Differential and Integral Inequalities: Functional Partial, Abstract and Complex Differential Equations v. 2: Theory and Applications: Functional

DIFFERENTIAL AND INTEGRAL INEQUALITIES Theory and Applications Volume I1

FUNCTIONAL, PARTIAL, ABSTRACT, AND COMPLEX DIFFERENTIAL EQUATIONS

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

DIFFERENTIAL AND

INTEGRAL INEQUALITIES Theory and Applications Volume I1

FUNCTIONAL, PARTIAL, ABSTRACT, AND COMPLEX DIFFERENTIAL EQUATIONS V. LAKSHMIKANTHAM and S. LEELA University of Rhode Island Kingston, Rhode Island

A CAD E MI C P R E SS

New York 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, RETRIEVAL SYSTEM, 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 X 6BA

LIBRARY OF CONGRESS CATALOG CARDNUMBER: 68-8425 AMS 1968 SUBJECT CLASSIFICATIONS 3401, 3501

PRINTED I N THE UNITED STATES OF AMERICA

Preface

T h e first volume of Differential and Integral Inequalities: Theory and Applications published in 1969 deals with ordinary differential equations and Volterra integral equations. It consists of five chapters and includes a systematic and fairly elaborate development of the theory and application of differential and integral inequalities. T h i s second volume is a continuation of the trend and is devoted to differential equations with delay or functional differential equations, partial differential equations of first order, parabolic and hyperbolic types respectively, differential equations in abstract spaces including nonlinear evolution equations and complex differential equations. T o cut down the length of the volume many parallel results are omitted as exercises. We extend our appreciation to Mrs. Rosalind Shumate, Mrs. June Chandronet, and Miss Sally Taylor for their excellent typing of the entire manuscript. V. LAKSHMIKANTHAM S. LEELA Kingston, Rhode Island August, 1969

V

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Contents

V

PREFACE

FUNCTIONAL DIFFERENTIAL EQUATIONS Introduction Existence Approximate Solutions and Uniqueness Upper Bounds Dependence on Initial Values and Parameters Stability Criteria Asymptotic Behavior A Topological Principle Systems with Repulsive Forces Functional Differential Inequalities Notes

3 4 9 13 18 21 24 29 32 34 42

7.0. 7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7.

Introduction Stability Criteria Converse Theoreins Autonomous Systems Perturbed Systems Extreme Stability Almost Periodic Systems Notes

43 43 49 58 62 66 72 80

8.0. 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7.

Introduction Basic Comparison Theorems Stability Criteria Perturbed Systems An Estimate of Time Lag Eventual Stability Asymptotic Behavior Notes

81 81 87 97 100 101 105 110

Chapter 6.

6.0. 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. 6.10.

Chapter 7.

Chapter 8.

vii

...

CONTENTS

Vlll

PARTIAL DIFFERENTIAL EQUATIONS Chapter 9.

9.0. 9. I . 9.2. 9.3. 9.4. 9.5. 9.6. 9.7.

Introduction Partial Differential Inequalities of First Order Comparison Theorems Upper Bounds Approximate Solutions and Uniqueness Systems of Partial Differential Inequalities of First Order Lyapunov-Like Function Notes

Chapter 10. 10.0. lntroduction

10. I . Parabolic Differential Inequaliies in Bounded Domains 10.2. Comparison Theorems 10.3. Bounds, Under and Over Functions 10.4. Approximate Solutions and Uniqueness 10.5. Stability of Steady-State Solutions 10.6. Systems of Parabolic Diffcrential inequalities in Bounded Domains 10.7. Lyapunov-Like Functions 10.8. Stahility and Boundedness 10.9. Conditional Stahility and Boundedness 10.10. Parabolic Differential Inequalities in Unbounded Domains 10.11. Uniqueness 10.12. Exterior Boundary-Value Problem and Uniqueness 10.13. Notes

Chapter 11. I 1.0. Introduction 1 1.1, 1 1.2. 11.3. 11.4.

Hyperbolic Diflerential Inequalities Uniqueness Criteria Upper Bounds and Error Estimates Notes

113 113 118 127 134 136 144 148 149 149 155 163 170 174 181 186 190 200 205 210 21 3 219

22 1 22 1 223 229 233

DIFFERENTIAL EQUATIONS IN ABSTRACT SPACES Chapter 12. 12.0. Introduction 12. I . 12.2. 12.3. 12.4. 12.5. 12.6. 12.7. 12.8. 12.9. 12.10.

Existence Norilocal Existence Uniqueness Continuous Dependence and the Method of Averaging Existence (continued) Approximate Solutions and Uniqueness Chaplygin’s Method Asymptotic Behavior Lyapunov Function and Comparison Theorems Stability and Boundedness 12.11. Notes

237 231 24 1 243 247 249 254 258 264 267 269 272

CONTENTS

ix

COMPLEX DIFFERENTIAL EQUATIONS Chapter 13. 13.0. Introduction 13.1. 13.2. 13.3. 13.4. 13.5.

Existence, Approximate Solutions, and Uniqueness Singularity-Free Regions and Growth Estimates Componentwise Bounds Lyapunov-like Functions and Comparison Theorems Notes

215 215 219 284 286 288

Bibliography

289

Author Index

315

Subject Index

318

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DIFFERENTIAL AND INTEGRAL INEQUALITIES Theory and Applications Volume I1

FUNCTIONAL, PARTIAL, ABSTRACT, AND COMPLEX DIFFERENTIAL EQUATIONS

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FUNCTIONAL DIFFERENTIAL EQUATIONS

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Chapter 6

6.0. Introduction

T h e future state of a physical system depends, in many circumstances, not only on the present state but also on its past history. Functional differential equations provide a mathematical model for such physical systems in which the rate of change of the system may depend on the influence of its hereditary effects. T h e simplest type of such a system is a differential-difference equation x’@) = .f(G x ( t ) , x(t

- T)),

where T > 0 is a constant. Obviously, for T = 0, this reduces to an ordinary differential equation. More general systems may be described by the following equation: x ’ ( 4 = f ( t , 4,

where f is a suitable functional. T h e symbol xi may be defined in several ways. For example, if x is a function defined on some interval [to- T , to a), a > 0, then, for each t E [ t o ,to a ) ,

+

+

(i) x i is the graph of x on [t - T, t] shifted to the interval (ii) x i is the graph of x on [to - T, t ] .

[-T,

01;

In case (ii), f is a functional of Volterra type which is determined by t and the values of x(s), to - T s t. Systems of this form are called delay-differential systems. I n what follows, we shall, however, consider the functional differential equations in which the symbol x L has the meaning described by case (i) and study some qualitative problems by means of the theory of differential inequalities. I n the present chapter, we consider existence, uniqueness, continuation, and continuous dependence of solutions and obtain a priori bounds

<
0, let V r L= C[[-T,O], R"] denote the space of continuous functions with domain [ - T , 01 and range in Rn. For any element $ E P,define the norm

R"]. For any t >, 0, we shall let x L denote Suppose that x E C[[--7, a),

a translation of the restriction of x to the interval [t - T , t ] ; more

specifically, x t is an element of V n defined by

I n other words, the graph of x t is the graph of x on [t - T, t] shifted to the interval [ - T , 01. Let p > 0 be a given constant, and let

With this notation, we may write a functional differential system in the form (6.1.1)

x'(f) = f ( t , XJ.

DEFINITION 6.1.1. A function x ( t , ,$0) is said to be a solution of (6. I .l) with the given initial function do E C, at t = t, >, 0, if there exists a number A > 0 such that (i) x(t, , $ J is defined and continuous on [to- T , to x,(t, , $0) E c, for t o to t o -tA ;

(4 X&o ,$0)

<
, 0, there exists an a > 0 such that there is a solution ~ ( t,+o) , of (6.1.1) existing on [to- T, to a).

THEOREM 6.1.1.

+o

E

Proof.

+

Let a

> 0 and

define y

E

C[[to- 7 , to

doe - to),

to to

r(t)= Id"(O),

-7

+ a ] , Rn] as follows:

< t < to

e t < to f

a

'1,

+

T h e n f ( t , y t ) is a continuous function of t on [ t o ,to a] and hence IIf(t ,yt)ll Ml . We shall show that there exists a constant




rt)

I/ < 1




t" ,

whereg t C [ J x R , , R,]. Assume that r ( t ) = r ( t , to , u,)is the maximal solution of the scalar differential equation u'

= g(t, u),

u(to) = U ( ,

existing for t >, t o . Then, provided 1 m,"

lo

< u0

m(f) .=, 4th ~

t

2 t"

7

20

(6.1.2)

6.1.

Proof.

7

EXISTENCE

T o prove the stated inequality, it is enough to prove that m(t) < u(t, f,

>

u"

, €1,

t

2 to,

where u(t, t o , u,, , c ) is any solution of u' = g(t, E

> 0 being

4 + E,

u(t0) = uo

+

€9

an arbitrarily small quantity, since lim u(t, to , uo , 6) e-30

r(t, to , uo).

=

T h e proof of this fact follows closely the proof of Theorem 1.2.1. We assume that the set

Since g ( t , u ) 3 0, u(t, t o ,u,, , c ) is nondecreasing in t , and this implies, from the preceding considerations, that

I mtl

lo = 4

4

7

to

, uo

9

4 = +l).

(6.1.4)

Thus, we are led to the inequality D-m(tl> G

>

I mtl I")

= R(tl

, U ( t , t, , uo , E ) ) , 7

which is incompatible with (6.1.3). T h e set 2 is therefore empty, and the lemma follows. With obvious changes, we can prove

COROLLARY 6.1.1. Let m E C[[t,- T , m), R,],and, for t D-m(t) 3 -g(t,

min

t&-TKS to ,

8

CHAPTER

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

Let p ( t ) = p(t, t o ,uo) >, 0 be the minimal

u' = -g(t, u),

existing for t

6

u(t0) = uo

>0

2 to . T h e n uo < min+GsGto mto(s)implies P(t)

< W), t 2 to -

T h e following variation of Lemma 6.1.1, whose assumptions are weaker, is also useful for later applications.

LEMMA 6.1.2. Let m E C[[to- 7 , co),R,], and let, for every t, for which I m t l lo = m(tl), the inequality D-Wd

> to

t o , the inequality

o-44 < s k 1 m, lo). Choosing I m," lo

=

/I

+,)

Ijo

I/ " ( t o , +o)(t)ll

< uo , we obtain, by Lemma 6.1.1, < r ( t , to , u,),

t"

< t < P.

(6.1.6)

Since the function g(t, u) 3 0, r(t, to , uu) is nondecreasing, and, hence, it follows from (6.1.6) that

11 % ( to ,+")ll"

< r ( t , t o , U"),

to

to , uo) - +I

>

t o , uo).

Letting t , , t, -+ p-, the foregoing relation shows that limi+o- ~ ( t, ,+ J ( t ) exists, because of Cauchy's criterion for convergence. We now define .(to ,+o)(p) = limL+o-.(to ,+O)(t)and consider lCIo = xs(to , do) as the new initial function at t = /3. An application of Theorem 6.1.1 shows that there exists a solution x(p, &) of (6.1.1) on [/3, p 011, 01 > 0. This means that the solution a f t o ,+,J can be continued to the right of p, which is contrary to the assumption that the value of /3 cannot be increased. Hence, the stated result follows.

+

COROLLARY 6.1.2. T h e conclusion of Theorem 6.1.2 remains true even when the condition (6.1.5) is assumed to hold only for t E J and 4 satisfying I1 4 I10 = I1 +(O)ll. 6.2. Approximate solutions and uniqueness

DEFINITION 6.2.1. A function x(to ,4, , 6) is said to be an €-approximate solution of (6.1.1) for t 2 to with the initial function +0 E C, at t = to if (i) x(to , + o , E) is defined and continuous on [to - T, a) and %(to 3 $0 ). E c, ; ( 4 .,(to 90 ) . = $0 ; (iii) .(to , +,, , E ) is differentiable on the interval [ t o , a),except for an at most countable set S and satisfies 9

9

for t

E

[ t o , CO)

7

II XYtO I 4 0 - S.

