear that V E C [ J x S o ,R,] and V ( t ,0) = 0, since x(0, t, 0) = 0 and u(t, 0,O) = 0. Furthermore, taking into account the fact that the solutions x ( t , t o , xo) and u(t, t, ,u,) are differentiable with respect to their arguments, we have V‘(t,3)

= u’(G 0, P(X(0,

= g(t,

t , .)I)

V ( t ,XI),

because of (3.6.11). Using the relations (3.10.6), (3.10.7), (3.10.8), and the definition of V(t,x), we obtain V ( t ,).

2J

W

u(s, 0, P(X(0, t ,

3 YP(X(0, t , 4) 3 ’YOLIl 40, t , X)ll” T h e proof is complete.

.)I)

ds

204

CHAPTER

3

We notice that the full force of assumption (ii), that is, LP-nature of solutions x ( t , 0, x,), is not used in the proof of Theorem 3.10.5. However, the linear character of the estimate in (3.10.6) is crucial in the proof. We give below a different type of converse theorem.

THEOREM 3.10.6. Let assumption (i) of Theorem 3.10.5 hold. Furthermore, suppose that t 3 to, (3.10.9) P& xu Il)Vt) < /I x ( t , t o xo)ll < Pi(11 Xn ll)x(t), where PI , p2 E 3” and h E C [ J ,R,] such that h E Lp. Then, there exists a function V ( t ,x) satisfying the following:

(1) V E C [ J x So , R,], V(t,x) possesses continuous partial derivatives with respect to t and the components of x, and V(t,x) is positive definite and decrescent.

(2) V’(t,X) = - 11 x I ’), ( t , X) Pmof. We define the function

E

J x So .

Then, it is evident that V E C [ J x S o ,R,], on the basis of assumption (i). Moreover, the differentiability of the solution x ( t, t o , xo) with respect to the initial values assures that V ( t ,x) is continuously differentiable. Hence, V ’ ( t ,x)

=

-11

x ( t , t , .Y)Il”

=

-11

x

/1’),

on account of the relation (3.6.1 1). Also, using the fact that h EL* and the estimate (3.10.9), we have, successively,

v(t,x)

1 I1 W

3

x(s, t , x)llnds

> P2p(/lx 11) =

b(jl x 11),

lW Ws)

ds

b E .x,

3.1 1.

20 5

PARTIAL STABILITY

3.1 1 + Partial stability Let us consider a differential system of the form

>!

x’ = G(t,x,y), Y’ = H ( t , X,Y),

(3.1 1.1)

x(to) = yxo o, Y(t0)

where G E C [ J x So x Rm, R”], H E C [ J x S, x R”,R”], and G(t,0, 0) = 0, H(t, 0,O) = 0, t E J. Let us denote a solution of (3.1 1 . I ) by 4 t ) = x(t, to 2 xo Yo),Y ( t ) = Y ( t , t o xo yo). 9

9

7

DEFINITION 3.11.1. The trivial solution x = 0, y = 0 of (3.11.1) is said to be ( P l )partially equistable with respect to components x if, for each E > 0, to E J , there exists a positive function 6 = 6 ( t o , E ) which is continuous in to for each E such that the inequality

/I xo II

implies

+ IIYO II < 8

I1 x(t, to , xo 3YO)ll <

€9

t

2 to .

Corresponding to the group of definitions (Sl)-(Slo), we may formulate (pl)-(plO).

THEOREM 3.1 I. 1. Assume that there exist functions V(t,x,y ) and g ( t , u) satisfying the following properties: (i)

V E C [ J x So x R”,R,], V(t,0, 0) = 0, and V(t,x,y ) is locally Lipschitzian in x and y.

(ii) g E C [ J x R, , R],g(t, 0) = 0, and D +v(t,X,Y)

=

+

1 limsup - [v(t h, x h+O+ h -

+ hG(t, x,y), y + hff(t,x , ~ ) )

w, .,Y)1

< g(t, v(tjx , ~ ) ) ,

( t , X,Y)

6

J

X

S

X

R”

.

206

3

CHAPTER

Then

( I ) if the solution u

=

b(ll 3L’ II)

0 of (3.2.3) is equistable and

e q t , x, Y),

(3.1

b E X ,

4

the trivial solution of (3.1 1.1) is partially equistable; (2) if the solution u = 0 of (3.2.3) is uniformly stable and

W11 x II)

< v(t,.,y) < 411x I1 + IIY 111,

a, b E X ,

(3.1 -3)

the trivial solution of (3.1 1.1) is partially uniformly stable; (3) if the solution u = 0 of (3.2.3) is equi-asymptotically stable and (3.1 I .2) holds, the trivial solution of (3.1 1.1) is partially equi-asymptotically stable; (4) if the solution u = 0 of (3.2.3) is uniformly asymptotically stable and (3.1 I .3) holds, the trivial solution of (3. I I . 1) is partially uniformly asymptotically stable. Proof. Let 0 < E < p and t,, E J . Assume that the solution u = 0 of (3.2.3) is equistable. Then, given b(e) > 0, to E J , there exists a positive function 6 = S ( t , , e ) that is continuous in to for each E such that, whenever uo 6, we have

<

u(t, t o , uo)

< b(E),

t

2 t,.

(3.11.4)

Choose uo = V ( t , , xo , y o ) . Since V ( t ,0 , O ) = 0 and V ( t ,x, y ) is continuous, there exists a 6, a1(t,, , e ) that is continuous in to for each E such that the inequalities :

I

l

L

d

V(to,xo,yo) G s

+ I I Y ~ I I GSi,

(3.11.5)

hold at the same time. We maintain that, with this 6, , (P,) is satisfied. For otherwise, suppose that there exists a t , > to for which

iI4tdll = E,

whenever /I xo 11

11 x(t)ll

<

E,

+ I] y o 11 < 6, , so that

t

E

[to, tll,

(3.1 1.6)

b(E) G V(t1 , 4tA Y(t1)).

This implies that / / x(t)ll i p for t E [t,,, t,], and hence condition (ii), together with the choice ug = V ( t o x,, , y o ) , yields, on the basis of Theorem 3.1.1,

v(t,.(t)>Y(t)) < r ( t , to

7

uo),

t

E

[to , tll,

(3.1 1.7)

3.1 1.

207

PARTIAL STABILITY

where r(t, t o ,u,) is the maximal solution of (3.2.3). Relations (3.1 1.4), (3.11.6), and (3.11.7) lead to the absurdity b(E)

< V(tl , .(tl),Y(tl)) < Y ( t 1

to uo)

9

9

< b(E),

proving the validity of (P,).This establishes (1). T o prove the statement concerning (2), we have to choose u, = u(II xo 11 IIyo 11) so that 6, may be taken equal to ~ ~ ( 6 Evidently, ) . 6, is independent of t o , and, as a result, (P2)is satisfied. Let us assume (S:), so that (SF) and (5’2)hold. Then, given b ( ~ > ) 0, to E J , there exist positive numbers 6, = 6,(to) and T = T(to, e ) satisfying t 2 to T, u(t, to uo) < b ( E ) ,

+

+

I

<

provided uo 8, . As previously, choosing uo = V ( t o ,xo ,yo), we can = so(tO)obeying the inequalities find a

so

II xo I1 + IIYO II < 80

V(to

9

9

Xll

,Yo)

< 80

simultaneously. It is easy to see that (P,) is true, which implies that the inequality (3.11.7) is valid for all t to . If we now suppose that there 00 as k + 00, such that exists a sequence {tk},t, >,.to T , and t,c 11 x(tlc)il2 E for some solution x ( t ) , y ( t ) of (3.11.1) with the property that (1 xo (1 I( y o (1 we are encountered with the following contradiction:

+

-j

< so,

+

46)

< V(tk ,

X(tk),

<

Y(tkc))

Y(t,

9

to , uo)

<

Thus, (P3) is true, which in turn shows the partial equi-asymptotic stability of the trivial solution of (3.11.1). Finally, analogous to the proof of (2), it is easy to verify the assertion occuring in (4).This completes the proof of the theorem.

THEOREM 3.11.2.

Suppose that the trivial solution of (3.11.1) is partially uniformly stable with respect to components x. Then, it is uniformly stable if the following conditions hold: (i) H(t , x,y ) satisfies a Lipschitz condition in x and y for a constant K > 0. (ii) T h e trivial solution of Y’

= H ( t , 0,Y

is uniformly asymptotically stable.

)

(3.1 1.8)

208

CHAPTER

Proof.

Consider the system Y' = H ( t , 0, Y )

3

+ [ H ( t ,x, Y )

-

fqt,

(3.1 1.9)

0, y ) ] .

By assumption (i), we have

I 1 H ( t , .,Y)

-

H ( t , 0,Y)ll

< KIIx 1 .

(3.1 1 .lo)

Treating (3.1 1.9) as a perturbed system of (3.1 1A), hypotheses (i) and (ii) assure, on the basis of Theorem 3.8.1, that the solution y = 0 of (3.1 1.8) is Tl-totally stable. It therefore follows that there exist positive numbers S,(E), S,(e) such that every solution y ( t ) of (3.11.9) verifies the inequality IIY(4 t o , xo 9Yo)ll

provided that /I y o Ij

< 6,

t

< a,(€) and

II q t , % Y ) - H(t7 0,Y)ll < 6,(€), If S,(E)

=

2 to,

< E,

IIY II

t

E

1.

(3.11.11)

S,(e)/K, we infer, from (3.1 l.lO), that relation (3.1 1.11) is

valid whenever /I x /I < a,(€). Since ( P 2 )is assumed to hold for the system (3.1 1.1), it follows that, given S,(E) > 0, t,, E J , there exists a Sq(e) > 0 such that 11 xo /I /I y o I/ < S,(E) implies

+

I/ x ( t , t o , xo ,Y")ll < 6 3 ( E ) ,

t

Choose S ( E ) = min[S,(e), S,(G)]. Then, I/ xo 11 (3.11.11), and

II x ( t , t o ql ,Yo)ll < 63(€) < 6 , /I ~ ( t o,, xn

7

< E,

t

~o)ll

+ I/ y o (1 t

7

so that

3to.

< S(E) guarantees

3to,

2 to .

Thus, the uniform stability of the trivial solution of (3.11.1) is proved.

THEOREM 3.11.3. If conditions (i) and (ii) of Theorem 3.11.2 hold, the partial uniform asymptotic stability of the trivial solution of (3.1 1.1) assures the uniform asymptotic stability of the same trivial solution. Proof. Since the trivial solution of (3.11.1) is partially uniformly asymptotically stable, we may write

I/ 4 for t

t T to

7

< P(II xn /I + II

xo > ~ o ) / l

Y O Il)u(t -

3 t o , p E .X, and u E 9. Let R(t,Y ) = H ( t , x(t)7 Y )

~

H ( t , 0, Y ) .

to)

(3.11.12)

3.12.

STABILITY OF DIFFERENTIAL INEQUALITIES

209

Then, because of (3.11.10) and (3.11.12), we deduce that

/I R(t,Y)ll ,< KP(ll xo /I + I1Yo Il)u(t - to).

(3.11.13)

Consider the perturbed system Y'

=

H(t, 0,Y)

+ R(t,Y),

(3.11.14)

and let V(t,y ) be the Lyapunov function constructed according to Theorem 3.6.9. If 11 y o 11 < S(8,) = po , it follows that Dfv(t,y)(3.11.14)

+

< -c[v(t,Y)l +

R(t,y)ll,

(3.11*15)

where C E X . Let 11 xo 11 11 y o 11 < So , where 8, is the number occuring in the definition of partial uniform asymptotic stability. It is easy to deduce, taking into account relations (3.11.13) and (3.1 1.15), that D+v(t, Y ) ( 3 . 1 1 . 1 4 ) < dt,v(t, Y))l

where

g(t, ).

= -C(U)

+ MKP(So)o(t).

Notice that R(t, y ) satisfies a Lipschitz condition in y because of conp < po be given, and let KI(ci,p) = $C(U). dition (i). Let 0 < 01 T h e fact that u E 2 shows that there exists a O(N, p) 3 0 such that

<

Consequently, if

N

< u < p, t

g(t, U ) = -C(4

3

O(a, p), we have

+ MfW,)u(t)

< -C(a) + &(a)

=

-K1(a,/3).

T h e hypotheses of Theorem 3.4.10 being verified, the conclusion follows as an application of Theorem 3.8.3.

3.12. Stability of differential inequalities I n this section, we shall be concerned with the differential inequality of the form (3.12.1) II x' - f ( G 4 11 < gl(4 I1 x II), which holds for I] x 11 < p , where f E C [ J x R", R"], f ( t , 0) g, E C [ J x R, R+1. 9

= 0, and

210

CHAPTER

3

DEFINITION 3.12.1. Let x ( t ) be a function defined and continuous for t 3 to 3 0. Suppose that x ( t ) has the derivative x’(t) and it satisfies (3.12.1) for t E [ t o , a)- S , where S is an atmost countable set of [ t o ,00). Then x ( t ) is said to be a solution of the differential inequality (3.1 2.1). If gl(t, u ) ~- 0, it is understood that S is empty and x ( t ) is a solution of the differential equation x ( t o ) = xo .

x’ = f ( t , x),

(3.12.2)

We wish to consider the stability properties of the differential inequality (3.12.1) with respect to origin.

THEOREM 3.12.1. (i) and

Let the following conditions hold:

V E C [ J x S o ,R,], V(t,0) = 0,

41x II) I v(t,x)

< V t , x), l’(tjY)I

~

b E .x,

< LII x

-

Y

( t ,).

II,

E

J x

A!,

(3.12.3)

,

( t ,x), ( t , ~E )J x S,

, (3.12.4)

L being a constant. (ii) R.- E C [ J x R, , R,], g2(t,0) = 0, and n+L’(t,x)(xiz.z)

< gz(t, v(t9x)),

( t , x) E J X S , ;

(3.12.5)

(iii) gl(t, 0) = 0, and gl(t, u ) is nondecreasing in u for t E J . Then, the stability properties of the trivial solution of (3.2.3) with

dt,4 = &(t,

b-’(u))

+ gz(t, 4

(3.12.6)

imply the same kind of stability properties of the differential inequality (3.12.1) with respect to origin.

Proof. Assume that (Sf)holds. We shall only prove the corresponding conclusion and omit the rest. Let x ( t ) be any solution of (3.12.1) such that V(t,,, x(to)) uo . Defining m(t) = V ( t ,x ( t ) ) ,we see, for small h > 0, that

<

m(t

+ h)

-

m(t)

< LII .K(t

+ h)

-X(t) -

hf(4 x(t))ll

+ v(t 4-h, + hf(t, x ( t ) ) )

-

V ( t ,+))

because of (3.12.4). Using (3.12.1), (3.12.3), (3.12.5), (3.12.6), and the monotonic character of gl(t, u ) in u,we obtain the inequality D+m(t)

< g(t, m(t)).

3.12.

STABILITY OF DIFFERENTIAL INEQUALITIES

21 1

By Theorem 1.4.1, it follows that =

V ( t ,X ( W

< r ( t , to 4, 9

(3.12.7)

for those values of t 3 t, for which 11 x(t)ll < p, r(t, t o , uo) being the maximal solution of (3.2.3). Let now E > 0, t, E J be given. If 11 x I/ = E, it follows from (3.12.3) that 6(€) < V ( t ,XI. (3.12.8) Since (ST) holds, given b ( ~ > ) 0, t, E J , there exists a positive function 6 = 8(t,, E) such that u, 6 implies

<

u(t, t o , %)

Choose u,

=L

/I x(t0)ll and 8

< b(€), =

t

8 ( t , , E)

2 t, .

=

(3.12.9)

6/L. Suppose now that

< 8 has the property that

a solution x ( t ) of (3.12.1) such that 11 x(t,)lI // x(tl)/l = E and / / x(t)/l E < p for t E [ t o ,t l ] , t,

<

> t, . This would mean, in view of relations (3.12.7), (3.12.8), and (3.12.9),

< V(tl ,X(t1)) < r(t1 , t o , %> < b(4.

b(E)

This contradiction proves that 11 x(t)ll and the proof is complete.

< E, t 3 t o ,whenever 11 x(to) < 8,

REMARK3.12.1. Theorem 3.12.1 includes many special cases. If gl(t, u) = 0, we obtain the stability theorems for the differential system (3.12.2), whereas, if II R(t,x)II < g l ( t , II x 11) for II x II < p, R E ~ [ J R", x R"], we deduce the stability properties of the trivial solution of (3.12.2) with respect to permanent perturbations R(t, x).

THEOREM 3.12.2.

Assume that conditions (i), (ii), and (iii) of Theorem 3.12.1 hold. Furthermore, suppose that the solutions u(t, t o ,u,) of (3.2.3) withg(t, u)given by (3.12.6) for 0 u, a have the property that liml+mu(t, t o ,uo) = 0. Then, every solution x(t) of (3.12.1) starting in the set

< <

D

tends to zero as t attraction.

-+

= [X E GO.

R" : V ( t ,X)

< a, t 2 01

I n other words, the set !2 is the domain of

Proof. Let x ( t ) be any solution of (3.12.1) such that x(t,) E a. Consider the function m ( t ) = V(t,x(t)).It is easy to obtain, as before, the differential inequality D+m(t)

< At, W)),

212

CHAPTER

3

and, consequently, the estimate " ( t , 44) = m ( t )

< r ( t , to

>

4,

t

3 to ,

where r ( t , t o ,N) is the maximal solution of (3.12.6), with u,, = a. T h e assumption that r ( t , to , a) + 0 as t 3 GO and V ( t ,x) is positive definite now assures that D is the domain of attraction. Notice that this theorem brings out an important feature of comparison principle which is overlooked at times, that is, the behavior of the particular solution u(t, to , uo), with V(t,, x,,) u,, determines for t 2 t o , not just the behavior of the particular solution of (3.12.1) with ~ ( t , ,= ) x o , but, indeed, of all the solutions of (3.12.1) for which V(t" XO) uo .

<

?

<

3.13. Boundedness and Lagrange stability We consider the differential system 2'

= f ( t , x),

x(tn) = xn

9

to

3 0,

(3.13.1)

where f~ C [ J x R", R"]. We shall assume, for convenience, that f is smooth enough to ensure global existence of solutions of (3.13.1). We shall not require that f ( t , 0) = 0. To the different types of stability, there correspond different types of boundedness. Some important types are defined in the following:

DEFIKITION 3.13.1.

T h e differential system (3.13.1) is said to be

( B J equibounded if, for each 01 0, to E J , there exists a positive function /3 = /3(to,N), which is continuous in t,, for each a, such that the inequality

I: xo :: G

implies

I/ x ( t , to

7

x0)Il

01

< P,

t

2 t" ;

(B,) un;form bounded if the /3 in (B,) is independent of t o ; (B,) quasi-eqzii-ziltinzately bounded if, for each a 2 0 and t,, E 1,there exist positive numbers N and T = T ( t o ,a ) such that the inequality

I/ .*'u II implies

<

I/ s(f, f" , xoy < N ,

01

t >, t" i- 7';

3.13.

213

BOUNDEDNESS AND LAGRANGE STABILITY

(B4)quasi-uniform-ultimately bounded if the T in (BJ is independent of t o ;

(B5)equi-ultimately bounded if (B,) and (B3)hold at the same time; (B,) uniform-ultimately bounded if (B,) and (B4)hold simultaneously ; (B,) equi-Lagrange stable if (23,) and (S,) hold simultaneously; (B,) uniform-Lagrange stable if (B,) and (S,) hold simultaneously;

PROPOSITION 3.13.1. If f ( t , 0) = 0, t E J , and p occurring in (B,) and (B,) has the property that /I+ 0 as 01 + 0, then the definitions (B,), (B,) imply the definitions (S,), (S,), respectively. T h e proof of the statement is obvious.

PROPOSITION 3.13.2. boundedness if

Quasi-equi-ultimate boundedness implies equi-

llf(t, .)I1

G g(t, II x

(3.13.2)

II)7

where g E C [J x R, , R,].

Proof. Consider the function m ( t ) = /I x ( t , t o ,xo)ll, where x ( t, t o , xo) is any solution of (3.13.1). Then, D+m(t)

< It

X'(t,

to , xo)ll

= Ilf(t,

x ( t , to , %))It

< g(t, m(t)),

using assumption (3.13.2). By Theorem 1.4.1, we have

whenever

1) xo 11

< a , where r ( t , t o , a ) is the maximal solution of u' = g ( t , u ) ,

(3.13.4)

u(to)= a.

By the quasi-equi-ultimate boundedness, given 3 0 and to E J , there exist two positive numbers N and T = T(to, a ) such that the inequality 11 xo 11 < 01 implies (I:

Since g(t, u ) 2 0, the solution r ( t , t o , a ) of (3.13.4) is monotonic nondecreasing in t , and therefore we have, from (3.13.3), that

/I x ( t , to

1

xo)ll

G r(to

+ T , to

7

a),

t

6

[to , to

+ 77-

214

3

CHAPTER

It then follows that

/I x ( t , to , xJl

+

< max",

~ ( t , T , t o , a)],

t

3 to,

and this proves (&). Analogous to the group of definitions (Ell)-(&), we can define the concepts of boundedness and Lagrange stability with respect to the scalar differential equation (3.2.3) and designate them by (BT)-(B$).

THEOREM 3.13.1. Assume that there exist functions V ( t ,x) and g(t, u ) with the following properties: (i) g E C [ J x R, , RI. (ii) V EC [ J x R", R,], V ( t ,0) = 0, V ( t ,x) is locally Lipschitzian in x, and, for ( t , x) E J x Rn, V ( t ,4 3 4 1x I/),

where h E N on the interval 0 u + 03.

(iii)

D+V(t,x)

< g(t, V ( t ,x)),

(3.13.5)

< u < co

and b(u) + co as

( t , x) E J x R".

Then, the equiboundedness of Eq. (3.2.3) implies the equiboundedness of the system (3.13.1).

<

Proof. Let (Y 3 0 and t, E J be given, and let I/ xo I / a. I n view of the hypotheses on V ( t , x), there exists a number 0 1 ~= al(t0 , a ) satisfying the inequalities

I/ xo It

< @4

Vt,

> 30)

< a1

together. Assume that Eq. (3.2.3) is equibounded. Then, given al 2 0 and t, E J , there exists a PI = Pl(to, a ) that is continuous in to for each rn such that r(t,t o ,

U")

<

< PI

t

7

provided u, ( X I . Moreover, as b(u) a L = L ( t , , a ) verifying the relation

--j

b(L) 3 B,(t,,

Now let uo = V ( t o ,x,). show that

3 to,

co as u

(3.13.6) --f

co, we can choose

4.

(3.13.7)

Then, assumption (iii) and Theorem 3.1.1

J q t ,x ( t , t o , 3"))

< r ( t , t o , U,),

t

3 t, ,

(3.13.8)

3.13.

BOUNDEDNESS AND LAGRANGE STABILITY

215

where r ( t , to , uo)is the maximal solution of (3.2.3). Suppose, if possible, 01 having the property that there is a solution x(t, t o , xo) with 1) x, (1 that, for some t, > t o ,

<

II 4 t l

7

to

7

%ll

= L.

Then, because of relations (3.13.5), (3.13.6), (3.13.7), and (3.13.8), there results the following absurdity:

d

Y t l 7 4 t l , to 7 xo))

<

Y(t1

, t o , u,,)

< Pi(tn

a)

< b(L)-

T h e proof is complete, since this contradiction implies that (B,) holds.

THEOREM 3.13.2. I n addition to the hypotheses of Theorem 3.13.1, let V ( t ,x) verify the inequality

v(t,x) < 41x II),

(3.13.9)

<

u < CO. Then, if Eq. (3.2.3) is uniform where a E .X on the interval 0 bounded, the system (3.13.1) is likewise uniform bounded.

Proof. T h e proof runs almost parallel to the proof of Theorem 3.13.1. = a(a),which is independent of t o . Since p1 = &(.I) in We choose this case, it is easy to see from the choice of L that it is also independent of to . Thus ( B 2 )is verified.

THEOREM 3.13.3.

Under the assumptions of Theorem 3.13.1, the quasi-equi-ultimate boundedness of Eq. (3.2.3) implies the quasi-equiultimate boundedness of the system (3.13.1).

Proof. If N 0 and to E J are given, then, as in the proof of Theorem 3.13.1, we can choose an a, = al(t0, a) satisfying

I/ xo I1 < QL,

v(tn

7

xu)

<

011

at the same time. From the quasi-equi-ultimate boundedness of (3.2.3), given a1 3 0 and to E J , there exist positive numbers Nl and T = T(t,, a) such that (3.13.10) t 2 to T ~ ( t o, , ug) < Ni

<

9

+

whenever u,, a1 . Since b(u) -+ co with u, it is possible to find a positive number N verifying b ( N ) b N1(3.13.11)

21 6

CHAPTER

3

We choose zi, = V(t,, x,) and obtain the estimate (3.13.8) as in Theorem 3.13.1. Now, let there exist a sequence {tic}, t, 3 to T , t,,. co as k co such that, for some solution x(t, t o ,xo) of (3.13.1) satisfying /I x,,11 (Y, we have

+

---f

<

---f

II X(f,

?

2 N-

to > .o)ll

We are led to the following contradiction, in view of relations (3.13.5), (3.13.8), (3.13.10), and (3.13.1 I): b(N)

< V(tk ,

X(t,

I

to 7x0))

e

y(1,

< Nl

7

to

, u0)

< b(N).

This proves that the system (3.13.1) is quasi-equi-ultimately bounded.

THEOREM 3.13.4. Under the assumptions of Theorem 3.13.1, the equiultimate boundedness of Eq. (3.2.3) implies the equi-ultimate boundedncss of the system (3.13.1). T h e proof of this theorem can be constructed by combining the proofs of Theorems 3.13.1 and 3.13.3.

THEOREM 3.13.5. Let the hypotheses of Theorem 3.13.2 hold. Then, the quasi-uniform-ultimate boundedness of Eq. (3.2.3) assures the quasi-nniform-ultimate boundedness of the system (3.13.1). Proof. As in Theorem 3.13.2, one can choose cyl = a(a) independent of t,, , and, consequently, quasi-uniform boundedness of Eq. (3.2.3) shows that T = T(m)is also independent of to . With these observations, the proof follows closely that of Theorem 3.13.3.

THEOREM 3.13.6. Let the hypotheses of Theorem 3.13.2 hold. Then the uniform-ultimate boundedness of Eq. (3.2.3) assures the uniform ultimate boundedness of the system (3.13.1). T h e following two theorems present weakening of the conditions of Theorems 3.13.2 and 3.13.6. Let Z, denote the set Z”

=

[x E Rn : I/ x // 3 p].

THEOREM 3.13.7. Assume that there exist functions V(t,x) and g(t, u) fulfilling the fo!lowing conditions: (i) g E C [ J x R, * RI. (ii) V E C [ J x Z,, , R,], V ( t ,x) is locally Lipschitzian in x and

satisfies

b(l! x II)

< L’(t, x) < .(I1

.Y

ll),

( t , x)

E

1 x 2,,

(3.13.12)

3.13.

217

BOUNDEDNESS AND LAGRANGE STABILITY

where u(u), b(u) > 0 are continuous and increasing for u as

b(u) + 00

2 p,

and

u + co;

<

g(t, V(t,x)), (t,x) E J x 2, . (iii) D+V(t,x) Then, the uniform boundedness of Eq. (3.2.3) implies the uniform boundedness of the system (3.13.1).

Proof. Let 01 > 0 (we may suppose a > p ) and to E J be given, and let I/ x,,I/ a. Define a , = a(01). From the uniform boundedness of (3.2.3), it follows that

<

r ( t , to

provided uo such that

<

01,.

111,)

< &(a),

2 to

t

(3.I 3 . 1 3)

7

Since b(u) 4 co as u + CO, there exists a

WP) 2 PI(.).

/3

=

/3(.1)

(3.13.14)

Now, if we suppose that, for some solution x(t, t, , x,) of (3.13.1) with 11 xo 11 a , we have

<

I/ X ( t , then there exists a t,

>

to

9

x0)Il

=

B

at

t

=

t,

> to >

< t, satisfying

(3.13.15)

Considering the function V(t,x ( t , t, , xo)), it follows that

Choose uo = a(l1 x, I\), where x, = x(t, , to , xo). Then, condition (iii) and Theorem 3.1.1 show that, because of (3.13.15),

<

Jqt, Y( t,t.2 ,.%>> r(t7 t, uo),

t

7

6 [ti? ,

tll?

(3.13.16)

where y ( t , t, , x2) is any solution through (tz, x,) of (3.13.1). Thus, (3.13.16) is true for x(t, t o , xo)on the interval t, t t, . We therefore obtain, from the foregoing considerations, using (3.13.1 3), (3.13.14), and (3.13.16),

< <

b(P)

< V(tl

> X(t1

, to xo)) < Yl(t1 t , 7

3

?

u*)

< PI(.)

< b(P).

This contradiction proves (B,), and the proof of the theorem is complete.

218

CHAPTER

COROLLARY 3.13.1. Theorcm 3.13.7. Proof. Let (Y We choose /3

The

3

function g(t, u ) = 0

> 0 and to E J be given as /3(&~)to satisfy the relation

is

admissible

before, and let

11 xo 11

<

in

01.

:

4.).

h(P)

(3.13.17)

T h e assumption that ( B 2 )does not hold for some solution x(t, to , xo) with I/ x ~(1, IY implies, as before, the inequality (3.13.15), and consequently, from (3.13.13), we have

<

and On the other hand, by condition (iii), it follows that r’(t, , 4f, to 7

> .yo))

< b7(ti

7

“(ti

7

to , x”)),

since thc function V ( t ,x(t, t, , x0)) is nonincreasing in t. T h e foregoing inequalities lead to a contradiction, in view of (3.13.17), thus proving that (B,) holds.

THEOREM 3.13.8.

Under the hypotheses of Theorem 3.13.7, if Eq. (3.2.3) is uniformly ultimately bounded, then the system (3.1 3.1) is likewise uniformly ultimately bounded.

Proof. By Theorem 3.13.7, the system (3.13.1) is uniformly bounded. Hence, there is a positive number B such that, if 11 xo 11 p, /I 4 4 to , %)I1 ,R t 3 t o . Let now > p and to E J be given, and let p /I xo 11 01. Define (yl U(W). From definition ( B z ) , it follows that, given 0 1 ~3 0, to E J , there exist positive numbers N , and T = T(tx) such that

<

<

<

:

~ ( t o, , uu)

<

provided uo I,et N* satisfy the inequality

< Ni ,

t 3 to

+ 1,

(3.13.18)

max(N, B), where N is chosen so as to

h ( N ) > N1.

(3.13.19)

Clearly, N* > p, and the choice of N is possible since 6(u) + co as u co. We claim that, with this N* and T ( ~ Ydefinition ), (B4) holds. Suppose that this is false. Since the solutions x(t, t o , x,,) starting in --f

3.13.

219

BOUNDEDNESS AND LAGRANGE STABILITY

<

11 xo I/ p remain in 11 x 11 < N , it is enough to consider only those 11 xo j / a. If uo = xo I]), solutions x ( t , to , xo) which start in p assumption (iii) yields, for such solutions, the inequality

<

+

<

Let there exist a sequence {tk},t, 3 to T , t, co as R -+ co,such that 11 x(tk , to , xo)lI 3 N* for some solution starting in p 11 xo 11 01. Then, the following inequality results, using relations (3.13.18), (3.13.19), and --f

<

<

(3. I 3.20) :

whence we have N* < N . This is absurd in view of the definition of N*, since N* 3 N . T h e proof is therefore complete.

3.13.2. T h e function g(t, u ) = -C(u), C E X , is adCOROLLARY missible in Theorem 3.13.8. With this choice of g(t, u), evidently (BZ) holds, and hence the corollary is a consequence of Theorem 3.13.8. COROLLARY 3.13.3. Theorem 3.13.7 by

The

replacement

of

assumption

(iii)

in

where a E X , is also admissible. Using the right inequality of (3.13.12), it follows that D+V(t,x)

< -a[a-lV(t,

x)] = -C( V ( t ,x)),

cE x,

and hence the truth of this corollary follows from Corollary 3.13.2.

THEOREM 3.13.9. Let the assumptions of Theorem 3.13. I hold. Then, equi-Lagrange stability of Eq. (3.2.3) assures the equi-Lagrange stability of the system (3.13.1).

Proof. By Theorem 3.13.1, equiboundedness of the system (3.13.1) follows, and hence (S,) remains to be proved. Let E > 0, a 3 0, and toE J be given, and let 11 xo (1 < a. As in the proof of Theorem 3.13.1, there exists an 0 1 ~= m l ( t , , m) satisfying II xo I/

<

019

V(t" xo) 9

<9

220

3

CHAPTER

simultaneously. Since (SF) holds, given 3 0, b ( ~ > ) 0, and to E J , (yl implies there exists a T = T ( t , , E , a) such that u,,

<

r ( t , io u,,) 9

< b(c),

t

3 t,

+ T.

(3.13.21)

Choose zq, : V(t, , xg). Then, condition (iii) and Theorem 3.1.1 yield the inequality (3.13.8). If possible, let there exist a sequence {tk}, t, 3 to I T , t , + co as k -+ GO, such that, for some solution x(t, to , xJ satisfying 11 x, 11 ( x , we have

<

I1 ~ ( t,,t n , xo)Il 2 h. This implies, in view of the inequalities (3.13.5), (3.13.8), and (3.13.21), the following absurdity:

<

< ~ ( t ,,to

b ( ~ ) 17(t, , ~ ( t, ,t o , ~ 0 ) )

>

uo)


which proves (S7).T h e proof is complete.

THEOREM 3.13.10. Let the assumptions of Theorem 3.13.2 hold.

Then, the uniform Lagrange stability of Eq. (3.2.3) implies the uniform Lagrange stability of the system (3.13.1).

Proof. Since uniform boundedness of the system (3.13.1) follows from Theorem 3.13.2, definition (S,) needs to be proved. Choosing a1 = a(.) and following the proof of Theorem 3.13.9, we observe that T = T ( E a, ) is independent oft,) because of (5'2).This shows that (S,) holds, and the theorem is proved. Using the similar techniques as employed in Sect. 3.6, we can construct Lyapunov functions in the case of boundedness also. T h e following theorem is a typical result in that direction.

THEOREM 3.13.11. Assume that: (i) the function f E C [ J x R", R"], af(t, x)/& tinuous for ( t ,x) E J x R", and

P,(II

xo II)

< I/ 4 t *0,

X0)ll

< P,(II xo

exists and is cont

Ill7

2 0,

(3.13.22)

where P1(u), P,(u) > 0 are continuous and increasing for u 3 0, and Pl(u) cc as zi + co; (ii) the function g E: C [ J x R, , R], ag(t, u ) / & exists and is continuous for ( t ,u ) E J x R,, and ---f

Yi(uo)

< u ( t , 0 , uo) < yz(uo),

t

2 0,

(3.13.23)

3.13.

22 1

BOUNDEDNESS AND LAGRANGE STABILITY

where y,(u), y2(u) > 0 are continuous and increasing for u y,(u) + co as u + co.

> 0 and

>0

satisfying

Then, there exists a function V ( t ,x) and a constant p the following conditions:

(1) V E C [ J x Z,, , R+],V ( t ,x) possesses continuous partial derivatives with respect to t and the components of x, and where a(u), b(u) > 0 are continuous and increasing for u b(u) + co as u + CO. (2) V ' ( t ,x)

=

aV(t,

av(t*x ) f ( t , x) +ax

at

= g(t,

3 0 and

V ( t ,x ) ) , ( t , x) E J x

z, ,

Proof. Let x ( t , 0, xo), u(t, 0, u,,) be the solutions of (3.13.1), (3.2.3) through (0, x,), (0, uo) satisfying (3.13.22), (3.13.23), respectively. Under the assumptions of the theorem, the continuity and differentiability of the solutions x ( t , t, , xo), u(t, t o , u,,) with respect to their arguments is guaranteed. Define p = p2(0), and observe that the common domain of definition of the inverse functions pr', pg' is [p, CO). Hence, denoting x ( t , 0, xo) by x, so that x, = x(0, t , x), we get from (3.13.22) the inequality

P;l(Il x 11)

< II 40, t , )1.I < P(Ilx II),

xE

z,.

We choose a continuous function p ( x ) for x E R",possessing continuous partial derivatives ap(x)/ax there and such that

%(I1 x 11)

< d x ) < %(I1

x

II),

< u < a,and

where a l , EX on the interval 0 u + a.For x E Z, , define the function u t ,).

= u ( t , 0, &(O,

t , x))).

Then, it is easy to obtain V ' ( t ,).

Furthermore, if

= g(t,

I( x 11 >, p,

V ( t ,x)), ( t , x ) E J x

we have

z,,

q ( u ) + co as

222

CHAPTER

3

and

Clearly a(u), b(u) > 0 are continuous and increasing for u 3 p, and b(u) + m as zi m. 'I'hc theorem is proved. --f

3.14. Eventual stability We shall now consider a notion that is a generalization of Lyapunov's stability. Let x ( t , t, , x,,)be any solution of (3.2.1). 'I'he set x

D E F I N I T I O N 3.14.1. system (3.2.1)]:

=

0 is said to be [with respect to the

( E l ) eventually uniformly stable if, for every T T ( E ) > 0 such that

6 - S(t) > 0 and

t

> 0, there exist a

~

I1 y ( t , f" , .%")I1 < t,

t d t"

T(E),

<

provided / / x,)j ( 6; ( E J eventually quasi-uniformly asymptotically stable if, for every there exist positive numbers 6,, T " , and T = T ( E )such that

/I y ( t ,

to,

x0)Il

€7

t

' f"

f T,

<

f,

E

> 0,

>70,

providcd 11 xo 11 6, ; (E,) eaentzially uniformly asymptotically stable if (E,) and ( E 2 ) hold simultancously; (E,) eventually exponentially asymptotically stable if there exist constants L 5 0, (Y > 0 such that

/I

r(f7

t o , .xo)II

-< LII xu /I c T b ( t

-

to)],

t 3 to,

(3.14.1)

d ( ~ ) where , d(r) is a monotonic Y provided 0 / ( x, I/ -, p and to decreasing function of Y for 0 . Y < p.

KFMARK 3.14.1. Notice that, if (E,) holds and if x = 0 is a trivial solution of (3.2. l ) , then the uniform Lyapunov stability (S,) results from the continuity of solutions with respect to the initial values, provided the unicity of solutions of (3.2.1) is assured. Similarly, ( E 3 )

3.14.

223

EVENTUAL STABILITY

implies, in such a case, uniform asymptotic stability of the trivial solution of (3.2.1). As usual, let us denote by (ET)-(E$) the corresponding notions of the set u = 0 with respect to the differential equation (3.2.3).

THEOREM 3.14.1. Assume that there exist functions V(t,x) and g(t, u ) verifying the following properties: (i)

I/ E

C [ J x S o ,R,], V ( t ,x) is locally Lipschitzian in x, and

4 x 11)

< v(t,4 < 4 x II),

for 0 < Y < 11 x 11 < p and t >, O(Y), where a, 6 E X and B(Y) is continuous and monotonic decreasing in Y for 0 < Y < p. (ii) g E C [ J x R,., R], and the set u stable with respect to (3.2.3).

=

0 is eventually uniformly

(iii) f E C[J x S, , R"],and D+V(t,4

for 0

< Y < 11 x I/ < p

Then, the set x system (3.2.1).

=

and t

< g(t, V ( t ,XI),

3 O(r).

0 is eventually uniformly stable with respect to the

Proof. Let 0 < E < p. Since the set u 0 is eventually uniformly stable, given b ( ~ > ) 0, there exist a 6, = 6 , ( ~ )> 0 and T , = T , ( E ) > 0 such that (3.14.2) u(t, t o , uo) < b ( E ) , t > t o > TI(€), 1

<

) T , ( E ) = d ( 8 ( ~ ) ) .Let T : T(E) = if u,, 6, . We define 6 = ~ ~ ( 6 ,and max[Tl(E), T ~ ( E ) ] . Then, ( E l ) holds with this choice of S(6) and T ( c ) . If this were not true, there exist numbers t, , t , such that t, > t, > to 3 T ,

II ~

( t, ,to , xo)ll = 6,

I/ x ( t 2 , to , x0)li = E,

and 8

< II x ( t , to > x0)Il < E ,

t

E

(tl

,tz).

Choose u,, = a(\\x1\I), where x1 = ~ ( t, ,to , x,,). Then, condition (iii) and Theorem 3.1 1.1 show that V ( t , y ( t ,tl

7

XI))

< r(t,t,

>

uo),

t

t

[tl

, f21,

(3.14.3)

where y ( t , t , , xl) is any solution of (3.2.1) through (t, , xl), r ( t , t , , u,,) being the maximal solution of (3.2.3) through ( t l , uo). Thus, (3.14.3)

224

CIIAPTER

3

is also true for x(t, t,, , x,,) on the interval t , obtain h(E)

< f’ftz,

< Y(t,

4 t , , t o , 4)

7

< t < t, . We

therefore

t , 3 uo) < @ E ) ,

taking into account the uniformity of the relation (3.14.2) and the fact t, t , > t, 3 T. This absurdity that we are led to prove ( E , ) is true, and the proof is complete. ‘ 2

COROLLAKY 3.14.1. If, instead of the eventual uniform stability of the set ti - 0, it is assumed that the trivial solution u = 0 is uniformly stable, the conclusion of Theorem 3.14.1 remains the same. I n particular, g ( t , ZL) -= 0 is admissible.

THFOREM 3.14.2. Suppose that there exists a function V ( t ,x) such that V E C [ J Y S,,, R-1, V ( t ,x) is Lipschitzian in x for a constant (1) L > 0, and h(l’

li)

+ 17(t, x) < .(I1

x II),

for 0 Y ,/ / s I/ <

<

(ii) f E (I[] x S , , R ” ] , and Di V ( t ,x) 0 for 0 < Y < 11 x I] < p and t 2 H(r). Then, the set s = 0 is eventually uniformly stable with respect to the perturbed system

(3.14.4)

r’ = f ( t , x) -l- R(t,x),

where R E C [ J x S, , R J t ]and, , for every continuous function x ( t ) such that /I .t(t>lI pi p, t 3 0,

<

l’rooj.

~

For a given

E

> 0, E

248)

< p*,

< b(€),

we choose a S(E) TI(€) =

0(8(E)).

> 0 so that (3.14.5)

Let $ ( t ) =: maxll,IIi ) c 11 R(t, x)li. Then, since +(t) is integrable, there exists a T , ( E ) 3‘ 0 such that, if t,, 3 T , ( c ) , we have

3.14.

225

EVENTUAL STABILITY

where L is the Lipschitz constant for V ( t ,x). Let T ( E ) = max[i-,(E), 5 - 4 6 ) ] . Then ( E l ) is true with T ( E ) and 6 ( ~ ) .For otherwise, there would exist a t, > to 2 T ( E ) such that

I1 x(t1)Il = E ,

I/ x(t)ll

< < p*,

t E [to , tll,

E

where x ( t ) = x(t, t o ,x,,) is some solution of (3.2.1). This implies, setting m ( t ) = V ( t ,x ( t ) ) , the inequality D+m(t)

and, consequently, m(t)

< Ld(t),

< m(to) + L f d(s) 4

t

2 to.

t0

Hence, at t

=

t , , there results

< m(t1) = V(t1 , x(t1)) e U t o , xo) + L J d(s) ds to < @) + = 2 4 3 , tl

b(E)

which is a contradiction in view of (3.14.5). T h e proof is complete.

THEOREM 3.14.3.

Suppose that there exists a function V ( t ,x) such that

V E C [ J x S, , R,], V ( t ,x) is locally Lipschitzian in x, and

(i)

b(ll x 11)

<

< V ( t ,x) < 4

x

11)

for 0 < Y 1) x 11 < p and t 3 O(r), where a, b E X and d(r) is continuous and monotonic decreasing in r for 0 < Y < p ; (ii) f

E

C [ J x S , , R"], and D + V ( t ,x)

for 0


< 11 x /I < p, t 3 O(Y),

Then, the set x

=

e -C(Il

x

11)

and C E X .

0 is eventually uniformly asymptotically stable.

Proof. By Corollary 3.14.1, eventual uniform stability of the set x

=

0

follows. Let E > 0 be given. Designate So = S(p), 5-o = T ( p ) , and T ( E )= .() a(p)/c[q6)1.Let to 3 7 0 and I1 xo II 6, . T o prove the theorem, it is sufficient to show that there exists a t* E [to T ( E ) , to T ( E ) ]such that /I x(t*, t o ,x,,)iI < S ( E ) , because the set x = 0 is eventually uniformly stable. Assume, if possible, that

+

<

+

+

S(E)

< II x ( t , to

9

x0)Il


t

[to

+

5-(€),

to

+ T(E11.

226

3

CHAPTER

Using assumptions (i) and (ii), we get

< -C(ii

D ' br(t, ~ ( tto, , x ~ ) )

for t E [to

+

I,'('(to

T(E), t,,

+ T ( E ) ]which, ,

+ U € )",( t , + T ( € ) , < C'(t" +

by integration, yields

t o , "0))

-1.

7 ( t ) , .V(t"

T(E),

to,

Xo)) -

It then follows, froin this relation, that 0

< b(S(E))

< -C[S(c)],

~ ( tt o, , .x0)ll)

< q t " -1-

qe), "(t"

G .(I1 "(4, t 4.), 1,

< n( p)

-

C [ S ( € ) ] [ T (€ ).(€)I.

+ T(c), 9

4 1 , 'yo))

- qs(,)l ___

4P)

"o)II)

crs(,)l

n(p) = 0.

+

This contradiction implies that there exists a t* E [to T ( E ) , to such that /I x ( t * , t o ,xo)il -:8 ( ~ and ) proves the theorem.

+ T(E)]

T h e following converse theorem for eventual uniform asymptotic stability may be proved, in the same way as in Theorem 3.6.9. T H E O R E M 3.14.4. Assume that the set x = 0 is eventually uniformly asymptotically stable with respect to the system (3.2.1). Suppose that

Ilf(h .y) -.f(CY)Il for t

3 0, x,y

E

< L(t)l/"

-

Y I/!

S, , and f ' " L ( s )ds t

< Lu,

u

3 0.

Then, there exist functions a, b, C E Z, O(u), and V ( t ,x) satisfying

for Y continuous and decreasing on (0, so);

,

:

continuous and decreasing in u for 0

(4)

M ( Y )being

O(r), where 8(u) is

< u < p;

3.14.

227

EVENTUAL STABILITY

P Y O O ~As . in the case of uniform asymptotic stability, we can find a(<), T ( E ) , and T ( E )such that 6 E X and T , T E 9. Let G(y) be the same function as in Theorem 3.6.9. Then, we define

for t 3 7 , , 11 x I/ < 6,. Clearly, V ( t ,x) 3 G(ll x II), and therefore b(u) = G(u).Let ~ ( 6 be ) the inverse function of 6 ( ~ )and , let O(u) = T(<(u)). We have

Ij x ( t

+

0,t ,

%)I1 < 4 x II),

0
< II x II < 6, ,

t

2 e(y).

As a consequence,

for 0 < r

< [I x 11 < a,,

t 3 O(Y), and thus it results that

v(t,x) < 4 x II),

0
< II x II < 8,

t

7

3 @(Y),

where a(.) = aG(e(u)). T h e rest of the proof is similar to that of Theorem 3.6.9. We omit the details.

THEOREM 3.14.5. Let the assumptions of Theorem 3.14.4 hold. Assume that the perturbation R(t, x) obeys

/I q t , x)ll

< 4t),

0-E

9, I1 x II

< 8" .

(3.14.6)

Then, the set x = 0 is eventually uniformly asymptotically stable with respect to the system (3.14.4). Proof. Let V ( t ,x) be the function constructed as in Theorem 3.14.4. For 0 < Y < 11 x 11 < 6, , t 3 O(Y), and h > 0 sufficiently small, we have

v(t + h, x + h[.f(t, x) + R(t,.)I)

~

v(t,).

< ~ w h iwi ,~ ) I+I v(t + h, x + ~

and hence, by (3.14.6), D+V(t,X)(3.14.4) G DtV(t7 Xh3.2.1)

< -C(ll

x 11)

( tx)),

v,

+ MPIl R(t7 .)I1

+M(Mt).

Since CT E 9, there exists a O1 E 9such that

-

XI,

228

CHAPTER

3

It then follows that

< -WI x 10,

D+ ’ ( t >x)t5.14.4)

provided 0 -’r . / / x I/ -, 6,, :and t 3 O,(Y), where O,(Y) T h e conclusion now follows from Theorem 3.14.3.

=

max[O(r), O1(~)].

THEOREM 3.14.6. Assume that the set x = 0 is eventually exponentially asymptotically stable with respect to the system (3.2.1) and that f ( t , x) is linear in x. Then, there exists a function V ( t ,x) satisfying

(1)

V E C [ J s S , , R , ] , and V ( t ,x) is Lipschitzian in x for a constant L > 0;

(2) /I x /I

< v(4 x) < L II x 11,

0

< ,

r

< 11 x 11

O(r) is the same function defined in ( E J ;

(3)

Proof.

I)’V(t,

x)(3.?.1)

< -avc


< It x II

x), 0 < r

t

2 O(r),

where

< P , t 3 O(+.

We define the function r’(t, x)

=

sup 11 x(t 02.0

+

U,

t , x)l/eau,

<

for 0 Y 11 $2: 11 .I p and t 3 O(r).Following the proof of Theorem 3.6.1, it is easy to prove this theorem with appropriate changes.

T I I E O R3.14.7. I~

Suppose that the hypotheses of Theorem 3.14.6 hold. I,et the perturbation R(t, x) satisfy

I1 R(t,x)Il

< Pll x /I

<

I/ x // i p and t O,(Y), where O,(u) is continuous, decreasing for 0 . Y p , and p is sufficiently small. Then, the set x = 0 in zc for 0 < zi -: is eventually exponentially asymptotically stable with respect to the system (3.14.4). Proof. By hypotheses, there exists a function V ( t ,x) satisfying the properties ( l ) , (2), and (3) of Theorem 3.14.6. For 0 < T < I\ xjl < p, O,(r), and h > 0 sufficiently small, we deduce that t

ni q t , q < -arqt,

+ L I I~ ( tx\l,, x) + LPll x ) I

< -aV(t, < ( - a +LP)V(t, x‘ < -Yr-(t, x),

3.15.

229

ASYMPTOTIC BEHAVIOR

where y > 0, y < 01 - I& and Q,(r) = max[O(r), O,(Y)]. This choice of y is possible, since is sufficiently small. T h e stated result is evident from the preceding inequali.ty, because of the properties of V(t,x).

3.15. Asymptotic behavior We shall discuss a number of results dealing with the asymptotic behavior of solutions. Let us begin with the following.

THEOREM 3.15.1. Let there exist functions V ( t ,x) and g(t, u)fulfilling the following conditions: (i)

V E C [ J x R", R,], V ( t ,0 ) = 0, V(t,x) is Lipschitzian in x for

a continuous function K ( t ) >, 0, and

@I1 x II) where b E Z such that b(u) -+ (ii) g E C [ J x

< V(t7

GO

as u -+GO.

R, , R], and

-

V ( t ,(x - Y))l

e g(t, V ( t ,x

- Y))l

wherefE C [ J x R",R"]. (iii)

Every solution u(t) of

tends to zero as t

+ GO.

Then, every solution x ( t ) of the system x' = f ( t , x),

tends to zero as t

+

x(to) = x,

(3.15.2)

00.

Proof. Let x ( t ) be any solution of (3.15.2) such that V(t, , x,,) Consider the function m ( t ) = V ( t ,r(t)).

< u, .

230

CHAPTER

3

where c(h)/Iz + 0 as h + 0. Employing assumption (ii) with y we arrive at the differential inequality D' m(t) < g(t, m(t))

=

0,

+ K(f)llf(t,0)lL

and this yields, by Theorem 1.4.1, the estimate t'(t, x ( t ) )

<

t" ,

t>:

Y(f),

r(t>being the maximal solution of (3.15.1). T h e statement of the theorem is now a direct consequence in view of conditions (i) and (iii).

THEOREM 3.15.2.

Assume that there exist functions V ( t ,x) and g(t, u ) enjoying the following properties:

0, and V ( t ,x) is positive definite (i) V E C [ j x R", R,], V ( t ,0) and locally Lipschitzian in x. (ii) g E C [ J x R,, R],and all solutions u(t) = u(t, to, uo),0 uo 01, of (3.2.3) have thc property that lim, u(t) = 0. V ( t ,x) satisfies the inequality (iii) T h e function D-'

< <

,%

I>+V(t,x )

< g(t, V ( t ,x))

for t E J and x E Z , where Z is the set defined by -= [-YE R"; r ( t )

<

+

17(t,X) < ~ ( t )

c,,

,t

> t,,],

r ( t ) being the maximal solution of (3.2.3) and c0 a certain small positive number.

Then, the domain of attraction for the solutions of (3.15.2) is the set Q

[x! E

R" : t y t , x)

< a, t E J ] ,

that is, all solutions x ( t ) of (3.15.2) such that xo E Q tend to zero as t + 00.

3.15.

23 1

ASYMPTOTIC BEHAVIOR

Proof. If x ( t ) is any solution of (3.15.2) such that x,,E Q, we choose u,,= V(to, x,,)and obtain, by Corollary 3.1.2, the estimate

J/(t, x ( t ) )

< r(t, to , u"),

t

2 to .

T h e positive definiteness of V ( t ,x) and assumption (ii) imply the stated result.

THEOREM 3.15.3. Assume that (i) f E C[J x So,R"],f ( t , 0) continuous on J x S, ; (ii)

I/ E

0, and af(t, x)/&

exists and is

C [J x S, , R,], V ( t ,0) = 0, V ( t ,x) is Lipschitzian in x for > 0, and

a constant K ,

Kill x II

< v(t,x),

K, > 0,

( t ,2) E

-/ x S o ;

(3.15.3) Then, the trivial solution of (3.15.2) is asymptotically stable. Proof. By assumption (i), given such that .f(t, ).

where llF(t, .)I1

=

E

> 0, it is possible

to find a 6 ( ~ )> 0

f d t , 0)x -tF ( t , 4,

< 4 x I1

if

/I x ll

<

S(C),

(3.15.4)

uniformly in t. It then follows by (iii) that D ' - V ( t ,Xh3.15.2)

< .(t)V(t,

x)

+ K,IlF(t, .)I.

(3.15.5)

Let E > 0 and to E J be given. Because of condition (3.15.3), we have, if E is small enough,

232

CHAPTER

If we choose

11 xo 11

J

3

6, , where K,S,B

< ,

K , ~ ( Eand )

then wc: get 11 .r(t)il <<6 ( c ) , t 3 t o . For, otherwise, there would exist a t , > to with the property that

I1 4t)Il

< qc),

t

E

11 x(t1)Il

[ t o tll, 7

=

qc).

Taking into account (3.15.4) and (3.15.5) and setting m ( t ) = V ( t ,x(t)), we obtain, by Theorem 1.4.1, the inequality (3.15.6) for t

E

[t,, , t l ] , which, in turn, yields, at t

t, ,

< IC28,B < K,S(c). This contradiction proves that, if 11 x,)11 < 6,, 11 x(t)l( < S ( E ) , t to , and so, the inequality (3.15.6) is true for all t 3 t o . It is now easy to see that limt+mx ( t ) = 0, establishing the theorem.

THEOREM 3.15.4. I n addition to the assumptions of Theorem 3.15.3, suppose that, for a constant L > 0,

IIfAG x) - f d t , 0111 < LII x II.

(3.1 5.7)

If y ( t ) = y ( t , t o ,x,,)is the solution of the variational system -Y’

=f,(f,X ( ~ ) ) Y >

to)

(3.15.8)

= xg,

where x ( t ) = x ( t , t,, , xo) is the solution of (3.15.2), I/ xo 11 being sufficiently small, then linit+Ty ( t ) = 0. L’yoof.

Let us first observe that 1)’ l’(f>Y)(3.15.8)

< Di b’(f>Y) + KZllfdf,4 t ) ) - f A t ,

0)Il IIY II.

If we now set m(t) = V ( t , y ( t ) ) ,we readily obtain, in view of (3.15.7) and condition (ii), the inequality D.44

< a ( t ) m ( t )+ LKII 4t)lI I/ Y(t)ll.

(3.15.9)

3.15.

233

ASYMPTOTIC BEHAVIOR

Since the hypotheses of Theorem 3.15.3 hold, if 11 xo 11 is small enough, we have // x(t)/\ < E, t >, t o . Consequently, choosing I/ x, // sufficiently small and using the relation Kl 11 x 11 V ( t ,x), it follows that

< D+m(t) < [a@) + L 551 m(t), Kl

which leads to the estimate

for t >, t o . If E is small enough, the condition (3.15.3) assures that lim,,,y(t) = 0. We shall next consider a theorem on the dependence of solutions on the initial values, which is useful in what follows.

THEOREM 3.15.5. Suppose that f ( t , x) is continuous on an open set D in J x Rn and that every solution of (3.15.2), (to , xo) E D is continuable to t = t, > 0. Let E be the set of all the points consisting of the solution curves for [ t o ,t J starting from ( t o ,xo), and let E be contained in a compact set in D.Then, to each E > 0, there exists a S > 0 such that every solution x*(t, t$, x;), t$ E [ t o ,t l ] , of x'

=At,).

+ g(th

(3.15.10)

< 8,

(3.1 5.1 1)

where g E C [ J ,R] and satisfies f ; d s ) ds

passing through [tf, t,] and obeys

p*

=

(t?, x?) such that d(p*, E )

I1 X * ( t ,

to*, x,+) - x ( t , to 9 xo)ll

< 8,

exists on

< c,

x ( t , t o ,xo) being a solution of (3.15.2) contained in E, which may ,):x depend on x * ( t , t z .

Proof. Suppose that, for some E > 0, there is no S such that it satisfies the condition in Theorem 3.15.5. We may assume that U ( E , c) C D, -~ where U ( E , 6) = [x: d(x, E ) < €1. Since U(E, c) is a compact set, there is a functionf*(t, x) that is continuous and bounded on (~ 03, 03) x Rn -~ and is equal to f ( t , x) on U ( E ,c). A solution of (3.15.2) remaining in U(E,E) is a solution of x' = f * ( t , x),

(3.15.12)

234

CHAPTER

3

and the set of all the points consisting of the solution curves for t E [to, tl] coincides with E. We may therefore assume that, for E > 0 and the equation of (3.15.12) through ( t , , x,)

s'

(3.15.13)

= f * ( t , x) + g ( t ) ,

the conclusion of the theorem is not verified. Every solution of (3.15.13) exists for all t. By hypothesis, there are a sequence of points {pr = (t,, , xJ} and a sequence of functions {gx(t)}such that d(p, , E ) tends to zero,

J;)mdt

+

0

as

k

+

m,

< t < t , , of

and a solution $ h ( t ) ,t,&

x' = f * ( t , x)

+ g&)

through p,> such that there is no solution curve of (3.15.12) lying in E with the property that the distances of all the points on the arc of the former to the latter are smaller than E . Since $,,(t)is defined on t,, t t , ,

< <

and thus (+/,(t):is uniformly bounded and equicontinuous on [to, tl]. Hence, wc: can select a uniformly convergent subsequence. Denote its index by h again, and let + ( t ) be its limit function. Because of (3.15.14), we havc + ( t ) = +(tn) - I

\' f * ( s , (~(s)) dsts,

" '0

and thus + ( t ) is a solution of (3.15.12). If t , is a point of accumulation of {t,,}, then ( t , , $(t2))E E. By +(t)and a solution joining (to, x,,) and ( t 2 , + ( t 2 ) ) ,we havc a solution x = +*(t) of (3.15.12) through ( t o ,xo). Therefore, +*(t)c I?, +*(t) = +(t), t I,. If k is sufficiently large and t,, is sufficiently close to t, , the distance between +,,(t) and +(t) is smaller than E , because + k ( t ) is uniformly Convergent to $(t). This contradicts our hypothesis. T h e theorem is thus proved.

3.15.

235

ASYMPTOTIC BEHAVIOR

Let us now consider a system of differential equations x' = f ( 4 x)

where f , R E C[]

+ R ( t ,x),

x E, R'"],E

(3.15.15)

x ( t o ) = xo ,

being an open set in Rn.

DEFINITION 3.15.1. A scalar function u(x) defined for x E E is said to be positive definite with respect to a set A C E if v(x) = 0 for x E A and, corresponding to every t > 0 and every compact set Q in E, there exists a positive number 6 = 8(Q, c) such that for x E Q n S(A, E

~ ( x2 ) 6

) ~ ,

where S(A, c)" denotes the complement of the set S (A , €)

=

[x : d(x,A ) < €1.

DEFINITION 3.15.2. A solution x ( t ) of (3.15.15) is said to approach a set A as t + 00 if, for each E > 0, there is a T > 0 with the property that, for all t > T , the points x(t) are contained in S ( A , c).

THEOREM 3.15.6.

Assume that the functions f ( t , x), R(t, x), and V ( t ,x) satisfy the following conditions for ( t ,x) E J x E:

(i) f E C [ J x E, Rn], and f ( t , x) is bounded for all t E J when x belongs to an arbitrary compact set in E. (ii) R E C [ J x E , R"], and, if x ( t ) is continuous and bounded on to t < co, that is, x ( t ) CQ, Q being a compact set in E, then R(t, x) satisfies the inequality

<

J;

(iii)

(3.15.1 6)

II R(s, x(s))ll ds < a-

V E C [ J x E , R,], V ( t ,x) is locally Lipschitzian in x, and

D+V(t,x)

= lim

1

sup - [V(t h

h-O+

< -C(4

+ h, x + h [ f ( t ,x) + R(t,x)])

+ g(4 V ( t ,x)),

-

V(t,x)]

(3.15.17)

where C(x) is positive definite with respect to a closed set L? in E, g E C[J x R, , R ] , and g(t, u)is monotonic nondecreasing in u for each t E J. Then, if all the solutions of (3.15.15) and (3.2.3) are bounded, every solution of (3.15.15) approaches Q as t -+ 00.

236

CIIAPTER

3

Proof. Let s ( t , t, , so) be a solution of (3.15.15). By assumption, ~ ( tt,,, , xo) is bounded, which implies that there is a compact set Q in E such that "\(t, t" , X(,)

€2,

> f,

t

.

If we suppose that this solution does not approach Q as t --t co, then, for some E > 0, there exists a sequence {tr>, t, co as k + 00 such that --f

.\(t,, t,, , x,) E S(Q, E)" n Q.

T h e assumption that f ( t , x) is bounded when x E Q assures that llf(t, .v)II

<

for some positive constant M . We may assume t , is sufficiently large so that, on intervals f,

t

:

r=

(3.1 5.18)

t , 1 k€/M ,

we have, bccause of condition (3.15.16), ,.tki : r l M

Thus, one gets, on the intervals (3.15.18), that n 0.

x ( t , t,, , so)E S(Q,')6

(3.15.19)

\Ye may assume that these intervals are disjoint, if necessary, by taking

a subsequence of ( t h } .T h e relation (3.15.17) and Theorem 3.1.3 give

the inequality

r'(t, ~ ( t o, ,

2.0))

:

-if

C[X(S,t o , xo)] ds -1- r ( t , t o , uo)

(3.15.20)

'0

for t 2 to , where zi,, = V(t O, xo) and r ( t , to , uo) is the maximal solution of (3.2.3). T h e positive definiteness of C(x) with respect to Q, together with the relation (3.15.19), shows the existence of a 6 = 6 ( ~ / 2 such ) that

hloreover, from the boundedness of all solutions of (3.2.3), it follows that r ( t , t,, , U") < P,

t

>to.

(3.15.22)

3.15.

231

ASYMPTOTIC BEHAVIOR

Thus, we arrive at the inequality

(tA

+ &i x Itk -t &i , to ,x.)

< - 8 g j k +P,

in view of relations (3.15.20), (3.15.21), and (3.15.22). Since V ( t ,x) >, 0, the foregoing relation leads to an absurdity as k + co. This proves that every solution x ( t , t o , xo) approaches Q as t + co, and the proof is complete.

DEFINITION 3.15.3. A point w E RTLis said to be a cluster point of a solution x ( t ) of (3.15.15) if there exists a sequence ( t 3 , t , -+ co as k + 00 such that x ( t k )+ w as k 4 co. T h e set of cluster points is called the positive limiting set and is denoted by If a solution x(t, t o , xo) of (3.15.15) is bounded for t 3 to , then its positive limiting set is a nonempty, compact set, and x ( t , to , x,,) -+ as t + 03. Furthermore, if x (t, t o , xo) is bounded for t 3 to and if A contains the positive limiting set of x(t, to , xo), then x(t, to , x,,)---t A as t co.

r+.

r+

r+

---f

LEMMA 3.15.1. Let 8 be a closed set in E, that is, Q is a closed set in the topology of E. Assume that a solution x ( t , to , x,,)is bounded and x ( t , to , x,,)-+ Q as t + a.Then the positive limiting set of x ( t , to , x,,) satisfies C 8. Let f ( t , x) satisfy the following hypotheses:

r+

r+

(a) f ( t , x) tends to a function H ( x ) for x E L?as t -+ co, where Q is a closed set in E, and, on any compact set in Q, this convergence is uniform. Consequently, H ( x ) is a continuous function on Q. (b) For each E > 0 and each y E Q, there exist positive numbers 6 ( y ) and T ( y ) such that, if 11 x - y (1 < 6 ( y ) and t 3 T ( y ) ,we have

I1 F ( t ,).

-

F (t, Y)ll

<

€ 0

If t E J , then we can choose 6 ( y ) so that (b) holds for all t >, 0. T h e following lemma can be proved in the same way as one proves the uniform continuity of a continuous function on a compact set. We merely state is an arbitrary compact set in 8, LEMMA 3.15.2. If y E GI, where the 6 and T of (b) are independent of y . Now consider the differential system x'

where H

E

=

H(x),

.(to)

= X"

,

t"

2 0,

C [ D ,Rn], D being an open subset in R".

(3.15.23)

238

3

CIIAPTER

DEFINITION 3.1 5.4. A set A C D is said to be a semi-invariant set of (3.15.23) if, for each point of A , there is at least one solution of (3.15.23) which remains in A for all future time. T h e following theorem is stated in a special form that is convenient for applications. However, the proof can be modified to apply to a more general situation.

THEORFM 3.15.7. Assume that a solution

n ( t , t,,, x,,) of (3.15.15) is bounded for t >- t,, and that it approaches a closed set Q in E. Let f ( t , x) satisfy hypotheses (a) and (b). Suppose that R(t, x) satisfies assumption (iii) of Theorem 3.15.6. 'Then, the positive limiting set of x ( t, to , x,,) is a semi-invariant set, contained in Q, of Eq. (3.15.23)

r+

Proof. By assumption, a solution x(t) = x ( t , t o ,x,,) of (3.15.15) is bounded in E, which implies that there is a compact set Q such that x ( t , to , X")

€0,

t

3 t" .

By 1,emma 3.15.1, we have

r+c Q n 0 = Q, Sincc 52, is a compact set in R", there exists a continuous, bounded function H * ( x ) on R" such that H * ( x ) = H ( x ) on Q,. Consider the system x' = I I * ( x ) . (3.15.24) Let w be a point of that ~(t,)

r+.T h e n w E Q, t,, , NO)

~((t,~,

, and there is a sequence ( t 3 such t , -* co

+W,

3s

k

+ CO.

(3.15.25)

Since (3.15.24) is an autonomous system, the behavior of the solutions of (3.15.24) through (tl, , w ) is the same as that of the solutions through (0, w ) . For an arbitrary h > 0, designate the interval t, t t, X by JI, . T h e boundedness of H * ( x ) shows that all the solutions of (3.15.24) exist on J/>. Now consider the system

< < +

x'

H*(r)

+f(t, s(t))

~

IZ*(r(t)) 1 R(t,x ( t ) ) .

(3.15.26)

Clearly, 3 ~ ' = x ( t ) is a solution of (3.15.26) through ( t x , x(t,)). As x ( t ) is bounded, from condition (3.15.16), it follows that, if k is sufficiently large, that is, k --k, , , we have, for a given S > 0, (3.15.27)

3.15.

239

ASYMPTOTIC BEHAVIOR

Since Q, is compact, for every point x(t) there is a point y ( t ) satisfying d ( 4 t ) ,Q,)

=

I1 x ( t ) - Y(t)ll.

By hypothesis (b) and Lemma 3.15.2, given 6/6h, there exist 8, T > 0 such that, if y E Q, , 11 x - y I/ < a,, and t 2 T , Ilf(t, X) - f ( t ,

> 0 and

~ ) l l< 6/6X.

On the other hand, since x(t) E Q, as t we have

-+

co, for sufficiently large t,

~ ( tC) S(Q1 ,a,) n 0.

Hence, if t is sufficiently large, that is, k

3 k, , it follows that

Ilf(4 44 -f(t,y(t))lI < 6/6h

on

Jk

.

(3.15.28)

Because of hypothesis (a), f(t, x) H ( x ) as t 00 for x E Q, and this convergence is uniform for x E Q, , and, for sufficiently large t , the inequality --j

--f

llf(t, x) - H(4Il < 6/6X holds. If k is sufficiently large, that is, k >, k, , we therefore have

I l f ( 4 x(t)>- f f ( Y ( t ) ) /< / S/6A

on

Jk

.