9

.>

( t ) - f @ , %(to

,40 .)I1 I

e

E

(6.2.1)

I n case E = 0, it is to be understood that S is empty and .(to , 40) is a solution of (6.1.1). We shall now give some comparison theorems on r-approximate solutions of (6.1.1).

10

6

CHAPTER

THEOREM 6.2.1.

x C, , R"],and, for ( t ,+), ( t , 4) E J x C, , Ilf(6 4) - f ( t , 4)ll < At, I1 4 4 llo), (6.2.2)

Letfg C[J

-

where g E C [ J x [0, 2 p ) , R,]. Assume that r ( t ) = r ( t , t o ,uo) is the maximal solution of u' = g(l, u )

+ + €1

€2

,

u(t0) = uo

20

existing for t >, t, . Let % ( t o+o , , el), y(t, , & , e n ) be proximate solutions of (6.1.1) such that

II 4 0

Then,

I1 4 f n do , ~i)(t)

-~

9

Proof.

d(t)ll

7

e2-ap(6.2.3)

< ~ ( tto,

7

2 t o . (6-2-4)

t

uo),

Consider the function

m(4

so that

=

I1 X ( t , 4, I

-

9

2 t o . Then, we have, for O-m(t)

d ( t ) - Y(t0

7

II %(to do > €1)

mt

for t

( f o40

< uo.

$" I10

-

el,

40

YdtO 3

7

%)(t)/l,

40 >

4 1

> to,

t

< I/ 4" 4 t < I/ "(to do 4 t

%)(t)ll ) - A t , %(to > 4 0 > 4)ll ll Y'(t0 , 4 0 > &> - J P , YdtO > 4 0 > %>>II

XYtO

9

9

)

-

r'(to

40

7

f

+ + /lf(t!%(to

40

4- f k

9

YdtO

$0

?

.2))ll.

Now, making use of the assumption (6.2.2) and the fact that ~ ( t, , , el), y(tO, I/J~ , 6.) are e l , €,-approximate solutions of (6.1.1), respectively, it follows, from the preceding inequality, that D-m(t)


0, is admissible in Theorem 6.2.1. In fact, Corollary 6.2. I implies the well-known inequaIity for approximate solutions

a" < I / dn

I1 4 t o

I

9

E d f )

-

-

40

Y(t"

/lo

4" 4(t)ll

eL(t--tl,)

>

I-

€1

1

€2 [eL(t--to)

7

--

I],

t

2 t o . (6.2.5)

6.2.

11

APPROXIMATE SOLUTIONS AND UNIQUENESS

It is possible to weaken the assumptions of Theorem 6.2.1 in that we need not assume the condition (6.2.2) for all +, $ E C, . T o do this, we require the subset C, C C, defined by

THEOREM 6.2.2. Let the assumptions of Theorem 6.2.1 hold except that the condition (6.2.2) is replaced by

for t E J ,

4,$ E C, . Then, (6.2.3)

implies the estimate (6.2.4).

Proof. Suppose that, for some t, > to , 1 mil ,1 = m ( t l ) , where m ( t ) , m , are the same functions as in Theorem 6.2.1. Setting = x,,(t, , 4,, el), ICI = Y#o $0 , 4, we see that $ E c, and 4 t l ) = II +(O) - +(O)Il. Hence, using (6.2.6), we get

+

$7

9

D-m(t1)

< g(t1

7

m(t1))

+ + €1

€2

,

as previously. T h e assumptions of Lemma 6.1.2 are verified, and therefore the conclusion follows.

A uniqueness result of Perron type may now be proved.

THEOREM 6.2.3.

Assume that

<
0, r ( t , t o ,uo) being the maximal solution of (6.1.2) existing for t 2 to . Then, if ~ ( t,+,J , is any solution of (6.1.1) such that 11 bo \lo = uo , we have

/I .(to

< r ( t , to

+o)(t)lI

7

9

(6.3.4)

24")

as far as ~ ( t, &) , exists to the right of to . Proof. Let [ t o ,T] be a given compact interval. Then, by Lemma 1.3.1, the maximal solution r ( t , C) = r ( t , t o , uo , E ) of u' = g(t, u )

+

u(to) = uo

E,

exists on [ t o ,TI, for all sufficiently small

E

>0

+

E

and

lim r ( t , C) = r ( t ) r+n

TI.

uniformly on [ t o ,

I n view of this, there exists an

r(t, .)

< r(t)

+

€0

,

t E [to

7

co

>0

such that

.I.

Furthermore, we have, by Theorem 1.2.1, y(t)

< r(t, C),

t

[to , 71,

E

which implies that r(t)

< r ( t , ). < r(t>

+

€0

f

3

E [to

, .I.

(6.3.5)

T o prove (6.3.4), it is enough to prove that

11 "(to

7

I/ < r ( t , E),

do)(t)

t

E

[ t o , TI.

Assuming the contrary and proceeding as in the proof of Lemma 6.1.1 we arrive at a t , > to such that (i)

I/ X(t"

(4 /I " ( t o (iii)

I/ "*'&

Y

9

+o)(t1)il = Y ( t 1

$o)(t)ll

, +,,)l,



= R(tl

I

Y(t1

41,

6.3.

15

UPPER BOUNDS

which contradicts the relation D-m(t,)

2

+,, €1

= g(t1 , r(t,

,4 )

+

6,

resulting from cases (i) and (ii) above. T h e proof is therefore complete. T h e following theorem gives a more useful estimate.

THEOREM 6.3.3. Let f~ C [ j x C, , R"], g E C [ J x [0, p), R,], and

(6.3.6)

/I 9 /lo 1 4 lo where A(t) >, 0 is continuous on of (6.1.1) such that

=

/I +(O)Il

[-T,

4 t h

a).Let ~ ( t, ,

/I 4 0 110 I Ato lo d

(6.3.7) be any solution

*a

and r(t, t o , go) be the maximal solution of (6.1.2) existing to the right of t o . Then,

/I %(to

>


4o)lI 4 *

I mto lo

=

II 40 110 I Ato lo

By hypothesis, we have

Let u(t, E)

=

u(t, to , uo , c) be any solution of u'

for

E

>0

< uo .

= g(t, u )

+

E,

u(to) = U"

sufficiently small. Since lim u ( t , E ) S i O

=

r ( t , to , uo)

(6.3.8)

16

6

CHAPTER

and u(t, E ) exists as far as r(t, t o ,uo) exists, it is enough to show that m(t)

< u(t, 4

t

3 to *

(6.3.9)

If this inequality is not true, let t , be the greatest lower bound of

numbers t > to for which (6.3.9) is false. T h e continuity of the functions m(t) and u(t, E) implies that (i) m ( t ) (ii) rn(tl)

< u(t, E), =

< t < ti ;

to

~ ( t,,E ) , t

=

t, .

By the relations (i) and (ii), we have D-m(t,) >, U Y t ,

7

4

= g(t1 , 4 t l

?

6))

+

+

E.

(6.3.10)

Since g(t, u ) E is positive, the solutions u(t, E ) are monotonic nondecreasing in t , and hence, by relation (ii),

Setting

+

I mtl In =

=

4ti)

=

4ti

6).

x t l ( t o $,,), , it follows that

/I4 /In I At1 In = I1d(0)Il A(t1)Thus, at t = t , , (6.3.7) holds true with this 4. Hence, using (6.3.6), there results, using the standard computations, the inequality D-44)

< g(t1 , 4 t l ) ) ,

which contradicts (6.3.10). It therefore follows that (6.3.9) is true, and this proves the stated result.

As a typical result, we shall prove next a theorem for componentwise bounds.

THEOREM 6.3.4.

I f i ( t , +)I for each i

=

Let f

< gi(t, 1

C [ J x C, , R"],and let

E $1

1, 2 ,..., n, t

I4i I"

I"

E =I

I 4i-1

10

, I $i(O)l~ 1 4i+1 1"

j...)

I4n lo) (6'3.11)

J , and $ E C, satisfying

I d,(O)I

(i = 1,

z-., a),

whereg E C [ J x R+",H.,."],g(t, u ) is quasi-monotone nondecreasing in u for each t t J . Then, if .x(t,,, 4,) is any solution of (6.1.1) with the initial function +n = ,...,I$,,,), such that 1 +io lo uio , we have


0. If (6.3.12) is false, let n

Z = (J [ t E J : mi(t) 2 u,(t, €)I i=l

be nonempty and t , = inf 2. Arguing as in Theorem 1.5.1, there exists an index j such that (i) mj(tl) = U j ( t l 9 E), (ii) mi(t) < uj(t,E), to d t (iii) and

"i(t,)

< %(t, ,

E),

D-w(t1)

< t, ,

# j,

2 "i'(t1 , €1 =

, U(t1 , .))

+

E.

(6.3.13)

Since u(t, E) is nondecreasing in t, it follows from (i), (ii), and (iii) that

18

CHAPTER

+

Setting == x,,(t, ,do),it results that (6.3.1 I ) , we arrive at

6

I +j lo

=

I +j(0)l, and hence, using

T h e quasi-monotone property of g(t, u) in u and the inequalities (6.3.15) yield

+

because of the definition of and (6.3.14). This inequality is incompatible with (6.3.13), and hence the set 2 is empty, which in turn proves the stated componentwise estimates. T h e theorem is proved.

6.4. Dependence on initial values and parameters We shall first prove the following lemma, which will be used subsequently.

LEMMA 6.4.1.

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

E

J , 4 E C, ,

Suppose that r * ( t , to , 0) is the maximal solution of U' =

G(t,U)

through ( t o ,0). Then, if ~ ( t ,,+,J is any solution of (6.1.1) with the initial function bn at t = t o , we have

on the common interval of existence of ~ ( t, ,+,J and ~ * ( t o, , 0).

Pmof.

Consider the function

6.4.

19

DEPENDENCE ON INITIAL VALUES AND PARAMETERS

and hence by Theorem 1.4.1. T h e function G being nonnegative, r*(t, t,, , 0) is nondecreasing in t , and therefore there results the desired inequality (6.4.1).

THEOREM 6.4.1. Let f~ C [ J x C, , R"],and, for t E J , 4,4 E C, , llf(t, $1

f ( t 7

~

$)I1

< At, II d(0)

-

$(O)lI),

where g E C [ J x [0, 2p), R,]. Assume that u(t) = 0 is the only solution of the scalar differential equation (6.1.2) through ( t o ,0). Then, if the solutions u(t, t o ,uo) of (6.1.2) through every point ( t o ,uo) exist for t 3 to and are continuous with respect to the initial values ( t o ,uo), the solutions %(to, of (6.1.1) are unique and continuous with respect to the initial values ( t o ,+o). Proof. T h e uniqueness of solutions of (6.1.1) is a consequence of Theorem 6.2.3, and hence we need only to prove the continuity with respect to initial values. Let ~ ( t ,+o), , ~ ( t, ,$,) be the solutions of (6.1.1) with the initial functions q50 , $o at t = to , respectively, existing in some interval to t to + a. Then, an application of Theorem 6.2.2 yields that

<
0, there exists a 6 > 0 such that r(t, to

7

I/ do

~

$0 110)

< E,

to

< t < t o + a,

provided I/ c$,,- $" /lo < 6. This, in view of (6.4.2), assures the continuity , ,) of (6.1.1) with respect to initial functions 4o . of the solutions ~ ( t,+

20

CHAPTER

6

We now prove the continuity of solutions with respect to the initial time t,, . IAett , > t, and " ( t o, #,), x(t, , #,) be the solutions of (6.1.1) through (to, #,,) and ( t , , do), respectively, existing in some interval to the right. Define m(t)

~

I1 4 t n

7

Co)(t)

?4o)(t)II,

- "(ti

< t < t i + a.

ti

Since, by Lemma 6.4.1, we have mtl

=

/I X t , ( t n

we obtain m(t)

14n)

< f(t),

4" II

~

t,

< r*(ti

9

t o , O),

< t < t , + a,

(6.4.3)

where F(t) = F(t, t , , r * ( t , , t , , 0)) is the maximal solution of (6.1.2) through ( t l , r*(tl , tn , 0)). Now, lim

tl+to+

f(t, t,

, ~ * ( t,,t , , 0)) = f ( t , t , , 0).