(3.15.29)

Moreover, since H*(x) is continuous on 0, there is a 6, > 0 such that, if (I x y I1 c,6, , (1 H * ( x ) - H*( y)II < 6/64 and, if y E Q, , H*( y ) = H ( y). From this it follows that, if y E Q, and 11 x - y /I -<. 6, , -

I1 H * ( x ) - f{(y)ll < S/6X. Consequently, if k

3 k, , we

have, on

Jk

,

Thus, if k 3 max[k, , k, , k , , k,], from the relations (3.15.28), (3.15.29), and (3.15.30) and the inequality

we have

240

3

CHAPTER

T h e relations (3.15.27) and (3.15.31) yield that

J:;

llf(S,

x(s))

-

H*(x(s))

+ R(s,

‘Z(S))Il

ds

< 6.

On the other hand, for sufficiently large k , (3.15.25) shows that (l((tk,X(tk)),

(tk

, w))

< 8.

Thus, by Theorem 3.15.6, there is a solution $/\.(t)of (3.15.24) through such that, for a given E > 0,

( t h . ,w)

d ( s ( t ) ,+ k i t ) )

< 6,

t

E /k

*

Since +/,.(t)is a solution (3.15.24) through (tk , w), we have $/;(t) = w

+ J‘ H*(+&)) ds,

< t < t , + A,

t,

(I

and, if we denote $/,.(t+

tl0, 0

Therefore, for a sequence $/,(t) of (3.15.24) satisfying

< t < A, by &(t) again, we deduce

{ E ~ , . } , el\.

+0

as

K

co, there exist solutions

+

,t

+/it)

c V ’4. ?

T h e sequence of functions {$!*(t)}is uniformly bounded and equicontinuous, and hence we can select a subsequence that is uniformly convergent. I,et $ ( t ) be its limit function. Then, it follows that fb(t) = w

+

and +(t)

Jt 0

H*(d,(s))cis,

el, + 0

H*(+(t))= +(t)

w

& t

c A,

c r+, o G t + A,

by (3.15.32) and the fact that implies and therefore

0

-+ jtZZ(d(s)) 0

as k

+

co. Since

WW), ds,

0

< t < A.

r+C Q,

, this

3.16.

RELATIVE STABILITY

24 1

This means that + ( t ) is a solution of (3.15.23) through (0, w ) , which remains in r+. Because of the arbitrary nature of A, one concludes that there is a solution of (3.15.23) defined for t 3 0 starting from w at t 0 and remaining in r+.Consequently, r+is a semi-invariant set of (3.15.23), and the proof is complete. :

COROLLARY 3.15.1. If, for a solution x ( t ) of (3.15.15) approaching Q, liml+,mx ( t ) = 0, then H ( x o ) = 0. T h e following theorem gives sufficient conditions for the asymptotic behavior of solutions of (3.15.15) whose proof follows by combining those of Theorems 3.15.6 and 3.15.7.

THEOREM 3.15.8. Let the hypotheses of Theorem 3.15.6 hold. Let f ( t , x) satisfy hypotheses (a) and (b). Then, all the solutions of (3.15.15) approach the largest semi-invariant set, contained in Q, of Eq. (3.15.23). 3.16. Relative stability

T h e concept of relative stability is concerned with the following two differential systems: (3.16.1)

where f , ,f2 E C [ J x R”, R”]. Let x ( t ) = x ( t , to , xo), y ( t ) = y ( t , t g ,y o ) be any two solutions of (3.16.1).

DEFINITION 3.16.1.

T h e two differential systems (3.16.1) are said to be E > 0 and to E J , there exists a 6 = S ( t , , 6) which is continuous in to for each E such that the inequality

(R1)relatively equi-stable if, for each 11x0-YOII

implies

I1 ~

( f -)~ ( t ) l l

< E,

<s t

2 tn.

Analogous to the definitions (S2)-(Sl,,),we can formulate (R2)-(Rl,,), following (R,).Iffi(t, y ) =f,(t, y ) , then the notion (R,) will be designated as extreme equistability of the product system (3.16.1). On the other hand, suppose that f 2 ( t ,y ) = 0 and y E M , where M is a nonempty subset in Rn. Let d(x, M ) denote the distance between a point x and the set M , defined by d(x, M ) = inf[//x - y 11, y E MI.

242

CHAPTER

<

3

Since d(x, M ) I[ x y 11, for all y E M , we can infer, as a special case, the stability with respect to a set M . If, furthermore, M = (0) and fi(t, 0) 0, the definitions (Rl)-(Rlo)coincide with (Sl)-(Slo). Thus, the study of relative stability is important in itself.

THEOREM 3.16.1.

~

Suppose that the following conditions hold:

(i) V E C [J x R1&x R",R,],V ( t ,x, x) = 0, and V(t,x,y ) is locally Lipschitzian in A and y . (ii) g

E

C [ J x R, , R],g(t,0)

D t l / ( f , .y, Y ) ( 3 . l I j . 1 )

7

0, and

:< g(4 q t , x, y ) ) ,

( t , x, y ) E

J

x'

x'

R".

Thcn (1) if the trivial solution of (3.2.3) is equistable and b ( ~ ~ x - y<~1'(t,x,y), ~ ) b E X ,

(3.16.2)

the two systems (3.16.1) are relatively equistablc; (2) if the trivial solution of (3.2.3) is uniform stable and b()l .Y -y

11)

< V(t,x , y ) < u(l1 x

-

y li),

a , b t .X,

(3.16.3)

the two systems are relatively uniform stable; (3) if (3.16.2.) holds, the equi-asymptotic stability of the trivial solution of (3.2.3) implies the relative equi-asymptotic stability of the two systems (3.16. I); (4) if (3.16.3) holds, the uniform asymptotic stability of the trivial solution of (3.2.3) implies the relative uniform asymptotic stability of the two systems (3.16.1).

Proof. (1) 1,et t > 0, to E J be given. By the equistability of the trivial solution of (3.2.3), we have, given 6(t) > 0, to E J , that there exists a S = S ( t , , t) > 0 such that u,) S implies

<

u ( t , f , , U")

< h(c),

t

2 t" -

(3.16.4)

I t follows from the continuity of the function V ( t ,x, y ) and the fact that V ( t ,x, x) :0 that it is possible to find 6, = S,(to, t) satisfying

Il xo

-

yo ll

.< 81 ,

v(t0 xn ,yo) 7

at the same time. Defining

44

=

q t , 4 t ) ,y ( t ) ) ,

<8

3.16.

243

RELATIVE STABILITY

and using assumptions (i) and (ii), we get the inequality

We choose u,

=

V(to, xo ,yo) and apply Theorem 1.4.1 to obtain

where ~ ( t o, , uo) is the maximal solution of (3.2.3). From (3.16.2), (3.16.4), and (3.16.5), it follows that

b(ll44

- Y(t)ll)

< b(+

t

2 to

%

implies (1 x(t) - y(t)ll < E , t >, t o , proving (1). (2) We choose uo = a(l1 xo - y o I[), and then it is enough to select 6, : ~ ~ ( 6 T)h.e proof proceeds as before. It is clear that 6, is independent of t o , since 6 does not depend on to . ) 0, t, E J , there ( 3 ) It follows from the hypotheses that, given b ( ~ > exist positive numbers 6, : ao(t0)and T = T(to, E ) such that uo 6, implies that Hence,

11 xo - yo (1 < 6,

<

u(t, t o 7 U O )

W e choose

< b ( ~ ) , t >, to

+ T.

so = so(to)such that

hold simultaneously and conclude, as before, that

Consequently, it follows that

Thus, whenever

I/ xo - y o jl

< so,we have

I1 4 t ) - ~ ( t ) l l <

€2

t

2 to

+ T,

proving (R3).Since, by (I), (R,) holds, equi-asymptotic stability results. ~ ) . (4) Choose uo = a(ll xo - yo I[), and then select = ~ ~ ( 6Clearly, 8, and T are independent of t o . We proceed as in ( 3 ) to establish ( R 6 ) . T h e theorem is proved.

so

244

CHAPTER

3

3.17. Stability with respect to a manifold

n, and let the set of points x satisfying the 1,et w E C[R”,K k ] , k ) 0 define an ( n - k) dimensional manifold. We define relation ~ ( x :

) ll?T(rtplt)(e)the sets [x E R” : 11 w(x)lI and denote by A i l ( 3 g - f t ) ( ~and and [x E R)&: jl w(.~)lI €1, respectively.

<

< €3

I~EFISITION 3.17. I . A set A C K“ is said to be (positively) self-invariant if x,, E A implies x ( t , t,, , xo)C A , t 2 t o , where x(t, t, , x,,) is any solution of (3.2.1). We shall assume that M ( n - l ~is) a sclf-invariant set with respect to the system (3.2.1). DEFINITION 3.17.2. T h e self-invariant manifold M(npk)is said to be (MI) epistable if, for each E > 0 and t, E J , there exists a positive function 6 8(t,, , 6 ) that is continuous in t, for each E such that 7

x, t Mcn-7m

implies x ( t , t” , xo)

c

~(T,-7c)(~),

t

2 t,, .

.lnalogously, the definitions (M2)-(M8)may be understood, parallel to the definitions (S2)-(S8).Obviously, if k = n and ~ ( x = ) x, these definitions (MI)-(M8) coincide with ( s l ) - ( s S ) . T h e following theorem gives sufficient conditions for the stability and asymptotic stability of the invariant manifold M(lL--li) , the proof of which is left to the reader as an easy exercise. l’ir~oru:ni 3.17.1. Assume that there exist functions V ( t ,x) and g(t, u) satisfying the following conditions:

(i) I’E C [ J x M(,tp/&), R+], V ( t ,x) = 0 if x E M(.n-k), and V ( t ,x) is locally Lipschitzian in x. (ii) ,q E C [ J :.: R , , K],and g(t, 0) = 0. (iii) f E C [ J s ~ ! ( ) ~ ~ / ~R”], ) ( p )where , M(n-ri) is a self-invariant manifold with respect to the system (3.2.1), and

n ’ I ’( t , x) s;g( t , l’(

t , ,Y)),

( t , x) E J x M(,-,)(p).

3.18.

245

ALMOST PERIODIC SYSTEMS

Then, the self-invariant manifold is (1)

equistable if the trivial solution of (3.2.3) is equistable and b(ll w(x)ll)

< v(t,x);

b E s,

( t , x) E J x

M(e-k)(p);

(3.17-1)

(2) uniformly stable if the trivial solution of (3.2.3) is uniformly stable and

aw(.)ll)

< V(’, ). < 4 w(x)II),

( t , x) E

Jx

Mb-/dP)> (3.17.2)

where, a, b E :X; (3) equi-asymptotically stable if the trivial solution of (3.2.3) is equi-asymptotically stable and (3.17.1) holds; (4) uniformly asymptotically stable if the trivial solution of (3.2.3) is uniformly asymptotically stable and (3.17.2) is satisfied.

3.18. Almost periodic systems

We shall consider, in this section, uniqueness of solutions, existence of almost periodic solutions, and stability results allowing the initial time to the perfect freedom of taking any value in the interval (~ a,co). However, the Lyapunov function that will be used is defined only for t 2 0. I n other words, we obtain results for any t, E (- a,a), although the conditions imposed in terms of Lyapunov function are only for t 0. This definite advantage inherent in periodic or almost periodic systems is exhibited in what follows. Let us begin by a uniqueness result.

>

THEOREM 3.18.1.

Assume that

(i) f~ C[(-a, a)x S , , R”], and f ( t , x) is almost periodic in t uniformly with respect to x E S , S being any compact set in S,, ; (ii) V E C [ J x S, x S, , R,], V ( t ,x, x) 5 0, V ( t ,x,y ) is Lipschitzian in x and y for a constant M = M(p) > 0, and

< Jqt,X,Y),

b(llX-Yll)

(iii) g E C [ J x R, , R ] ,g(t, 0)

+

= 0, and, for ( t , x,y ) E J x S, x S, ,

1 D + v ( ~x,, y ) = lim sup - [ ~ [ t h, x h-Of

h

< g(t, q t , x,y));

b E z ;

+ h f ( t ,x),y + h f ( t , ~ ) l

-

v(t,~ , Y ) I

246

CHAPTER

(iv)

3

the maximal solution of (3.2.3) through the point ,0), T~ >, 0, is identically zero.

( T ~

Then, the almost periodic system

x' = f ( t , x),

X(t") =

xo ,

to E (-03, co)

(3.1 8.1)

has at most one solution to the right of t o .

Proof. Suppose that, for some (to , xo), to E (- co, co), xo E S, , there exist two solutions x ( t ) = x ( t , t o , xo), y ( t ) = y ( t , t o ,xo). Then, at a certain t , > t, , we have

/I 4 t l ) -Y(tdll where, we may assume, B -: p such that

E

=

< t , , there exists a constant

< p. For to < t

II 4t)ll

< B,

€7

IIY(t)ll

< B.

By Lemma 1.3.1, given $I(€) and a compact set a(€) > 0 such that p(t,

T o , 0 , 6)

< Me),

t

E [To,

, TI, there exists

[ T ~

a6 =

TI,

(3.18.2)

where r ( t , T~ , 0, 8) is the maximal solution of u' = g(t, u )

+ 6,

U(T0) = 0.

(3.18.3)

Let 8 be a 6(~)/2M-translationnumber for f ( t , x) such that to + 6 0, that is, Ilf(t 87-4 - A t , .)I1 < 6(€)/2M, (3.1 8.4)

+

provided x E a compact set S C S,. Consider the function m(t) = V ( t 8, x ( t ) , y ( t ) ) for t E [ t o , t,]. For small h > 0, we have, using Lipschitz condition on V ( t ,x,y ) ,

+

m(t -1- 12)

< Mh [lIJ(t'x(t)) - f ( t

+ Y [ t + 8 + h,

X(t)

+ 0, x(t))ll

+ hf(t + 0, x ( t ) ) , Y ( t ) + hf(t + 0, y(t))],

where t,(h)/h,~ , ( h ) / h+ 0 as h -+ 0. It then follows, because of assumption (iii) and relation (3.18.4), that D'm(t)

< g(t + 0, m ( t ) ) + 6,

t

E

[t" , tll.

3.18.

Defining ro = to inequality

+

ALMOST PERIODIC SYSTEMS

247

+ 8, we get, on the strength of

Theorem 1.4.1, the

4 t )

< r(t + 8,

70

0, a),

9

t

[to , tll,

E

where r(t 8, T ~ 0,, 6) is the maximal solution of (3.18.3). At t we obtain, using relations (3.18.2) and (3.18.5), the estimate 4tl)

d

r(t1

+ 8,

70

9

0,q

(3.18.5) =

t,

,

< ib(€),

which is a contradiction to the fact that m(t1)

3b(E).

Hence, it follows that the system (3.18.1) has at most one solution to the right of to .

COROLLARY 3.18.1. T h e function g(t, u) = 0 is admissible in Theorem 3.18.1. As observed earlier, in the stability results that follow, we allow to E (- GO, GO). Then, the corresponding notions will be designated as perfect stability concepts to distinguish them from the previous stability definitions. We need the following notions with respect to the scalar differential equation (3.2.3).

DEFINITION 3.18.1. The trivial solution of (3.2.3) is said to be strongly equistable if, given any E > 0, r,,E J , and any compact interval K = [T,,,tl], there exist an 7 = V ( E ) > 0 and a positive function 6 = 6(r0 , E ) that is continuous in T,,for each E such that, if u,, 6,

<

u(t, 7 0 , U o 9 17)

< E,

t

E [To

, tll,

where u(t, ro , u,, , 7)is any solution of U' = g ( t , U)

+ 7,

U(70) = Ug

3 0.

(3.18.6)

If, in addition, 6 is independent of r,, , it is strongly uniformly stable.

DEFINITION 3.18.2. The trivial solution of (3.2.3) is said to be strongly equi-asymptotically stable if it is strongly equistable and if, for any E > 0, T,,E J , there exist positive numbers 6, = So(r0), 7 = T ( E ) , and T = T ( r 0 ,E ) such that provided

4 4 70

9

Uo

9

17)

< Uo

€9

< 60,

t 2

70

+ T,

248

CHAPTER

3

where u(t, T, , u,, q) is any solution of (3.18.6). If the numbers 6, 6, , and T are independent of T, , the trivial solution is said to be strongly tinvormly asymptotically stable.

THEOREM 3.18.2. L

(i)

Assume that

V E C [ J x S o ,R,], V ( t ,x) is Lipschitzian in x for a constant > 0, and V ( t ,x) is positive definite;

= L(p)

(ii) g

E

C [ J x R , , R],g(t, 0) = 0, and D+V(t,x)

< g(t, V(t,x)),

( t , x) E J

x so;

(iii) f E C[(-co, co) x S, , R"], f ( t , 0 ) = 0, and f ( t , x) is almost periodic in t uniformly with respect to x E S, S being any compact set in S , . Then, the strong equistability of the trivial solution of (3.2.3) implies that the null solution of (3.18.1) is perfectly equistable.

Proof. Let 0 <' E < p and t, E (-a, co) be given. Since V ( t ,x) is positivc definite, there exists a function b E X' such that b(11 .x!

11)

< q t , x),

( t ,).

E

J x

s,.

(3.18.7)

Assume that the trivial solution of (3.2.3) is strongly equistable. Then, given h ( ~ > ) 0, T, E J , and any compact interval K = [T, , t,], there exist an 7 = V ( E ) > 0 and a 6 = S(7, , c) > 0 such that 4 t >70 7 un ?I) < b ( c ) ,

t

E [To

9

(3.18.8)

ti],

<

providcd zi,, 6, where u(t, T, , u, , q) is any solution of (3.18.6). Choose LS, = S and u,, = L /I xo 11, L being the Lipschitz constant for V(t,x). This choice implies that u(, S and 11 x,,11 6, are satisfied at the same time. Suppose now that there exists a solution x ( t ) = x(t, t, , x,), with 11 x, j j < 6, and t, E (- co, co)such that, for some t, > t, , we have

<

<

I/ x(t)ll

<

E

< P,

to

< t < t,

ll x(tz)ll

9

=

€ 0

Let 8 be an V/L-translation number for f ( t , x) such that to that is, llf(t

+ 0, ).

-f(t,

)1.I


t

E

(-a), a),

(3.18.9)

+ 8 > 0, (3.18.10)

if x E S, any compact set in S o . We consider the function m(t) = V ( t 8, x(t)),t E [to, t2].If h > 0 is small, we obtain, using assumption

+

3.18.

249

ALMOST PERIODIC SYSTEMS

(ii), the Lipschitzian character of V(t,x) and the relations (3.18.9) and (3.18. lo),

Define T,, = to yields that

+ 8 and t , = t , + 8. An application of Theorem 1.4.1 m(t>

+

<~

+

( t 6,

70

210

t

7)s

E

[to t z l , 1

where r ( t 8, T,, , uo , 7)is the maximal solution of (3.18.6). At t there results an absurdity b(c)

< V(tz + 8,x(tz)) < Y ( t , + 8,

70,

*o

7

=

t, ,

7) < W E ) ,

because of relations (3.18.7), (3.18.8), and (3.18.9). This proves the perfect equistability of the trivial solution of (3.18.1). COROLLARY 3.18.2. Under the assumptions of Theorem 3.18.2, the strong uniform stability of the trivial solution of (3.2.3) assures the perfect uniform stability of the solution x = 0 of (3.18.1). In particular, the function g(t, u ) = 0 is admissible.

THEOREM 3.18.3. Suppose the trivial solution of (3.2.3) is strongly equi-asymptotically stable and that assumptions (i), (ii), and (iii) of Theorem 3.18.2 hold. Then, the perfect equi-asymptotic stability of the trivial solution of (3.18.1) follows. Proof. Since, by Theorem 3.18.2, the trivial solution of (3.18.1) is perfectly equistable, it remains to be proved that it is perfectly quasiequi-asymptotically stable. For this purpose, let 0 < E < p and to E ( - G O , a).Then, given b ( ~ > ) 0 and T,, E J , there exist positive numbers 6, = 6,,(~,,),7 = V ( E ) , and T = T(T,,,C ) such that u(t, 70 7 un, 7)

<

< WE),

t

3 70

+ T,

(3.18.11)

whenever u,, 6. Choose u,, = L 11 xo 11 and I,&, = 6,. Let 6: = min[8,, , where = S(T,,, p). Thus, if 11 x,,11 82, it follows that 11 x(t)ll B < p for some B. As before, let 8 be an y/L-translation number so that (3.18.10) is satisfied. Then, by defining m(t) = V ( t 8, x ( t ) ) , where x ( t ) is any solution of (3.18.1) such that (1 x,, 11 88, we obtain

so], <

<

so

+

<

< s(t + 8, m ( t ) ) + 7,

D-+m(t)

250

CHAPTER

3

which implies, by Theorem 1.4.1, the inequality m ( t ) ;2 r ( t

choosing

T~ =

t 6, T o , ~

t 2 to,

0 T), ,

to + 8. It then follows that

b(11 x(t)ll)

< l'7(t + 8, x ( t ) ) 6 r ( t + 6,
t

2 t,

To

+ T,

9

Uo

17)

in view of (3.18.11). We thus have

ll x(f)ll < 6, provided /I xo 11

< S$.

t

3 t,

+ T,

T h e proof is therefore complete.

COROLLARY 3.18.3. Under the assumptions of Theorem 3.18.2, the strong uniform asymptotic stability of the trivial solution of (3.2.3) implies the perfect uniform asymptotic stability of the null solution of (3.18.1). I n particular, the functiong(t, u) - w , N > 0, is admissible. Suppose that the functionf(t, x) is not almost periodic in t and that f E C [ J x S , , K"].Then, from the strong stability notions of the scalar differential equation (3.2.3), we may infer the strong stability of the system (3.18.1), which we now define. 7

DEFINITION 3.18.3. T h e trivial solution of (3.18.1) is said to be strongly equistahle if, for any E > 0, t, E J , and any compact interval [ t o ,t J , there exist an 7 = ~ ( e > ) 0 and a positive function S = S(to , E) that is continuous in t,)for each E such that, if 11 xo I/ 6,

<

where x ( t , to , x,), 7 ) is an 7-approximate solution of (3.18.1) on [ t o ,tl]. If, in addition, S is independent of to , the trivial solution is said to be strongly uniform stable. T h e notion of strong asymptotic stability may be defined similar to Definition 3.18.2. We may now state the following

THEOREM 3.18.4. Suppose that assumptions (i) and (ii) of Theorem 3.18.2 hold. Let f E C [ J x S o ,P ] and f ( t , 0 ) = 0. Then, one of the strong stability definitions of the trivial solution of (3.2.3) yields the corresponding one of the strong stability notions of the trivial solution of (3.18.1).

3.18.

251

ALMOST PERIODIC SYSTEMS

Proof. We shall indicate the proof corresponding to the statement that the strong equistability of u = 0 of (3.2.3) implies the strong equistability of the trivial solution of (3.18.1). We follow the proof of Theorem 3.18.2 and choose 6,, vl such that LS, = 6 and Lv, = 7. We can, then, claim that, with the numbers S , , q1 so chosen, the trivial solution of (3.18.1) is strongly equistable. Supposing the contrary and proceeding as in Theorem 3.18.2, we obtain, in the present case, the inequality D+m(t)

< g(t, m ( t ) ) + 7,

where m ( t ) = V(t,x ( t , t o , xo , 7)). With these changes, we can mimic the rest of the arguments to prove the stated result. Regarding the existence of almost periodic solutions for the system (3.18. I), we have the following

THEOREM 3.18.5. Assume that (i) f E C[(-co, co) x S , , R"], and f ( t , x) is almost periodic in t uniformly with respect to x E S, S being any compact set in S, ;

(ii) V E C [J x S, x S, , R,], V ( t ,x,y ) is Lipschitzian in x and y for a constant L = L ( p ) > 0, and, for ( t , x, y ) E J x Sox S, ,

< v t 9 x,Y ) < 4x D+V(t, x,y ) < g(t, V ( t ,x,y ) ) , b(ll x

-

Y

II)

~

Y

ll),

a, ZJ

E

-x;

where t E J , x, y E S, , (iii) g E C [ J x R, , R l ; (iv) there exists a solution x ( t ) = x(t, t o , xo) of (3.18.1) such that

I/ x(t)II < B,

t

3to,

t o E (-a, a), B


) 0, Q: > 0, and T~ E J , there exist positive numbers (v) given b ( ~ > T ( E ) , T = T ( Ea , ) such that, if uo N and T T~ T,

7 =

<

47,7 0 , U O , 17) <

+

ZJ(E)

(3.18.12)

where U(T,7 0 , uo , 7) is any solution of (3.18.6). Then, the almost periodic system (3.18.1) admits a bounded almost periodic solution, with a bounded B. Proof. We can prove this theorem following the proof of Theorem 2.15.3. Hence, we shall indicate it briefly. Let x ( t ) = x ( t , to , xo), to E (- co, co) be the bounded solution such that 11 x(t)ll B , t 3 to . Let { T ~ }be any sequence of real numbers such that T~ -+ 03 as k + co and

<

252

CHAPTER

3

+

x) - f ( t , x) -+ 0 as k -+ co, uniformly for t E (- 00, co) and x E S , S being any compact set in S, . Let p be any number, and let U

f(t

be any compact subset of [p, a).Let 0 < E < p be given. Choose (Y = a(2B). Then, let q and T be the numbers given in assumption (v), for this choice of N . Let k, = KO@) be the smallest value of k such that ,8 -1- T , ~ .3 ~ to ?'. Choose an integer 7, = T,(E, p) >, k, so large that, for k, 3 k, 3 ? l o ,

+

+

llf(t

,). - f ( t

7tL

+

Tk2,

x)ll ,< d3L,

(3.18.13)

for all t E (- co,co),x E S. Let 6 be an q/3L-translation number for f ( t , x) such that t,, B 3 0, that is,

+

Ilf(t

for t

E

+ 0, x) - A t ,

x)ll

< r1/3L,

(3.1 8.14)

(- co, GO) and x E S. Consider the function

where t, D+m(t)

=

t

+

m(t)

-=

v(t + 0, ~ ( t )%(ti),),

t

3 to ,

. Then,

T,,., - T , ~ ~

-< lim sypp h1 [ V ( t + B + h, x(t + h), x(t, + h ) ) h-0

-

--

v(t -1Q -4- 12, x ( t ) + h f ( t + 0, m, +,)

+ ~ (+ tB , x ( ~ ) ) ) I

using the Lipschitzian character of V ( t ,x,y ) and assumption (iii). Since t + T ~ ~2 . , t o , €or t E U , we obtain, by virtue of the relations (3.18.13) and (3.18.14), the inequality

which, by Theorem I .4.1, yields

3.18.

253

ALMOST PERIODIC SYSTEMS

provided m(t,,) = uo , where ~ ( rT,,, , u,,, q) is the maximal solution of (3.18.6). But, for all t E CJ, t r k , >, to T . Hence, identifying T = t r k , 8, T,, = to 0, we get

+

+

+

+

m(t

+

+

< WE),

Tk,)

t

E

u,

according to relation (3.18.12). Consequently, for all t

II x ( t

+

TkJ

-4 t

+

TkJIl

<

k,

€7

E

2 kl >, ?fo

U , we have *

This proves the existence of a function w ( t ) defined on [p, co) and bounded by B. Since /3 is arbitrary, w ( t ) is defined for t E (- co, a), and we have as k, + CO, x(t r k l )- ~ ( t+ )0

+

uniformly on all compact subsets of (- co, GO). Using the same arguments as in Theorem 2.15.3, it is easy to show that w ( t ) is differentiable and satisfies (3.18.1). T o show that w(t) is almost periodic, it is sufficient to show that, for any sequence { T k } for which {f(t r k ,x)} converges uniformly for t E (~ co, GO), x E S , the sequence {w(t r k ) }converges uniformly for t E (- 00, co), where rk tends to a finite number or infinity. We may assume that rk approaches either -00 or 00. Assume now that rlC+ co as k + co. For any E > 0, there exists an no = no(<) > 0 such that, if R, >, k, >, n o , (3.18.13) holds. We choose R , 3 n,, so large that 7,., >, T. For each t E (-a, a),let 6' be an q/31-translation number such that t 6' >, 0, that is, (3.18.14) holds. For t s t r k 1 ,we consider the function

+

+

< < +

m(s) = V(s

where s,

=

s

+ rlC2

-

D+m(s)

r k , .Then

CT

= s

-

+ 6, w(s),

+ l f(. +

Tk2 9

W(SI)),

we obtain, as before,

< g(s + 6, 4 s ) ) + L[llf(s,

+ Ilf(Q + where

+

w(s1))

7k19 7 4 4 )

+ 4 4s))ll

w(4) - f ( s

+ w(s1))ll + + 8, w(s1))111,

-f(.

7k1

-f(u

9

Tfr,

r k 1 .This implies, using (3.18.13) and (3.18.14), that

Dtm(s)

< g(s + 8, m(s)) + 7,

t

<s
By Theorem 1.4.1, it follows that m(t) < r(t

+ 8, t + 0, % , 71,

t 3 t,

Tkl

.

254

CHAPTER

if m(t) = uo . As previously, choosing (3.18.12),

@ -t 8,t If we set, therefore, 4 Wl(t

+ 8,uo

=

t

-1

+

Tkl)

7

7)

3 =

01

a(2B), we get, by relation

6

< qc),

> t + T.

, there results

T ~ ,

< b(€),

t

E

(-a, 0).

Accordingly, it follows that

+

) ) convergent for all t E (- co, a).Thus, the sequence ( ~ ( t T ~ ~ is uniformly on ( - co, a). In the case when T / ; + - co, we can prove in the same manner that ( ~ (+t ~,<.)jis also convergent uniformly on (-GO, a).T h e proof is therefore complete.

COROLLARY 3.18.4. Let hypotheses (i), (ii), (iii), and (iv) of Theorem 3.18.5 hold, and let the trivial solution of (3.2.3) be strongly uniformly asymptotically stable. Then, the system (3.18.1) admits an almost periodic solution that is uniformly asymptotically stable. I n particular, the function g(t, u ) = -nu, lx > 0, is admissible. 3.19. Uniqueness and estimates It is naturally possible to give very general conditions for the uniqueness and the growth of solutions by employing Lyapunov functions. Let us begin with a uniqueness result of Perron type.

THEOREM 3.19.1. Assume that

< < +

t to a, u 0, (i) the function ~ ( tu ), is continuous for to and, for every t , , t,, -:. t, to 4-a, u ( t ) -= 0 is the only differentiable function on t,, t -: t , , which satisfies

<

u'

for to ' t

-

R(t, u),

(3.19.1)

u(4,) = 0 ,

,f , ;

(ii) f E C[R,,, PI], where R , : t"

:t

t"

+ a,

/I x

- X"

I/ < : 6;

3.19.

255

UNIQUENESS AND ESTIMATES

(iii) V E C[R, , R,], V ( t ,0) = 0, V ( t ,x) is positive definite, continuously differentiable on R, , and, for ( t , x), ( t ,y ) E R, , V’(t,x

-

< g(t, J q t ,x

-

(3.19.2)

Y)).

Then, the differential system

x’ = f ( t , x), admits a unique solution on t,

(3.19.3)

x(t,) = x,

< t < t, + a.

Proof. Let us suppose that there are two solutions x(t), y ( t ) of the system (3.19.3) on t, t t, a. Consider the function

< < +

m(t) = V ( t ,x(t) - y ( t ) ) .

We have, in view of (3.19.2) and the continuous differentiability of V ( t ,x), the inequality m’(4

< g(t, 4 t ) ) -

Also, m(to) = 0. For any t, such that t, Theorem 1.4.1, the estimate m(t)

< r(t),

to

< t, < t, + a, we

et

obtain, by

< tl ,

where r ( t ) is the maximal solution of (3.19.1). Assumption (i), together with the positive definiteness of V , assures that x(t) = y ( t ) ,t, t < t, . T h e proof is therefore complete.

<

We can state a uniqueness result analogous to Kamke’s theorem in terms of Lyapunov functions as follows.

THEOREM 3.19.2. Assume that

< +

t, a, u 2 0, (i) t h e function g(t, u) is continuous for t, < t and, for every t , , t, < t, < t, a, u ( t ) r= 0 is the only function differentiable on t, < t < t, and continuous on t, t < t, , for which

+

<

and u(t,) = u&)

= 0;

256

3

CHAPTER

(ii) hypotheses (ii) and (iii) of Theorem 3.19.1 hold except that the condition (3.19.2) is satisfied only for ( t , x), ( t ,y ) E R, and t fl t, . Then. the conclusion of Theorem 3.19.1 remains true.

Proof.

Define the function g,(t, u )

=

sup

V(t,Z-~)--U

V ( t ,x - y )

< <

t t, + a and u 3 0. Since f ( t , x), aV/at, aV/ax are all for t, continuous on R, , it follows that g,(t, u ) is continuous on t, t t,+a and ZL 0. Following the proof of Theorem 2.2.2, it is now easy to establish this theorem. We leave the details.

< <

Much the samc way, we can state and prove a uniqueness theorem analogous to Theorem 2.2.4. T h e next two theorems are concerned with estimating the difference between a solution and an approximate solution.

THEOREM 3.19.3.

Suppose that

(i) g E C [ J X R t , R l , u, 8 E C [ J ,R+1, and D-u(t) 3, g(t, u ( t ) )

+ M(t)8(t),

t

> to ;

(ii) V E C [ j x R n , R,], V ( t ,x) satisfies the Lipschitz condition in x for a function M ( t ) , where M E C [J , R,], and

(3.19.4) for t

E

J , x,y

E in, where

st

[x, -? E R'l : V ( t ,x

-

y)

--

~ ( t )f , > t o ] ;

(iii) f E C [ J x RT1, R"],x ( t ) is a 6-approximate solution, and y ( t ) a solution of (3.19.3), defined for t 2 t o . Then, V ( t , , x,,

~

y o ) < u(to)implies V ( t ,~ ( t- )y ( t ) )

cc ~ ( t ) ,

t

to

.

3.20.

CONTINUOUS DEPENDENCE AND THE METHOD OF AVERAGING

257

Proof. Consider the function m(t) = V(t,x(t) - y(t)). If, for a t = t, , x ( t l ) , y(tl) E 8, then we obtain, using assumption (ii), the differential inequality D+m(t,)


+ M(tl)Vl),

9

which, by Lemma I .2.3, implies D-m(t1) G A t 1 7 N t l ) )

+~ ( t , ) W -

T h e desired result now follows on account of Theorem 1.2.2.

THEOREM 3.19.4. Assume that g E C [ J x R, , R] and that r ( t ) is the maximal solution of u' = g ( t ,

4 + M(t)S(t),

U ( t 0 ) = Uo

>, 0,

defined for t 2 t o . Suppose further that assumptions (ii) and (iii) of Theorem 3.19.3 hold except that the condition (3.19.4) is satisfied for t E J , x,y E 8*,where, for a certain > 0, 9* = [x, y

E

Rn : r ( t ) < V(t,x

-

y)

< r ( t ) + c0 , t 3 to].

Then, V(to, xo - y o ) = uo implies V ( t ,x ( t )

- y(t))

< r(t),

t

b

to.

T h e proof of this theorem can be constructed following the arguments of Theorem I .4.2.

3.20. Continuous dependence and the method of averaging Consider the differential system x' =f ( t , x, Y ) ,

(3.20.1)

where f E C [ J x S , x Rwl,Rn]. Assume that, for every y o E Rm, there exists a solution xo(t) of the system x'

defined for t

3 0.

LEMMA 3.20.1.

=f

( t , X, Yo),

(3.20.2)

Suppose that

(i) V E C [ J x Rn, R,], V ( t ,0 ) = 0, and V ( t ,x) is positive definite and satisfies the Lipschitz condition in x for a constant M > 0;

258

CHAPTER

3

(ii) g E C[J x R , , R],g(t, 0) = 0, and, for any step function v(t) on J with values in So and for every t E J , x E So , 1 D+U(t," , y o ) = lim s y - [V(t h-0 h

+W

+ h,F(t, v(t),yo)

x

t , 4 t ) , Yo) -f(4 x,Yo)))- V(t7

< g(t, V ( t ,4 t )

where

-

-

w, 4 t h Yo)

-41

x)),

(3.20.3)

+ Jh, n(s),Y ) ds.

(3.20.4)

F(t>4 t ) ,Y ) = u(0)

Then, given any compact interval [0, To]contained in J and any E > 0, there is a S = S ( E ) > 0 such that, for every step function v(t) in [0, To], with v(0) xo(0) and I/ v(t) - xo(t)lj < S in [0, T,,], there follows T

1' J i0 [ f b ?4 s ) , Yo) - f ( s , for every t

E

xo(s), Yo)]ds

/I <

€9

(3.20.5)

[0, To],xO(t) being any solution of (3.20.2).

Proof. Since V ( t ,x) is positive definite and V ( t ,0) = 0, given any > 0, there exists a p = p(t) > 0 such that

E

V(t,x)


/ / x 11 < E .

implies

Moreover, by the assumptions on g(t, a), there exists a y such that

I At, 4 < P/To

= Y(E)

>0

?

< <

whenever t E [0, To] and 0 ZL y . Let v ( t ) be a step function in [0, To],with values in S, such that v(0) == xo(0) and /I v(t) - x,(t)ll < 6 in [0, To],where S = y / M , M being the Lipschitz constant for V(t,x). It then follows that

I~

( f L'(t, ,

v ( t ) - xo(t)))l

< p/To

9

t

E

[O, To].

(3.20.6)

Hence, considering the function

4 4 = V(t,F ( t , v ( t ) ,Yo) - xo(t)), we obtain, in view of the relations (3.20.3) and (3.20.6) and the Lipschitzian character of V ( t ,x), D + m ( t )< p/1',, ,

t

E

[O, To].

3.20.

CONTINUOUS DEPENDENCE AND THE METHOD OF AVERAGING

259

This implies that, for every t E [0, To],

which proves the assertion (3.20.5) of the lemma. Assume that, for each t E J and x E S, , (3.20.7)

It then follows that, given any compact interval [0, To] and any step function v(t) in [0, To]with values in S, ,

uniformly in [0, To].Hence, if the assumptions of Lemma 3.20.1 hold, there exists, for every E > 0, a constant 6 = S ( E ) > 0 such that, whenever v ( t ) is a step function in [0, To] with v(0) = xo(0) and 11 v(t) - x,(t)ll < 6 in [0, To],there is a neighborhood F = F ( E )C Rm of y o for which y E F implies

I n view of (3.20.4), this means that, for each y

ll F ( 4 v(t),Y ) - xo(t)l/ < E ,

t

E

r,

E

10, To].

This fact will be used prominently in the following

THEOREM 3.20. I .

Assume that

(i) V E C [ J x R’l, R,], V ( t ,0) = 0, and V ( t ,x) is positive definite and satisfies the Lipschitz condition in x for a constant M > 0; (ii) g E C [ J x R+ , R ] ,g(t, 0) = 0, and r ( t ) = 0 is the maximal solution of u’ = g(t, u ) , (3.20.8) passing through (0, 0);

260 t

3

CHAPTER

(iii) for any step function v(t) on J , with values in S, , and for every J , x E S, ,y E RJJ1,

E

' U(t,x, r) < g(tt q t , v ( t ) - 4);

(3.20.9)

I)

(iv) the relation (3.20.7) holds. Then, given any compact interval [0, To]C J and any E > 0, there exists a neighborhood T ( E )of yo such that, for every y E (3.20.1) admits a unique solution x ( t ) with x(0) = xo(0), which is defined in [0, To]and satisfies

r,

II x ( t )

~

<

x,(t)ll

t

€9

E

(3.20.10)

[o, To].

Proof. T h e assumptions on V and g, together with (3.20.9), imply, on the basis of Theorem 3.19.1, that there exists a unique solution x ( t ) of (3.20.1) with x(0) = xo(0), which is defined in some interval J ( Y >= T(y)l c J. From hypothesis (ii) and Lemma 1.3.1, we deduce that, given any compact interval [0, To]C J and any p > 0, there is an q = q(p) > 0 such that the maximal solution r ( t , 0, 0, q) of u' = s(t, u )

exists for t

E

+ $7

(3.20.1 1)

[0, T , ] and satisfies r(4 o,o, 7)

<

t E [O, T"1.

cL9

Let E > 0 and [0, To] be an arbitrary compact interval. Since V ( t ,x) is positive definite on J x R", we can find a p = p ( ~ > ) 0 such that, whenever V ( t ,x) p, we have 11 x 11 < Let q(c) > 0 be the constant referred to previously. Choose a constant OL > ME. By the continuity of g on x R , , there exists a S ( E ) > 0 such that

4.

1

I

%)

- R(t3

%)I

< '27

for t E [0, T o ]u1 , , u2 E [0, n], and I u1 - u2 1 < S(E). For every y E Rm and every step function v in J with values in S, , we have, for every t E J and x t P i ,

I

L'(t,

U ( t ) - 2) -

V(t,r;.(t,.(t),y)

Mlli 4 t ) - %(t)ll

~

)I.

+ II q t , .(t), Y )

-

%(~)lll9

where F ( t , v(t), y ) is defined as in (3.20.4). Hence, as observed earlier, we can select a positive constant P ( c ) < E and a step function v in [O, To]

3.20.

CONTINUOUS DEPENDENCE AND THE METHOD OF AVERAGING

261

with v(0) = xo(0) and 11 v(t) - xo(t)lI < fi in [0, To] such that there is a neighborhood T ( E of ) y o for which y E T ( Eimplies ) and

II F(t, 4 t h Y ) - xo(t)ll < BE,

t

E

[O, To],

(3.20.12)

I q t , v ( t ) - 4 - V(t,F(t,W , Y ) - 4 < 8, for every t E [0, To] and x E Rn. Let us now take some y E T ( E )and , let us consider the unique solution x ( t ) of (3.20.1), with x(0) = xo(0), which exists on some interval J ( y ) = [0, T ( y ) ]contained in J . Defining for t E J ( y ) n [0, To], m(t)

=

V ( t ,F(t, +), Y ) - x(t)).

We deduce from (3.20.9) that DWt)

< g(t, q t , u ( t )

-

x(t))).

Hence, for all those t E J ( y )n [0, To] for which

This implies, by Theorem 1.4. I, that

where r ( t , 0, 0, 7) is the maximal solution of (3.20.1 I ) through (0, 0). Since r(t, 0, 0, 7) < p for every t E [0, To], we infer that m ( t ) < p as long as (3.20.13) holds, and therefore

Thus, using (3.20.12), we obtain, for sufficiently small t E J ( y ) n [O, To],

1144 - xo(t)ll <

€7

(3.20.14)

and so (1 v(t) - x(t)ll < 2~ < a/&!, for these values of t. Consequently, (3.20.14) holds for every t E J ( y ) n [0, To].All that remains to be shown is that [0, To]C J ( y ) . Suppose the contrary, and let J* = [0, T*), with T* < T o , be the

262

CHAPTER

3

maximal interval in which x ( t ) exists. Since x,(t) is bounded in [0, To], (3.20.14) shows that x ( t ) is bounded in /*. It follows that


llf(t, .(t),Y)II

in

I*

for some N = N ( y ) > 0. Hence, x ( t ) has a limit C as t T", and, by continuity,

11

c

-

<:

X"(T*)l/

E

]*

tends to

E

in view of (3.20.14). Since there exists a constant y > 0 such that, for each t t [0, To],the open ball B , in R" of center x,(t) and radius y is contained in S o , we shall have C E S, provided E > 0 was chosen sufficiently small. Therefore, x ( t ) can be continued as a solution of (3.20.1) to the compact interval [0, T * ] ,which contradicts the definition of ]*. This completes the proof. If the assumption (3.20.7) is replaced by the stronger requirement lim f ( 4 x,Y ) = f ( t ,x,Y") T,,

Y

uniformly in J x S , , then we can prove the conclusion (3.20.10) without the use of approximating step functions. This we state in the form of a corollary, observing that it is a generalization of Theorem 2.5.2. COROLLARY 3.20.1. Let assumptions (i) and (ii) of Theorem 3.20.1 hold. Suppose that, for each t E 1,x1 , x2E S , , and y E Rm,

Then, the conclusion of Theorem 3.20.1 is true. \.lie next consider the problem of continuity of solutions with respect to initial values. We first prove the following

LEMMA 3.20.2.

Suppose that

(i) I;E C [ ] x R", R+],and V ( t ,x) satisfies a Lipschitz condition in x locally; (ii) f E C [ ] x R", R"], and G(t,m )

=

max

Y(t,x-r")

m

D+V(t,x

-

x,,),

3.20.

CONTINUOUS DEPENDENCE AND THE METHOD OF AVERAGING

263

where D+V(t,x

(iii)

-

1 h

x,,) = lim sup - [V(t + h, x h-Of

x,,

~

+ hf(t, x)))

-

V(t,x

- xO)];

r * ( t , t o , 0) is the maximal solution of U' =

G(t,u),

~ ( t ,= ) 0

existing for t >, t o . Then, if x ( t ) is any solution of x),

x' = f ( t ,

existing for t

x(to) = xg

(3.20.16)

2 t o , we have V(t,x(t)

- xo)

< Y*(t,

Proof. Define m ( t ) = V ( t ,x ( t )

-

t" , 01,

=

*

xo). Then, it is readily seen that

D+m(t) < D+V(t, x ( t )

<

2 to

t

max,

x,,)

~

Y ( t , Z - - Z o )c,m( t )

D+V(t, x

-

xo)

G(t,m(t)),

which implies, by Theorem 1.4.1, that m(t)

< Y*(t,

t o , 01,

12 to.

This proves the lemma. We now state the following theorem on continuity of solutions to , x,,) with respect to initial values, whose proof may be constructed by combining the arguments of Theorems 2.5.1,3.1.4,3.19. I , and 3.20.1.

x(t,

THEOREM 3.20.2.

Assume that

(i) V E C [ J x R", R,], V ( t ,0) = 0, and V ( t ,x) is positive definite, mildly unbounded, and satisfies a Lipschitz condition in x locally; (ii) g E C [ J x R, , R ] , and u ( t ) = 0 is the unique solution of (3.20.17)

u' = g(t, U )

passing through ( t o ,0); (iii) f E C[J x R", R"], and, for ( t , x), ( t ,y ) E J x R", D+V(t, x

~

Y)

e g(t, V ( t ,x

- Y)).

264

CHAPTER

3

Then, if the solutions u(t, t o , uo) of (3.20.17) through every point 3 to and are continuous with respect to ( t o , uo), the solutions x ( t , t,) , xo) of (3.20.16) exist for t 3 to and are unique and continuous with respect to initial values (to , xo). ( t o , uo) exist for t

3,21. Notes

A result of the type given in Theorem 3.1.1 is due to Conti [l]. Corollary 3.1.2 is new. Theorem 3.1.2 is adopted from Lakshmikantham [6, 101. Theorem 3.1.3 is also new and is useful in certain applications. For the result contained in Theorem 3.1.4, see Brauer [3], Conti [l], Lakshmikantham [6, 101, and Strauss [l]. See also Wintner [l]. Instead of D+V(t,x) given in (3.1.2), it is more general to consider D-V(t, x). T h e proofs do not require any changes (see Corduneanu [ I l l and Yoshizawa [2]). Section 3.2 introduces various definitions (see Antosiewicz [4], Hahn [I, 3, 41, LaSalle and Lefschetz [l], Lakshmikantham [6], Massera [4], and Yoshizawa [ 161). For relationships between various kinds of stability and boundedness, see Antosiewicz [4], Massera [4], and Yoshizawa 1161. T h e results of Sect. 3.3 are adapted from the work of Antosiewicz [4,6], Rrauer [3, 81, Carduneanu [ll], and Lakshmikantham [6, 101. Theorem 3.3.5 is taken from Hahn [I, 3, 41, whereas Theorem 3.3.6 is due to Corduneanu [ I l l . Theorems 3.3.7 and 3.3.8 are adapted from Halanay [2]. Most of the results of Sect. 3.4 are based on the work of Antosiewicz [4,61, Brauer [3, 81, Corduneanu [11], and Lakshmikantham [6, 101. See also Hahn [ I , 3, 41, Persidskii [4], and Yoshizawa [16]. Section 3.5 deals with the results concerning the preservation of stability properties of unperturbed systems under certain classes of perturbations. Theorem 3.5.1 is due to Corduneanu [8], whereas Theorem 3.5.2 is new. See also Corduneanu [ I l l . Theorems 3.6.1 and 3.6.2 are based on the work of Yoshizawa [2, 161. T h e proof of Theorem 3.6.3 is new. Theorems 3.6.4-3.6.8 are due to 1,akshmikantham and Leela [2]. See also Corduneanu [13]. Theorem 3.6.9 is due to Massera [4]. T h e proof in the text is taken from Halanay [2]. T h e condition J y L ( s )ds Ku, u 3 0, is not more general than L(t) K. Theorem 3.6.10 is due to Corduneanu [ I l l , which is more useful than Theorem 3.6.9, while considering the stability of perturbed systems. See also Yoshizawa [16]. Theorems 3.7.1 and 3.7.2 arc taken from Halanay [2]. T h e short proofs given in the text are new. Theorem 3.7.3 is due to Strauss and

<

<

3.21.

NOTES

265

Yorke [l], whereas Theorem 3.7.4 is new. Theorem 3.7.5 is based on the work of Hale [2]. For Theorems 3.7.6 and 3.7.7, see Halanay [2]. Theorem 3.7.8 is due to Corduneanu [15]. See also Halanay [2], Krasovskii “141, and Malkin [8]. For transformation of time in the problem of stability by the first approximation, see Bylov [ 11. T h e results on total stability given in Sec 3.8 are adapted from Halanay [2]. T h e notion of integral stability is introduced by Vrkoc [2]. For an equivalent notion, see Hayashy [I]. T h e results of Sect. 3.9 are based on Halanay [2]. Section 3.10 consists of results adapted from the work of Strauss [3]; for a generalization, see Hahn [3]. T h e results of Sect. 3.11 are due to Corduneanu [16]. See also Halanay [2]. Section 3.12 contains the work of Lakshmikantham [l I]. Results of Sect. 3.1 3 are based on the work of Antosiewicz [4], Lakshmikantham [6, 101 and Yoshizawa [2, 161. Theorem 3.13.1 I is new. T h e concept of eventual stability is due to LaSalle and Rath [I]. For a different version of this concept, see Lakshmikantham and Leela [I]. T h e results of Sect. 3.14 are based on the work of Wexler [l] and Yoshizawa [12, 151. Theorems 3.15.1 and 3.15.2 are new, whereas Theorems 3.15.3 and 3.15.4 are adopted from Brauer [lo, 121. T h e rest of the results of Sect. 3.15 are due to Yoshizawa [lo, 161. Section 3.16 consists of the results due to Lakshmikantham [6], whereas the contents of Sect. 3.17 are due to Bhatia and Lakshmikantham [I]. See also Ling [l]. Section 3.18 contains results due to Lakshmikantham and Leela [3]. See also Deysach and Sell [I], Hale [2], Miller [l-31, Sell [4], Seifert [3, 5 , 61, and Yoshizawa [16]. Uniqueness theorems 3.19.1 and 3.19.2 are based on the work of Brauer and Sternberg [I] and Olech [4]. Theorems 3.19.3 and 3.19.4 are new. Lemma 3.20.1 and Theorem 3.20.1 are due to Antosiewicz [7], whereas Lemma 3.20.2 and Theorem 3.20.2 are new.

This page is intentionally left blank

Chapter 4

4.0. Introduction As we have seen, using a single Lyapunov function, it was possible to study a variety of problems in a unified way. I t is natural to ask whether it might be more advantageous, in some situations, to use several Lyapunov functions. T h e answer is positive, and this approach leads to a more flexible mechanism. Moreover, each function can satisfy less rigid requirements. I n this chapter, we attempt to obtain criteria for stability, instability, boundedness of solutions, and existence of stationary points, in terms of several Lyapunov functions.

4.1. Main comparison theorem Let us consider the differential system %(to)= xo,

x’ = f ( t , x),

(4.1.1)

to 2 0.

Let V E C [J x S , , R+N].We define the vector function D+V(t,x)

= lim

1 sup - [V(t

h-O+

h

+ h, x + /zf(t, x))

-

V(t,x)]

(4.1.2)

for ( t , x) E J x S o . T h e following theorem is an extension to systems of the corresponding theorem 3.1.1 and plays an important role whenever we use vector Lyapunov functions. Let V E C [ J x S o ,R+Y and V ( t ,2) be locally Lipschitzian in x. Assume that the vector function D+V(t,x) defined by (4.1.2) satisfies the inequality

THEOREM 4.1.1.

D+Jqt,x)

< g(t, V ( t ,x)), 267

( t , x) E

1 x so,

(4.1.3)

268

CHAPTER

4

where g E C [J x R, N, R N ] ,and the vector function g ( t , u) is quasimonotone nondecreasing in u , for each fixed t E J . Let r ( t , t, , u,,) be the maximal solution of the differential system U' = ~

( tu), ,

2 0,

u(tn) = U"

to

3 0,

(4.1.4)

existing to the right of t, . If x ( t ) = x ( t , 2, , x,,) is any solution of (4.1.1) such that (4.1.5) Y t " -2"") < U " , 1

then, as far as x ( t ) exists to the right of t o ,we have

v(t,~ ( tto,

7

xt.0))

< r(t, t o ,

(4.1.6)

~ 0 ) -

Proof. Let x ( t , to ,x,,)be any solution of (4.1.1) such that V(to,x,) Define the vector function m ( t ) by

< u,,.

m(t) == V(t,x(t, t o ,X")).

Then, using the hypothesis that V ( t ,x) satisfies Lipschitz's condition in x,we obtain, for small positive h, the inequality m(i

-th ) - m(t) 5; KII

x(t

+ h)

+ v(t + h,

-x(t) -

x(t)

hf(t, x(t))ll

+ hf(t, x(t)))

-

q t , x(t)),

where K is the local Lipschitz constant. This, together with (4.1.1) and (4.1.3), implies the inequality D+m(t)

Moreover, m(tJ

< At, m(t>).

< un . Hence, by Corollary 1.7.1, we have m(t)

< r ( t , to

7

uo)

as far as x ( t ) exists to the right of to , proving the desired relation (4.1.6). We can now state a global existence theorem analogous to Theorem 3. I .4.

4.1.2. Assume that b' E C [ J x Rn, R + N ] , V ( t ,x) is locally Lipschitzian in x, and Cr=l Vi(t, x) is mildly unbounded. Suppose that g E C [ J x R+N,R N ] ,g(t, u ) is quasi-monotone nondecreasing in u for each fixed t E J , and r ( t , t, , uo) is the maximal solution of (4.1.4) existing for t >, to . I f f € C [ J x R7L,Rn] and 'rlEOREhl

D+V(t,x)

< g(t, I'(t, x)),

( t , x) E J x R",

4.2.

ASYMPTOTIC STABILITY

269

then every solution x ( t ) == x ( t , to , x), of (4.1.1) exists in the future, and (4.1.5) implies (4.1.6) for all t 3 to . By repeating the arguments used in the proof of Theorem 3.1.4, with appropriate changes, this theorem can be established. On the basis of Corollary 1.7.1 and the remark that follows, we can prove the following:

THEOREM 4.1.3. Let V E C [ J x S, , R+N] and V ( t ,x) be locally Lipschitizan in x.Suppose that g, ,g, E C [J x R+N,RN], gl(t, u), g2(t,u ) possess quasi-monotone nondecreasing property in u for each t E J , and, for ( t , x) E J x S , ,

< D-'V ( t ,x) ,< gz(t, V ( t ,4).

gdt, V ( t ,4)

Let r ( t , to , uo),p(t, t o , vo) be the maximal, minimal solutions of u' = gz(t,

4,

v' = gl(C v),

4 t " ) = uo , v(t0) = U o ,

respectively, such that

Then, as far as x ( t ) = x(t, t o , xo) exists to the right of to , we have p(t, t o

,4

< V ( t ,4 t ) ) <

y(t,

t o , UO),

where x ( t ) is any solution of (4.1.1).

4,2, Asymptotic stability

An approach that is extremely fruitful in proving asymptotic stability is to modify Lyapunov's original theorem without demanding D+V ( t ,x) to be negative definite. As we have seen, Theorem 3.15.8 is a very general result of this nature, although it covers a particular situation of the function f ( t , x). T h e theorem that follows takes care of the general case of f ( t , x) and requires two Lyapunov functions.

THEOREM 4.2.1.

J

(i) f x Sp-

E

Suppose that the following conditions hold:

C [ J x S, , R"],f(t,0) = 0, and f ( t , x) is bounded on

270

CHAPTER

4

(ii) V , E C [ J x S, , R,], Vl(t,x) is positive definite, decrescent, locally Lipschitzian in x, and D+c;(t, x)

< w(x) < 0,

( t , x)

E

1 x s,,

where ~ ( x is) continuous for x E S, . (iii) V 2E C [ j x S, , R,], and V z ( t ,x) is bounded on J x S, and is locally 1,ipschitzian in x. Furthermore, given any number a , 0 < 01 < p, there exist positive numbers [ = ((a) > 0, 7 = ~ ( D I> ) 0, 7 < a , such that D+G',(t, x) > 8 for

tt

/ / x 11

p and d(x, E ) < 7, t

,

R

=

2 0, where

[x E s, : W(.)

= 01

and d(s, E) is the distance between the point x and the set E. Then, the trivial solution of (4.1.1) is uniformly asymptotically stable.

0 and t,, E J be given. Since V l ( t ,x) is positive definite Proof. Let t and decrescent, there exist functions a, b E .f such that A ;

We choose 6

6 ( ~ so ) that

:

h ( € ) > a(6).

(4.2.2)

Then, arguing as in the first part of the proof of Theorem 3.4.9, we can conclude that the trivial solution of (4.1.1) is uniformly stable. 1,ct u s now fix t p and define 6, = S(p). Let 0 'c E p, to E J , and S -= S(t) be the same 6 obtained in (4.2.2) for uniform stability. Assume that 11 A,, 11 .S,, . T o prove uniform asymptotic stability of the solution .T = 0, it is enough to show that there exists a T = T ( E )such that, for some t* E [t,,, t,, 7'1, we have ~

+

I1 x(t*, t"

9

x0)ll

< 8.

This we achieve in a number of stages: (1)

If d[x(t,), x ( t 2 ) ] > Y

> 0, t , > t , , then Y

< M?P(t,

-

tl),

(4.2.3)

4.2. where IIf(t, x)II

ASYMPTOTIC STABILITY

< M , ( t , x) E J

x S, . For, consider

I "Atl) - xi(tz)/< fz I x:(s)l

ds ,<

ti

< M(tz

and therefore

27 1

-

Itz tl

Ifds, x(s))l ds

( i = 1, 2, ..., n),

t,)

Let us consider the set

u = [x E s,: 6 < Ij x I/ < p, d(x, E ) < 171, and let sup Vz(t,x)

= L.

IIXII-CP

t2O

Assume that, at t = t , , x(tl) = x(t, , t o ,xo) E U. Then, for t we have, letting m ( t ) = V2(t,x(t)),

> t, ,

D+m(t) b D Vz(t,x ( t ) ) > 6,

because of condition (iii) and the fact that V2(t,x) satisfies a Lipschitz condition in x locally. Thus, m(t) - m(tl) =

j t D-Im(s)ds, tl

and hence m(t)

+ m(Q

2

it

D'-m(s)ds

fl

>at

2 f lD+V,(s, x(s))ds

- tl)

as long as x ( t ) remains in U. This inequality can simultaneously be realized with m ( t ) L only if

<

t

< t,

+ 2Llt.

272

CHAPTER

4

< +

It therefore follows that there exists a t, , t , < t , t, 2L/f such that x ( t 2 ) is on the boundary of the set U. I n other words, x ( t ) cannot stay permanently in the set U .

( 3 ) Consider the sequence t,&== t"

Set n(t)

~

{tJ

such that

2L

-t k -

5

(k = 0, 1, 2,...).

V,(t, x(t)). Thcn, by assumption (ii), we have /Pn(t)

We let =

<- DbV1(t, x ( t ) ) ,< 0.

s < I1 "X I1 < f ,

inf[l w(x)I,

and

d(x, E ) 2 11/21,

< <

Suppose that x ( t ) is such that, for t , t t,,, , 6 / / x(t)/l < p. If, for t, t t, , we have S -: /I ' x(t)ll < p and d(x,E ) >, 67, then, using assumption (ii) together with the definition of the set E , we obtain

< < ,,

j

-tk I 2

"(t,,,)

~

n(tn) =

/ _

D+n(s)ds

'Ir

(4.2.4)

On the other hand, if it happens that, for t ,

s < /I .Y(!Jl/

\

p,

< t, < t,,, ,

dlx(t,), B ]

< < +

< .ty,

then there exists a t, , t , t:, t , 2 L / f such that d[x(t,), El = 7, in view of (2). It follows that there also exists a t, , t, t, t, satisfying d[x(t4),El = 8.1. These considerations lead to d[x(t3),x(t4)] Bq, and hence we obtain, because of ( I ) ,

i-q =, Mn""(t,

<

~

t4),

% .

4.3.

273

INSTABILITY

which implies (4.2.5)

Moreover, n(t,)

-

n(tl)

<[

t4

D+V,(s, x(s)) ds

tl

+

D+V,(s, x(s)) ds

[13 64

< -A(t3

-

t4)

-A7 < 2Mn1I2 ~

-A,.

-

Since n(t)is a nonincreasing function, we have fi(t,+,)

<

fi(t3)

<4tl) < n(tJ

-

A,

-

A,

.

Also, on the basis of (4.2.5), we obtain from (4.2.4) that 4t,+2)

G n(td

-

4

*

Thus, in any case,

<

l71(4C+27 +€+3))

>

4tk))

-

A,

> ~ ( 6 , )and T = T ( E )= 4k *L/f ( € ) .

Choose an integer k* such that A$* Assume that, for to t to T,

< < +

I/ 4 4 t o

L’dtlC

7

x0)Il

2 6.

I t then results from the preceding considerations that Vl(t0 f-1’7 4 t ”

+ T ) ) e Vl(t0 < a@,) < 0,

3

Xo)

-

-

k*Al

K*A,

which is incompatible with the positive definiteness of V,(t, x). Thus, there exists a t* E [ t o , to TI satisfying

+

II “ (t* , to , X0)ll < 6 , and the proof is complete.

4.3. Instability I n Sect. 3.3, we proved a theorem on instability by means of a single Lyapunov function. We give below an instability theorem in which two Lyapunov functions are used.

274

CHAPTER

‘I’HEOREnI

4.3.1.

4

Suppose that the following conditions hold:

(i) f E C [ J Y S,, R”],f ( t , 0) = 0, and f ( t , x) is bounded on S,. (ii) V , E C [ J x S o ,R,], V,(t, x) is locally Lipschitzian in x, decrescent, and, for any t >- 0, it is possible to find points x lying in any given small neighborhood of the origin such that Vl(t,x) > 0. 0, (iii) D+V,(t, x) 3 0, ( t , x) E J x S o , and, in each domain t 11 .T /I . p, D+V,(t, x) 2 +a(t)w ( x ) , where w(x) 0 is continuous for x E S o and +Jt) 3 0 is continuous in t such that, for any infinite system S of closed, nonintersecting intervals of J of an identical fixed inter1 al, we have

J x

(4.3.1) and V z ( t x) , is bounded on J x So and is (iv) V , E C [ J x S o , R,], locally Lipschitzian in A . Furthermore, given any number 01, 0 < 01 < p , U , and a continuous function f a ( t )> 0 it is possible to find 7 : 7 ( ~ )7, such that

J

and, in the set z

11 x 11

00

t

p,

G..

L(t>

d(x, E )

=

J

(4.3.2)

a, 7, t

E

J, (4.3.3)

where Then, thc trivial solution of (4.1.1) is unstable.

Proof. T h e proof of this theorem closely resembles that of Theorem 4.2.1, and hence we shall be brief. Suppose that, under the conditions of thc theorem, the trivial solution is stable. T h a t is, given 0 E p, t , E J , there exists a 6 > 0 such that I] x, I/ < 6 implies 6, t 3 to . II x( t , t,, , %,)I1 According to assumption (ii), a point ( t o ,x$) can be found such that I/ .x(,’ / / 6 and tT1(t,,, x$) > 0. We shall consider the motion x(t) = a ( t , t,, , x;) and its properties: I

‘

( I ) d ( A ( t ) , x ( T ) ) 3 7 , t > T ; then t from ( 1 ) in the proof of Theorem 4.2.1. (2)

For every t

- T

2 q/Mn1/2.This

is clear

3 t, , there will be a positive number 01 such that 01

< [I x(t)ll < E < p .

(4.3.4)

4.3. INSTABILITY

275

This is compatible with the assumption of stability, that is, 11 x(t)ll t 3 to . However, since D+V,(t, x) 3 0, it follows that V1(4

< E,

40) 2 Vl(t,, .,*I > 0.

Since Vl(t,x) is decrescent, for numbers V l ( t o x$) , > 0, a number a: > 0 can be found such that, for all t 3 t o , (1 x 11 a, we shall have

<

vlct, x) < V1(to

7

x,*).

Consequently, 11 x 1) < a is not possible. According to (iv), there exists ) 0 such a number 7 = ~ ( aE),, 7 < a , and a continuous function ( & ( t > that (4.3.2) and (4.3.3) hold. ) ,) < 7, then a t* > 7 can be found such that (3) If ~ ( x ( T E d(x(t*),E )

=

(4.3.5)

7.

Suppose that d(x(t), E ) < 7 for all t 3 7. Letting m ( t ) = Vz(t,x(t)), we obtain, using the Lipschitzian character of V2(t,x) in x,the inequality D+m(t) 2 D'Vz(f,

44) >, U t ) ,

and hence

Since Vz(t,x) is assumed to be bounded, the relation (4.3.2) shows that d(x(t),E ) < 7 cannot hold for all t 3 7. Hence, there exists a t* > T such that (4.3.5) is satisfied. ) ,) < 7 / 2 , then, for t = t*, when d(x(t*), E ) = 7, (4) If ~ ( x ( T E we have Vl(t*,x ( t * ) ) 2

where 7

= inf[w(x),

~

01

< 11 x /I < p, d(x, E ) 2 $71 > 0.

In fact, under the given conditions,

T

d(x(f*),E )

and, for t ,

ds,

rl < t** === t* - 2Mn1I2

and E

4.))+ 6 J t** C&) t*

Vl(T,

< t < t*, we shall have 4,

< t , < t* can be found such that = $7,

< d(x(t),E ) < 7.

216

4

CHAPTER

Hence, by (iii), it follows that 4 t ) ) 3 4At) w ( x ( t ) )3 E # b ( t ) ,

D"'l(t,

using the fact that V,(t, x) is locally Lipschitzian in x, and, consequently,

Observing, however, that d(x(t*),x(t*)) >, $77, we get, in view of (I), that

( 5 ) There is no number t,

have

t,, such that, for all t

d(x(t),E ) 3

=

would

11.

> t, , we should have

Indeed, if such a t , exists, then, for all t 1 71(t,x(t))

> t, , we

+ J" n+Vl(s, x(s)) ds

V,(tl , ~ ( t , ) )

tl

Ry (4.3.1), this implies that V,(t, x ( t ) )+ co as t -+ co, which is absurd because of the relation (4.3.4) and the fact that V,(t, x) is decrescent. Thus it follows that, for any t:, a T ~ > , t: can be found such that

,

d(x(72+1),

< frl,

and, according to ( 3 ) , there corresponds a tz, > T ~ + ,satisfying d(x(tz*,,),E )

= rl-

1,et us consider the infinite sequence of numbers to

< T I < t: < '.' < Tz < t,* < "'.

I n view of assumption (iii) and (4), we have

<

where rj tF* = t,* ~ / 2 M n l /T ~ h. e infinite system of segments [tT*, t:] satisfies condition (iii), and therefore the last sum increases ~

4.4.

CONDITIONAL STABILITY AND BOUNDEDNESS

277

indefinitely with i. In other words, Vl(tF,x ( t 7 ) )-+ co as i+ CO. This is not compatible with the boundedness of Vl(t,x(t)). T h e contradiction shows that the assumption of stability is wrong, and the theorem is proved.

4.4. Conditional stability and boundedness Let, for k < n, M ( n p ! ~denote ) a manifold of (n - k) dimensions containing the origin. Let S(a), S(O)represent the sets S(a) = [x E s,: / / x // S(01) =

[x E

s, : 11 x 11

< 011,

< a],

respectively. Suppose that x ( t ) = x ( t , t o ,xo) is any solution of (4.1.1). Then, corresponding to the stability and boundedness definitions (Sl)-(S8)and (Bl)-(B8),we shall designate the concepts of conditional stability and boundedness by (C,)-(C,,). We shall define (C,) only, since, on that basis, other definitions may be formulated. DEFINITION 4.4.1. T h e trivial solution of (4.1.1) is said to be (C,) conditionally equistable if, for each E > 0 and to E J , there exists a positive function 6 = 6 ( t o ,6) that is continuous in to for each E such that x ( t , to , xo)

provided

c

S(C),

t

3 to ,

x0 E S(8)n MG-k) .