Since lim,l,to r * ( t , , t, ,0) = 0 and by assumption, F ( t , t o , 0) is identically zero. This fact, in view of the relation (6.4.3), proves the continuity of solutions of (6. I . 1) with respect to initial time. T h e proof is complete.

COROLLARY 6.4. I . Theorem 6.4. I .

T h e function g(t, u)

= Lu,

L

> 0, is

admissible in

Using the arguments of Theorems 2.5.2 and 6.2.1, we can prove the following theorem on dependence on parameters. We merely state

THEOREM 6.4.2. Let f~ C [ J x C, x R", R"], and, for p xo(t) = xo(to, #o , pn)(t)be a solution of

= po,

let

x' = f ( t , X t Po), 1

with an initial function #o at t

=

t o , existing for t >, to . Assume that

limf(t2 4, P )

!-+PO

uniformly in ( t , #) F J x C, , and, for t

I/ f(t,4, P )

-

4, Pn)

=f(t,

E

J, #,

f(t, $>PII< I~

$J

E

C, , p

E

R",

( t/I ,4 - $ IIJ,

where g F C [ J x R, , R , ] . Suppose that u(t) = 0 is the maximal solution of (6.1.2) such that u(t,) = 0. Then, given E > 0, there exists

6.5.

21

STABILITY CRITERIA

a a(€) > 0 such that, for every p, 11 p - po I/ system x’ = f ( t , X t P)



d0)Iln


, t” *

DEFINITION 6.5.2. T h e trivial solution of (6.1.1) is said to be asymptotically stable if it is stable and, in addition, for any > 0, to E J , there exist positive numbers S o , T such that / / $, < 6, implies

II xt(to > do)lIo < E ,

t

3 to

+ T.

Simple criteria for stability and asymptotic stability of the trivial solution of (6. I . 1) are given in the following theorems.

THEOREM 6.5.1. LetfE C [ J x C, ,R ” ] , gE C [ J x [0, p ) , R ,],g(t, 0) and, for t E J, E C, such that

+

~

0,

(6.5.1)

holds. If the trivial solution of the scalar differential equation (6.1.2) is stable, then the trivial solution of (6.1.1) is stable.

Proof.

By Theorem 6.3.1, we have

I1 4 t n > dn)(t)ll < ~

( ttn,

9

II do llo),

t

2 tn ,

22

CHAPTER

6

where r ( t , to , 11 $, I ,J is the maximal solution of (6.1.2) through ( t o ,I( $o I l o ) . T h e fact g 3 0 implies that r ( t , to , (1 $o )1, is nondecreasing in t , and therefore it follows that

< r(t, t n

I1 "dto ,4o)l&

3

/I $0

(6.5.2)

2 to .

t

Iln),

Assume that the trivial solution of (6.1.2) is stable. Then, given E -. p, to c J , there exists a 6 > 0 such that 0 r ( t , to

I1 (fu !In) < 6 ,

9

t

2 to

< 6. T h e conclusion is immediate from (6.5.2).

provided

11 do /I,

TIiEoRm

6.5.2. 1,et f~ C [ J x C,, Rn], g E C [ J x [O, 0. Assume that

g(t, 0)

... :

a t >lim inf Wl d(0) h+O-

+ hf(t> $)I1 - II +(O)lll

p),

R,], and

-t- 11 4(0>llD W t )

< s(t2 I1 C(0)lI J J ( t ) ) for t

> to and $ E C,, satisfying II 4 1In I At

I1 $(o)il A ( t ) ,

lo

(6.5.3)

where A ( t ) 2 1 is continuous on [to- 7,a)and A(t)+ GO as t + co. Then, the stability of the trivial solution of (6.1.2) implies the asymptotic stability of the trivial solution of (6.1 .I). Proof.

By Theorem 6.3.3, it follows that

II "(ti]

4o)(t)!l -d(t> ,< r ( t , to

>

ll (fu Ilo I Ato lo),

t

2 tn ,

and, arguing as in Theorem 6.5.1, we have

11 .yt(t,

I

a,,,;',,1

24,

I"

< r ( t , to , I/

$0

/lo I

t B to.

lu),

Since A ( t ) 23 I , the stability of the trivial solution of (6.1.1) is a consequence of 'I'heorem 6.5.1. Now let E = p and designate by 6,, the S-obtained corresponding to p. Suppose that (1 $(,1Io S,, . Assume that, if possible, there exists a sequence {t,J, t,. co as h ca and a solution (lo : : S,, such that x(t,,, +,,) of (6. I . 1 ) with 11


to and t > t , , we have

Because g(t, u ) 3 0 and consequently r ( t ) is monotonic nondecreasing in t, it results from the inequality (6.6.2) that

This, together with (6.6.3) and the monotonicity of g in u, implies that

Moreover, by the assumption of boundedness of all solutions of (6.1.2), we deduce that r ( t ) tends to a finite limit as t + m. This means that, given an E > 0, it is possible to find a t , > 0 sufficiently large such that

0

e r(t) -

Y(tJ

< E,

t

> t, .

Consequently, we obtain, as a result of (6.6.4),

This completes the proof. T h e next theorem deals with the asymptotic equivalence of two functional differential systems.

THEOREM 6.6.2.

Let u(t) be a positive solution of u'

> g(t, u )

26

6

CHAPTER

for t 2 to such that limt-,mu(t) = 0, where g E C [ J x R, , R]. Suppose further thatf, ,f 2 E C [ J x V n ,R"] and lim inf h h-n-

for t

"I1 440)

ItfO) iWdt,4

~

< d t , I1 4(0)

-

-

f & 4lll

-

/I d(0) - lcr(0)tll (6.6.5)

lcr(0)ll)

> t o and a, II, E Q, where l2

=

[+, 4 E V": 11 &(O)

$(O)ll

~

=

Then, if the existence of solutions for all t

4th t

2 to].

3 t, of

the systems

(6.6.6) (6.6.7) is assured, the systems (6.6.6) and (6.6.7) are asymptotically equivalent. Proof. Let us first suppose that y(to , II,J is a solution of (6.6.7) defined for t 2 ; t , . Let ~ ( t, , be some solution of (6.6.6) such that

/I $0

-

lcro

110

< 4to)-

Define m(t) = II 4 t " > a d t )

Then,

m(t) < u ( t ) ,

-

Y(t" lcr")(t)ll. 9

t 3 t".

If this is false, let t , be the greatest lower bound of numbers t >, to for which m ( t ) z i ( t ) is not satisfied. T h e continuity of the functions m(t ) and ~ ( tguarantees ) that, at t = t , , < .

(6.6.8) and

(6.6.9)

This implies that D_m(t,) 3

On the other hand, defining $

444

= N(t,

1

‘Go)(

and therefore ??&(ti)

=

)I(

6.6.

ASYMPTOTIC

BEHAVIOR

27

because of (6.6.8). Thus, 4, It E Q at t = t, , and hence, using the condition (6.6.5), it is easy to deduce that D-m(t,)

< g(t1

m(t1)).

>

This being incompatible with (6.6.10), we conclude that m ( t ) < u(t), t 3 t o . T h e assumption that limt+mu(t) = 0 now implies that

pt 4 t o

9

+ow

-

Y(to M t ) ?

= 0-

(6.6.11)

If x(t, ,+,) is a given solution of (6.6.6), defined on [ t o , a), arguing as before, we can assert that there is a solution y ( t o ,Ito) of (6.6.7) such that (6.6.11) is satisfied. It therefore follows that the systems (6.6.6) and (6.6.7) are asymptotically equivalent. We shall now give an analog of Theorem 2.6.3 with respect to the following two systems: x' =f&, Y'

4,

=f d t , Y d .

(6.6.12) (6.6.13)

THEOREM 6.6.3. Let fi E C [ J x 'P,R"] and y(to,#o) be any solution of (6.6.13) defined for t 3 to . Assume that fi E C [ J x R", R"], afl(t, x)/ax exists and is continuous on J x Rn.If x(t, t o ,+,,(O)) is the solution of (6.6.12) such that %(to, to ,+,,(O)) = #,,(O) existing for t 3 to , then y(to ,+,J satisfies the integral equation Yt" = 4 0

where @(t,to , xo) = ax(t, to , x o ) / a x o and y ( t ) = y(to , #o)(t). T h e proof of this theorem is very much the same as that of Theorem 2.6.3. It is important to note, however, that (6.6.12) is an ordinary differential system, whereas (6.6.13) is a functional differential system. As an application of Theorem 6.6.3, we have

THEOREM 6.6.4. Assume that (i) fi E C [ J x R", R"], 8f1(t,x)/ax exists and is continuous on E C [ J x Vn, R"];

J x R", and f2

28

CHAPTER

6

(ii) @(t,t o , xu) is the fundamental matrix solution of the variational system %'

=

q-dt, x(t, t o , xo)) ax

such that @ ( t o ,t o , x,,) = identity matrix I ; (iii) for a given solution y ( t ) = y ( t O+,,)(t) , of (6.6.13), existing on [t" a), I

J=q t ,

$7

Y ( W f & , YS) -fi(s?

Y(4)l

ds

+

0

as

t

-j

a*

Then, there exists a solution x ( t ) of (6.6.12), on [to , a), satisfying the relation lim [ x ( t ) - Y(to ,M t ) l = 0. tim T h e proof can be constructed using the arguments employed in Theorem 2.10.3. Finally, we may mention a result parallel to Theorem 2.14.10.

THEOREM 6.6.5.

Assume that

(i) f E C [ J x K", R"], f ( t , 0) = 0, and fz(t, x) exists and is continuous on J x R"; -0, 0 > 0, t E J ; (ii) p[fX(t,O)] (iii) R E C [ J x V n ,R n ] ,R(t, 0) = 0, and there exists an 01 > 0 such that, if / / (b /lo Lt, t E J ,


0 depending on ~ ( t )there ] , is an E > 0,0 < E < u, such that ( t , x(t)) E E for to - E t r, t o . We shall denote by S* the set of points of strict egress. We see from the preceding definitions that Po = ( t o ,x,,)E S* implies that Po E S if there is at least one do E WL, $,,(0) = x,,, such that the solution ~ ( t,,$,,)(t) can be extended to the left. These definitions coincide with the definitions given in Sect. 2.9, if r = 0. ~

< +


to such that Q = (tl , x ( t l ) ) E S and ( t , x ( t ) )E E for to t < t , . We denote the point Q by ~ ( t ,,,+,,). For every to > 0, let E(t,,) be the set of points (to, x) E E. Again, for t, > 0, $,, E E7L, define


E 2 - w. Then, r(to, xo + $,) exists for every (to , xo) E 2 - w , and hence

+

+

2 = (2 n W )u (2 - W ) C w

LJ

G(t, ,do).

This implies that the restriction of the mapping T to 2 is a retraction from Z into w n Z , which is a contradiction. T h e proof is therefore complete. Corresponding to Theorem 2.9.2, we now state the following result, which gives sufficient conditions to ensure that S C S".