Evidently, if k = 0 so that M(7L-!L) = R",definitions (Cl)-(C16)coincide with the stability and boundedness notions ( S1)-(S,) and (B1)-(B8). Analogous to the definitions (C,)-(C,,), we need some kind of of conditional stability and boundedness concepts with respect to the auxiliary differential system (4.1.4). Perhaps the simplest type of definition is the following. DEFINITION 4.4.2. T h e trivial solution of the system (4. I .4) is said to be (CF) conditionally equistable if, for each E > 0, to E J , there exists a positive function S = S ( t o , 6) that is continuous in t,, for each E such that the condition

C uio < 6, N

i=l

and

uio = O

(i = I , 2,..., K)

218

CHAPTER

implies

c N

% ( t , t o , %)

4

< 6,

t

2=1

> to

'

Definitions (Ca )-( C&) are to be understood in a similar way.

THEOREM 4.4.1.

I'\ssume that

(i) g E C[/ x R+N,K N ] ,g(t, 0) = 0, and g(t, u) is quasi-monotone nondecreasing in 11 for each t E 1; (ii) V E C [ J x S o ,R+N], V ( t ,x) is locally Lipschitzian in x, CL, Vi(t,x) is positive definite, and

C T'i(t, x) N

--f

0

as

/ / x / ---f (

0 for each t~ J ;

i-I

(iii) Vi(t, x) = 0 (z' = I , 2,..., k), k < n, if x E M(n--k), where M(,,_,, is an ( n -- k) dimensional manifold containing the origin; (iv) f E C [ J x S o ,R " ] , f ( t ,0) = 0, and I ) ; q t , x)

< g(t, V ( t ,x ) ) ,

( t , x) E

J x

s,.

Then, if the trivial solution of (4.1.4) is conditianally equistable, the trivial solution of the system (4.1.1) is conditionally equistable. Proof. Let 0 :' E < p and t , , J~ be given. Since positive definite, there exists a b E .X such that b(l/x 11)

N

<: 1 Q t , x),

( t , x) E J x S, .

Vi(t,x) is (4.4.1)

i=l

Assume that the trivial solution of the auxiliary system (4.1.4) is conditionally equistable. Then, given b(6) > 0 and t,, E ], there exists a 6: 6 ( t , , E ) that is continuous in to for each E , so that (4.4.2)

provided

c u,o < 8, v

u,o

=0

(i

=

1 , 2 )...,k ) .

(4.4.3)

a=1

Let us choose ui0 = V , ( t o ,xo) (i = I, 2, ..., N ) and x,,E M(n-fc)so that 0 (z' = 1, 2, ..., k), by condition (iii). Furthermore, since Cf=l V L ( tx,) + 0 as 11 x 11 + 0 for each t E J , and V ( t ,x) is continuous,

uio =

4.4.

279

CONDITIONAL STABILITY AND BOUNDEDNESS

it is possible to find a 6, verifying the inequalities

=

S,(t, , e ) that is continuous in to for each

E,

(4.4.4) simultaneously. With this choice, it certainly follows that x0 E S(8,) n M(n-k)

>

implies x(t, to , xo)C S(E),t to . If this were not true, there would exist a t, > to and a solution x ( t , t o ,xo) of (4.1.1) such that, whenever xo E s(S,) n M(n-lc), we have x(t, t o , xo) C S ( E ) , t E [to , t,), and x ( t , , to , xo) lies on the boundary of S ( E ) This . means that

II 4 4

to

7

> x0)Il

<

II x ( t , to > .")I1

= €7

t

P7

E

[to

9

t119

and, consequently, N

(4.4.5)

Moreover, for t E [to , t J , we can apply Theorem 4.1 .I to obtain V( t,x ( t , t o , xo))

< r ( t , to ,uo),

[to

t

7

tll,

where r(t, to , uo) is the maximal solution of (4.1.4), which implies that N

N

1 V,(t,x ( t , t ,

7

xo))

< 1 rdt, to ,uo),

t

i-1

i-1

6

[4J,tll.

(4.4.6)

Notice that, from the choice uio = V i ( t o ,xo) and the relation (4.4.4), xo E s(6,) n M(n-k)assures that (4.4.3) is satisfied. Hence, (4.4.2) and (4.4.6) yield the inequality

c N


Vi(t1 9 X(t1

1

i=l

to > xo))

ri(t1

i=l

, t"

7

uo)


which is incompatible with (4.4.5). Thus, x ( t , t o ,xo) C S ( E ) ,t >, t o , provided xo E s(6,) n M(n-k), and the theorem is proved.

THEOREM 4.4.2.

Let assumptions (i), (ii), (iii), and (iv) of Theorem 4.4.1 hold. Suppose further that

c Vi(t, N

i=l

x) + 0

as

11 x 11

--t

0 uniformly in t .

(4.4.7)

280

CHAPTER

4

Then the conditional uniform stability of the solution u = 0 of (4.1.4) guarantees the conditional uniform stability of the trivial solution of (4.1.1).

Proof. By definition ( C z ) , it is evident that 6 occurring in (4.4.3) is independent of t o . I n view of (4.4.7), this makes it possible to choose 6, also independent of t o , according to (4.4.4). Noting these changes, the theorem can be proved as in Theorem 4.4.1.

THEOREM 4.4.3.

Under assumptions (i), (ii), (iii), and (iv) of Theorem 4.4.1, the conditional equi-asymptotic stability of the trivial solution of (4.1.4) implies the conditional equi-asymptotic stability of the trivial solution of the system (4.1.1).

Proof. Assume that the trivial solution of the auxiliary system (4.1.4) is conditionally equi-asymptotically stable. Then, it is conditionally equistable and conditionally quasi-equi-asymptotically stable. Since, by Theorem 4.4.1, the conditional equistability of the trivial solution of (4.1.1) is guaranteed, we need only to prove the conditional quasi-equi0 of (4.1.1). For this purpose, asymptotic stability of the solution x suppose that we are given 0 < E < p and to E J . Then, given 6 ( ~ )> 0 and t,, E J , there exist two positive numbers 6, = S,(t,) and T = T ( t o ,6) such that, if the condition

c N

1

=I

U,"

.< 8" ,

U," = 0

(2-

=

1 , 2 ,..., k)

(4.4.8)

As previously, the choice uio = Vi(t,, x,,) and x,,E implies uio == 0 (i = 1, 2, ..., k). Also, there exists a So = S,(t,) satisfying

II xo /I <:

-

an,

c K(to N

i=l

so)

< 6,

(4.4.10)

at the same time. Let 8,, = min[S,, 6$], where S$ = 8(t,, p) is the n MG-/,-), we number obtained by taking E = p. Thus, if x,)E notice that the condition (4.4.8) is fulfilled. Furthermore, since (C,) holds, the inequality (4.4.6) is valid for all t 3 t o . We can now assert n M(n-jt). For otherthat x ( t ) C S ( E ) ,t >, to -1 T whenever xo E s(8,) wise, suppose that there exists a sequence {tx-},t, > to + T , and t,<-+ 00

s(8,)

4.4.

CONDITIONAL STABILITY AND BOUNDEDNESS

28 1

as k -+ 00 such that, for some solution x ( t , t o , xo) of (4.1.1) with xo E S($) n M(n-k), we have

II 4t7C

to

7

%)I1 2 E .

t

This leads to an absurdity,

< i=lc Vi(tl,,X ( t , N

b(c)

<

N Yi(t,

9

i-1

I

to,

t o , uo)

&I))

< b(c),

in view of relations (4.4.1), (4.4.6), and (4.4.9). We thus have (C3),and consequently the theorem is established.

THEOREM 4.4.4.

Suppose that assumptions (i), (ii), (iii), and (iv) of Theorem 4.4.1 hold, together with (4.4.7). Then, the conditional uniform asymptotic stability of the trivial solution of (4.1.4) implies that the trivial solution of (4.1.1) is conditionally uniformly asymptotically stable. Proof. We proceed as in Theorem 4.4.3, observing that the uniformity of conditional stability is assured by Theorem 4.4.2. As the numbers 6, and T are independent of t o , resulting from (4.4.10) is certainly independent of t o , because of (4.4.7).

so

THEOREM 4.4.5. Assume that (i) g E C[J x RtN, RN]andg(t, u ) is quasi-monotone nondecreasing u in for each t E J ; (ii) V E C [J x R?',R N ] , V(t,x) is locally Lipschitzian in x,and

411x 11)

N

< i=l 1 Vi(4 4 6 .(it

where a, b E .X on the interval 0 b(u) + co

x ID,

(t,

4E J

x R",

< u < 00, and as

u+

00;

(iii) Vi(t, x) = 0, i = 1, 2,..., k, k < n, if x E M(n-k), where M(n-k) is an (n - k) dimensional manifold containing the origin; (iv) f E C[/ x Rn, R"], and D'V(t, x)

< g(t, V ( t ,x)),

( 4 ).

E

J x

Rn.

282

CHAPTER

4

Then, if the auxiliary system (4.1.4) satisfies one of the definitions (C$)-(C&), the system (4.1.1) verifies the corresponding one of the definitions (C9)-(C16).On the basis of parallel theorems of Sect. 3.13 and the proofs given previously, the proof of the respective statements of this theorem can be constructed. Let us now indicate the modifications necessary in order to obtain the usual stability and boundedness results, using several Lyapunov functions. Designate by (Cf*)-(C$*) the parallel definitions obtained by dropping the conditional character in (CT)-(C$). For example, the definitions (CF*) would run as follows:

8

(CT*) For each E > 0 and to E J , there exists a positive function 8(t,) e ) that is continuous in to for each E such that the inequality

=

implies

As a typical example, we shall merely state a theorem that gives sufficient conditions, in terms of any Lyapunov function, for the equistability of the trivial solution of (4.1.1). THEOREM 4.4.6.

Suppose that

(i) g E C [ J x R+N,R N ] g(t, , 0) = 0, and g(t, u) is quasi-monotone nondecreasing in u for each t E J; (ii) V E C [ J x S o ,R + N ] ,V ( t ,x) is locally Lipschitzian in x, C;"=, Vi(t,x) is positive definite, and N

L'*(t, X)

+

0

as

(1 x 11 -+ 0

for each t

E

J;

2=1

(iii) f

E

C [ J x So,R n ] , f ( t0) , = 0, and

D+F7(f,).

< g(t, V ( t ,x)),

( t , x)

E

J x

s,.

Then the definition (Cf*) implies that the trivial solution of (4.1.1) is equistable. T o exhibit the fruitfulness of using vector Lyapunov function, even in the case of ordinary stability, we give the following example.

4.4.

283

CONDITIONAL STABILITY AND BOUNDEDNESS

Let us consider the two systems

Example.

+ y sin t y’ = x sin t + e+y x’ = e+x

+ xy2)sin2 t,

-

(x3

-

(x2y

+ y 3 )sin2 t .

( 4 . 4 . 1 1)

Suppose we choose a single Lyapunov function V given by V ( t ,x)

= x2

+

y2.

Then, it is evident that D+V(t,x)

< 2 ( r t + j sin t 1)

V ( t ,x),

< u2 + b2 and observing that [x2 + y2I2sin2 t 3 0.

using the inequality 21 ab I

Clearly, the trivial solution of the scalar differential equation u’ = 2(e+

+ I sin t I) u,

u(to) = uo 3 0

is not stable, and so we cannot deduce any information about the stability of the trivial solution of (4.4.1 1) from Theorem 3.3.1, although it is easy to check that it is stable. On the other hand, let us attempt to seek a Lyapunov function as a quadratic form with constant coefficients V(t,X)

=

+[x2

+ 2Bxy + Ay2].

(4.4.12)

Then, the function D+V(t,x) with respect to (4.4.1 1) is equal to the sum of two functions q ( t ,x), w,(t, x), where q ( t , x) = x2[e+

+ B sin t] + xy[2Be+ + (A + 1 ) sin t ]

+ y2[Aect+ B sin t ] , wz(t,x) = -sin2 t [ ( x 2+ y2)(x2+ 2Bxy + Ay2)].

For arbitrary A and B, the functions V ( t ,x) defined in (4.4.12) does not satisfy Lyapunov’s theorem (Corollary 3.3.2) on the stability of motion. Let us try to satisfy the conditions of Theorem 3.3.3 by assuming w,(t, x) = h(t) V(t,x). This equality can occur in two cases:

+

+

1, B , = 1, h,(t) = 2[c1 sin t] when V,(t, x) = &(x Y)~. 1, B, = -1, h,(t) = 2[ect - sin t] when V,(t, x) = +(x - y)2.

(i) A, (ii) A,

=

=

284

CHAPTER

4

T h e functions V, , V, are not positive definite and hence do not satisfy Theorem 3.3.3. However, they do fulfill the conditions of Theorem 4.4.6. I n fact,

x:=l

(a) the functions Vl(t,x) >, 0, V,(t, x) >, 0, and Vi(t,x) = x2 + y2, and therefore C Z , Vi(t,x) is positive definite as well as decrescent;

< +

(b) the vectorial inequality D+V(t,x) g(t, V ( t ,x)) is satisfied with the functions g l ( t , u1 , uz) = 2(e+ sin t ) u1 , g2(t,uI, u2) = 2(e+

-

sin t ) uz .

I t is clear that g(t, u ) is quasi-monotone nondecreasing in u, and the null solution of u' = g(t, u ) is stable. Consequently, the trivial solution of (4.4.11) is stable by Theorem 4.4.6.

4.5, Converse theorems We shall consider the converse problem of showing the existence of several Lyapunov functions, whenever the motion is conditionally stable or asymptotically stable. T h e techniques employed in the construction of a single Lyapunov function earlier in Sect. 3.6 do not right away extend to this situation. As will be seen, the results rest heavily on the choice of special solutions of a certain differential system and the chain of inequalities among them, a kind of diagonal selection of the components of these solutions, and the quasi-monotone property. With a view to avoid interruption in the proofs, let us first exhibit some properties of certain solutions of the system (4.1.4) and its related system u' = g*(t, u ) , (4.5.1)

Assume that g E C [ J x R+N,A"], g(t, 0 ) E 0, ag(t, u ) / & exists and is continuous for ( t ,u)E J x R+N,and g(t, u ) is quasi-monotone nondecreasing in u for each t E /. Evidently, g*(t, u ) also satisfies these

4.5.

285

CONVERSE THEOREMS

assumptions. Moreover, since ui > 0 (i = 1, 2,..., N ) , it follows, in view of the quasi-monotone property of g(t, u), that

< g(t, 4.

g*(t, u )

(4.5.2)

Observe that the hypothesis on g(t, u ) guarantees the existence and uniqueness of solutions of (4.1.4) as well as their continuous dependence on initial values. Also, the solutions u(t, to , uo) are continuously differentiable with respect to the initial values. Furthermore, u s 0 is the trivial solution of (4.1.4). Clearly, similar assertions can be made with respect to the related system (4.5.1). If U ( t) = U(t,0, uo) and U*(t) = U*(t, 0, uo) are the solutions of (4.1.4) and (4.5.1), through the same point (0, uo),respectively, it follows, from Corollary 1.7.1, that U*(t)

< U(t>,

t

(4.5.3)

2 0,

in view of (4.5.2). Consider next the N initial vectors, with uio > 0 (i = 1, 2, ..., N ) defined by

o,..., 01, u20 o,..., 01,

Pl

= (u10 9

P,

=h10

Pi

= (u10 u20

,*-., uio, o,..., O ) ,

= (%I3

,..-,U N O ) .

P,

9

, 9

9

...

...

a20

<

It is easy to see that pi pi+l , for each i = 1, 2,..., N - 1. Let us denote the solution of the system (4.5.1) through the point (O,p,) by

Ui*(t)= Ui*(t,0 , p J

UiAt,

07 P i )

Uinr(t7

0, P i )

=

for each fixed i, i = 1, 2,..., N . By Corollary 1.7.1, it follows that

where UF(t), U$l(t) are the solutions of the system (4.5.1) through (0, pi) and (0, pi+l), respectively.

286

CIIAPTER

This implies that, for each j 0, P I )

Ul,(t,

-2U Z i ( t ,

=

4

1, 2,..., N and t 3 0,

< ." <

0, p 2 )

UNj(t,

0, p N ) .

(4.5.4)

We may now have the following:

THEOREM 4.5.1.

Assume that

(i) the functionfg C [ J x So,R"],f(t,0) = 0, and af(t, %)/axexists and is continuous for ( t , x) E J x So; (ii) the solution x ( t , 0, xo) of the system (4.1.1) satisfies the estimate

Pi(1' xo ll) where

-c

ll x ( t , 0, xn)ll

< Pz(ll xn ll),

t

2 0,

xn E M(n--k)9

(4.5.5)

, PL E H ;

(iii) the function g E C [J x R t N ,R N ] ,g(t, 0 ) = 0, ag(t, u)/au exists and is continuous for ( t , u ) E J x R+N, and g(t, u ) is quasi-monotone nondccreasing in ii for each t E J ; (i~7) the solution U ( t ,O,p,) of the system (4.1.4) verifies the inequality

provided uiD 0 (i = 1 , 2,..., k), where y 2 E X ; (v) the solution U$(t, 0, p N ) of the related system (4.5.1) is such that

if ziio =- 0 (i I , 2 ,..., k), where y1 E X . Then, there exists a vector function V ( t ,x) with thc following properties: ( I ) I/ E C [ J s S, , and V ( t ,x) possesses continuous partial derivatives with respect to t and the components of x for ( t ,x) E J x S,,;

4.5.

287

CONVERSE THEOREMS

Proof. Let us first observe that assumption (i) implies the existence and uniqueness of solutions of (4.1.I), as well as their continuous dependence on the initial values. Also, the solutions x(t, t o , xo) are continuously differentiable functions with respect to the initial values ( t o ,xo), and the system (4.1.1) possesses the trivial solution. Let us denote x(t, 0, xo) by x so that, by uniqueness of solutions, we have xo = x(0, t, x). Choose any continuous function p(x) E R + N , possessing continuous partial derivatives a p ( x ) / a x for x E S, , such that a1(II

where

JI)

, a2 E X and

N

< C pi(-%)< aAIJ x II),

(4.5.8)

i=l

( Y ~

pi(%)= 0

(i = 1, 2,..., K)

if

XE

M(n-k).

(4.5.9)

We then define the vector function V ( t ,x) as follows:

VI(4 x)

= Ull(t,

0, Pl(40,t , x)),

OY.9

01,

V d t , 2 ) = " z d t , 0, PcLI(4-2 4 x)), P2(40,

...

l"(4

2) = U N N ( t ,

0, PI(x(0, t , x)),...,

t 9

XI>,

CLN(X(0,

0,... 01, ?

(4.5.10)

t , x))).

Because of the continuity of the functions x(0, t , x), p(x), Up(t),..., U$(t), with respect to their arguments, it is clear that Vi(t, x) (i = I, 2, ..., N ) is defined and continuous for ( t , x) E J x S, . Since the functions f and g (and hence g*) satisfy hypotheses (i) and (iii), the functions UT(t), U$(t),..., U$(t), and x(0, t, x) are all continuously differentiable with respect to their arguments. This, together with the choice of p(x), shows that V ( t ,x) possesses continuous partial derivatives with respect to t and the component of x. Thus, for each i = 1, 2,..., N , Vi'(t, x) = &(t, 0, p1(x(O, t , x)),..., CLi(X(0, t , x)), 0,..., 0 )

since, by relation (3.6.1 I),

288

CHAPTER CHAPTER

4

Using the quasi-monotone nondecreasing character of g(t, u) in u,the , U $ ( t) are all nonnegative, and relations fact that the solutions U f ( t ) ..., (4.5.4), we obtain

This proves (1) and (2). To show that (3) holds, we observe that, if x belongs to M(n-k) , then x,,= x(0, t , x) also belongs to the manifold M(T!+X) . Now, by the definition (4.5.10), the choice of p(x) satisfying (4.5.9), and the fact that the system (4.5.1) has the identically zero solution, it follows that, if x E M(rL+IT), 0 (z' = 1, 2,..., k). V,(t, X) Since .1c = x(t, 0, x,J and xu = x(0, t , x), we get, from (4.5.5), that (4.5.11)

where p;',

p ~ E' X . T h e definition (4.5.10) and the relations (4.5.4) yield

which, by virtue of (4.5.3), leads to

4.5.

289

CONVERSE THEOREMS

where u l , u2 ,..., uN are the components of the solution U(t,O,p,) of the system (4.1.4). In view of (4.5.9) and the fact that x,,E M(n--k), using the relation (4.5.6) and the upper estimates in (4.5.8) and (4.5.1 I), we get

c Vi(4 4 G [ c N

N

Yz

i-1

G

i=l

PLi(40, 4

Y 2 [ 4 40, t , x)ll)l x lD>l

G r2[.,(8;'(11 =

4)j

4 x Ill,

a E3 - 0

Finally, as the solution U$(t) is nonnegative, we have

c vi(t, N

i=l

x)

2 ~ " [ t 0, , ~ i ( x ( 0t,, x)),--., E " N ( x ( O ,

t , x))I,

which, by using the inequality (4.5.7) and the lower estimates in (4.5.8) and (4.5. I I), yields

3 Y l [ ~ l ( I 1 4 0 ,t , 4l)l

T he proof is complete.

x ll))l

2 Yl[%(P;l(ll = b(ll x II),

b E 3--

It is to be observed that the upper estimate in (4.5.5) and the inequality (4.5.6) ascertain the conditional stability of the null solutions of (4.1.1) and (4.1.4), respectively. The lower estimate in (4.5.5) and the estimate (4.5.7) are compatible with the conditional stability of the null solutions of (4. I. 1) and (4.1.4), respectively.

THEOREM 4.5.2. Let assumptions (i) and (iii) of Theorem 4.5.1 hold. Suppose further that (a)

the solution x( t , 0, xo) of (4.1.1) satisfies the inequality

Pl(ll xo 11) where

Ul(t)

< /I 4 4 0, %)I1 < Bz(ll xo 11) az(t),

, p2 E X and

u1 , u2 E 2;

t

3 0,

xu E M(7L-k) >

(4.5.12)

290

CHAPTER

4

(b) the solution U(t,0, p,) of (4.1.4) verifies the estimate (4.5.13) where y, E X , 6, E 27, and, whenever uio = 0, i = 1 , 2,..., I z ; (c) the solution U,$(t,0, p,) of (4.5.1) is such that (4.5.14) and whenever ui0 = 0, i = I, 2,..., k ; where y1 E X ,6,E 9, (d) y l ( y ) is differentiable, and y;(r) 3 m 3 0; (e) 8,(t) and a2(t)are such that 6,(t) 3 mla2(t), m, > 0. Then, there exists a function V ( t ,x) with the properties (I), (2), (3) of Theorem 4.5.1 and

41x 11) ,<

c N

VZ(t>

z=1

4

< a ( t , II x ll),

( 4 x) E 1 x

s,,

where 0 t X and a(t, Y) belongs to class X for each fixed t E J and is continuous in t for each Y.

Proof. Let x ( t , 0, x"), U ( t ,0, p,), and U$(t, 0, p,) be the solutions of (4.1.1), (4.1.4), and (4.5.1) satisfying (4.5.12), (4.5.13), and (4.5.14), respectively. Choose any continuous function p ( x ) E R,N possessing continuous partial derivatives with respect to the components of x, such that (4.5.9) and

all x II)

< c P Z ( 4 < 41x Ill? N

1 1

B2

1

01

E

.x,

(4.5.15)

hold. Using the same definition (4.5.10) for V(t,x) and proceeding as in Theorem 4.5. I , it can be easily shown that (l), (2), and (3) are valid. Assumption (d) implies that YI(YlY2)

2 mYlYZ.

(4.5.16)

T h e inequality (4.5.12), in view of the fact that x = x(t, 0, xo) and x,,= x(0, t , x), yields that (4.5.17) where p ~ ' ,1,3;1 both belong to class X .

4.5.

29 1

CONVERSE THEOREMS

As in Theorem 4.5.1, using the definition (4.5.10) and the nonnegative character of U$(t), we get N

1vZ(t,

i=l

).

2 u N N [ t , O, P1(x(O,t ,

x))>***7

PN(x(o,

t , x))],

which, by virtue of (4.5.14), the lower estimates in (4.5.15) and (4.5.17), the relation (4.5.16), and the assumption ( e ) , gives successively

Again, as before, making use of the definition of V(t,x) and the relation (4.5.4) and (4.5.3), we obtain N

1

Z=l

N

vi(t,

< 1 uZ(t, i-1

Pl(x(o,t , x)),***, PN(x(o, t , x))],

which, in its turn, allows the following estimates successively,

because of (4.5.13) and the upper estimates in (4.5.15) and (4.5.17). The theorem is proved. Under the general assumptions of Theorem 4.5.2, it is not possible Vi(t,x) .(\I x 11). This can, to prove the stronger requirement that however, be done if the estimates (4.5.12). (4.5.13), and (4.5.14) are modified as in the following:

xr=l

<

292

CHAPTER

4

THEOREM 4.5.3. Let assumptions (i) and (iii) of Theorem 4.5.1 hold. Assume that the inequalities (4.5.12), (4.5.13), and (4.5.14) in assumptions (a), (b), (c) of Theorem 4.5.2 are replaced by

81 /I x’o /la

4 t ) < II 4 4 0, x’0)ll

< 8, II ~o I/= ~ ( t ) ,

/3, , p2 , 01 > 0 being constants, N

1

t

3 0,

i=l

O, $ N )

(4.5.18)

> 0,

(4.5.19)

and u E 9; N

ui(t,

xo E M ( n - k ) ,

< y2 1

*iO

i=l

s(t),

where y 2 > 0 is a constant, 6 E 9, and, whenever uio = 0, i N

~ “ ( t 0, , pjv)

2 yi C uio s(t),

t

i=l

2 0,

=

1, 2 ,..., k; (4.5.20)

where y1 > 0 is a constant and uio = 0, i = 1, 2, ..., k; respectively. Furthermore, let the functions 6 ( t ) and a ( t ) be related by S=(t)

=

d(t),

for some constant p > 0. Then, there exists a function V ( t ,x) with the properties (I), (2), (3) of Theorem 4.5.1, and M I

/I x’ /I”

N

< 1 Vi(t,). < M , II x’ llP, 2=1

where MI = ylA1/3;”, M , = y2A,/3yp, p suitable positive constants.

=

Pla,

and A,,

A, are some

PToof. By choosing the continuous function p(x) E R,N that satisfies (4.5.9) and

4I1 x’ /ID <

c N

i-1

Pi(4

< A,

II x 1 °,

A,, A,, p being constants greater than zero and following the proof of Theorem 4.5.2, with necessary changes, it is easy to construct the proof of the theorem. It may be remarked that the conditional asymptotic stability of the null solutions (4.1.1) and (4.1.4) is expressed in terms of the upper estimate in (4.5.12) or (4.5.18) and (4.5.13) or (4.5.19), respectively. Also, the lower estimate in (4.5.12) or (4.5.18) and the inequality (4.5.14) or (4.5.20) are compatible with the conditional asymptotic stability of the trivial solutions of (4.1.1) and (4. I .4).

4.6.

STABILITY IN TUBE-LIKE DOMAIN

293

T h e conditional character of the stability notions in Theorems 4.5.1, and 4.5.2, and 4.5.3 are due to the requirements that xo E uio = 0, i = 1 , 2,..., k. By dropping these conditions and modifying the technique suitably, it is easy to get a set of necessary conditions for the stability concepts, in terms of several Lyapunov functions.

4.6. Stability in tube-like domain Lyapunov stability of the invariant set of a differential system does not rule out the possibility of asymptotic stability of the set, nor does the asymptotic stability of the invariant set guarantee any information about the rate of decay of the solution. Various definitions of stability and boundedness are, so to speak, one-sided estimates, and thus they are not strict concepts in a sense. It is natural to expect that an estimation of the lower bound for the rate at which the solutions approach the invariant set would yield interesting refinements of stability notions. We introduce below the concepts of strict stability and boundedness of solutions. represent the sets Let Z ( a ) and

z(a)

s : II x I/ > a ] , Z(a) = [x E s : [I x I/ 2 a], Z(a) = [x E

respectively, and let S ( a ) , S(ol), and M(n--k)have the same meaning as in Sect. 4.4. Let x(t, to , xo) be any solution of (4.1.1).

DEFINITION 4.6.1.

T h e trivial solution of (4.1.1) is said to be

(CS,) conditionally strictly equistable if, for any E , > 0, to E 1,it is possible t o find positive functions 6, = 6,(t0, el), 6, = ?&(to,E , ) , and E, = €,(to , el) that are continuous in to for each el , such that €2

provided

< 8,

< 6,

< €1 ,

x(4 t o , xo>c q . 1 ) n Z ( E 2 ) , xo E

W,) n Z(8,) n

t

2 to,

M(T2-k) ;

(CS,) conditionally strictly unijormly stable if 6, , 6, , and E, in (CS,) are independent of to ; (CS,) conditionally quasi-equi-asymptotically stable if, given E , > 0, 01, > 0, and to E 1, it is possible to find, for every 01, satisfying 0 < 01, < 01, ,

294

4

CHAPTER

positive numbers c2 , T , that Ti

<

r 2

~ ( tt,,, , .yo) c s(€,)

whenever

T I ( &, E , , al),and T ,

=

€2

j

c €1

nz ( E ~ ) ,

to

€2

=

T2(t0, c2 , a,) such

a2

+ T , e t G to + T , ,

x0 E S(al)n Z(.(,) n M ( ~ - ;~ )

(CS,) conditionally quasi-uniform-asymptotically stable if T , and T , in (CS,) are independent of t, ; (CS,) conditionally attracting if it is conditionally equistable and, in addition, (CS,) holds; (CS,) conditionally uniformly attracting if it is conditionally uniformly stable and, in addition, (CS,) holds. T h e system (4.1.1) is said to be (CS,) conditionally strictly equi-bounded if, given a1 > 0, to E J , it is possible to find, for every rx, satisfying 0 < a, a1 , positive functions PI = &(t,, , q),p2 = &(tn, q) that are continuous in to for each al , such that

<

P2

x(t, t"

9

XO)

provided X" E

< Pl

P2

< 012

t

c S(P1) n Z(P2)>

t

3 to ,

S(al) n Z(a2)n M ( ~ -;~ )

(CS,) conditionally strictly uniform bounded if p1 , p, in (CS,) are indepcndent of to . We observe that the foregoing notions assure that the motion remains in tube-like domains. I n order to obtain the sufficient conditions for the stability of motion in tube-like domains, we have to estimate simultaneously both lower and upper bounds of the derivatives of Lyapunov functions and use the theory of differential inequalities. We are thus led to consider the two auxiliary systems uo

u'

=: g l ( t , u),

up,)

v'

= g2(t, ?I),

v(to)= uo

=

3 0, 2 0,

(4.6.1)

(4.6.2)

<

where g , > g , E C [ J x R+N,R N ) ,g2(t, 4 g d t , u), and g1(t, u), gz(4 u> possess the quasi-monotone nondecreasing property in u for each t E J. Then as a consequence of Corollary 1.7.1, we deduce that p(t9 to

7

< ~ ( tto,

~ 0 )

7

uo),

t

>, to

9

4.6.

295

STABILITY IN TUBE-LIKE DOMAIN

provided where r ( t , t o , uo), p ( t , to , vo) are the maximal, minimal solutions of (4.6.1), (4.6.2), respectively. Corresponding to definitions (CS,)-( CS,), we may formulate (CSf)-(CSZ) with respect to the system (4.6.1) and (4.6.2). For example, (CSF) would imply the following:

(CSf) Given el > 0, to E J , there exist positive functions 6, = a1(t0, el), 6, = S,(to , el), eg = e z ( t o , el) that are continuous in to for each el such that €2 < 6 , e 6, < €1 , €2

<

c N

i=l


P i ( 4 t o , vo>

N

yz(t9

i=l

t o , uo> < €1

,

t b to

7

if uio = vi0 = 0 (i = 1, 2, ..., k) and

Let us restrict ourselves to proving conditional strict equistability only. Similar arguments with necessary modifications yield any desired result.

THEOREM 4.6.1. Assume that (9 g,

gz(t,0)

, g2 E C [ J x R+N,RN1,

gz(4

4 < gdt, 4,

gdt, 0) = 0,

= 0, and g l ( t , u), g2(t,u ) possess the quasi-monotone nondecreasing property in u for each t E J ; (ii) V E C [ J x S,, , R+N],V(t,x) is locally Lipschitzian in x, and, for ( t , x) E J x S , ,

(iii) Vi(t,x) = 0 (i = 1, 2,..., k), k < n, if x E M(n-k), where M(n-k) is an ( n - k) dimensional manifold containing the origin; (iv) f E C [ J x S,, , R n ] , f ( t ,0 ) = 0, and, for ( t , x) E J x S,, ,

<

gz(4 Jqt,4) D+V(t,4

< gAt, V ( t , 4).

296

CHAPTER

4

Then, if the auxiliary system satisfies condition (CST), the trivial solution of (4.1.1) is conditionally strictly equistable.

< p and to E J be given. Assume that (CST) holds. Proof. Let 0 :c Then, given !I(€,) > 0, t, E J , there exist positive functions 8, = %Po 4, 8, = s2(t0, E ~ ) ,and i , = <,(to , el) such that 5

2,

< 8,

< 8,

< b(t,),

(4.6.3) provided uio = vin== 0

(i = 1, 2,..., K),

(4.6.4)

= Vi(to, xo) = uio ( i = 1, 2 ,..., N ) and x E M(%-,()so We choose that v?,,: uio = 0 (i : 1, 2, ..., k), by condition (iii). Let us make the following choice:

s, = b-1(8,),

6,

=

u-y81),

.(€,)

< i,,

€2

< 6,.

< 6, < and that < I/ xo 11 < 6, implies

Then, it is easy to verify that < 6, depend on to and . Furthermore, 6,

8,

E,

, a,, 6,

< c Vdto, ~XO) < 8 1 , N

z=1

and vice versa. With this choice of E, , 6,, and e l , the trivial solution of (4.1.1) is conditionally strictly equistable. Suppose that this is false. Then, there is a solution x(t, to , xo) of (4.1.1) satisfying xu E

S(8,) n Z(8,) n M ( ~ - ~ )

such that, for some t = t , > to , it reaches the boundary of S(el) n Z(e2). This means that either 11 x(t1 , t o ,xo)ll = or I/ x(t, , to , x,)lI = e2 . Also, I! x ( t , to , xg)lI -: p, t E [t,,, t l ] , and therefore, for t E [to, tl], we can apply Theorem 4.1.3 to obtain

4.7.