THEOREM 6.7.2. Let (i) E

=

LetfE C [ J x C,, R"], u E C[B, Rp],and ZI E C[Q,Rq].

[ ( t , x) EQ: uj(t,x)

=

[ ( t , x)

EQ:

I<j, is asymptotic with respect to E,,. By taking a convergent subsequence of {xrtt, ynt>with limit (xo ,yo),it follows that x(t) . y ( t ) 0, t >, to , where x(t) = X ( t 0 Y ) ( t ) , Y ( t ) = Y(to 7 Y ) ( t ) , and Y = (Xo $0 Yo $0). Since 5,n Zbl is empty for b, # b, , we see that it is possible to take (xo,yo) depending on n parameters. T h e proof is complete.

+

+

+

9

DEFINITION 6.8.2.

forces if

+

+

9


0,

t

2 T,

474 E wl-

COROLLARY 6.8.1. If Eq. (6.8.1) is a system with repulsive forces, then the conclusion of Theorem 6.8.1 is valid.

6.9. Functional differential inequalities

It is natural to expect that a comparison theorem for functional dif-f'erential equations, analogous to Theorem 1.4.1, would be equally useful in some situations. T o prove such a result, we need to show the existence of maximal solution for functional differential equations. We are thus led to the study of functional differential inequalities. With this motive, we shall consider a functional differential equation of thc form (6.9.1) x' = f ( t , x,4, which is convenient for some later applications. T h e existence theorems 6.1.1 and 6.1.2 are valid for such an equation, with obvious changes. Let us begin with the following basic theorem on fundamental differential inequalities, recalling that V = C[[-T, 01, R].

'I'HEOHEM 6.9.1. Let f~ C [ J x R x Y , R] and f ( t , x,+) be nondecrcasing in d, for each ( t , x). Let x,y E C[[to T , m), R] and ~

Xi"

(6.9.2)

Yt, '

Assume further that (6.9.3)

(6.9.4)

Pyoof.

If the assertion (6.9.4) is false, then the set

is nonempty. Let t , Moreover, .?(tl)

z

=

=

inf Z. It is clear from (6.9.2) that t ,

[ t E [t,, , 02): x ( t ) 3 y ( t ) ]

and

=Y(tJ

x(t)

< Y(t),

t E [ t o , tl).

>to. (6.9.5)

'I'hus, we obtain for small h

> f"

+4

then r(t,, ,$(J is said to be the maximal solution of (6.9.1). A similar definition may be given for the minimal solution by reversing the preceding inequality.

THEOREM 6.9.3. Let f E C [ J x R x Y , R] and f ( t , x, 4) be nondecreasing in 4 for each ( t , x). Then, given an initial function 4,E Y at t t,, , there exists an m1 > 0 such that Eq. (6.9.1) admits a unique maximal solution r(t0 , 4") defined on [to , to q).

+

~

Proof. Following the proof of Theorem 6.1.1, we obtain a , b Suppose that 0 :E b/2. Consider the equation


'2))

, a0 , )1'

+ +

'2

,

€2

.

We can now apply Theorem 6.9.1 to get .$to

14"

>

4 t )

< "(t"

7

$,I

1

E,)(f),

t

6

[t" , to

+

011).

Since the family of functions x(t, ,4" , ~ ) ( t are ) equicontinuous and uniformly bounded on [ t o ,t, (Y~),it follows by Ascoli-Arzela's

+

6.9.

37

FUNCTIONAL DIFFERENTIAL INEQUALITIES

theorem that there exists a decreasing sequence ( e n } , E , ~ such that

---f

lim

n+m X ( t 0

7

do > E d t )

= r(t0

1

0 as n

-+

00,

d")(t)

+

uniformly on [to, to al). Clearly, ~ ( , ( t,,+o) = . T h e uniform continuity of f implies that f ( t , ~ ( t,, , ~ , ) ( t ) , xl(t, , , en)) tends uniformly to f ( t , r(to , +o)(t),y l ( t 0 , $o)) as n 00, and, thus, term-byterm integration is applicable to

+,

---f

4 t o Id0

7

%At)= dO(0) + E n t

.cI [ f h

do

X(t0

to

9

E

m

, Xdt"

> $0

7

4 )+ %I ds,

which, in turn, shows that the limit ~ ( t ,$o)(t) , is a solution of (6.9.1) 4. on [ t o 7 t o We shall now show that ~ ( t, $o) , is the desired maximal solution be any solution of (6.9.1) defined of (6.9.1) on [to, to q).Let "(to, on [ t o ,t, al). We then have

+

+

+

4") < Xf,(tO > d o , €1, do E ) ( t ) 2 f ( t > X(t0 40 , E ) ( t ) ,

%,(to XYtO

7

XYt,

for 0

7

7

40x4 < f ( h "(to

7

%(to , $0 , 4)

do)(th %(to

> $0))

+

+

6,

c,

< E < 6/2. By Remark 6.9.1, it follows that X(t0

Since

9

9

40)(t)

< X ( t 0 do 7

?

lim r+U X ( t " , do , .)(t)

uniformly on [ t o ,to

t t

E)(t),

=

[to to ?

+ 4.

'(to ,d">(t)

+ al), the theorem is proved.

REMARK 6.9.2. Under the assumption of Theorem 6.9.3, we can show the existence of the minimal solution also. T h e proof requires obvious changes. We are now in a position to prove the following comparison theorem for functional differential inequalities. Let %+ denote the set of all nonnegative functions belonging to %'.

THEOREM 6.9.4. Let m E C[[to-r>-m(t)

T,

a), R+],and satisfy the inequality

< j ( t , m(t), m,),

t

>- t" ,

38

CHAPTER

6

where f E C[J x R, x V+ , R]. Assume that f ( t , x, 4)is nondecreasing in 4 for each ( t , x) and that ~ ( t ,,do), , E Y+ , is the maximal solution of (6.9.1) existing for t 3 t o . Then, mto $(, implies




7

t J'I@(hs, V ( N f ( S , Y ( 4 Y ( S - 7)) (0

for t

3 t o . Moreover, on the basis

-fh Y(+ Y ( 4 l 4 (6.9.15)

of Theorem 2.6.4, we deduce that

Also, using assumption (i), we have

Thus, we obtain, on account of (6.9.15), (6.9.16), and (6.9.17),

because of condition (ii). Observe that

6.9.

FUNCTIONAL DIFFERENTIAL INEQUALITIES

41

Hence,

where M

= LNr.

We now define

so that This, together with the fact that

yields the functional differential inequality u’(t)

< -au(t) + M

sup

Ut

-27 G.9GO

.

By Theorem 6.9.4, we therefore get u(t)

provided u l , = defined on -27 of

< r(to

t

u)(t),

9

> to ,

(6.9.19)

< u, where is the initial function at t = t o , < s < 0, and r ( t ) = r(to , u ) ( t ) is the maximal solution

11 &,I/

(r

+M

U’ = - a ~

SUP -27
0, provided a > M , which implies that 0 0, to €1,there exists a positive function S = S ( t , , E ) , which is continuous in to for each E, such that the inequality

ll4ollo

implies

/I "dt,

4o)Ilo

?


)1*

(7.1.2)

Sometimes, we also define D+V(t,4) = lim sup h-l[V(t h+O+

+ h, ~ ~ + ~ ( t , V+ ()t , + ) ] . -

(7.1.3)

where it is understood that x(t,#) is any solution of (7.1.1) with an initial function # at time t.

REMARK7.1.1. If the uniqueness of solutions of (7.1.1) is assured, both the definitions (7.1.2) and (7.1.3) are identical, since letting # xt(t, , &) and noting that ~ ~ + ~,A)) ~ (= t , , ,,,(t,41, h 2 0, because of uniqueness, (7.1.2) reduces to (7.1.3), and vice versa. We now state certain fundamental propositions regarding Lyapunov stability of the trivial solution of (7. I . I). ~

THEOREM 7.1.1. Assume that there exist a functional V ( t ,4) and a function g(t, u ) fulfilling the following properties: (i) V EC [ ] x C,, R , ] , and, for t 3 t o , I)+ G-(t, X , ( t ,

9

$0))

-<s(4 L'(4

%(to

,$0)));

7.1.

45

STABILITY CRITERIA

(ii) g E C [ J x R, , R], and g ( t , 0) = 0; (iii) there exist functions b E Z and a E C [ J x [0, p), R,],a(t, u ) E X for each t E J , such that

Wld 110)

< V(4d) < 4 4 II d llo),

(4d) E J x

c,

*

Then, the trivial solution of the functional differential system (7. I. 1) is (1") equistable if the trivial solution of (6.1.2) is equistable; (2") equi-asymptotically stable if the trivial solution of (6.1.2) is equi-asymptotically stable.

Proof. Suppose that the trivial solution of (6.1.2) is equistable. Let 0 < E < p , to E J be given. Then, given b(c) > 0, to E J , there exists a positive function 6 = S ( t , , c) that is continuous in to for each E such 6 implies that uo


(t

-


I.

I t is now easy to see from Theorem 7.1.3 that I b I a, I b I < a imply uniform stability and uniform asymptotic stability of the trivial solution of (7.1.8), respectively. It is observed that the particular Lyapunov functional just used has given a region of stability which is independent of the lag T and the sign of b. If, on the other hand, we take the functional

and assume that b 3 0, we see that this functional verifies condition (i) of Theorem 7.1.3. T o verify condition (ii), we compute V ( t ,x l ( t , , 4,)) using (7.1.9). After some calculations, we obtain the following relation:

b"(4 .,(to

,$0))


II 41 - 4 2

/I d 110 < v(t,4) G K ( t ) It # 110 > t E J, # E Coo; (3") D+V(t,#) < -(l - q) p'(t) V ( t ,#), J,#

I10

;

(2") Proof.

'Do

'

Let q, T be given satisfying (7.2.3). Define

Since K ( t ) is assumed to be bounded, let po = p / M , where = supteJK(t). Then, it is clear that V E C[J x C o o R,]. , The relations (2") and (3") can be proved, following the proof of Theorem 7.2.1.

M

52

7

CHAPTER

T o show that V ( t ,+) satisfies the stated Lipschitz condition, notice that

'I

exp[U

+)llo

Xtta(4

-

4)Mt

+

:k ' ( t ) exP[-dP(t

+4

0) -

P(W P ( W II C I10 > -

and, because of (7.2.3), b7(t,+)

=

Consequently, for

I

f (t7

SUP

O o T

+,,

$2

II rt+a(t! 4)IIn e x ~ [ ( l- q ) { P ( t E

+

0)

-

Cooand t E J ,

91) v(t!(bdl ~

< oSUP II 9 t o r eLT

SUP

I

~(t)>l-

"(t>41) - X t + o ( 4

exp[(l

-

4 d O exp[(I

+

q ) { ~ ( t u)

-

-

4){P(t

+4

-

P(t))l

~ ( t ) ) I/l (61 - C z /In ,

on the basis of the estimate (6.2.5). This proves the stated result.

T h e next theorem is a result similar to Theorem 7.2.2, whose conditions and the arguments of proof are slightly different.

THEOREM 7.2.3. Assume that (i) for any two solutions x(0, $0), x(0, &) of (7.1 .I), the lower estimate

!I +o

-

$0 110

cxp [-Jlpl(s) ds]

i II Xt(07

40)

-

Xt(O,

+o)llo

t

2 0,

holds, where p , E C[/, R ] ; (ii) there exists a p E .K for t E J , p ( t ) 4 co as t + CO, $ ( t ) exists, and

11 .rt(O,4o)ll(,

K

11 4 0 110 e-"(t),

t

2 0,

K > 0,

+,,

where x(0, +(J is a solution of (7.1.1) with an initial function at t = 0; (iii) the system (7.1.1) is smooth enough to ensure uniqueness and continuous dependence of solutions. Then, there exists a functional V ( t ,+) satisfying the following properties:

(1 ') I/ t C[/ x C, , R,], V ( t ,4) is Lipschitzian in function K ( t ) >, 0;




t h, 0, /I X"(4 +)ll").