STABILITY OF ASYMPTOTICALLY SELF-INVARIANT SETS

297

where r ( t , t o ,u,,),p(t, t, , vo) are the maximal, minimal solutions of (4.6.1), (4.6.2), respectively, such that no = V(to, xo) = u, . This implies that

c Pi(4 to

N

N

i=l

* uo)

< i=l1 Vdt, 4 4 t o ,

.ON

(4.6.5)

for t E [ t o ,t J . I n the first instance, if 11 x( t, , t o ,xo)lI = e l , using the right side inequality in (4.6.3) and (4.6.5), we arrive at the contradiction N

b(4

< 1 Vztt, i=l

7

X(t,

7

to xo>> Y

because of the left side inequalities in (4.6.3) and (4.6.5). This shows that (CS,) follows from (CST), and the proof of the theorem is complete.

4.7. Stability of asymptotically self-invariant sets One has to consider, in many concrete problems like adaptive control systems, the stability of sets that are not self-invariant; this rules out Lyapunov stability, because those definitions of stability imply the existence of a self-invariant set. T o describe such situations, the notion of eventual stability has been introduced in Sect. 3.14. It is easy to observe that, although such sets are not self-invariant in the usual sense, they are so in the asymptotic sense. This leads us to a new concept of asymptotically self-invariant sets. Evidently, asymptotically self-invariant sets form a special subclass of self-invariant sets, and therefore it is natural to expect that their stability properties closely resemble those of invariant sets.

298

CHAPTER

4

Let zu E C[Rn,R"]. Define (4.7.1) We shall denote the sets

[x E R" : I/ 4411

< €1

by G, S(G, E ) , S(G, E ) , respectively. Suppose that x ( t ) = x ( t , to , xo) is any solution of (4.1.1). DEFINITION 4.7.1. A set G is said to be asymptotically self-invariant with respect to the system (4.1.1) if, given any monotonic decreasing sequence { ep} , e p ---f 0 as p + CO, there exists a monotonic increasing sequence {tJE)}, t , ( ~ -+ ) 00 as p + 00, such that xo E G, to 3 t p ( E ) , implies x(t)

C S(G, cD),

t

2 to,

p

=

1, 2,... .

be an (n - k) dimensional manifold containing the set G. Let We shall assume that G is an asymptotically self-invariant set with respect to the system (4.1.1). DEFINITION 4.7.2. T h e asymptotically self-invariant set G of the system (4.1.1) is said to be (AS,) conditionally equistable if, for each E > 0, there exists a tl(e), tl(e) + 00 as E + 0, and a S = S(t, , E ) , to 2 t , ( e ) , which is continuous in to for each e such that x(t)

provided

c S(G, €1, xo E

t

3 to 2 tl(.),

S(G, 6) n M n - - k )

*

On the basis of this definition, it is easy to formulate the remaining notions (AS,)-(AS,) corresponding to (Cl)-(C8) of Sect. 4.4. The following theorem gives sufficient conditions for the set G to be asymptotically self-invariant with respect to the system (4.1.1).

THEOREM 4.7.1.

Assume that

(i) g E C[J x R+N,R N ] , and decreasing in u for each t E J ;

g ( t , u ) is

quasi-monotone

non-

4.7.

STABILITY OF ASYMPTOTICALLY SELF-INVARIANT SETS

299

V I EC [ J x S(G,p), R+N],V ( t ,x) is locally Lipschitzian in x,

(ii)

< c Vi(t, x), N

b(ll w(x)ll)

( t , x)

E

i=l

and

c Vi(t,x) N

=

J

a(t)

X

S(G,p),

if

x E G,

6 E .X,

(4.7.2)

(4.7.3)

2-1

where u E 9; (iii) Vi(t,x) = 0 (i = 1, 2,..., k), k < n, if x E M(n--li), where M(n-k) is an (n - k) dimensional manifold containing the set G; (iv) f E C [ J x S(G,p), R"],and

o+v(t, x) < g(t, v(t,x)),

( t , x) E

J x S(G,p ) ;

(v) for any function P(t, u), which is continuous for t 2 0, u 2 0, decreasing in t for each fixed u, increasing in u for each fixed t such that lim lim/3(t, u ) t-m

u-0

=

0,

(4.7.4)

we have

provided uiO = 0 (i = 1, 2, ..., k), where u(t, t o , uo) is any solution of (4.1.4). Then, the set G = [x E Rn : 11 w(x)lI invariant with respect to (4.1.1).

=

01 is asymptotically self-

Proof. Let x0 E G. Since G C M(n-,c. , it follows that x0 E M(n--li). As a consequence, we have, by (iii), Vi(to, XJ = 0 (i = 1, 2,..., k), k < n. We choose uiO= V i ( t o xO) , (i = 1, 2 ,..., N ) . Then, because of (4.7.3), we obtain N

1

c N

ui0

i=l

=

i=l

Vi(t0 9

(4.7.6)

xo) = O(t0).

Consider the function y ( t ) = p(t, u(t)), which decreases to zero as t+ co because of the assured monotonic properties of the functions /3 and u. Let now {E,} be a decreasing sequence such that E , + 0 a s p -+ 00. Then, the sequence {b(e,)} is a similar sequence. Since y(to)+ 0 as to + m, it is possible to find an increasing sequence { t p ( e ) } ,t p ( e ) + co as p + 00, such that y(t0)


to

2 tD(E),

P

=

1 , 2,-

*

(4.7.7)

300

CHAPTER

4

We claim that xo E G implies that x ( t ) C S(G, el,), t 2 to 3 t p ( c ) , for each p = I , 2, .... Suppose, on the contrary, that there exists a solution x ( t ) of (4.1.1) such that xo E G, to >, t , ( ~ )for a certain p , (1 w(x(t,))I(= c p €or some t = t, > to >, t , , ( ~ )and ,

II 44t))ll G ~p < P , For t

E

t

E

[to 9 t i l e

[t,, , t J , we obtain, on account of Theorem 4.1.1, the inequality

where r ( t , t, , uo) is the maximal solution of (4.1.4). At t = t, , we arrive at the contradiction

< c Vdtl N

b(%)

N

7

+l))

i=l

< 1Y d t l < i=l

>

to

> %I)

P(t0 1 4 t O ) )

= At,)


( 4

making use of the relations (4.7.5), (4.7.6), (4.7.7), and (4.7.8). This proves that the set G is asymptotically self-invariant with respect to the system (4.1.1). If we assume that the set u = 0 is asymptotically self-invariant with respect to the auxiliary system (4.1.4), we have the following definition parallel to Definition 4.7.2.

DEFINITION 4.7.3. T h e asymptotically self-invariant set u = 0 of the system (4.1.4) is said to be (ACT) conditionally equi-stable if, for each E > 0, there exists a t , ( ~ ) t, , ( ~-+ ) 00 as E 4 0, and a 6 == S ( t o , E ) , to >, t l ( c ) , which is continuous in to for each E , such that

provided N

1uio < 8, i=l

ui0 = 0

(i = 1 , 2 )...,k).

T h e following theorem assures the conditional equistability of the asymptotically self-invariant set G.

4.7.

STABILITY OF ASYMPTOTICALLY SELF-INVARIANT SETS

301

THEOREM 4.7.2.

Suppose that hypotheses (i), (ii), (iii), and (iv) of Theorem 4.7.1 hold, except (4.7.3). Assume further that the set G is asymptotically self-invariant and

c N

i=l

< a(t, I1 w(x)II),

Vi(t,x)

(4 "4E J x S(G, P),

(4.7.9)

where the function a(t, r ) is defined and continuous for t >, 0, r 2 0, monotonic decreasing in t for each fixed r, monotonic increasing in r for each fixed t, and lim lim a(t, T ) = 0. t-m

r-0

Then (ACT)implies (AC,). Proof. Let 0 < E < p be given. Assume that the definition (ACT)holds. Then, given b ( ~ > ) 0, there exist a t , ( ~ ) t,l ( e ) -+ co as E -+ 0, and a 6 = S ( t o , E ) , to >, t , ( ~ such ) that N

1 udt, t o , uo) <

2 to 3 tl(4,

t

WE),

i=l

(4.7.10)

provided N

1 uio < 6,

uio = 0

(i

=

1, 2 ,...)k).

(4.7.11)

i=l

Choose uiO = V i ( t o ,x,,), i = 1, 2,..., k, and xo E M(n-k)so that ui,, = 0 (i = 1, 2, ..., R ) , by condition (iii). If we now make the choice that CF=l uio = a(to , 11 w(xo)[l),the assumptions on a(t, r ) imply the existence of positive numbers tz(.) and 6, = Sl(t0 , E ) , to >, t Z ( e ) ,such that 4to

9

ll w(x0)ll)

< 8,

I/ w(x0)lI G 6,

9

(4.7.12)

provided to 3 t2(e). Let t3(c) = max[t,(E), t 2 ( c ) ] .I t can then be claimed that, if xo E S(G, S,) n M(n--k), we have x(t, to , xo)C S(G, E ) for t >, to >, t3(c), where x(t, to , xo) is any solution of (4.1.1). Let us assume that this is not true. Then, there exists a solution x ( t ) of (4.1.1) such that, , x(t) C S(G, c) for t E [ t o , t,], whenever xo E S(G, 6,) n t, > to 2 t 3 ( e ) , and x(tl) lies on the boundary of S(G, c). This implies that

I/ w(x(t))ll

and

<

€9

t

E

[to tll, Y

11 w(x(t,))ll = E. Thus, there results (4.7.13)

302

CHAPTER

Furthermore, for t

E

4

[to, t J , we obtain the inequality

in view of Theorem 4.1.1, r ( t , t o , uo) being the maximal solution of and the relation (4.7.12) guarantee that, (4.1.4). Since the choice of us,, whencver x,,E S(G, 8,) n M(%-,J, the condition (4.7.1 I ) is satisfied, it is easy to derive, from (4.7.10) and (4.7.14), the inequality

Th i s relation is incompatible with (4.7. I3), thereby establishing (AC,).

COROLLARY 4.7.1. Under the assumptions of Theorem 4.7.2, the conditional equistability of the trivial solution of (4.1.4) assures the definition (AC,). We can easily prove the statements corresponding to the definitions (AC,)-(AC,), on the basis of Theorem 4.7.2. T o show the close relationship between theorems of this section and Sect. 3.14, we shall merely state below a theorem parallel to Theorem 3.14.1.

THEOREM 4.7.3.

Assume that

(i) g E C [ J x R , N , R N ] g, ( t , u ) is quasi-monotone nondecreasing in u for each t E J , and the asymptotically self-invariant set u = 0 of (4.1.4) is conditionally uniformly stable; V ( t ,x) is locally Lipschitzian in x, and (ii) V E C [ J x S(G,p ) ,

4 441) <

c Vdt, x) < 4 N

w(x)II),

i=l

-, Y 11 w(x)ll < p and t 3 O(Y), where a, b E X and the function B(r) 2 0 is monotonic decreasing in Y for 0 < r < p ;

for 0

(iii) V,(t,x) 0 (i 1, 2,..., k ) , k < n if x E M ( l L - k,~where M(n-k) is an ( n k ) dimensional manifold containing the set G; ~

:

~

(iv) f~ C[/ >\ S(G,p ) , R"],the set G is asymptotically self-invariant with respect to the system (4.1. l), and D-i-V(t,x)

< g(t, q t , x)),

4.7.

STABILITY OF ASYMPTOTICALLY SELF-INVARIANT SETS

303

Then, the asymptotically self-invariant set G is conditionally uniformly stable. Analogous to the boundedness concepts (Bl)-(B8)defined in Sect. 3.13, we have the following weaker notions. DEFINITION4.7.4. The system (4.1.1) is said to be, with respect to the set G, (EB,) conditionally eventually equi-bounded if, given a 2 0, there exist tl(a) > 0 and /3 = ,f3(t,, a ) , to 2 tl(a), which is continuous in to for each a , such that x(t)

c S(G, 81,

provided x,, E

t

3 to 3 tl(a),

S(G,a ) n M(+-k) .

The remaining definitions (El?,)-(EB,) may be easily formulated. As previously, the definitions (EBf)-(EB$) refer to the conditional boundedness concepts with respect to the system (4.1.4). A typical theorem on eventual boundedness is the following:

THEOREM 4.7.4.

Suppose that

(i) g E C [ J x R+N, decreasing in u for each t (ii) V E C [ J x R",

and g(t, u) is quasi-monotone

J;

E

non-

V ( t , x) is locally Lipschitzian in x, and N

b(II w(x>II)

< i=l1 vi(t, < 44 I1 w(x)II),

t

3 0,

x E R",

where a(t, r ) is continuous for t >, 0, r 2 0, montonic decreasing in t for each Y, monotonic increasing in r for each t , and lim lim a(t, r )

2-02

7-0

= 0,

<

and b E Z on the interval 0 Y < GO such that b(r) + co as r -+ co; ; (iii) Vi(t,x) = 0 (i = 1, 2,..., k ) , k < n, if x E (iv) f

E

C[J x R", R"], and D+V(t,x) ,< g(t, v(t,x)),

Then the condition (EBf) implies (EB,).

( t ,x)

E

1x

R".

304

CHAPTER

4

Proof. Let (L: 3 0 be given. Suppose that x" E S(G, a ) n M(n--h-) . Because of the assumptions on a ( t , Y), it is possible to find two positive numbers y = y(m) and t l ( m ) such that 4t" a) 9


if

to

3 tl(a).

(4.7.15)

Assume that (El?:) holds. Then, given y > 0, there exist two numbers and p = B ( t o , a), t , 3 t2(n),such that

whenever N

1 Ui" < y,

(i = 1 , 2)...)k).

Ui" = 0

i=l

(4.7.17)

t3(a)= max[t,(cu), tz(a)]. Choose uio = Vi(t,, xo), to 3 t3(n) = a(t,,,I] w(x,)ll).Since x, E S(G, a ) n M(n--k), this implies, in view of condition (iii), that ui0 = 0 (i = I, 2, ..., k). Moreover, the condition (4.7.17) is satisfied, in view of this choice, and consequently (4.7.16) is true. Since h(r) + 03 as Y -+ CO, there exists a p1 = & ( t o , a) such that

Let

(i = 1 , 2,..., A), and Cf=lui,

W l )

3 P*

(4.7.18)

We can now conclude that (EB,) holds with and t3(a).T h e assumption that this is false leads to the existence of a t , > t, 3 t3(a) and a solution x ( t ) with xo E S(G, m) n M(n-,~) , such that

/I w(x(t1))lI = P1 at t = t, > to 3 tB(a).By assumption (iv) and Theorem 4.1.1, we can infer that

1 Vdt, 4 t ) ) s c N

N

2-1

z=1

Y,(t,

4,

7

Uo),

> t" 3 t3(a),

t

which, because of the relation (4.7.16) and assumption (ii), shows that

-= c N

4Pl)

Z-1

VL(tl

4tl))

G

c N

2=1

rz(t1

7

to

3

Uo)

< P.

This is a contradiction to the choice of in (4.7.18), and hence we claim that (EB,) holds. T h e proof is complete.

4.8.

STABILITY OF CONDITIONALLY INVARIANT SETS

305

4.8. Stability of conditionally invariant sets We shall introduce in this section the concept of a conditionally invariant set with respect to a given set and consider the stability properties of such sets.

DEFINITION 4.8.1. Let A and B be any two subsets of Rn such that A C B. Then, the set B is said to be conditionally invariant with respect to the set A for the differential system (4.1.1) if xo E A implies that x ( t , t o ,xo) C B for all t 2 to 3 0. Let w E C[Rn,Rm],and let 11 w(x)ll mean the same norm of w defined by (4.7.1). Let us continue to use the sets G

= [X E

R" : 1) w(~)ll= 01,

S(G, c)

= [X E

R" : [I w ( ~ ) )< l €1,

S(G,6)

= [X E

Rn : 1)

ZU(X)~I

< €1,

and let us designate the set S(G, a) by B. Suppose that the set B = S(G, a) is conditionally invariant with respect to G, for some a > 0. Let M(npk)denote, as before, an ( n - k) dimensional manifold containing the set G. We define S(B, C)

=

S(G, 01

+ c),

E

> 0.

DEFINITION 4.8.2. T h e conditionally invariant set B with respect to the set G and the system (4.1.1) is said to be (CC,) conditionally equistable if, for each E > 0 and to E 1, there exists a positive function S = S ( t o , E ) , which is continuous in to for each E , such that

Evidently, on the strength of (CC,), we can define (CC,)-(CC,) corresponding to (Cz)-(C8).

REMARK4.8.1. We observe that the set B need not be self-invariant. If 01 = 0, these definitions coincide with (C,)-(C,), that is, the conditional stability concepts of the self-invariant G.

306

4

CHAPTER

T o define the corresponding definitions (CCF)-( CC:) for the auxiliary system (4.1.4), let us define the set, for some p > 0, N

1 ui < 81,

u E R+N:

(4.8.1)

iL1

and assume that B* is conditionally invariant with respect to the set 0 and the system (4.1.4).

zi =

DEFIKITION 4.8.3. T h e conditionally invariant set B* with respect to the set 11 - 0 and the system (4.1.4) is said to be (CCF) conditionally equistahle if, for each E > 0 and t,, E J , there exists a positive function S = S ( t , , E ) , which is continuous in t, for each E , such that N

1 ~ t ( ttn,

7

un)

z=l

provided N

1

Ui"

1-1

THPORERI 4.8.1.

< 6,

ui,

B

=

+ 0

t

E,

2 to

7

(i = 1 , 2 )...,K).

Assume that

, = 0, and g(t, a) is quasi-monotone (i) g E C [ J x R+N,R N ] , g ( t0) nondecreasing in u for each t E J ; (ii) V E C [ J x R", R . , N ] ,V ( t ,x) is locally Lipschitzian in x, and N

b(Il 4.z)lI)

< 1 Vdt, x) < 4w(x)II),

( t , x) E

J x R",

2-1

where a , h E f on the interval [0, a)and h(r) + co

(iii) f

E

as

Y + co;

C [ J Y R", R"], and D - l ' ( t , x)

< g ( t , V ( t ,x)),

( t ,x) E J x R".

Then, if the set B* is conditionally invariant with respect to the set 0 and the system (4.1.4), the set B = S(G, a ) , where N = b-l(P), is conditionally invariant with respect to the set C and the system (4.1.1).

zi =

PFoof. Assume that the set B* defined by (4.8.1) is a conditionally invariant set. This implies that, if ui0 = 0 (i = 1, 2 ,..., N ) , N

1 ~ , ( tt ,o , 0 ) < P,

7=1

t

2 t o 3 0.

(4.8.2)

4.8.

STABILITY OF CONDITIONALLY INVARIANT SETS

307

Let us choose uin = V i ( t o x ,,,) (i = 1, 2, ..., N ) . Then, it follows that xo E G and Vi(to, xo) == 0 (i = 1, 2, ..., N ) hold simultaneously. By Theorem 4.1.1, we obtain

where r ( t , to , u,,) is the maximal solution of (4.1.4) through ( t o ,u,,). Since b(ll w(x)ll) CC, Vi(t,x), we readily get the inequality

<

in view of (4.8.2) and (4.8.3). As a consequence, we deduce that, if xo E G, x(t, t,, , xo)C S(G, a ) , t 3 to , where 01 = h-l(P). T h e Conditional invariancy of the set B is immediate, and the proof is complete. REMARK 4.8.2. Notice that the p occurring in (4.8.2) may depend on to , in which case cx depends on t o , and, as a result, the set B depends on t o . This suggests that the invariant sets we generally consider are, in a sense, uniform invariant sets, and perhaps a classification of invariant sets and the study of their stability properties may be of some interest. Regarding the stability behavior of the conditionally invariant set B, we have the following:

THEOREM 4.8.2.

Assume that conditions (i), (ii), and (iii) of Theorem 4.8.1 hold. Suppose further that Vi(t,x) = 0 (i = 1, 2,..., A ) , k < n, if x E MG-,;) . Then, if one of the conditions (CCT)-(CC$) is satisfied, the corresponding one of the conditions (CC,)-(CC,) is assured.

Proof. We shall only indicate the proof corresponding to the statement (CC,), that is, the conditional quasi-uniform asymptotic stability of the conditional invariant set B. Let E > 0, y > 0, and to E J be given. Suppose that

V i ( t o xo) , = 0 (i = 1, 2, ..., k). Choose uio = Vi(t, , x,,)(i = 1, 2,..., N ) . Then, we have by Theorem 4.1.2 that every solution x(t, t,, , x,,) of (4.1.1) exists for t 3 to and satisfies

so that we can infer that

"(4 x ( t , t o , xo))

< Y ( t , t" , 4,

t

2 to,

where r ( t , t o ,u,,) is the maximal solution of (4.1.4). Define y1

(4.8.4)

= a(y),

308

CHAPTER

4

+

and assume that (CC,") holds. Let 01 = h-'(P). Then, given b(oc C) > 0, > 0, and to E 1, there exists a positive number T = T ( y , e ) such that

y,

V

1

U , ( f , f,,

, UO)

t=1

< b(a

+

+ T,

(4.8.5)

(i = I , 2,..., k).

(4.8.6)

t

c),

2 4,

provided V

2 U," < y1 ,

/

u,,, = 0

I

Clearly, by the choice of y1 and u i O , the condition (4.8.6) is satisfied. Hence, wc obtain, using (4.8.4), (4.8.5), and the fact that

WI w(4Il)

c V,(f, N

d

4

7

2-1

the relation b(ll zu(.v(t, t o , ~ 0 ) ) l l )

< b(a

+

e),

t

3 f0

+ T,

whenever xo E S(G, y ) n M(,L--I,) . Evidently, this implies that the conditionally invariant set 13 is conditionally quasi-uniform asymptotically stable. T h e proof of the theorem is thus complete.

4.9. Existence and stability of stationary points This section is concerned with the conditions sufficient to assure the existence of yo satisfying (4.9.1)

f(Yo) = 0

and the stability of the solution x ( t ) = y o of the autonomous differential system x' = f ( x ) , x(0) = x0 , (4.9.2) where f E C[R",R"].

TIIEORENI 4.9. I . (i)

Assume that

I/ E C[R",K + N ] , V(x)is locally Lipschitzian in N

C

V ( ( x )4

as

11 x 11

---f

x,and

co;

t=1

, is quasi-monotone nondecreasing in u , and (ii) g E C [ R ,N , R N ] g(u) D+V(x) g( V ( x ) ) ,x E RtI;

<

4.9.

EXISTENCE AND STABILITY OF STATIONARY POINTS

309

(iii) Q E C[R+N,R+], Q(v) is monotone nondecreasing in w, and Q(v(x)) = 0 only if f(x) = 0; (iv) for a certain u o , the system U'

u(0) = u g

= g(u),

>0

(4.9.3)

possesses the maximal solution r ( t , 0, uo) defined for 0 that r ( t , 0, uo) is bounded and satisfies Q(r(t,0,U,]))

--f

0

as

t

+

co.


such (4.9.4)

Then, if x ( t ) is any solution of (4.9.2), it exists and is bounded for t E J , and every cluster point (a-limit point) yo of x(t) satisfies (4.9.1).

Proof. Let x ( t ) be a solution of (4.9.2). Then, by Theorem 4.1.2, x ( t ) exists for 0 t < m. Furthermore, if V(t,, xo) uo ,

<

<

< r(4

V(x(t))

O,UO),

t

2 0,

(4.9.5)

where r(t, 0, u,,) is the maximal solution of (4.9.3). T h e assumptions that r ( t , 0, uo) is bounded for t 0 implies, in view of (i), the boundedness of x(t). Also, the function Q being monotonic nondecreasing, we have, by (4.9.5),

8(Q ( t ) ) ) G Q(r(t, 0, uo)), which, on account of (4.9.4), guarantees that

-

lim O ( V ( x ( t ) ) )= 0.

t-m

Hence, every w-limit point y o of x ( t ) satisfies Q(V(YoN

= 0,

and (4.9.1) follows, because of assumption (iii). This proves the theorem.

COROLLARY 4.9.1. Let the hypotheses of Theorem 4.9.1 hold, except that Q( V ( x ) )= 0 only if f(x) = 0 is replaced by Q(V(x))= 0

only if

U ( x ) = 0,

where U E C[Rn,R n ] . T h e n the assertion of Theorem 4.9.1 remains valid if (4.9.1) is replaced by V Y n ) = 0.

310

CHAPTER

4

THEOREM 4.9.2. Suppose that the conditions of Theorem 4.9.1 are satisfied. In addition, assume that, for every uo > 0, the maximal solution r ( t , 0, u,,) of (4.9.3) exists on 0 < t -: a3 and is uniformly bounded for t 3 0 and bounded uo and satisfies (4.9.4) uniformly for bounded ZL,, . Then the set Z

=

[x : f ( x ) = 01

is nonempty and connected.

Proof. It is clear that, under the assumptions of the theorem, every solution x ( t ) of (4.9.2) exists for 0 t a,and, given any 01 > 0, there exists a P ( N ) such that

<

I1 x(t)ll

<<

t

&a),

2 0,

<

provided 11 xo I / n. Furthermore, every w-limit point y o satisfies (4.9.1). Ry ‘I’heorem 4.9.1, it follows that the set Z is nonempty. Hence, only connectedness remains to be proved. Let E, (Y be arbitrarily positive numbers. Then, it follows from (ii) that there exists a 6 = a(€, a ) > 0 such that

p(I+)) 2 6

if

d(x, Z )

2 E, /I x 1) < p(a).

Hence, by thc uniformity of (4.9.4) and by (4.9.5), it is possible to find a T - T ( E <x) , such that d(.x(t),Z )

<E

if

t

3 T , 1) so(1

< a.

(4.9.6)

<

Since ail solutions ~ ( tof) (4.9.2), for arbitrary xo , exist on 0 t < GO, it follows by a generalization of H. Kneser’s theorem that the set Za(t) of points z reached by some solution of (4.9.2) at a time t 3 0, when 11 x,) j j 1, that is,

<

ZJt) = [z : z = x ( t ) , I/ XI) 1)

< a],

is closed and connected. We notice that the set

Zm= Z n [ x : I I . Y / ~

< a]

is contained in Zn(t)for t >, 0 [for, if yo E 2, then x ( t ) = y o is a solution of (4.9.2)j. I,ct s,, , xt,be two arbitrary points of .Z;, C Zm(T ) . Then, there exists a finite set of points x , ~= x,,,x1 ,..., xN+l xb in Z,(T) such that E, if i / I s, - s~~~/I 0, 1,..., N . I n view of (4.9.6), there is a point

4.10.

NOTES

311

xi E 2 satisfying 11 xi - xi 11 < E for i = 1, 2,..., N . Hence, there is a finite set of points x , = x , = xu, x l , ..., x N , xN+I = xN+l = xb in 2 such that /I xi - xi+l 11 < 3 ~ for , i = 0, I , ..., N . Since c is arbitrary and 2 = n 2, , 01 3 0, the set 2 is connected. This proves the theorem.

COROLLARY 4.9.2. Assume that the conditions of Theorem 4.9.2 hold. Suppose that the zeros off(.) are isolated. Thenf(x) has a unique zeroy, , and the solution x ( t ) = y o of (4.9.2) is completely uniformly asymptotically stable. 4.10. Notes

Section 4.1 introduces comparison theorems that are useful when several Lyapunov functions are employed (see Lakshmikantham [ 131). T h e results of Sects. 4.2 and 4.3 have been taken from the work of Matrosov [l]. For the results contained in Sect. 4.4, see Lakshmikantham [13] and Matrosov [2]. Converse theorems of Sect. 4.5 are due to Lakshmikantham et al. [I], whereas the results of Sect. 4.6 concerning the stability in tube-like domains are taken from the work of Charlu et al. [l]. T h e notion of asymptotically self-invariant sets is introduced by Lakshmikantham and Leela [l], and the contents of Sect. 4.7 are based on their work. T h e results of Sect. 4.8 dealing with the criteria for the stability of conditionally invariant sets are due to Kayande and Lakshmikantham [l]. Section 4.9 deals with the results due to Hartman [6]. For related work using several Lyapunov functions, see Antosiewicz [4], Bellman [4], D’Ambrosio and Lakshmikantham [l], and Lakshmikantham and Verma [I].

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VOLTERRA INTEGRAL EQUATIONS

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Chapter 5

5.0. Introduction We treat in this chapter certain problems concerning Volterra integral equations. We consider successively basic integral inequalities, local and global existence theorems, the existence of extremal solutions, uniqueness of solutions, bounds, and error estimates of approximate solutions. We discuss the asymptotic behavior of solutions by suitably choosing Lyapunov-like functions and examining their properties with respect to the integral equations. Using functional analytic methods and the concept of admissibility, we obtain certain general results concerning the behavior of solutions of integral equations from which a number of results may be deduced as particular cases regarding existence, uniqueness, boundedness, and asymptotic behavior. Finally, we indicate some results on a class of general integrodifferential inequalities.

5.1. Integral inequalities We have considered in Sect. 1.10 those integral inequalities that are reducible to differential inequalities. We shall now discuss general integral inequalities. A principle result in integral inequalities is the following.

THEOREM 5.1.1.

Assume that

(i) K E C [ J x J x R, R ] ,K ( t , s, x) is monotone nondecreasing in x for each fixed ( t , s), and one of the inequalities (5.1 . I )

is strict, where x, y , h E C [ J ,R ] ; 315

316

(ii)

5

CHAPTER

4 t O ) . Y(t">.

Then, we have


s(t)

t

(5.1.2)

2 to.

Pyoof. Assume that the conclusion (5.1.2) is false. Then, there exists a t, such that 4 t l ) = Atl)!

%(t)



t,

(5.1.3)

< t, .

Clearly, by (ii), t , > t o . Since K is monotone nondecreasing in x, it follows from (5.1.3) that K(t,

S,

7

44) < K(t, > S, Y(S)),

and consequently, using (5.1.l), we arrive at the inequality x(Q

< h(tJ -t-

1

tl

to

K(tl , S,

< h(t,) + J to K(11

4s)) ds

tl

9

S,Y(S))

ds

< Y(t1). This is a contradiction to the fact that x ( t J = y(tl).Hence, the inequality (5.1.2) is true. Let us now consider the integral operator defined by K+

=

i' K ( t ,

S,

4(~))dS.

(5.1.4)

t0

DEFINITION 5.1. I . We shall say that the integral operator K is monotone nondecrcasing if, for any $2 E C [ J ,R] such that, for any t , > t o ,

<

to G t

< tl

9

implies K#l(tl) G K+ztt,)*

5.1.2. Let the integral operator K defined by (5.1.4) be monotone nondecrcasing. Suppose further that, for t > t o ,

'rHEORER.1

x

-

Ks < y

-

Ky,

(5.1.5)

5.1. where x, y

E

INTEGRAL INEQUALITIES

317

C [ ] ,R]. Then ~ ( t , < ) y(t,) implies x(t)
t

2 to.

Proof. We proceed as in Theorem 5.1.1 and obtain the relations (5.1.3). Since the integral operator K is assumed monotone nondecreasing, we have, by (5.1.3), W t l ) G KY(t1). (5.1.6) As a result, (5.1.5) and (5.1.6) yield x(t1) = x(t1) - Kx(t1)


~

KY(t1)

< Y(tl>. This contradicts the fact that, at t proof is complete.

=

+ Kx(t1)

+W t l )

t, , x(tl) = y ( t l ) , and hence the

DEFINITION 5.1.2. A function u E C [ ] ,R] is said to be an under function of the integral equation x=h+Kx (5. I .7) if it satisfies the inequality u
Similarly, u is said to be an over function of (5.1.7) if it verifies the inequality u>h+Ku,

whereas if u satisfies Eq. (5.1.7), it is said to be a solution of (5.1.7). T h e following theorem, whose proof is a consequence of Theorem 5.1.2, shows the relation between solutions, under and over functions of the integral equation (5.1.7).

THEOREM 5.1.3.

Let the integral operator K defined by (5.1.4) be monotone nondecreasing. Suppose that x, y , z E C [ ] ,R] be an under function, a solution, and an over function of (5.1.7), respectively on [ t o , co). Then x(t0) < Y ( t d < 4 t a ) implies t 2 to * x(t)
w,

T h e foregoing results can easily be extended to systems of integral inequalities. We prove a result parallel to Theorem 5.1.1 only.

318

CIIAPTER

THEOREM 5.1.4.

5

Assume that

I+, R"], K ( t , s, x) is monotone nondecreasing (i) K E C [ J x J in x for each ( t , s), and one of the inequalities

< h(t)+

X(f)

Y(f)

is strict, where x,y, Iz (4

z'(to)

E

-f

4t)

+

st

K(t, s, .+)) ds,

fn

Jt

K ( t , S , Y ( S ) ) ds

t0

C [ J , R?,];

. Y(4,)-

'l'hese conditions imply r(t) < y ( t ) ,

f

2 to.

Proof, If the assertion of the theorem is not true, then the set

u 7,

=

[t E [ t o , a): .%(f)

2 y,(t)l

1=1

is nonempty. 1,et t , inf %. By (ii), it is clear that t , > to . Furthermore, sincc 2 is closed, t , E 2, and consequently there exists an i n d e x j such that ~

.t,(td = Y A f A u,(f) XL(f)

.%

t -I f, ,

Y,(t),

f,,

1 :

Y,(f),

f,,

'i

t

:t , ,

#j.

From the monotonicity of K ( t , s, x) in x, it results that

< h,(t,) + J

11

K,(t, , S , Y ( S ) ) ds

'1,

.Y,(tl),

which is an absurdity, since xj(t,) empty, and the theorem is proved.

= y j ( t l ) .This

shows that the set 2 is

5.2.

319

LOCAL AND GLOBAL EXISTENCE

5.2. Local and global existence Let us consider the integral equation x ( t ) = x,(t)

+ J' K ( t ,

s,

to

4s)) ds.

(5.2.1)

We shall, first of all, prove a local existence theorem analogous to Caratheodory's existence theorem for ordinary differential equations. Assume that xo E C[[to, to

THEOREM 5.2.1.

K E C [ [ h, t o

+ a)

X

[to , to

+ a ) , R"],

+ a ) x R", Rn],

and the function

is summable on [ t o ,to ,f3 =

+ a ) , where sup

to':t/-to+a

11 x,(t)ll

and

<

t , >, 0.

Then, there exists a number 0 < 01 a such that the integral equation (5.2.1) has at least one solution on [ t o ,to 011.

+

Proof. Since the proof is similar to the existence theorem of Caratheodory, we shall be brief. Consider the sequence of approximations {$j} defined by + , ( t ) = xo(t),

d,(t) = xo(t) =

to

< t < to + (ah)

Ct,,-(n'i)

J to

(j

=

K ( t - (ah),s, a+)) cis,

1,2,...)7 to

+ (ah) < t < to + (.j

=

01

1, 2)...).