Thus, for ( t ,4)E J x C,, , Dl

i ? ( t ,+)

~-

lim sup h-'[ I ' ( f h-O+

lim sup I/ ' [ ~ ( t t 12, 0, 11 r,,(t,+)llo I1-0

+

2 ~ ' ( 1 ,0, Ii -

d f$)lid ,

d r , 4 4 0, 'I % ( t , $)ll",) g(t,

proving (3").

+ h, x,+,[(t,+))

E

(f,

a,>,

['(t,+)]

~

~

~ ( t0,, 11 xo(f,d)ll,)]

7.2.

Since

+

=

55

CONVERSE THEOREMS

x,(O, b0) and do = xo(t,+), the relation (7.2.5) yields

A,,' A;' being the inverse functions of A, , A, , respectively. Hence, using this inequality and (7.2.7) successively, we obtain

and

Evidently a, b E X , and hence (2") is verified. Finally, for t E J , ,d2 E C , ,

I

w

$1)

-

v(4d2)l

=

I 4 4 0, /I xo(4 c1)lIo)

< /I xo(4 41) - xo(4

-4

d2)lIo

4 0, II xo(4 d2)llo)l

exp

[St

0

P2(4

4,

using the condition (7.2.6). Furthermore, as observed in the proof of Theorem 7.2.3, we have, as a consequence of assumption (i),

These considerations imply that

I v(441) - J+, where K ( t ) = exp[lh [p,(s)

42)l

< K ( t )I1

$1 - $2 110 Y

+ p,(s)] ds]. T h e proof is complete.

REMARK 7.2.1. We note that, since p2(t)need not be nonnegative, there is a possibility that K ( t ) may be bounded by a constant. On the basis of Theorem 7.2.4, it is possible to state and prove other converse theorems involving differential inequalities, parallel to certain theorems in Sect. 3.6. We shall only state two converse theorems with respect to uniform asymptotic stability.

56

CHAPTER

7

THEOREM 7.2.5. Assume that (i) the system (7.1.1) is smooth enough to ensure the uniqueness and continuous dependence of solutions; (4 P1 /I do I10 U l ( t - t o ) II X L t O 7 d0)lIo G P 2 /I do I10 0 2 ( t - to), t b t o 7 where P1 ,P2 > 0 are constants.


0, which we shall choose later, define 9

Since, by assumption (ii), we have

it follows that

(7.2.8)

Moreover, similar arguments yield

We have thus proved (lo). T o prove the validity of (2"), notice that

7.2.

57

CONVERSE THEOREMS

Hence,

Let us now fix T so that u2(T ) < (2Pz)p1. This choice is possible, since uz E 2. It then results that

This, together with (7.2.8), yields, setting (2~x-l = ~ B z u z 2 T ( 0, )

and proves the theorem.

THEOREM 7.2.6. Let the trivial solution of (7.1.1) be uniformly asymptotically stable. Suppose that

for ( t , +), ( t , 4) E J x C,, , where L(t) >, 0 is continuous on J and t+u

t

L(s)ds

< Ku,

u

2 0.

Then, there exists a functional V(t,+) with the following properties: (1") V E C[J x C, , R+],and V ( t ,+) satisfies

I for t

E

J , 4,4E

CS(S,)

<
I

< M II c

~

$110

?

+

(2") b(ll+ 110) V ( t ,4) d 4 Ilo), 4 b E 3? ; (3") D+V(t,4) -C[V(t, 411, c E 3Y. T h e proof of this theorem can be constructed parallel to Theorem

3.6.9 with essential changes. We leave the details.

58

CHAPTER

7

7.3. Autonomous systems

I n this section, we consider some stability and instability results for autonomous systems of the form x ' ( t ) =f ( 4 , t E

I,

(7.3.1)

where f~ C[C,, Rn] and f(+) is locally Lipschitzian in 4. It is quite natural to consider the system (7.3.1) as defining motions or paths in $9. I n fact, we can define a motion through 4 as the set of functions in %?n given by UteJxt(O,+), assuming that the solutions xt(O,+) exist on J. We shall, in what follows, abbreviate xi(O, +) by xi(+).

DEFINITION 7.3.1. An element $ E V nis said to be in the w-limit set of [-T, co) and there is a sequence of nonnegative real numbers {tn}, t , + co as n ---t co,such that

+, Q(+) if ~ ~ ( is4 defined ) for

DEFINITION 7.3.2. A set M C W Lis said to be an invariant set if, for any 4 in M , there exists a function x(+) depending on 4, defined on (-00, a), xi(+)E M for t E (- co, co), xo(+) = +, such that, if x*(u, xu) is the solution of (7.3.1) with the initial function xu at u, then x*(cr, xD) = xt(+) for all t 3 u. We notice that to any element of an invariant set there corresponds a solution that must be defined on (- co, a).

LEMMA 7.3.1. Let x(+) be a solution of the system (7.3.1) with an initial function at t = 0, defined on [ - T , a), and let

+

/I 4 d ) l l n

< pi < P

1-

t~

for

Then, the family of functions {xi(4))},t E J , belongs to a compact subset of Vn,that is, the motion through 4 belongs to a compact subset of en. T h e proof of this lemma follows from the fact that, for any p1 < p, there exists a constant L > 0 such that Ilf(+)ll L for all satisfying

I1

+

110

G

P1


> P as long as xt(+,) E E n C,, . If xl(+,) leaves E n C,, , then it must cross the boundary aC,, of C,, . I n fact, it must cross either aE or aC, , but it cannot cross aE inside C, since V = 0 on that part of aE inside C, and V(x,(+,,)) 3 V(+,,)> 0, t >, 0. Now, suppose that xt(+,) never reaches aC, . Then, xi(+*) belongs to a compact subset of the closure of E n C,, , for t 3 0. Consequently, xl(+,) approaches Q(+,), the w-limit set of

+,,

62

CHAPTER

7

and .Q(dO) C closure of E n C,, . Since V(xl(b0))is nondecreasing and bounded above, it follows that V(xl(+,))+ B, a constant, as t + co,and, implies that Since 4 E thus, D+V(xl(+))= 0 for 4 E a(&).

/I $(O)ll

< .-'(V(40>> > 07

we have a contradiction to hypothesis (iv). Consequently, there is a t , > 0 such that jl x((bo)(tl)ll = y. Hypothesis (ii) implies instability, since do can be chosen arbitrarily close to zero. This completes the proof of the theorem. 7+4. Perturbed systems

We shall be interested, in this section, in the perturbed functional differential system (7.4.1) 4 9 = f ( t , 4 R(4 4,

+

where f , R E C [ J x C, , R"] and f ( t , +), R(t, (6) satisfy a Lipschitz condition in 4 for each t E J.

THEOREM 7.4.1. Suppose that the trivial solution of (7.1.1) is exponentially asymptotically stable andf(t, 4)is linear in 4. Assume further that

/I R(t?4111 < 711 c 110 >

t6

/t

4E c,

(7.4.2)

9

7 being a sufficiently small positive number. Then, the trivial solution of the perturbed system also enjoys the exponential asymptotic stability.

By Theorem 7.2.1, there exists a V ( t ,4) such that, for d) E J x C,

Proof. ( t 7

7

(i) I/ E C [ J x C, , R,], and V is Lipschitzian in K>O; ( 4 II d 110 v(t,4) K 114 110 ; (iii) D+V(t,(b) < --(xV(t,(b), a > 0.


> + h, Yt,+h(h

V ( t , Yt,(to > do))]

-

-

I d > )

!

V(t2

+ h,

,4 4 - V t 2 +)I,

Xt,+h(f,

9

Xt,+h@,

,4))

+

where ~ ( t,4) , is the solution of (7.1.1) with an initial function at t = t, . Using the Lipschitzian character of V , the assumption (7.4.2), and the preceding relation, we obtain (7.4.4)

D+m(t2) G K.111d 110 - ""(t2).

Since q > 0 is sufficiently small, there exists a y > 0 such that Kq < a - y , and, hence, the fact that 11 (I, V ( t ,d), together with the inequality (7.4.4), implies that


0, if II 4 1l0


c0)Ilo

< B(Ild0

- $0

Ilo),

EX

t >to

such that I

because the system (7.1.1) is extremely uniformly stable. We thus have

c, $1 < g ( W Gk(B(II c exp[W-l, < P(ll c $ M(lld - $

V7C(t,

- $110))

-

110)

+,

II d

~

$ Il0)l

(7.5.9)

110).

We shall next show that Vk(t, $) satisfies a Lipschitz condition with respect to 4 and $. For any 4, , $, E C, ,

7.5.

EXTREME STABILITY

69

By (7.5.10) and the continuity of x t ( t 0 , c $ ~ in ) t , it follows that V k ( t ,4, $) is continuous in ( t , 4,$). Let us now consider D+V,(t, 4,$) with respect to the product system (7.5.1). Since

T h e desired Lyapunov functional may now be defined by

I n view of (7.5.8), this V ( t ,4,$) can be defined for all t E J , 4,$ E P. From (7.5.7) and (7.5.91, it is easy to see that there exist functions a, b E 3-satisfying (2"). Furthermore, because of the inequality

we obtain, for 4,41 , #, t,bl E C, ,

which proves that V ( t ,4,$) satisfies the Lipschitz condition as described in (1'). Finally, we shall show that (3")is satisfied. By the definition,

70

CHAPTER

7

if h is small enough,


0, to E J are given, there exists a 6 = 6 ( ~ > ) 0 such that uo 6 implies


do)),

t

E

[ t o 7 tll.

where x * ( t 8, xt), y*(t 8, y t ) are the solutions of (7.1.1) such that 4++o(t 8, x t ) = X f = X d t , > 4 0 ) and YTfdt 8, rt) = Y t = Yt(& , 4 0 ) , respectively. We thus have

+

+

74

CHAPTER

because of (7.6.3). Defining inequality m(t)

T,,

=

< r(t + 6,

to

70

7

+ 8, we get, by Theorem 1.4.1, the

, 0, a),

t E [ t o , tll, (7.6.4)

Assumption (ii) and (7.6.1), on the other hand, lead to 4tl)

2 w,

contradicting (7.6.4). It therefore follows that there is a unique solution for the system (7.1.1) to the right of to . COROLLARY 7.6.1. 7.6.1.

T h e function g(t, u) = 0 is admissible in Theorem

DEFINITION7.6.2. If, for any p that l l f ( t , #)]I < M(p), whenever bounded.

> 0, there exists an M ( p ) > 0 such # E C, , we shall say that f ( t , 4) is

The next theorem gives the sufficient conditions for perfect stability criteria of the trivial solution of (7.1.1).

THEOREM 7.6.2. L

Suppose that

(i) V E C [ J x C, , R,],V ( t ,#) is Lipschitzian in > 0, and, for ( t ,#) E J x C, ,

# for a constant

= L(p)

b(ll9 110)

< q t , 41,

(ii) g E C [ J x R,, R],g(t,0)

b E .f;

(7.6.5)

= 0, and, for ( t ,4) E J x C, ,

D+V(t>9)(7.1.1) G A4 V ( t ,4));

(iii) f E C [ ( - CO, CO) x C, , R"],f ( t , 0) = 0, f ( t , #) is bounded, and 4)is almost periodic in t uniformly with respect to 4 E S, S being any compact set in C, .

f(t,

Then, the null solution of (7.1.1) is (I") perfectly equistable if the trivial solution of (6.1.2) is strongly equistable; (2") perfectly uniform stable if the trivial solution of (6.1.2) is strongly uniform stable;

7.6.

75

ALMOST PERIODIC SYSTEMS

(3") perfectly equi-asymptotically stable if the trivial solution of (6.1.2) is strongly equi-asymptotically stable; (4") perfectly u n i f o r d y asymptotically stable if the trivial solution of (6.1.2) is strongly uniformly asymptotically stable. We shall prove only the statement corresponding to (4"). Let us suppose that the trivial solution of (6.1.2) is strongly uniformly asymptotically stable. Let 0 i E < p and to E (- 03, 03) be given. Then, given b ( ~> ) 0, T,,E J , and any compact interval K = [T,,, t*], there exist an 7 = q ( ~> ) 0 and a 6 = S(T,, E ) > 0 such that Proof.

u(t, 70 uo I 7)

< b(t-),

t s[To

7

t*l,

(7.6.6)

whenever u,,< 6, where u(t, T,,,u o , q) is any solution of (3.18.6). Choose LS, = 6 and u,, = L 11 +o I/, , L being the Lipschitz constant for V(t,$). Consider a solution x(t0 ,#,,) of (7.1.1) such that to E (- 03, a), I/ +o I/, < 6, . Suppose that, at some t , we have

/I X d t , > 9o)llo

=

6.

Then, there exist t , and t, such that to < t , II x&o , +o)llo = and that

< t , , (1 x t , ( t , , +,,)\lo

=

a,,

€9

81

< II %(to 9o)llo < 6, 7

tl

< t < t2

*

Clearly, there exists a compact set S C C, such that xt(t, , +,) E S for t t, . Let 0 be an q/L translation number off ( t , +) for E S such that to 0 3 0, that is, to

< < +

Consider the function

Then we have

+

76

CHAPTER

where xl*,,(t

x *( t

+ 8, x l ( t , , &))

is a

7

(7.1.1) such

solution of

+ 0, x t ( t , , do)) = xt(t, , do).It, therefore, follows that

Since x , ( t o , +,J E S for to D+m(t)

that

< t < t, , we obtain, using

< g(t + 8, m ( t ) ) + 7,

t

E

[to,

bl,

and hence, by Theorem 1.4.1,

< r(t + 8, *, 4,

m(t>

letting T~ : t, of (3.18.6). At t

+ 0, =

h(6) s; q t ,

70,

t

9

E

+

(7.6.8)

[ t o , 421,

where r(t 0, T ~ a,, , 7) is the maximal solution t, , we are led to an absurdity

+ 8, %,(to

7

40))

< r ( t , + 8,

70

, *o , 7 )

< 44,

in view of relations (7.6.5), (7.6.6), and (7.6.8). Thus, the perfect uniform stability of the trivial solution of (7.1.1) is proved. By assumption, given b ( c ) > 0, T , >, 0, there exist positive numbers S o , 7 = T ( E ) and T = T ( E )such that u(t, 7 0 , * o , ~ )< b(c),

3 70

t

+ T,

(7.6.9)

whenever uo 6 6 , . Choose u, = L (1 4, )lo and L8, = 8, , and let 6,* min[S,, So], where 8, = 6(p). Suppose now that x(t, ,4,)is any solution of (7.1.1) such that 11 4,1, So*, t, E (-a, a). Since [ / x l ( t , ,+o) l o ,< p for all t 2 to a n d f ( t , 4)is bounded and consequently ~ ' ( t , +,)(t) , is bounded by some constant, there exists a compact set S such that xt(t,, E S for all t 3 t, . As before, let 6 be an v/l-translation number of f ( t ,+) so that (7.6.7) is satisfied. Considering the function


0, is admissible in Theorem 7.6.2 to yield perfect uniform asymptotic stability of the trivial solution of (7.1.1). As remarked in Sect. 3.18, if the functional f ( t ,+) is not almost periodic and f E C [ J x C, ,R"],then, from the strong stability properties of the trivial solution of (6.1.2), we may deduce strong stability properties of the trivial solution of (7.1. l), on the basis of Theorem 7.6.2. Finally, the following theorem assures the existence of an almost periodic solution.

THEOREM 7.6.3.

Suppose that

(i) V E C [ J x C, x C, , R,],V ( t ,+, +) is Lipschitzian in for a constant L = L ( p ) > 0, and, for t E J , +, E C, ,

+

b(II 4 - 1cI IIo)

< v(t,4,$) < .(I1 4

-

(ii) g E C [ J x R, , R ] ,and, for t E J , +, t,h

D+q4 4,$1

a , b E S;

$ llo), E

+ and + (7.6.10)

C, ,

< g(t, U t ,4,$1);

(iii) f E C [ ( - co, 00) x C, , R n ] , f ( t +) , is bounded, almost periodic in t uniformly with respect to E S, S being any compact subset in C, , and f ( t ,+) is smooth enough to ensure the existence and uniqueness of solutions of (7.1.1); (iv) for any b(c) > 0, CY > 0, and 5, E I , there exist positive numbers 7 = q ( c ) , T = T ( E CY) , such that, if uo CY and 5 3 5, T,

+




4111 < 7 / 3 4

(7.6.13)

for all t E (- GO, co), q5 E S. Let 8 be an y/3Ltranslation number for f ( t , 6) such that t, 8 >, 0, that is,

+

a), 4 E S. for t E (-a, Consider the function, for t

where t ,

=

where x*(t

t

+

3 to,

and x t = xt(to, +o). Then,

T~~ - T ~ ,

+ 8, xt),y*(t + 0, xt,) are the solutions of (7.1.1) such that

7.6.

79

ALMOST PERIODIC SYSTEMS

+

+ 6, x t ) = x t ,y&(t 6, xt,) = x t , , respectively. Thus, in view of the Lipschitzian character of V ( t ,4, $) and assumption (ii), we get

x&(t

o+m(t) < g ( t + e, fl(t>>+

+ /I

lim sup h4’

< g(t + 0, 4 t ) ) + L[ll %’(to

+ II

X’(t0

9

+ 6,

< g(t + 8, d t > + L[llf(t, Tkz

+

T7cl

Xt,)(t

-f(4

+ e,

Xt)l10

+ 6, %)(t + 6)ll

+ e)lll

+ e,xt)ll + Ilf(4

-f(t

XtJ

?

do)(t>- x*’(t

7

+o)(G) - Y*’(t

+ llf(t + - f ( t + 6, %,)Il .

X t + h - X?+-ts+h(t

+ 6, ~t,>iioI

- Y?+e+h(t

Xtl+h

h-’[l l

XtJ

Xt,)

Since t rkl 2 to + T , for t E U , we obtain, using the relations (7.6.13) and (7.6.14), D+m(t

+ < g(t + + 0, m(t + Tkl

.k,>

TkJ)

+ 17,

which implies, by Theorem 1.4.1, if uo = m(to), m(t

+

< r(t +

Tkl)

Tkl

+ 6, + 6, uo, 4, to

where r(4, t o ,u,, ,7)is the maximal solution of (7.6.12). By assumption (iv), it follows that r(6, t o ,

But,

for

all

UO,

t E U, t

5 = t + Tk, + 6, to= to m(t

17)

+

< &)

> to + + T .

+ 0, we get +

t 3 t o + T.

if

T ~ ,

Tkl)

Consequently, for all t E U , k,

II Xt+.rkl

Hence,

T

< b(€),

t

E

identifying

u.

> k, 3 no , we have, in view of (7.6.10), - X t + T k Z /lo

< E,

which, in turn, leads to the inequality

II 4%> do)(t

+

T k J ~X(t0

9

+o>(t

+ %z)lI
0, is admissible.

7.7. Notes T h e results of Sect. 7.1 are adapted from the work of Driver [3]. See also Halanay [22] and Krasovskii [5]. Theorem 7.1.4 is new. Theorems 7.2.1 and 7.2.2 are taken from Hale [l]. See also Yoshizawa [3]. Theorems 7.2.3 and 7.2.4 are new. Theorems 7.2.5 and 7.2.6 are based on Halanay [221T h e results on autonomous systems in Sect. 7.3 are taken from the work of Hale [S], which may also be referred to for a number of illustrative examples. For the results on perturbed systems of Sect. 7.4, see Corduneanu [2], Halanay [22], and Hale [l]. Theorem 7.5.1 is due to Yoshizawa [I], whereas Theorem 7.5.2 is new. Section 7.6 contains the work of Lakshmikantham and Leela [3]. See also Hale [6] and Yoshizawa [2, 31. For closely related results, see Driver [3], Halanay [22], Hale [5], J. Kato [l], Krasovskii [5], Lakshmikantham and Leela [2], Liberman [I], Miller [l], Razumikhin [2, 61, Reklishkii [l-51, Seifert [I], Sugiyama [S], and Yoshizawa [3].

Chapter 8

8.0. Introduction I n what follows, we wish to treat the solutions of the functional differential system (7.1.1) as elements of euclidean space for all future time except at the initial moment. Our main tool, in this chapter, is therefore a Lyapunov function instead of a functional. T h e derivative of a Lyapunov function with respect to the functional differential system will be a functional, which may be estimated either by means of a function or a functional. While estimating the derivative of the Lyapunov function in terms of a function, a basic question is to select a minimal class of functions for which this can be done. Thus, by using the theory of ordinary differential inequalities and choosing the minimal sets of functions suitably, several results are obtained. If, on the other hand, the estimation of the derivative of the Lyapunov function by means of a functional is considered, the selection of a minimal set of functions is unnecessary. Nevertheless, this technique crucially depends on the notion of maximal solution for functional differential equations and the theory of functional differential inequalities. This method also offers a unified approach, analogous to the use of general comparison principle in ordinary differential equations. Moreover, it is important to note that the knowledge of solutions is not demanded in either case.

8.1. Basic comparison theorems

+

Let V E C[[-T, co) x S o , R+], and let E C,, . We define D+V(t,+(O), +), D-V(t, #(O), +) with respect to the functional differential system (7.1.1) as follows:

+ h, 4(0) + w, 4)) lim inf W V ( t + h, 4(0) + hf(t, 4))

D+V(t,4(0),4) = lim SUP h - V ( t

D-V(t, +(0),4) =

~

h-O+

-

h-0-

81

V ( 44(0))1, V ( t ,d(0))l.

(8.1.1)

82

8

CHAPTER

We need, subsequently, the following subsets of qn,defined by Q, =

[+E c, : I

Qo = [+ E

and Q,

=

v,lo = w, +(ON,

t E J1,

(8.1.2)

c, : v(t + s, W)) < W V ,+(O))),

[d E c, : I VtA, lo =

where A(t) > 0 is continuous on

+

w d(O)A(t)),t

t E 11,

(8.1.3)

J1,

(8.1.4)

E

a),

[-7,

(i) I Vt lo = sup,- 0; and (iii) I VtAt ( 0 = sup V(t s, +(s))A(t s).

+

-T<s u, for (8.1.5)

We now state a few fundamental comparison results.

THEOREM 8.1.1.

Let V E C[[-T, co) x So , R,] and V(t,3) be locally Lipschitzian in x. Assume that the functional D-V(t, +(O), +), defined by (8.1. I), verifies the inequality

o-v(t,+(O), 4) < g(t, l’(t, +(O))),

t

> to

9

4E Q,,

(8.1.6)

where g E C[J x R , , R,], and r ( t , to , uo) is the maximal solution of the 0. Let scalar differential equation (6.1.2), existing to the right of to %(to,+o) be any solution of (7.1.1) defined in the future, satisfying vto

SUP

--7<s

< r(t, t o

>

< uo uo),

(8.1.7)

t 2 to

-

(8.1.8)

Proof. Let %(to, +o) be any solution of (7.1.1) with an initial function E C, at t = to . Define the function

49 For

E

=

V(t,

4 t O 1 #O)(t)).