I t is easy to show that the sequence {+j} forms a family of uniformly bounded and equicontinuous functions on [ t o ,t,, N], where (Y is defined by the relation P(a) p , where

<

+

(5.2.2) Since P(0) = 0, P ( t ) is continuous and monotonic nondecreasing, the existence of such an IY is clear. T h e n {+j} contains a subsequence con-

320

CHAPTER

5

+

verging uniformly on [t,, , t, A) to a limit function ~ ( t )which , can be shown to satisfy (5.2.1), using the usual techniques. T h e proof is complete.

A4global existence theorem that includes Theorem 2.1.2 can be proved using 'I'ychonoff's Theorem 2.1.1. 5.2.2. 1,et K E C [ J x J x R", R"],G E C [ J x J x R, , R,], G(t,s , z/) be monotone nondecreasing in u for each ( t , s), and

'rHkom;Li

k'(C

1;

J,

.Il + G(t,s, II x 11).

.\ssume that, for every continuous function u,(t) ~ ( f = )

[' G(t,

u,(t) 1

to

S,

(5.2.3)

> 0, (5.2.4)

u ( s ) ) ds

possesses a solution u(t) existing on [to , a).Then, for any x0 E C [ J ,R"] such that I/ k,,(t)[l u(,(t), there exists a solution x ( t ) of the integral equation (5.2. I .) on [to, a)satisfying

<

I1

l(t)l!

:u(t)>

t

3to.

Pyoof. The proof is very much the same as that of Theorem 2.1.2. I n the present cdsc, the integral operator T defined by (2.1.5) takes the form ~ ( v ) ( t=) l o ( t )

+ J~~ ( t ,

s, x(5))

U'

'l'he space of continuous functions B, the topology on B , and the closed,

con\e\, and bounded set B, remain the same as in the proof of Theorem_ 2.1.2. ~ _ 'I'he operator T is compact in the topology of B , and hcnce T(B,,) is compact, sincc the set B, is bounded. T o show T(B,,)C B,) , we notice that 11 T(x)(t)ll

!I %dt)lI+ J t II k'(4 s,

ds

.(s))ll

'I,

..u,(t)

+ It G(f, s, /I n(s)Il) ds

u,(t) 1

I,,

1' G(t,

-

J,

u ( s ) ) ds

=

~(t),

'0

using the monotony of G. We can conclude the validity of the theorem, on the basis uf Theorem 2. I .2.

T h e notion of maximal and minimal solutions may be introduced.

5.2.

32 1

LOCAL AND GLOBAL EXISTENCE

DEFINITION 5.2.1. Let r ( t )be a solution of the integral equation (5.2.1) existing on [ t o ,to u). Then r ( t ) is said to be the maximal solution of (5.2.1) if, for every solution x(t) of (5.2.1) existing on [ t o ,to a), the inequality

+

+

x(t)

< r(q,

t

[to , to

E

+

a)

is verified. By reversing the preceding inequality, we may define the minimal solution of (5.2.1). T h e existence of maximal and minimal solutions may be proved under the hypothesis of Theorem 5.2.1.

THEOREM 5.2.3. Let the hypotheses of Theorem 5.2.1 be satisfied. Suppose that K ( t , s, x) is monotone nondecreasing in x for each ( t , s).

Then there exists a maximal solution and a minimal solution on [to , to a:] for a certain a: > 0.

+

Proof. We shall indicate the proof of the existence of maximal solution only. Consider, for some arbitrarily small vector > 0, the integral equation

On the basis of Theorem 5.2.1, there exists an a: solution x(t, eo) on [ t o ,to a:]. Let 0 < E~ <

+

“(to

>

> 0 such that there is a

< c0 . Then, we have

4 < x(to , El),

x(t, 4

< xo(t) + f 2 + J‘

x ( t , fl)

> xo(t)

K ( t , s, x(s, c2)) ds,

to

+ + €2

Jl

q t , s, x(s, €1)) ds.

An application of Theorem 5.1.1 yields x(t,

€2)

< x ( t , 4,

t

E [ t o , to

+ .I.

Since the family of functions {x(t, E ) } are equicontinuous and uniformly bounded on [ t o ,to a ] , it follows by Theorem 1.1.1 that there exists a decreasing sequence { c n } tending to zero as a+ co, and the uniform limit r ( t ) = lim x ( t , c,)

+

+

n -m

exists on [ t o ,to a].I t can be easily shown that r ( t ) is a solution of (5.2.1). Furthermore, to show that r(t) is the desired maximal solution

322

CHAPTER

5

+

of (5.2.1) on [ t o ,to 'Y], let x ( t ) be any solution of (5.2.1) defined on [t,, , to (21. Then, on the strength of Theorem 5.1.1, it follows that,

for

E

+ <

E,,

,

r(t)< w(t,

t E [ t o , t,,

E))

+ a].

T h e uniqueness of the maximal solution shows that ~ ( tE ), tends uniformly to r ( t ) on [ t o ,t, a ] , and therefore the proof is complete.

+

5.3. Comparison theorems

As in ordinary differential equations, an important technique is concerned with comparing a function satisfying an integral inequality by the maximal solution of the corresponding integral equation. T h e following theorem is a result of this type. 5.3.1. Let G E C [ J x J x R, , R ] , G(t,s, u ) be monotone nondecreasing in u for each ( t , s), and

THEOREM

4 t ) ,< mo(t)

+

G(t,s, 4 s ) ) ds,

Jt

t

2 to

1

t0

where m E C [ J ,R,]. Suppose that r ( t ) is the maximal solution of the scalar integral equation

+ J" G(t,

u ( t ) = u,(t)

s, 4 s ) )

ds

(5.3.1)

tU

existing on J . Then, the inequality m(to) m(t)

Proof.

<

t

y(t),

< uo(t,,)implies 2 to.

(5.3.2)

Let u ( t , c ) be any solution of the integral equation ~ ( t= ) u,(t)

+

E

Jl

4-

G(t, S, u(s)) d.r

for E > 0 sufficiently small. Since lim6+ou ( t , E ) = r ( t ) , it is enough to show that m ( t ) < u(t, E ) , t 3to. (5.3.3) Observe that m(to) < u(t0

)

E)

and

5.3.

323

COMPARISON THEOREMS

Hence, an application of Theorem 5.1.1 shows that the inequality (5.3.3) is valid. This establishes the theorem. We shall next prove an extension of the result of Theorem 5.3.1 to systems of integral inequalities. T h e proof that will be presented is simple and short and makes use of the partial ordering in Rn. Let us introduce the relation in Rn,namely, we set, for any two elements x, y E R",

<

x

iff


xi .
=

1, 2 ,..., n.

(5.3.4)

This relation induces a partial ordering in R",and it is easy to see that, for any bounded set A C R", there exists the sup A with respect to the relation (5.3.4), which is sup A

=

min[x E Rn : x


for each x

E

A].

(5.3.5)

I n fact, we need (5.3.5) only for two elements sets, in which case we have (5.3.6) x i ,yi being the components of x and y , respectively. We are now in a

position to prove the following:

THEOREM 5.3.2. Let K E C [ J x J x R",Rn],K ( t , s, x) be monotonic nondecreasing in x for each (t, s), and (5.3.7)

where x, x,,E C[/, R"].Assume that r ( t ) is the maximal solution of (5.3.8) (5.3.9) (5.3.10)

<

By (5.3.6), x ( t ) sup[y, x(t)], and therefore it follows, from the monotonicity of K and (5.3.10), that

324

CHAPTER

5

Let r * ( t ) be the maximal solution of u(t) = %(t)

+ j t F(t, s, 4s)) ds tll

existing on [t,,, a). Then, using (5.3.11) and (5.3.7), we get r*(t)

=

xn(t)

+ f F(t,

S,

r * ( s ) )ds

tU

(5.3.12)

2 x(t). It then results from (5.3.12) and (5.3.6) that sup[r*(t), x ( t ) ] = r * ( t ) ,

and consequently, by (5.3.10), F ( t , s, r * ( t ) ) : K ( t , s, r * ( t ) ) .

Thus, Y " t ) is also the maximal solution of (5.3.8). Hence, (5.3.12) proves the desired result (5.3.9). T h e proof is complete.

COROLLARY 5.3.1. Let f E C [ / x R",R"], f ( t , x) be monotonic nondecreasing in x for each t and x(t)

< xn + j t f(s, tll

x(s)) ds,

where x E C [ / , R"]. Suppose that r ( t ) is the maximal solution of Y' - f ( t , ~ ) , Y(tn)

= XO,

existing on [t,,, m). Then, s(t)

< r(t),

t

to.

5.4. Approximate solutions, bounds, and uniqueness Let us define an approximate solution of the integral equation (5.2.1).

DEFINITION 5.4.1.

Let x E C [ / , R"], and satisfy

5.4.

APPROXIMATE

SOLUTIONS,

BOUNDS, AND UNIQUENESS

325

where 6 E C [ J ,R,]. Then x ( t ) is said to be a &approximate solution of (5.2.1). T h e difference between an approximate solution and a solution is given by the following result.

THEOREM 5.4.1. Assume that (i) K E C [ J x J x R”,R”],G E C [ J x J x R, , R,], G(t,s, u ) is monotonic nondecreasing in u for each ( t , s), and

II K ( t , s, x)

-

< G(t, s, I/

K ( t , s, Y)ll

x’ - y

il);

(5.4.2)

(ii) x ( t , 6) is a kipproximate solution of (5.2.1), where 6 E C [ J ,R,]; (iii) r ( t ) is the maximal solution of u(t) = 8(t)

+ j’G(t,

S, u(s))ds

(5.4.3)

tU

existing on [to , co). Then, if y ( t ) is any solution of (5.2.1) existing on [ t o , a), we have

/I x ( t , 8) - Yiqll Proof.

< r(t),

t >, t o .

(5.4.4)

Consider the function

where x(t, 8) and -y(t) are &approximate solution and solution of (5.2.1), respectively. Then, using (5.4.1) and (5.4.2), we get m(t)

=

I1 x(t, S) - x d t )

< S(1) +

1’

~

G(t, S ,

Jt to

WZ(S))

K ( t ,s, x(s, 8)) ds /I

ds.

to

An application of Theorem 5.3.1 now yields

<

m(t) = I1 4 4 8) - r(t)l/ r ( t ) ,

f

2 t” ,

and the proof is complete. T h e next theorem offers an estimate of the growth of solutions of (5.2.1).

326

CHAPTER

T m o n m I 5.4.2.

5

Suppose that

(i) I< E c'[J x J >, I<", H i ' ] , G E C [ J x J x H , , R,], G(t,s, u ) is monotonic nondecreasing in ZL for each ( t , s), and (5.4.5)

.G(t,s, II .y 11);

I1 K ( t , s, 1)I.

(ii) r ( t ) is the inaxiinal solution of (5.3.1) existing on [ t o , co); (iii) ~ ( t is) any solution of (5.2.1) existing on [to, co) such that I/ -vo(t)li 4 t ) .

<

Then, we have 11 .v(t)il s. r ( t ) ,

Proof.

If m(t)

~

11 s ( t ) ( /we , have, I1

m(t)

V"(f)il

t

3 t" .

(5.4.6)

by (5.4.5), the integral inequality

i

to

I1 k'(4 s, .(s))lI

ds

- u,,(t) I- it G(t,s, m(.s)) ds, to

and, conscquently, Theorem 5.3.1 assures (5.4.6).

'rmom1 5.4.3.

Assume that

(i) K 1 , K , t C [ J x J x R'I, R"],G t C [ J x J x R, , R , ] , G(t,s, u ) is monotonic nondecreasing in ZL for each ( t , s), and

li k-l(t,S, (ii)

,\,)

~

k-dt,5, Y)II

-- G(t,s, I I x

-

Y

11) ;

, yo E c'[J,R"],and x(t), y ( t ) are any two solutions of

i'l I q t ,

v ( t ) = q l ( t ) -f

s,

x ( s ) ) cls,

7,

y ( s ) )ds,

tll

.t

j

Y " ( t ) I-

.Y(f)

t.

k'dt,

rcspcctivcly ; (iii) r ( t ) is the maximal solution of (5.3.1) such that

I1 % ( t )

~

Y"(t)Ii

'

IJnder these assumptions, we have

u"(%

t 2 41 '

(5.4.7)

5.5.

327

ASYMPTOTIC BEHAVIOR

Proof. T h e proof is an easy modification of the proof of Theorem 5.4.2. For, setting m ( t ) = I/ x ( t ) - y(t)/land using (5.4.7), we obtain

< uO(t)+ J"

G(t,s, m(s))ds.

t0

T h e desired result follows from Theorem 5.3.1.

A uniqueness theorem of Perron type may now be stated.

THEOREM 5.4.4.

Suppose that

+

+

0, to a] X [to , to a] x R+ , R+], G(t, s, 0 ) (i) G E c[[to, G(t,s, u ) is monotone nondecreasing in u for each ( t ,s), and u(t) = 0 is the only solution of the integral equation (5.4.8)

(ii)

K

E

+ u] x

C [ [ t , ,t ,

I/ K ( t ,s, 4

-

[ t o ,t ,

+ a] x R",R"],and

K ( t , S,Y)ll

< G(4 s, II x

-

3'

11).

< t < t, 4-a. Let x ( t ) ,y ( t ) be two solutions of (5.2.1) existing on [to, t, + u ] .

Then, there exists at most one solution of (5.2.1) on to Proof. Setting m(t) = 11 x ( t )

~

y(t)lj and arguing as before, we get m ( t ) ,<

Jl,

G(t,s, m(s)) d.7,

which implies, in view of Theorem 5.3.1, that m(t)

< r(t),

t

3 t" ,

where r ( t ) is the maximal solution of (5.4.8). Since m(to) = 0 and u ( t ) fi 0 is the only solution of (5.4.8), the assertion of the theorem is immediate.

5.5. Asymptotic behavior I n this section, we shall investigate the asymptotic behavior of solutions of a Volterra integral equation of the form x'(t) =

-

0

a ( t - s),y(x(s)) ds.

(5.5.1)

328

5

CIIAPTER

This is equivalent to considering the integral equation s(t)

~

j

-1

s(0)

-

J ( t - s)R(x(s)) ds,

0

where

Before proceeding further, we shall prove some elementary lemmas. f E C[J, K , ] ,f ” ( t ) exist on 0 t < co, and k > ; -co for 0 t co, k being some positive f ’ ( t ) < 0, f ” ( t ) , constant. T h en f ’ ( t ) + 0 as t co.

I,EAIR.IA 5.5.1.

I,pt

-

,<

% ,

--f

Proof. Suppose that the conclusion is false. Then, by hypothesis, there exists a X ‘ 0 and a sequence [ t , , ; ,t,, + co as n + co,such that f ’(t,,)

-: 0.

A

Consider the intervals I,

n h e r c f,,

-

(h/2k)

~

for

(A/2k),t,]

[t,)

n

:,N ,

0 for n 3 N . Using mean value theorem, we obtain f‘(t)

~

f’(t,?) r-f”(mt

-A

uhcre t F I,,, IZ 2 N , and t 19 theoreni again, it follows that

4 -

A;

~

Lvhich clearly contradicts the fact that t + xs,and thus completes the proof.

;A,

-

Q ( t ,t,,)

t,)

-

I,

f(t)

t,, . Applying mean value

decreases to f ( m ) 3 0 as

I ~ I ~ N A I K5.5. I . l f f ” ( t ) is bounded from above rather than from below, 1,emnia 5.5.1 remains valid. Also, instead of f ” ( t ) 3 - - k , we may ask t h e right second derivative only. I,EAIXI\ 5.5.2. .‘lssuinc that a E C [ J , R],(- l)La.‘(t)>, 0 for 0 f: 00 (i 0, 1 , 2, 3 ) , and u ( t ) 4 a(0). The n, either -a’(t), a”(t) > 0 for 0 t w, or there exists a t,, 0 such that -a’(t), a”(t) I 0 for 0 ’f :< to and n ( t ) = n(to) = a ( m ) >, 0 for t,, C t co. < _

-

% ‘

5.5.

329

ASYMPTOTIC BEHAVIOR

If there exists a to >, 0 such that a’(to)= 0, then, since -a’(t) 3 0, a”(t) >, 0, it follows that a ( t ) = a(t,,) 0 for to t < co. This implies t, > 0 in view of the fact a ( t ) ji a(0). Hence, there exists a t, > 0 such that a’(tl) <: 0, and thus a’(t) a’(tl) 0 for 0 < t t, . Suppose that a”(tl) = 0. Then, as -a”’(t), a”(t) >, 0, this means that a”(t) =L 0 for t , t << GO. Hence, PToof.

>

<

<

<

a f t ) = aft1)

+ a’(tJ(t

-

<

t,

t,),

f

< 00,

which contradicts a ( t ) 3 0 for t sufficiently large. Consequently, a”(tl) > 0, and thus a”(t) 3 a”(tl) > 0 for 0 t 5 t , . T h e conclusion of the lemma is immediate.

>

REMARK5.5.2. Although a( CO) 0 is not necessarily zero, as a result of Lemma 5.5.1, we derive that a’(t) increases to zero and a”(t)decreases to zero as t + co.

LEMMA 5.5.3.

If a

E

C [ J ,R ] , ( - l ) i a i ( t )

30

t”a”(t)+ 0

ta’(t) + 0,

as

for 0 <
< co, then

+ Of,

(5.5.2)

t%z”’(t)tL,(O, a).

(5.5.3)

t

and a’(t),

ta”(t),

Proof. By the mean value theorem and the monotonicity of a‘(t), we deduce that a ( f )- a(0) = fa‘(() : ’ ta’(t) < 0,

0

\

< t < co,

from which lim,,,+ ta’(t) = 0 follows. By the second differences and the monotonicity of a”(t), we have 2a

cf, + a(0) = j2)

t ?

-

a”(()

b - a”(t) 2 0,

0 -1 ( < t

< co,

which yields lim,,,+ tza”(t)= 0. As the integrand is of constant sign, a’(t) E ZA,(O,m) follows from 00

0

E

+o+

= u(o3)

a‘(s) ds

=

lim [ a ( t ) - a(.)]

t-a t

-

-o+

a(0).

By a similar reasoning, we can show the other two statements in (5.5.3). Hence, the proof is complete.

330

5

CHAPTER

<

LEMMA 5.5.4. Let b(t) be defined on 0 < t T , b'(t) exist and be finite on 0 < t T , and b'(t) E&(E, T ) for each 0 < E < T. Let q(t, s), aq(t, s ) / a t be continuous on 0 < s, t T in t , s. Suppose that

<

b ( ~q(t )

where y that

E

E,

t)

---f

y(t)

as

< $,

E

+Of

on

0

< t < T,

C[[O, TI, R ] . Assume that there exists a +(() eL1(O,T ) such

df,t - 01, 1 b ' ( 8 q ( t , t

I for 0

+

<

-

4)19

< t < T. Then, f(t) =

fb(t n

- S)

I 43 adt, at E ) I G W) t

-

q(t, S) ds

is continuously differentiable, and f ' ( t ) = h(t), where h(t) = y ( t )

for 0

< t < T.

+ 1'b'(t 0

- S)

q ( t , s) ds

+

at

ds

(5.5.4)

Proof. Define h ( t ) by (5.5.4). I t follows readily from the hypothesis that h E C[[O, TI, R].Also,

(5.5.5)

where the interchange of order of integration is easily justified by the hypothesis and Fubini's theorem. We note that the assumption of the t T , and lemma implies that b(t) is absolutely continuous on 0 < E this yields the second equality in

< <

j'b'(s

- T)

q(s, T) ds

=

lim

<-Of

T

=

1' [

lim b(s

*-0+

b'(s - T ) q(s, T) ds

TfE

- T)

q(s, T)

0

< T < t.

(5.5.6)

5.5.

33 1

ASYMPTOTIC BEHAVIOR

Combining the relations (5.5.5) and (5.5.6), we obtain

after an interchange of the order of integration. The conclusion of the lemma now follows readily. We now prove

THEOREM 5.5.1. Assume that (i) a E C [ J ,R], ( - l ) W ( t ) >, 0, 0 < t a(t) f 4 0 ) ; (ii) g E C[R,R ] , xg(x) > 0, x # 0, and G(x) = l z g ( f )d f

as I x

co

-+

0

< co, i

I

+

(iii) u ( t ) is any solution of (5.5.1) existing on 0 Under these assumptions, lim ui(t) = 0

Proof.

Differentiating (5.5.1), we get x"(t)

+ a(0)g(x(t)) =

= 0,

(i

t-m

-

=

0, 1, 2, 3, and

a;

< t < co.

1 , 2).

(5.5.7)

it 0

a'(t - s) g(x(s)) ds.

(5.5.8)

Whenever we refer to (5.5.1) and (5.5.8), we mean the identities that result from substituting u(t) into them. The possibility of none of a'(O),a"(O),a"'(0) being finite necessitates that a little care be exercised in handling certain integrals that arise. In all the cases, the arguments used in the preceding lemmas supply the rigor, and hence, in this proof, we tacitly assume such considerations whenever they are relevant. Consider the function

-

$

st 0

a'(t - s)

[It

g(U(T))

dT]' ds

S

Using (5.5.1) and integrating by parts, we obtain

0.

(5.5.9)

332

CHAPTER

5

which implies that

< v(t)< v(0)= G(un),

G(u(t))

where u,, = u(0). I t then follows from assumption (ii) that

<

I 4t)I

t

B 9

E

J,

(5.5.1 1)

where /3 = P(u,,) 0 as un + O . I n succeeding formulas, will not necessarily be the same as in (5.5.11). However, it will have the same property. From (5.5.3), (5.5.8), and (5.5.11), we derive that ---f

< 8,

I u”(t)l

t

E

J.

(5.5.12)

T h e inequalities (5.5.11), (5.5.12), and the mean value theorem show that (5.5.13)

Integration by parts and (5.5.8) yield V”(t)= + ” ( t )

st

$

-

[Jt

n

g(u(s))

d”(t - S)

0

-

&I2 [Jt g(U(T))

dT]’dS

S

+

g(u(t))[u”(t) a(O)g.(u(t))l.

By Lemma 5.5.3, tza”(t)€L1(O,00). This, together with (5.5.1 l), (5.5.12), and Lemma 5.5.3, implies that

< 8,

I Vt)I

tE

J.

By Lemma 5.5.1, it follows that V ( t )+ 0

as

t

---f

co.

Hence, lim

t-m

J‘,,a”(t

s)

~

[J‘g ( U ( T ) ) dT]’ds = 0, S

which assures, in view of -a”’(t), a”(t) 3 0,

-

fim U”(1‘) rli

Jt t--T

[it

g(U(7))

s

dT]’dS

=

0,

5.6.

PERTURBED INTEGRAL

333

EQUATIONS

for every 0 < T < co. Choose To > 0 arbitrarily if the first alternative of Lemma 5.5.2 holds, and choose 0 < T , < to if the second one does. Then, clearly (5.5.1 4)

0
<

T o .Suppose that limt+mu(t) # 0. Then, there exists a X and a sequence {tlL}, t, > 0, t , -+ co as n + co, such that

>0

I4tn)i 3 A. This, together with the relations (5.5.8), (5.5.13), and the mean value theorem, implies the existence of a 6 > 0 and a p > 0, where 0 < 6 min( To , t l ) , such that

<

for

Ig(u(7))l 3 CL

t,

-

6

< r < t, .

As a result, we have

=

+pW

>0

(12

=

1, 2,...),

which contradicts (5.5.14). Thus, limt+mu(t) = 0 is established. Formula (5.5.7), i = 1, follows from (5.5.7), i = 0, (5.5.12), and the mean value theorem by employing an argument similar to the proof of Lemma 5.5.1. Similarly, formula (5.5.7), i = 2, follows from (5.5.7), i = 0, assumption (ii), (5.5.8), and the fact that a'(t) E L ~ ( O a , ).This completes the proof of the theorem.

5.6. Perturbed integral equations Corresponding to the integral equation (5.5. I), let us first consider the perturbed equation x'(t)

=

-Jt

n

a(t

-

s ) g ( x ( s ) )ds

-

b(t) +f(t).

(5.6.1)

As in the previous section, the letter ,l3 denotes a finite a priori bound that may vary from time to time. Concerning Eq. (5.6.1), we have the following result.

334

CHAPTER

5

THEORIN 5.6.1. Assume that t -< m, i = 0, 1, 2; (i) a E C [ J ,121, (- I)”Q(Q(t)2 0 for 0 (ii) g E C [ R ,K], xg(x) 3 0, G(x) = J:g([) d [ -+ co as I x I --t m, and 1 g(s)/ Kl(l G(x)) for some K , > 0;

+

<

(iii) 11 E C [ J ,R ] , b‘(t) exists and is continuous on 0 -.: t <<m; (iv) there exists a y E C [ J ,R] such that y ( t ) is continuously differentiable on 0 .t ‘,. co and ‘

h’(f) (v)

< a ( t )y ( t ) ,

(h’(t))?

< a ‘ ( t )y’(t),

0 < t < co;

f~ C [ J ,I?], and f~ &(O, a).

Then, if

s(t)

is a solution of (5.6.1) on 0 ,< t j r(t)l

< K,

,<

GO,

we have (5.6.2)

t E J.

Suppose that, in addition: t co, xg(x) > 0, x f 0, g ( . ) is differ(vi) -a”’(t) 0, 0 entiable on R, and b”(t),y “ ( t ) exist on 0 t -, co; (vii) cither ( l ~ ” ( t ) ) a“(t) ~ y”(t),0 t -. CO, or I b’(t)l, 1 tb”(t)l, I y”(t)l K , for some I< > 0, on p t < m, p > 0 being some n um b e r ; and bounded. (viii) f(t) is continuously differentiable on p t \ ‘I‘hen, ~

<

<

<

<

1 x‘’(t)l

< K,

p

<

f

< a,

and, if also a ( t ) 2. a(O), lim xL((t)= 0

I

(i

=-

0, I).

(5.6.3)

P ~ o o f . \. eI shall first prove (5.6.2). For t E J , tlcfine

+ 6 ( t ) [‘g(.v(s)) ds -k f y ( t ) ‘ 0

(5.6.4) (5.6.5)

5.6.

PERTURBED INTEGRAL EQUATIONS

and V ( t ) = [l

+ E(t)]exp(--K,F(t)).

335

(5.6.6)

From assumptions (i), (ii), and (iv), it is evident that E ( t ) , V ( t ) 3 0. Differentiation of (5.6.6) yields, after some calculation involving an integration by parts,

x exp( -K,F(t)).

(5.6.7)

Hence, by (i), (iv), and (5.6.7), we see that

rw G {--K~

-

+

- ~ , ~ ( x ( t ) )I g ( x ( t ) ) i i if(t)i ~XP(--K,WL

which, together with assumption (ii), implies that

<

V ( t ) 0.

Therefore, it follows that

and so

From this inequality, the truth of (5.6.2) is clear in view of the assumptions on f and G. T o prove the second part, we differentiate (5.6.1) to obtain x ” ( t ) = -a(O)B(x(t)) -

it

U ’ ( t - T ) g ( X ( T ) ) dr

- b’(t) +-f’(t).

0

<

<

(5.6.8)

Because of assumptions (vii), I b’(t)I K, p t :. GO if the second alternative holds. If the first alternative holds, we proceed as follows. Since a”(t) >, 0, we see that y ” ( t ) >, 0, and therefore -y’(t) is nonincreasing, which, together with the last condition in (iv), proves that I h’(t)l K. Noting that a’(t) E / , ~ ( ~ , c o ) ,we see, from (5.6.2), (v), (5.6.8), and the GO. This, together with (5.6.2) hypothesis, that I x”(t)l K, p < t

<

<

336

5

CHAPTER

< K , p < t < co. < K, 0 < t <

and the mean value theorem, yields that j x’(t)l Furthermore, from (5.6.1), (iii) and (v) imply 1 x’(t)l I t is easy to get, after some calculation, lqt)

~

Q,(t) I

ta”(t)

0

g(x(s))ds),

00.

+ h”(t)J t g(.x(s))ds 0

(5.6.9)

where V t ( t )is the right-hand derivative of V(t),

QAf) =

-k; I f(t)’ “’(t)

-

KlJqt)/f(t)l;

+ IX.’(.W x’(t).f(t) -1+ 2f’(t)

- X”(t) -

-

!”’

(~af(t)

dS

R(X(S))

0

If ( 9

a(O)g(x(t))]

(’jf

- k-1 i f(t)i

-1- h’(t)

dy(t))[--K1f(t)

g(s(.s)) ds12

+ iy’(t) 5 J‘ a”(,? -

-T)

0

1‘ d”(t

T )

~~

* n

ds)‘

(/‘g(.X(.C))

(jt

g(X(S)) dS)’

dT)

T

dT/ eX[3(-KlF(t)).

T

<

There exists a K such that Q,(t) 3 -K > - G O , p t < co. This follows from the condition V ( t ) 3 0, V’(t) 0, the boundedness of s ( t ) , ~ ’ ( t ).x”(t) , on p t .-: a,the relation 1 f ( t ) l ; = / f ’ ( t ) l , and the t << co. hypothesis. Hence, (iv) and (5.6.9) imply that V;(t) 3 --K, p Ry 1,cmma 5.5.1 and Remark 5.5.1, we have V ( t )+ 0 as t + co. Iieturning to (5.6.7), we find, as a result of V ( t ) 0 as t -+ co, (i), (iii), (iv), and (v), that

<

<

<

-j

.-,J

lim

f

-1

o”(t

~

T ) (J‘g(.x(s))

ds)’ dT = 0.

(5.6.10)

T h e arguments used in the proof of Theorem 5.5.1 enable us to conclude from (5.6.10) that . ~ ( t+ ) 0 as t co. From this property, the boundedness of .r”(t),and the mean value theorem, we deduce that x’(t) 3 0 as t + ,m. ‘I’he proof is thus complete. --f

‘I’he nest theorem concerns the perturbed equation

5.6.

337

PERTURBED INTEGRAL EQUATIONS

(5.6.12)

where

Observe that, if g(x) is odd and nondecreasing, then (5.6.12) reduces to !(X) = g(x), M ( x ) == g(.), m,(x) = mz(.) = G(x).

THEOREM 5.6.2. Assume that a E C [ J ,R ] ,( - l ) i a ( i ) ( t )3 0 for 0 < t < GO, i = 0, 1, 2; (ii) g E C [ R ,R], xg(x) 3 0 for I x j p, 0 < p < a,and g(x) is not identically zero in any neighborhood of the origin; (i)

<

<

<

t < a,1 x 1 p; (iii) p E C[J x R, R ] ,xp(t, x) 3 0 for 0 (iv) f~ C [ J x R, R ] , and, for each E > 0, there exists a 6 = 6 ( c ) > 0, where S ( E ) + 0 as E 0, and a P ( t ) = P(t, c) > 0, where J:P(t) dt < E , such that 1 f(t, x)I P ( t ) whenever 0 t < 00 and 1x1 < 5 ; > e. (v) for sufficiently small c > 0, rn,(8)/~M(S) -j

<

<

<

Then, for any 0 < 7 p, there exists a 6, = 6 , ( ~ ) such that every solution x(t) of (5.6.1 I ) defined on 0 t < GO with 1 x(0)l xo satisfies

<

I .(t)l

< 7,

<

t

3 0.

Suppose, in addition, that (vi) p ( t , x) is continuously differentiable, and

I p ( t , x)l, I P d t , (vii) (viii)

a,I p,(t, .)I < K ,

0


< a, 1 x I

< p;

-a”(t) >, 0, 0 < t < a, a(t) eL1(O, a), a(t) s 0 ; xg(x) > 0, x f 0, and g(x) is differentiable for 1 x 1

< p;

338

CHAPTER

5

(ix) f ( t , x) is continuously differentiable, and

If,(!, (x)

X)l,

lf,(f,

s k-,

.y)l

0


< a, I x I

sp ;

Pi@), P ( t ) e K(E),0 < t < co, 0 < < 1 . E

Then, lim

s Z ( t )=

/-*a

0

(i

=

0, 1 ) .

(5.6.13)

<

Pmof. T o prove the first assertion of the theorem, let 0 < 7 p. Choose E ~(q),so that 0 < S ( E ) < q, and the assumption (v) is satisficd. T h i s means that 6 and P ( t ) = P ( t, E ) are fixed for the remainder of the proof. Now choose 6, = 6,(y) so that E M @ )-1- m,(S,,) e < rn,(S),

(5.6.14)

which, by the definition of m, and m 2 , implies that 6, < 6. Let x ( t ) be a solution of (5.6.1 1) defined on J with I x(0)I by continuity, I ~ ( t ) l 6 for sufficiently small t. Define

-

E(t)

=

G ( x ( t ) )t Aa(t)

< 6, . Then,

(1'

g ( x ( s ) ) ds)'

(I

(5.6.15)

(5.6.16)

Thus, by hypothesis, we have, as long as I x(t)l

< 6, the relation

5.6.

and hence also

339

PERTURBED INTEGRAL EQUATIONS

< [cM + G(x(0))l exp (fIt P(T) d < (EM + m,(S,)) e < m,(S).

G(x(t))

Suppose that there exists a t , such that I x(t,)l and (5.6.18), it follows that

=

~ )

(5.6.18)

8. Then, from (5.6.14)

e G ( x ( t ) )< m,(S),

m m

which is impossible. Hence, no such t , exists, and

l4t)l < 8

< 7,

3 0.

t

(5.6.19)

We shall now prove the second part of the assertion. Since a ( t ) E L,(O, a), we deduce from (5.6.19), (5.6.11), and the hypothesis the inequality

I x’(t)l S K ,

t

2 0.

This, together with x ” ( t ) = -Pt(C -

Jl

4 4 ) - P x ( t , 49 X ’ ( t ) a’(t

~

T )g(4.))

dT

-

a(O)g(x(t))

+f d t , 44)

+f d f ,4 t ) ) x’(t> and the hypothesis, implies that

1 x”(t)i < K ,

0


< co.

Taking the right derivative of (5.6.17), we obtain the formula

340

CHAPTER

5

Arguing as in the proof of Theorem 5.6.1, it is easy to deduce that V:(t) 2 -A7 > - co, t E J , and V’(t)+ 0 as t -+ co. Hence, it follows from (5.6.17) and the assumptions that (5.6.10) is true. T o conclude (5.6.13) from this, we have to repeat the corresponding reasoning as in Theorem 5.6.1. T h e proof is therefore complete

5.7. Admissibility and asymptotic behavior In this section, we shall be concerned with an integral equation of the (5.7.1)

In order to obtain better results concerning Eq. (5.7.1), we shall need thc concept of admissihility of a pair of subspaces with respect to an operator. ‘I’he underlying space will be the space C [ J ,R“], of all continuous functions from J to R’l, with the topology of uniform convergence on every compact interval of J . This topology may be defined by means of seminorms, namely,

It is easy to see that this topology is metrizable and that C [ J ,Rn] is complete. Suppose that l3, D are Banach spaces of functions from J to Rn such such that B, D C C [ J ,IP].We shall assume that the topologies of B, D are stronger than the topology of C [ J ,R”].