> 0 sufficiently small, consider the differential equation u‘

= g(t, u)

+

E,

u p o ) = uo 2 0,

(8.1.9)

whose solutions u(t, E) = u(t, to , uo , E ) exist as far as ~ ( t o, , uo) exists,

8.1.

83

BASIC COMPARISON THEOREMS

to the right of t o . Since lim u(t, e )

= r(t, to ,uo),

c-0

the truth of the desired inequality (8.1.8) is immediate, if we can establish that t 2 to. m(t) < u(t, E ) , Supposing that this is not true and proceeding as in the proof of Theorem 6.3.3, we can see that there exists a t, > to such that


to ,q5 E .R, , D-V(t, 4(0),4)

V(t, x) be

< 0.

Let x(t, , be any solution of (7.1.1) such that % ( t o +,)(t) , E So for t E [to , t,] C J. Then,

84

CHAPTER

8

Proceeding as in Theorem 8.1.1 with g = 0, we arrive at the inequality Proof.

w,4 t o

?

G

+o)(tN

w z

9

4 t o do>(tz)>, 7

where t, E (to , t l ) . Since V ( t , , ~ ( t,&)(t,)) , > 0, the assumptions L(u) imply that

< < t, . T h e rest of the proof is

which shows that x l ( t 0 , +,) E Qo , to t similar to the proof of Theorem 8.1.1.

T h e next comparison theorem gives a better estimate.

THEOREM 8.1.2. Let the assumptions of Theorem 8.1.1 hold except that the inequality (8.1.6) is replaced by

+

+ C(ll d(0)lI) < g(t, V t ,+(O)>>,

D+V(t,d(O), 4)

(8.1.11)

for t 3 t o , E C, , where the function C E X . Assume further that g ( t , u ) is monotone nondecreasing in 2c for each t . T h e n (8.1.7) implies

+

Set == x,(t, ,+o) so that +(O) = x(to ,+,,)(t). We then obtain, using the condition (8.1.1 I), the inequality D+m(t1)

t o ,4 E SZ, Then,

, where A(t) > 0 is continuous on [-T, a).

and therefore, in view of the assumption (8.1.13), it follows that D-L(t, +(O),C)

e g(t, L(t, C(O))),

for t > t o , 4 E Q, , where Q, , in this case, is to be defined with L(t, x) replacing Y(t,x) in (8.1.2). It is clear that L(t, x) is locally Lipschitzian in x, and, thus, all the assumptions of Theorem 8.1.1 are satisfied, with L(t, x) in place of V ( t ,x). T h e conclusion is now immediate from Theorem 8.1.1. On the basis of the comparison theorem for functional differential inequalities developed in Sect. 6.10, we are now in a position to prove the following result, which plays an equally vital role in studying the behavior of solutions of functional differential systems.

THEOREM 8.1.4. Y E C[-T, a)x S , , R,], and V(t,x) is locally Lipschitzian in x. Assume that, for t E 1,4 E C, , D+V(t,(b(O),

where

Y t = Y(t

+ s, 4(s)),

4)< g(4 w,+(ON, -T

< s < 0,

g

E

Vt),

(8.1.15)

C [ J x R, x % + , R],

86

CHAPTER

8

g(t, u , a) is nondecreasing in u for each ( t , u ) , and r ( t o ,o0)is the maximal

solution of the functional differential equation

(8.1.16)

u‘ = g(t, u, U t )

with an initial function uo E %?+, at t = t o , existing for t >, t o . If x(t0 ,40) is any solution of (7.1.1) defined in the future such that Vt0 = J”t0

+ s, 4o(s))


then we have

Proof.

Let x(t0 ,q50) be any solution of (7.1.1) such that

Set 4 = xt(t,, $,), which implies that +(O) m ( t ) = C'(t, X ( t ,

9

=

~ ( t,+,)(t). , Define

+o)(t)),

so that

v(t + s, d(s)). Since (8.1.17) holds, we have mto < uo . Moreover, for small h > 0, "1

=

because of the fact that V ( t ,x) satisfies a Lipschitz condition in x. This, together with (8.1.15), yields the inequality (8.1.19)


0, to E J , there exists a positive function

6

=

8 ( t o , c) that is continuous in to for each

E,

such that, whenever

I14 0 /lo < 6,

we have

I/ 4 t O do)(t)ll < 1

€7

t

2to.

With this understanding, we can prove the following results.

THEOREM 8.2.1. Let there exist functions V ( t ,x) and g ( t , u) enjoying the following properties : (i) V E C [ - T , GO) x S, , R,], V ( t ,x) is positive Lipschitzian in x, and V ( t ,).

< 4, II x ll),

definite,

( 4 4 E J x S" ,

where a E C[J x [0, p), R,], and a E 3'for each t E J ; (ii) g E C [ J x R, , R,],g ( t , 0) = 0, and, for t > to , D-Vt,

d(O>,4

locally (8.2.1)

E

SZ, ,

< g ( 4 q t , d(0"

Then the trivial solution of (7.1.1) is (1") equistable if the trivial solution of (6.1.2) is equistable; (2") uniform stable if the trivial solution of (6.1.2) is uniform stable and, in addition, V(t,x) is decrescent.

Proof.

Let x(t, , &) be any solution of (7.1.1). Choose

88

CHAPTER

< u,, , by

so that V(to,+,,) yields the estimate

8

(8.2.1). An application of Theorem 8.1.1

< r(t, t u , 4,

L'(t, 4 t o + o ) ( t ) ) 7

t 3 tu ,

(8.2.2)

where r ( t , t , , uo) is the maximal solution of (6.1.2). Also, because of the positive definiteness of V ( t ,x), we have

411x 11)

< t'(4

( 4 4E

x),

J x

s,,

bEx-.

(8.2.3)

Let 0 < E < p and to E ] be given. Assume that the null solution of (6.1.2) is equistable. Then, given b ( ~ > ) 0, to E J , there exists a 8 = 8(t,, E ) > 0 satisfying 4 4 t o , uo) < &),




=

a1(t,,,e) such that


0 is continuous on

[-T,

a),and A(t)-+ a as t -+a;

8.2.

89

STABILITY CRITERIA

(ii) V E C [ [ - T , co) x S o , R,], V(t,x) is positive definite, locally Lipschitzian in x, and verifies (8.2.1); (iii) g E C [ J x R, , R + ] , g ( t ,0) = 0, and, for t > t o , E SZ, ,

4)D - W

4)+ v(4 $(ON D - W

+(O),

+

< g(4 U t ,+(ON W).

Then, the trivial solution of (7.1.1) is equi-asymptotically stable if the trivial solution of (6.1.2) is equistable. Proof. If x(to,+o) is any solution of (7.1.1) such that 4 t o )4

0

9

I1$0

110)

=

uo 9

we have, by Theorem 8.1.3,

4 4 q t , 4 t O ,+o>(t)) < y ( t , t o

7

uoh

t

2 to

(8.2.6)

Let 0 < E < p and to E J be given. Let 01 = min-7Gi<mA(t).By assumption on A(t),it is clear that 01 > 0. Set q = 01b(~).Then, proceeding as in the proof of Theorem 8.2.1 with this q instead of b ( ~ )it, is easy to prove that the trivial solution of (7.1.1) is equistable. To prove equi-asymptotic stability, let q* = ab(p). Let S l ( t o , p ) be such that 11 #Io [lo 6, implies 11 %(to,#Io)(t)ll p, t 3 to . This is possible by equistability. Designate 8,(t0) = 8,(t, ,p), and suppose that (1 +o 8, . Since A(t)-+ co as t -+ 00, there exists a positive number T = T ( t o ,E ) such that

5


, to

+ T.

(8.2.7)

We then have, using (8.2.3), (8.2.6), and the fact that u ( t , t o , uo) < q* if uo




+o)(t)ll)

w,

%(to > +o)(t)> G G r(t, to uo) < T * = ab(p), t 3 t o . 9

If t >, to + T , it follows, from the foregoing inequality and (8.2.7), that provided 11 +o

[lo

< 8,.

This concludes the proof of the theorem.

COROLLARY 8.2.2. The functions g ( t , u ) are admissible in Theorem 8.2.2.

= 0 and A(t) = eat,

01

> 0,

90

8

CHAPTER

THEOREM 8.2.3. Assume that there exists a function V(t,x) satisfying the following conditions: (i) V t C[[-T, CO) x S, , R,], V ( t ,x) is positive definite, decrescent, and locally Lipschitzian in x; (ii) for t > t o , E Q, ,

+

u-If, m,4)< -C(ll d@)ll),

c: E .x.

Then, the trivial solution of (7.1 . l ) is uniformly asymptotically stable.

Proof. Since V is positive definite and decrescent, there exist functions a, b E .f satisfying

4 11) < V ( t ,4 < 4 x ll), Let 0

s :’

E

< p, t,

E

(4 ).

J be given. Choose 6

=

E

J x

s, .

(8.2.8)

> 0 such that

6(e)

a(S) < b(€).

(8.2.9)


J

=

6

E

[to >

4, (8.2.10)

3

because of (8.2.8). Furthermore, this means that x(t, , +,,)(t)E S, t E [t,,, t.1, Hence, the choice u,,= a(ll4, ),1 and the condition

,

give the estimate

If,44l

9

$o)(tN

< 41

$0

lid,

t

E

[to 9

f21,

(8.2.11)

because of Corollary 8.1.1. Now the relations (8.2.10), (8.2.1 l), and (8.2.9) lead t o the contradiction

This proves that the trivial solution of (7.1 . I ) is uniformly stable.

8.2.

91

STABILITY CRITERIA

T o prove uniform asymptotic stability, we have yet to show that the null solution of (7.1.1) is quasi-uniform asymptotically stable. For this , be any solution of (7.1.1) such that 11 +o (lo a,, purpose, let ~ ( t,+o) where 6, = 6(p). It then follows from uniform stability that


0 such that

+ /3

L(u) > u

if

b(6)

< u < a(6,).

Moreover, there exists a positive integer N inequality b(€)

If, for some t

+ N/3 >

=

(8.2.12)

N ( E ) satisfying the (8.2.13)

480).

t o , we have

w4to

do>(t))2 b ( 4 ,

it follows that there exists a 6, = a,(€) > 0 such that 11 ~ ( t, cjo)(t)lI , >, 6, because of (8.2.8). This, in turn, implies that C(lI X ( t 0 do)(t)ll) ?

2 C@,) = 8 2 -

(8.2.I 4)

Obviously, 6, depends on E. With the positive integer N chosen previously, let us construct N numbers t,i = t k ( t o ,E), k = 0, 1, 2,..., N , such that 9

€)

=

tk+l(tO

7

9

t k ( t O , €)

€)

+ @is,) +

,

+1

T*

It then turns out that

+ and, consequently, letting T ( E )= N[(,8/8,)+ tk(tO

3

+

€1=

'[(/3/'2)

T],

tN(t0,

4 = t o + T(+

we have

Now, to prove quasi-uniform asymptotic stability, we have to show that

II X ( t ,

P

90>(t)II


V(t* t- s, $(S))>

-7

< s < 0,

where 4 = xl,(t,, , +J, so that +(O) = %(to, $J(t,). Hence, 4 E Qo . It therefore follows from condition (ii) and the relation (8.2.14) that

n- V ( t ,4(0),4)
6 ( t o , e ) > 0 such that 1 uo lo 6 implies

Proof.

0 <E 6

=

< :


0; A(t)400 as t 00, and, (ii) A ( t ) > 0 is continuous on [ - T , a), fort>t,,+EQ,, ---f

+ V t ,4(0))D - 4 t ) < 0; C [ J x R, , R,],w(t, 0) 0, and, for t > t o ,+ A f t )D-V(t, 4(0),6)

(iii) w

E

E

A ( 4 I/ R(t,6111

< 4 4 JV,

E

Q, ,

A@)).