DEFIKITION 5.7.1. T h e pair of spaces ( B , D) is said to be admissible with respect to the operator T : C [ J ,RrL]+ C [J , R“] iff TB C D.

5.7.

34 1

ADMISSIBILITY AND ASYMPTOTIC BEHAVIOR

LEMMA 5.7.1. Let T be a continuous operator from C [ J ,R"] into itself. Suppose that R, D are Banach spaces that are stronger than C [ J ,Rn] and the pair ( B , D) is admissible with respect to T.Then, T is a continuous operator from B to D. Proof. It is sufficient to show that T is a closed operator from B to D. Then, on the basis of the closed graph theorem, wc can conclude that T is a continuous operator from B to D. Let x, 5 x and Tx, L y . We must prove that y = Tx. From the fact that x, 5x, it follows that x, CIJ,Rnl + x. Consequently, Tx, C I J ' R " l t Tx. On the other hand, Tx, % y , and this implies T~

_C[J,R"I

11

+

Y.

Hence, y = Tx,and this means that T is a closed operator, because the graph is closed in B x D. It follows that one can find a constant k > 0 such that XEB. 1 T x ID < k I x I B , We can now prove an existence theorem for the Volterra integral equation (5.7.1).

THEOREM 5.7.1.

Consider Eq. (5.7.1) under the following conditions:

(i) B and D are Banach spaces stronger than C [ J ,R"] such that ( B , D) is admissible with respect to the operator

J" K ( t ,s) x ( s ) ds, where K ( t , s) is a continuous function for 0 < s < t -:I a. ( T x ) ( t )=

(ii)

x(t) + f ( t ,

0

(5.7.2)

x(t)) is a continuous operator on S = [x(t) : x ( t ) E

D and 1 x I D

< p],

with values in B such that

lf(t,

4 t ) ) -f(t,y(t))le

<

I 4 t ) -Y(t)lD,

x,y

E

s,

(5.7.3)

X being a positive constant. (iii) h(t) E D. Then, there exists a unique solution of the Eq. (5.7.1), provided that kX

< 1,

I h(t)lD

+ fl lf(40)lB

< P(1

-

Xk).

(5.7.4)

342

CHAPTEK

Pmof.

5

Consider the following operator on S :

(5.7.5) JYe can write

By 1,emina 5.7.1, we gct

taking into account the condition (5.7.3). By the assumption (5.7.4), is a contraction operator. it follows that I t no\\ suffices to prol e that U S C S, in order to conclude the existence and uniqueness of the solution, by means of Ranach fixed point theorem. \Ye have I(uy)(t)lD

But,

1 f ( / ,L ( t ) ) i f l

*Is a result,

IVC

2 1 ' z ( t ) l D -1

:I f ( t *

I f('?,v ( t ) ) l B .

-f(t> 0)iB 1

if(t>

(5.7.7)

0)iS

obtain

bccausc of thc condition (1 3.7.4).

'l'he proof is complete.

I<EhlAl
'ViiEoREni

5.7.2.

Consider Eq. (5.7.1) under the following conditions:

(i) T h e same as in Theorem 5.7.1.

5.7.

ADMISSIBILITY AND ASYMPTOTIC BEHAVIOR

(ii) x ( t ) -.f(t, x ( t ) )is a continuous operator from in C [ J ,R"]) into B such that

where (iii)

Y

343

(the closure of S

is a positive constant and h(t) is continuous on R , T h e same as in Theorem 5.7.1.

Then, there exists at least one solution x(t) E S of Eq. (5,7.1), whenever I h(t)l, and Y satisfy the inequality

I h(t)lD

+ kr s

p.

I n applying Theorems 5.7.1 and 5.7.2 to concrete situations, it is only the admissibility condition (i) that is difficult to be verified. I t is therefore important to obtain necessary and sufficient conditions in order that a given pair of spaces is admissible with respect to an integral operator. For this purpose, we shall introduce the space Cg defined as follows. Let g(t) > 0 be continuous on [0, m). Then we designate by C, the space given by

c, = [ x ( t )E C[/, R"1 : II r(t)ll< &(t),

t

2 01,

where A depends on the function x(t). I n the space C, , we introduce the topology by means of the norm

Then it is easy to check that the space C,, is stronger than the space C [ J ,R"1. When the spaces B, D are C,/ spaces with different g, criteria for their admissibility may be given. T h e following theorem is to that effect.

THEOREM 5.7.3.

Let us consider the pair of spaces (C, , C,) and the integral operator T , defined previously. Then the pair ( C , , C,) is admissible with respect to T iff t

for some L

> 0 depending only on g, G, and K.

2 0,

(5.7.8)

344

CIIAI'TER

5

Pmof. Since the sufficiency part of the proof is obvious, we shall only prove that the condition (5.7.8) is necessary. First of all, let us treat the scalar case, that is, n = 1. Suppose that (5.7.8) is not satisfied. Thcn, we can find a sequence (t,J such that t,,, > 0 and t,,,+ u3 as m -+ GO, for which (5.7.9) 1,et us define a new function (5.7.10) This is a measurable function, and

From (5.7.9) and (5.7.10), we have (5.7.1 1)

'l'he theorem of Lucin concerning the structure of measurable functions shows that there exists a continuous function fTrL(t),defined on 0 5t I,,, , such thdt

<

If,n(t)l

x At),

0i t < tm ,

nnct

!- qf?)! , (ts : mG(t,), . fm

m

\)fi,JS)

0

2 1.

(5.7.12)

iyithout any difficulty, we can extend the function f,,,(t) on the whole h,ilf-linc > 0 such that it remains continuous and

Suppose now that T ( b C C , . Hy Lemma 5.7.1 and the fact that I , 2 ,..., it follows that I, m

f,,,(t)lc, -5

~

1

Tf,,I(t)l

LG(t),

tt

J,

for some conkenient I, > 0. This contradicts (5.7.12). Consequently, the condition (5.7.8) is necessxy for the admissibility of the pair

(C,,

1

CG).

5.I.

ADMISSIBILITY

I n the case when n

> 1,

AND ASYMPTOTIC BEHAVIOR

345

we shall represent the kernel as a sum n

s,

K(t7

_1

1

KZJ(t>

s)7

2.)-1

where Kij(t,s) is a matrix kernel whose elements are all zero excepting the (i,j)th element. It is easy to see that such a kernel acts as a scalar kernel. As a result, for every K f j ( t s), , we can obtain an admissibility of the type (5.7.8). Since 2.3=1

it follows that, for K ( t , s) also, we can obtain the condition (5.7.8). T h e proof is therefore complete. We shall now derive a particular case of Theorem 5.7.1 as an application.

THEOREM 5.7.4.

Consider Eq. (5.7.1) under the following conditions:

<

<
a,k, (X > 0.

(i) 11 K ( t , s)ll k exp[-a(t - s)], 0 s (ii) f E C [ / x S o ,R"],f ( t , 0) = 0, and

ll.f(t? x) -f(f,Y)lI

(iii) 11 h(t)l/ such that 0 <:

< h, exp[-Pt], p < a.

<

/I x

Y

11.

P

are positive numbers

-

where h, and

Then, there exists a unique solution of Eq. (5.7.1) satisfying

!I x(t)ll

< p exp[--Ptl,

t tJ,

whenever h,, and X are small enough.

Proof.

From condition (i), we obtain

This implies that the pair of spaces ( C , , CJ, with g ( t ) = exp[-Pt], is admissible with respect to the operator T , in view of Theorem 5.7.3. Considition (iii) nieans that h(t) E C, , and, from (ii), it follows that

l f ( 4 4 t ) ) -f(4r(t))lcg-='

Ix

-

Y Ic,, .

Thus, we have verified all the assumptions of Theorem 5.7.1, and hence the proof is complete.

346

CHAPTER

5

Let us now consider the linear integral equation of the type (5.7.14)

< <

with continuous kernel for 0 s t < co and continuous f ( t ) on J. T h e unique solution of (13.7.14) is given by (5.7.15) where R ( t , s) is the resolvent kernel of K ( t , s), that is,

1q (U

R(t,s)

=

(5.7.16)

t , s),

n -I

K J t , s)

=

q t , s),

K,,+l(t,s)

=

.i:k;,(t,

u ) K(u, s) du,

2 1.

(5.7.17)

Corresponding to (5.7. I4), we consider the perturbed integral equation .v(t) =

j

.t

K ( t , s) s(s) ds +f(t, x).

0

(5.7.18)

~ I E F I ~ I T5.7.2. I O N Let B, D be Banach spaces such that B, D C C [ J ,Rn] and are stronger than (.[I,R"]. Then, the pair ( B , D) is said to be admissible for (5.7.14) if the solution x ( t ) E D whenever f(t) E B. Wc note that this concept of admissibility differs from the one given in Definition 5.7.1. An argument similar to that utilized in the proof of Lemma 5.7.1 shows that, for every admissible pair ( B , D),the mapping f - x is continuous from B to D. Consequently, for any admissible pair ( B , D), we can find a number h > 0 such that

We are now in a position to prove a result concerning the perturbed equation (5.7.18).

THEOREM 5.7.5. (i)

Assume that

the pair ( B , 0)is admissible with respect to (5.7.14), where B

a n d D arc Banach spaces stronger than C [ J ,Rn];

5.7.

ADMISSIBILITY AND ASYMPTOTIC BEHAVIOR

347

(ii) x ( t ) +f ( t , x) is a mapping from D to B such that if(t,

x) - f ( t > Y ) / B

<

I

-y

ID

x,yE

9

D m

Then, there exists a unique solution x ( t ) E D of Eq. (5.7.18) provided h is small enough. Proof. T h e proof is very simple. Indeed, consider the mapping A : u(t)+ x ( t ) from D to D, where x ( t ) is the solution of the linear equation (5.7.19)

From the admissibility of the pair ( B , D),it follows that

1 -Y

ID

,< If(t,

u,

-f(t,

.)IB

where x = Au, y = Av. Consequently, for X contractive, and the proof is complete.

< k-l,

the mapping is

Another general theorem that aseerts only the existence of a solution of (5.7.18) may be proved in the same way, making use of Schauder's fixed point theorem.

THEOREM 5.7.6.

Let us assume that condition (i) of Theorem 5.7.5 holds. Suppose further that f( t , x) is an operator from D to B such that (a) f ( t , x) is completely continuous; (b) there exists a number Y > 0 with the property If(t,

u)IB

<~k-l

for

I u ID

< Y.

<

Then, Eq. (5.7.18) has at least one solution x ( t ) E D such that I x I D r. For the proof, it is enough to observe that the mapping A : u(t)-+ x(t) defined by (5.7.19) is completely continuous from D to D and carries the ball into itself. S = [u : u E D,I u ID < Y] We can derive some concrete results concerning the existence and the behavior of the solutions of perturbed integral equations, as an application of Theorems 5.7.5 and 5.7.6.

348

CHAPTER

5

Let us suppose that the solution of Eq. (5.7.14) is in C, for every C, . From (5.7.15), it follows that this situation occurs if and only if

f(t) E

whenever f ( t ) E C, . Hence, on the basis of Theorem 5.7.3, we obtain the necessary and sufficient condition for the pair (C,, C,) to be admissible with respect to (5.7.14), namely,

(5.7.20) Suppose, for example, that the operatorf(t, x) is given by

< <

s t < 03, x E R where f ( t ) E C,, and K,(t, s, x) is continuous for 0 such that K,(t, s, 0) -= 0. In order for the operator defined by (5.7.21) to act from C, to C,, it is enough to impose appropriate conditions on K,(t, s, x). We shall suppose that

where K"(t, s) is a positive continuous function for 0 satisfying

<s
03,

(5.7.23)

and this proves thatf(t, x) E C, iff(t) E C, . T h e foregoing considerations prove the following result.

5.7.

THEOREM 5.7.7.

349

ADMISSIBILITY AND ASYMPTOTIC BEHAVIOR

Consider the perturbed integral equation

subjected to the following conditions: (i) T h e pair (C, , C,) is admissible for (5.7.14), that is, the resolvent kernel R(t, s) satisfies (5.7.20). (ii) T h e function R,(t, s, x) is continuous for 0 s t < co, K,(t, s, 0) = 0 and obeys (5.7.22) and (5.7.23). (iii) f ( t ) E C, .

< <

Then, there exists a unique solution of (5.7.24), belonging to C,, , provided Mo is small enough. As particular cases of Theorem 5.7.7, we mention the following results concerning the boundedness and the exponential decay of the solutions of the perturbed integral equations. T h e first case corresponds to the choice g(t) = 1 , whereas the second one corresponds to g (t) = exp[-at],

01

> 0.

COROLLARY 5.7.1. Let us suppose that Eq. (5.7.14) has a unique bounded solution for every bounded f ( t ) . Then, the perturbed integral equation (5.7.25)

has a unique bounded solution for every bounded f ( t )if

where Mo is small enough. COROLLARY 5.7.2.

Assume that (5.7.14) has a solution x ( t ) verifying

for every f(t) satisfying

350

CHAPTER

5

where n/l and A depend on f. Moreover, let K,,(t,s) be a continuous function for 0 s t CG with the property that

< <

sl

1 &(t,

I

s)l exp[ol(t

- s)]

ds

< M, ,

t

2 0.

'Then, Eq. (5.7.25) has a unique solution satisfying (5.7.26), whenever f ( t ) verifies (5.7.27), provided Ml is small enough.

5.8. Integrodifferential inequalities Lct 1; be an operator from C [ J ,R] into C [ J ,R].We shall consider the integrodifferential equation f ( t , m',

X,FX) =

x(0)

0,

= X"

,

(5.8.1)

where f E C [ J x R3,R].Let us first prove a basic theorem on integrodifferential inequalities.

THEOREM 5.8.1.

Let us assume that

(i) f E C [ J x K",R], and f ( t , x , y , x) is nondecreasing in x for fixed ( t , y , 2) and nonincreasing z for fixed ( t , x, y ) ; (ii) the operator F maps C [ J ,R] into C [ J ,R],and, for any two functions u l , u2 E C [ J ,R],the inequality %(t)

implies

t.;' %!(t), Fu &Fv

0

< t < t, , for

t,

(0, co)

t = t, ;

(iii) a),w t C [ J , R],o, w are continuously differentiable on (0, co), and the inequalities f ( t , v', v,Fv) < 0, f ( t ,WI,w , F w ) 2 0

hold for t

E

(0, co), one of them being strict.

Then, v(0) .-Iw(0) implies v(t) < w(t),

Pyoof.

t

2 0.

Suppose that the set

z== [t E 1 : v ( t ) 2 w(t)]

(5.8.2)

5.8. is nonempty. Let t , more, we have

=

INTEGRODIFFERENTIAL INEQUALITIES

351

inf Z. Then t ,

> 0, because v(0) < w(0). Further-

v(t1) = W(tl)>

(5.8.3)

v(t)

and

<4th

0

< t < t, , (5.8.4)

v’(t1) 3 w’(t1).

It then follows from assumption (ii) that FV

< FW

for

t

=

t,.

(5.8.5)

T h e monotonicity of the function f now yields f ( t l V’(tl),V(tl),F v ) 9

2f(tl , W’(tl>,w ( t J , Fw) because of the relations (5.8.3), (5.8.4), and (5.8.5). This implies a contradiction in view of the strictness of one of the inequalities assumed in (iii). Consequently, the set 2 is empty, and (5.8.2) is true. T h e proof is complete.

REMARK 5.8.1. If the function f ( t , x, y , z ) is independent of x, then the operator-differential equation (5.8.1) reduces to pure operator equation. Then, for the validity of Theorem 5.8.1, the continuous differentiability of v, w is not necessary. Remark 5.8.1 may be used to prove the following: COROLLARY 5.8.1. Let v, h E C [ J ,R,],and suppose that

where ZI,,

Proof. and

> 0 is a constant. Then,

T o apply Theorem 5.8.1, we set

3 52

CHAPTER

5

Consider the function w ( t ) = (v,, f c ) exp[l: A(s) ds] for arbitrary small > 0. Then, it is easy to check that

E

< 0,

f ( t ,v , W

f ( t ,w,Fw) 0,

and

< w(0).

vo

Since the assumptions of Theorem 5.8.1 hold, we have v(t) < W ( t ) ,

t 23 0.

As this inequality is true for all E > 0, we deduce, letting E + 0, the desired result. I t is not difficult to see that Theorem 5.8.1 includes integrodifferential equations of the form X'(t)

=-.f(t, x ( t ) )

+ Jt K(t,s, x(s)) ds, 0

where the kernel K is monotone nondecreasing.

DEFINITION 5.8.1. Let

for t E (0, a).If

21

u E C [ J ,R ] , and ~ ' ( texist ) and be continuous satisfies the inequality

f ( t , v', v,F V )

3, 0,

t

€

(0,m),

then v(t) is said to be an over function with respect to the integrodifferential equation (5.8.1). O n the other hand, if z, satisfies f ( t , v', v ,Fv) < 0,

t E (0, a),

then v ( t )is said to be an under function. As a consequence of Theorem 5.8.1, we have the following result.

THEOREM 5.8.2. Let u ( t ) , w ( t ) be under and over functions with respect to Eq. (5.8.1) and v(t) be a solution of (5.8.1) existing on [O, a). Then u(0)

< v(0)< w(0)

implies u(t) < v ( t ) < w(t),

t

3 0.

5.8.

353

INTEGRODIFFERENTIAL INEQUALITIES

DEFINITION 5.8.2. Let v E C [ J ,R], and v’(t) exist and be continuous for 0 < t < co. Then v(t) is said to be a S-approximate solution of the integrodifferential equation (5.8. l), if v(t) satisfies the inequality If(C v’(t),u(t),Fv)l


t E (0, a),

where 6 E C [ J ,R,]. A result that gives an error estimation of the &approximate solution is the following:

THEOREM 5.8.3. Let v(t) be a S-approximate solution of (5.8.1). Suppose further that f ( t 9

x1 ,Y1 FYI) - f(t,x2 Y2 FY,) 9

3 dt,x1 - x2 ,Y1 - Y2 G(Y1 - Y2)h

XI

7

3%

Y1

1

3 Y2

?

where g E C [ J x R3, R], and G is an operator that maps C [ J ,R] into C [ J ,R].Assume that the function g(t, x, y , z ) is nondecreasing in x for fixed ( t ,y , 2) and nonincreasing z for fixed ( t ,x,y ) , and, for any two functions u, v E C [ J ,R],the inequality aft)

implies

<4th Gu

0
< Gv

< t, , for t

f, 6

=

(0, a),

t,.

Then, if u(t) is any solution of (5.8.1) such that u(0) = xo a n d 1 v(0) - xu 1 p o , we have

<

I 4 t ) - 4t)I < p(t), where p ( t ) satisfies

>0

t

2 0,

is continuously differentiable for 0 g(t, p’, p, Gp)

> qt),

t

E

< t < 00

and

(0, a).

Proof. We shall first show that v(t) - u(t) < p(t), t 3 0. Setting z(t) = v(t) - u(t) and proceeding as in the proof of Theorem 5.8.1, we arrive at a t , > 0 with the properties

and

GZ .< Gp,

t

=

t,

354

5

CHAPTER

Since p ( t J > 0, p’(tl) 3 0, we have v(tl) 3 u(tl), v‘(tl) 3 u’(tl), and, consequently, S(t1)

3 f(tl

7

U’(tl), V ( t l ) >

3 R ( t 1 , Z’(tl),

Fv) - f ( h

7

u’(t1)y

U(tl),

Fu)

GZ).

Z(tl),

Now, using the monotonicity properties of g, it follows that

s(t1 > Z’(tl), G g(t1

Z(tl), 7

Gz)

f’(tl)9

P(tl)? GP)

< s(tl), which implies 8 ( t l ) < 8(tl). This absurdity proves v(t)- u(t) < p(t),

t

2 0.

A similar argument shows that u(t) - v ( t ) < p ( t ) , t is therefore proved.

3 0. T h e theorem

5.9, Notes

See Walter [3] for the type of results in Sect. 5.1 (see Jones [l]). Theorems 5.2.1 and 5.2.3 are due to Nohel [l]. Theorem 5.2.2 is new. For Theorem 5.3.1, see Nohel [I]. T h e proof of Theorem 5.3.2 is adopted from Olech [9], whereas Corollary 5.3.1 is due to Olech [9]. See also Cafiero [I]. For the results of the type given in Sect. 5.4, see Walter [3]. Sections 5.5 and 5.6 consist of the work of Levin and Nohel [2, 31. See also Friedman [ l , 21, Halanay [3], Levin [2, 31, Miller [5], and Padmavally [I]. T h e results of Sect. 5.7 are due to Corduneanu [18, 211. Section 5.8 contains results adopted from Nickel [l]. See also Azbelev and Tzaliuk [I], Barbu [I], Baumann [I], BeneS [ l , 21, Cameron and Shapiro [I], Corduneanu [19], Erdelyi [l], Goldenhershel [I], Iwasaki and Sato [I], Krasnosclskii [I], Krein [2], Levin and Nohel [4], Mann and Roberts [I], Miller [6, 71, Mitryakov [l], Nohel [2-51, Petrovanu [I], Ramamohana Kao [2], Sato [ l , 41, Volterra [l-31, and Willett [I].

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Author Index

Conti, R., 130, 264, 359 Coppel, If’. A , 130, 359

A Alekseev, V. M., 130, 355 Antosiewicz, H. A , 130, 264, 265, 311, 355, 356 Azbelcv, N. V., 354, 356

Corduneanu, C., 44, 129, 264, 265, 354, 359

D Dahlquist, G., 360 D’Ambrosio, Ll., 311, 360 Davis, Y., 356, 361 Ilcysach, L. G., 130, 265, 361 Diaz, J. B., 130, 361 Dicudonne, J., 130, 361 Duboshin, G. N., 361 Dugundji, J., 356, 361

B Babkin, B. N., 44, 356 Baiada, E., 44, 356 Barbslat, I., 356 Barbashin, E. A., 356 Barbu, V., 354, 356 Bass, R. W., 356 Baumann, V., 354 356. Bellman, K., 44, 311, 356 BeneS, V. E., 354, 357 Bertram, J. E., 357 Bhatia, N. Y., 265, 357 Bihari, I., 44, 130, 357. Brauer, F., 129, 130, 264, 765, 357, 358. Burton, L. P., 44, 358 Bylov, B. F., 265, 358.

E Ehrinann, 14. H., 361 Erdelyi, A, 354, 361 Erugin, K.I’.. 361 Eacilo, Jo 0. C., 362

F Filippo\-, .I. F., 362 354, 362 Fricdman, .I.,

C Cafiero, F., 44, 130, 354, 358 Cameron, K. H., 354, 358 Cartwright, M. L., 130, 358 Cesari, L., 130, 358, 359 Chandra, J., 359 Chaplygin, S. A., 44, 130, 359 Charlu, A. S. N., 311, 359 Chetaev, N. G., 359 Coddington, E. A , 130, 359

G Gambill, I<. h., 362 Ganikrelidze, R. V., 362 Gerniaidzc, V. E., 362 Giuliano, L., 44, 362 Goldenhershel, E. I., 354, 362 Gorsin, S., 362

385

386

AUTHOR INDEX

Grobman, D. XI., 362 Gronwall, T. H., 44, 363

H Hahn, W’., 264, 265, 363 Halanay, A . , 130, 264, 265, 354, 363. Hale, J. K., 264, 265, 363 Hale, S. K . , 363. Hartman, P., 44, 130, 311, 363, 364 Hayash?., K., 265, 364 Hukuhara, >I., 364

I Infante, E. F., 364 Iwasaki, A , 354, 364

J Jones, G. S., 354, 365 Jones, IT.R., 365

K Kalman, R. E., 357, 365 Kamenkov, G. V., 365 Kamke, E., 44, 129, 365 Kato, J., 365 Kayande, A. A , 311, 365 Knobloch, H. W., 365 Kooi, O., 130, 366 Krasnosel’skii, M . X., 129, 354, 366, 367 Krasovskii, N. N., 130, 265, 366 Krein, 31. G., 354, 367 Krein, S. G., 129, 366, 367 Kudaev, &I. B., 367 Kurzweil, J., 367

L Lakshmikantham, V., 44, 129, 130, 264 265, 311, 367, 368, 369 Langenhop, C. E., 44, 130, 369 LaSalle, J. P., 130, 264, 265, 369 Lasota, A., 369 Lebedev, A. A., 369 Leela, S., 44, 264, 265, 311, 368, 370 Lefschetz, S., 264, 369, 370 Levin, J. J., 311, 370 Levinson, Ti.,130, 370

Lewis, D. C., 370 Li, Y., 371 Ling, H., 265, 371 Lozinskii, S. M., 371 Lusin, N. N., 130, 371 Luxemburg, W. A. J., 130, 371 Lyapunov, A. A , , 371 Lyascenko, N. Ya., 371

M Malkin, I. G., 265, 371 Mamedov, Ja. D., 44, 366, 372 Mann, W.R., 354, 372 Marachkov, V., 372 Markus, L., 130, 372 Massera, J. L., 130, 264, 372 Matrosov, V. M., 311, 373 llcShane, E. J., 84, 371 Miller, R. K., 130, 265, 354, 373 Minorsky, N., 373 Mitryakov, A. P., 354, 373 Mlak, W., 44, 373 Maser, J., 374 Moyer, R. D., 374 Muley, D. B., 365, 374

N Narendra, K. S., 374 Nemytskii, V. V., 374 I\-eumann, C. P., 374 Nickel, K., 354, 374 Nohel, J. A , 354, 370, 374

0 Okamura, H., 375 Olech, C., 44, 129, 265, 354, 375 Onuchic, N.,130, 363, 364, 368, 375 Opial, Z., 44, 369, 375, 376 P

Padmavally, K., 354, 376 Peano, G., 44, 376 Perron, O., 44, 129, 376 Persidskii, K. P., 264, 376 Persidskii, S. K., 377 Petrovanu, D., 354, 377

387

AUTHOR INDEX

Picone, M., 377 Plii, A,, 130, 375, 377 Pliss, V. h., 377

R Ramamohana Rao, M., 354, 377 Rath, R. J., 265, 369, 377 Redhcffer, R. M., 378 Reissig, R., 359, 378 Rohcrts, J. H., 354, 372, 378 Roxin, E. O . , 378

S Sadovskii, B. N., 378 Sansone, G., 130, 359, 378 SdStry, T., 368, 378 Sato, T., 354, 364, 378 Schechter, E., 378 Scifert, G., 130, 363, 369, 378 Scll, G. R., 130, 265, 361, 379 Shapiro, J. NI., 354, 358, 379 Skalkina, hl. A., 356, 379 Spinadel, V. W., 378, 379 Stepanov, \-.V., 374, 379 Sternberg, S., 265, 358, 379 Stokes, A. P., 129, 363, 379 Strauss, A , , 130, 264, 265, 379 Szarski, J., 44, 379 Szepo, c. P., 379

Tsokos, C . P., 368, 370, 380 Turowicz, 130, 380 Tzaliuk, Z . B., 354, 356, 380

v Van Kampen, E. R., 380 Verma, G. R., 311, 368, 380 Vinokurov, V. R., 380 Viswanatham, B., 44, 130, 369, 380 Volterra, V., 354, 380 Vorel, Z., 367, 381 Vrkoc, I., 367, 381

W Walter, W., 44, 129, 130, 354, 361, 381 Wazewski, T., 44, 130, 381 Weiss, L., 364, 381 Wexler, D., 265, 381 Whyhurn, W. M., 44, 358, 381 Willett, D., 354, 382 Wintner, A., 129, 130, 264, 364, 382 Wong, J. S. W., 382

Y Yakuhovic, V. A,, 382 Yamahe, H., 130, 372, 383 Yorke, J . A., 130, 265, 379, 383 Yoshizaua, T., 264, 265, 365, 383

T Taani, C . T., 379 Tonelli, L., 44, 380 Tricomi, F. G., 380

Z

Zabreiko, P. P., 383. Zubov, V. I., 130, 383

Subject Index

A Admissibility of spaces, 340 Almost periodic solutions, 124, 128, 251 existence, 124, 251 Approximate solutions, 79, 82, 324, 353 Ascoli-Arzela theorem, 4 Asymptotic behavior, 108, 229, 327, 340 Asymptotic equilibrium, 88, 89 Asymptotic equivalence, 91, 92, 94 Autonomous systems, 308

B Bounds componentwise, 84 lower, 79, 82 upper, 79, 82, 324 Boundedness conditional, 277 equi, 212, 214 equi-ultimate, 212, 216 quasi-equi-ultimate, 212, 215 quasi-uniform ultimate, 212, 216 uniform, 212, 215, 217 uniform-ultimate, 212, 216

Continuation of solutions, 5 Continuous dependence with respect to initial conditions, 69, 70 to parameters, 69, 72, 257 Converse theorems for asymptotic stability, 168 for boundedness, 220 for conditional stability, 284 for eventual stability, 226 for exponential stability, 158 for generalized exponential stability, 158 for LP-stability, 202 for stability, 163

D Differentiability with respect to initial conditions, 74 Dini’s Derivatives, 7 Domain of attraction, 230

E

C Caratheodory type inequalities, 42 Chaplygin’s method, 64 Chaplygin’s sequence, 66, 68 Comparison theorems, 15, 27, 131, 267, 322

Egress points, 96 strict, 97 Equi-continuity, 4 Error estimates, 254 Existence global, 45, 135, 319 local, 4, 319 Existence theorems for ordinary difierential equations, 4 for Volterra integral equations, 319, 320

388

389

SUBJECT INDEX

F Fixed point theorem, Tychonoff’s, 45 Fundamental matrix solution, 76, 109

I Infinite systems, 31 Integral equations perturbed, 333 Volterra type, 313, 314 Integral inequalities, 37, 3 15 Intcpro-diflerential inequalities, 350 Instahility, 142, 273 Invariant sets asymptotically, 298 conditionally, 305 semi, 238

K Kamkc’s uniqueness theorem, 50 Krasnoselski-Krein condition, 55

L Logarithmic norm, 104

M 11Iasinial and minimal solutions, 11, 321 continuation, I2 existence, 11 Method of averaging, 257 Mild unboundedness, 134 Mini-mas solution, 25 existence, 25

N Negative definite, 137 Nonuniqueness, 55

P Partially ordered sets, 32 Peano’s existence theorem, 4 Periodic solutions, existence, 120 Perturbed systems, 155 Perturbations bounded, 187 in mean, 188

constantly acting, 187 tending to zero, 190 Positive definite, 137 with respect to set, 235 strongly, I37

Q Quasi-monotone property, 21 mixed, 21

R Retract, 97

S Several Lyapunov functions, 267 Stability asymptotic, 103, 113, 136, 269 of asymptotically invariant sets, 297 complete, 136 conditional, 277 of conditionally invariant sets, 305, 306 of differential inequalities, 209 equi, 135, 138 equi-asymptotic, 136, 145 eventual, 222, 223 exponential, 158 generalized, 158 by first approximation, 177 integral, 191 L’I8, I99 Lagrange, 21 2 of manifolds, 244 partial, 205 perfect, 247 quasi-equi asymptotic, I36 quasi-uniform asymptotic, 136 relative, 241, 242 strict, 293 strong, 247 total, 186, 187, 189, 190 in tubelike domains, 293 uniform, 136 uniform-asymptotic, 136, 151 Stationary points, 308 Successive approximations, 14, 60

390

SUBJECT INDEX

T Topological principle, 96 applications, 100

U Under and over functions, 7, 21, 317, 352 Uniqueness criteria, 48, 50, 53, 245, 254,

327

V Variation of parameters, 76, 78 Vector Lyapunov function, 267

Z Zygmund’s lemma, 9

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28. N. P. ERUGIN.Linear Systems of Ordinary Differential Equations. 1966 29. SOLOMON MARCUS.Algebraic Linguistics ; Analytical Models. 1967 30. A. M. LIAPUNOV. Stability of Motion. 1966

31. GEORGE LEITMANNt ed.) . Topics in Optimization. 1967 AOKI. Optimization of Stochastic Systems. 1967 32. MASANAO J. KUSHNER.Stochastic Stability and Control. 1967 33. HAROLD

34. MINORUURABE.Nonlinear Autonomous Oscillations. 1967 Variable Phase Approach to Potential Scattering. 1967 35. F. CALOGERO. Graphs, Dynamic Programming, and Finite Games. 1967 36. A. KAUFMANN. and R. CRUON.Dynamic Programming: Sequential Scientific Man37. A. KAUFMANN agement. 1967

E. N. NILSON,and J. L. WALSH.The Theory of Splines and Their 38. J. H. AHLBERC, Applications. 1967

39.

P.SAWARAGI, >-. SUNAHARA, and T. NAKAMIZO. Statistical Decision Theory in

Adaptive Control Systems. 1967 BELLMAN. Introduction to the Mathematical Theory of Control Processes 40. RICHARD Volume I. 1967 (Volumes I1 and I11 in preparation) LEE. Quasilinearization and Invariant Imbedding. 1968 41. E. STANLEY 42. WILLIAMAMES.Nonlinear Ordinary Differential Equations i n Transport Processes. 1968 MILLER,JR. Lie Theory and Special Functions. 1968 43. WILLARD F. SHAMPINE, and PAULE. WALTMAN. Nonlinear 11. P.tr.1. B. RAILE>-, LAWRENCE Two Point Boundary Value Problems. 1968 \-ariationnl Methods in Optimum Control Theory. 1968 45. Ir.. J’. I’ETRO\. 0. A. LADYZHENSKAYA and N. N. URAL’TSEVA. Linear and Quasilinear Elliptic 46. Equations. 1968 and R. FAURE. 47. A. KAUFMANN Introduction to Operations Research. 1968 Comparison and Oscillation Theory of Linear Differential Equa48. C. A. SWANSON. tions. 1968 49. ROBERT HERMANN. Differential Geometry and the Calculus of Variations. 1968 50. N. K. JAISWAL. Priority Queues. 1968 51. HUKUKANE NIKAIDO.Convex Structures and Economic Theory. 1968 52. K. S. Fu. Sequential Methods in Pattern Recognition and Machine Learning. 1968 53. Y I . n E L L L. LUKE.The Special Functions and Their Approximations ( I n Two Volumes). 1969 P. GILBERT.Function Theoretic Methods in Partial Differential Equations. 54. RORERT 1969 5.5. V. IAKSHMIKANTHAM and S. LEELA.Differential and Integral Inequalities (In Two Volumes). 1969

In preparation

MASAOIRI.Network Flow, Transportation, and Scheduling: Theory and Algorithms

S. HENRYHERMESand JOSEPHP. LASALLE.Functional Analysis and Time Optimal Control

BUSTINBL~ Q ~ ~ I E RFRANCOISE E, GER4RD, and GEORGE LEITMANN. Quantitative and Qualitative Games