Then, the trivial solution of (7.4.1) is equi-asymptotically stable if the null solution of (6.1.2) with g ( t , u ) = Lw(t, u ) is equistable.

Proof.

If t

> to , + E QA , it follows that

which implies the inequality a t ) D-V(4 +(Oh

d4(7.4.1)

+ v(t,(b(0))D - 4 ) < Lw(4 w, $((I)) A@)).

Consequently, the conclusion follows by Theorem 8.2.2.

THEOREM 8.3.3.

Suppose that

(i) V E C[[-T, 00) x S, , R,], V ( t ,x) is positive definite and satisfies a Lipschitz condition in x for a constant L = L(p) > 0; (ii) for t >, to , E C, , and C E X ,

+

D'V(t3

4h.l.l)

< -C(ll4(0)11);

(iii) w E C [ J x R, , R,.], w(t, 0 ) = 0, w(t, u) is nondecreasing in u for each t E J , and, for t 3 t o , E C,, ,

+

/I R(t, 4111 < w(t, V ( t ,4(0))). Then, the uniform stability of the trivial solution of (6.1.2) with g ( t , u ) = Lw(t, u ) assures the uniform asymptotic stability of the trivial solution of (7.4.1).

Proof,

Let t

3 to and + E C, . Then, as previously,

D+V(t,+('),

$)(7.4.1)

d(o)li) + Lw(t, V(t,+(O))),

and, therefore, the uniform asymptotic stability of the trivial solution of (7.4.1) follows by Theorem 8.2.4.

8.3.

THEOREM 8.3.4.

99

PERTURBED SYSTEMS

Assume that

(i) V EC[--7, CO) x S o ,R,], V(t,x) satisfies a Lipschitz condition in x for a constant L = L ( p ) > 0, and b(ll x II)

in

< V t , 4,

(4 4 E J x

s,,

b E .x;

(ii) g, E C [ J xR, x V, , R],gl(t, 0, 0) 3 0, gl(t, u, u) is nondecreasing u for each ( t , u), and, for (t,4) E J x C, , D+V(t,#('),

gl(t, v(t,#(O)), ;)6'

#)(7.1.1)

(iii) g, E C[J x V+ , R,],gz(t,0) = 0, gz(t,u) is nondecreasing in u for each t E J , and II R(4 #)I1 < gz(4 II # 11)Then the stability properties of the trivial solution of (8.1.16) with g(4 u, 4 = gl(4 u, 4

+ Lg&, b - W

imply the corresponding stability properties of the trivial solution of (7.4.1). Proof.

Let t E J and #J D'v(t, #(O),

E

#)(7.4.1)

Co . Then, D'v(t, #(O), #)(7.1.1) < gl(C v(4#(O)), V t ) < gl(4 v(4#(O)), V t ) = g(4 W)), Vt),

w,

+ 11 R ( f ,#>I1 + Lgz(4 II 4II) + Lgz(4 b-YVt))

because of assumptions (i), (ii), and (iii). Now, Theorem 8.2.5 can be applied to yield the stated results.

COROLLARY 8.3.1.

T h e functions gl(4 u, 0) =

--01u

+

PT

sup -T<S 0, p > 0, and g2(t, 0) = y sup-,GsGo u(s), y being sufficiently small, are admissible in Theorem 8.3.4, provided b(u) = u and 0 < T < (01 - y ) / P , to guarantee that the trivial solution of (7.4.1) is exponentially asymptotically stable. 01

Proof. Under the assumptions, it is easy to see, as in Theorem 6.10.7, that u' = -01u (PT y ) sup-,GsGo ut(s) admits a solution r ( t o ,uo) that tends to zero exponentially as t + CO, and, therefore, the conclusion follows from Theorem 8.3.4.

+

+

100

CHAPTER

8

8.4. An estimate of time lag We wish to estimate the time lag r in order that the solutions of an ordinary differential system (8.4.1)

x’ = f ( t , x)

and a functional differential system Y’

=

(8.4.2)

F(4 Y t )

may have the same behavior, namely, exponential decay. Since Eq. (8.4.2) may also be written as x’

where

=f

( 4 x)

R(4 x, 4

+ R ( t , x, 4,

= F(4

(8.4.3)

4 - f ( k 4,

it is sufficient to consider the perturbed system (8.4.3).

THEOREM 8.4.1. Suppose that (i) I/ E C [ S , , R,], V ( x ) is positive definite and satisfies a Lipschitz condition in x for a constant L = L(p) > 0 ; (ii) f E C[/ x S , , R”], and, for (t, x) E J x S,, ,

< -aL’(x),

D+V(X)(&4.1)

(iii) R

E

01

> 0;

C [ / x S, x C, , R”],and for 4 E C, ,

/I R ( f >+(Oh +)I1

< NT

SUP

--7<sY ( t l +l)(t))

t21,

(8.5.2)

where y(tl , is any solution of (7.1 . I ) through (tl , +1), and Y ( t , , a,) is the maximal solution of (8.1.16) through ( t l , a,,). I t turns out that (8.5.2) is also true for ~ ( t, +,J , on the interval t, t t, . Hence, we get

<
t, > to >, T ~ ( E )and the uniformity of the relation (8.5.1) with respect to t o . This contradiction shows that ( E l ) is valid, and the theorem is proved. COROLLARY 8.5.1. The uniform stability of the trivial solution u = 0 of (8.1.16) is admissible in Theorem 8.5.1 in place of the eventual uniform stability of the set u = 0. I n particular, g ( t , u, a) = 0 is admissible.

THEOREM 8.5.2. Assume that (i) V EC[[-T, 00) x S, , R,], V(t,x) is Lipschitzian in x for a constant L = L(p) > 0, and

&ll x II)

e v(44 < 4 x ll),

for 0 < 01 < 11 x 11 < p and t >, O(a), where a,b E X and O(u) is continuous and monotonic decreasing in u for 0 < u < p ; (ii) f E C [ J x C, , R"],and Df

w,W),4)e 0,

for every 4 E C, such that 0 < 01 < 11 +(O)lI < p and t 2 O(cx); (iii) R E C[J x C,, R"], and, for every 4 E C,*, p* < p and t >, 0,

Then the set 4 = 0 is eventually uniformly stable with respect to the perturbed system (8.5.3) x' = f ( 4 X t ) R(t, 4.

+

Proof. Let 0 T ~ ( E )such that

< E < p* 248)

be given. Choose the numbers 6

< b(e)

11 R(t, $)I[. Define h(t) = max,,+,,oGDt to find a T ~ ( E ) > 0 such that rrca

and

T~(E= )

0(8(e)),

=

a(€) and (8.5.4)

Since h(t) is integrable, it is possible (8.5.5)

provided to >, T ~ ( c ) where , L is the Lipschitz constant for V ( t , x). Let = m a x [ ~ ~ (T~(E)I. ~),

TO(€)

104

CHAPTER

8

Suppose that there exists a solution x(t0 , of the perturbed system (8.5.3) and two numbers t , , t, such that t , > t , >, to >, T,,(E),

At t where = xl,(to, (8.5.5) and the fact that I/

< V(tz ,

X(t,

7

t,

=

< 6,

, we therefore obtain, in view of

Co)(tz)>

< 4 8 ) + 4) = 24%

which is incompatible with (8.5.4). This shows that ( E l ) holds, and the theorem is established.

THEOREM 8.5.3. Let assumption (i) of Theorem 8.5.1 hold. Suppose further that f E C [ J x C, , R"] and

+

D+

$(O),

4 ) < -C(ll4(0)ll),

for every E C, such that 0 < 01 < I/ +(O)lI < p and t >, e(u) and C E %. Then, the set = 0 is eventually uniformly asymptotically stable.

+

Proof. T h e eventual uniform stability of the set C$ = 0 follows by Corollary 8.5.1. Let 0 < E < p be given. Choose 6, = 6(p), T~ = ~ ( p ) , and T ( E )= T ( E ) {a(p)/C[G(e)]). Assume that to >, T~ and 11 I(o 6,. It is sufficient to show that there is a t , E [to T ( E ) , to T ( E )such ] that

+

+

+

II X ( t 0 4 O ) ( ~ l ) I l < Y E ) , Y

in order to complete the proof. Suppose, if possible, that

< /I "(tn

8(~)

4n)(t)ll
)

< r(to

3

oo)(t),

t b to >

(8.6.2)

where r ( t , , u0) is the maximal solution of the functional differential equation (8.6.1). T h e stated result is now a direct consequence of the hypotheses (i) and (iii). T h e next theorem is very useful in applications, since it does not demand V ( t ,x) to be positive definite.

THEOREM 8.6.2. Assume that (i) f E C[J x C,, R"], and

llf(f, 4111 < M ,

tE

1,

I1 4 I10

< P*

, to . Then, every solution of (7.1,l) approaches the set 52 as t + CO.

Proof. Let x ( t 0 , 40) be any solution of (7.1.1). By assumption (iii), it is bounded, and, hence, there exists a compact set Q in S, , such that t 2 to to ,do)(t) E Q , Moreover, it also follows that 11 x , ( t o , +o)llo < p* < p , t 3 t o , and

therefore, by assumption (i), we have lI.f(t9

%(to

9

d0))ll

< M-

Suppose that this solution does not approach 52 as t + CO. Then, for some E > 0, there exists a sequence {trc},t, + co as k + CO, such that 4 t n > ddtd E

S(Q3

€1" n 8,

where S(Q, E)" is the complement of the set S(52, e ) = [x : d(x,Q) < €1. We may assume that t, is sufficiently large so that, on the intervals t, t t, ( E / ~ Mwe ) , have

< < +

4 t o 2 do)(t) E S(Q,

4v nQ-

(8.6.3)

These intervals may be supposed to be disjoint, by taking a subsequence of (t,), if necessary. By Theorem 8.1.2 and assumption (ii), we get V

9

4 t O do)(t))


, to . Since C(x) is positive definite with respect to 52, the relation (8.6.3) shows that there exists a 6 = 6 ( ~ / 2 )> 0 such that C[x(tO

~ d O ) ( ~ ) l2

It therefore turns out that

tk

<
, 0. As a con! as t -+ 00, and the sequence, any solution x(t, , $o) tends to the set 2 theorem is proved. Making use of two Lyapunov functions, we can extend Theorem 4.2.1 to functional differential system (7.1.1). ---f

THEOREM 8.6.3.

Let the following assumptions hold:

(i) f~ C [ J x C,, , R n ] , f ( t ,0) = 0, andf(t, 4)is bounded on J x C,, ; (ii) V , E C[[-T, CO) x S o ,R,], Vl(t,x) is positive definite, decrescent, locally Lipschitzian in x, and, for t E J , (b E C, ,

< 44(0)) < 0,

D+Li(t,4(0i 4)

where ~ ( x is) continuous for x E So ; (iii) V , E C[[-T, co) x S o ,R,], Vz(t,x) is bounded on J x S, and is locally Lipschitzian in x. Furthermore, given any number a, 0 < 01 < p, there exist positive numbers [ = ,$(a)> 0, 7 = ~ ( a> ) 0, 7 < 01 such that

d(O), 4)> 4 that < II $(O)ll < p

D’L72(t,

for every t where

3 0, $ E C,, such E

OL

= [x €

so: a(.)

=

and d(+(O),E )

< 7,

01,

and d(x,E ) is the distance between the point x and the set E. Then, the trivial solution of (7.1.1) is uniformly asymptotically stable.

Pmof. As the proof requires appropriate changes in the proof of Theorem 4.2.1, we shall indicate only the modifications. Let 0 < E < p and to E J. Since Vl(t,x) is positive definite and decrescent, there exist functions a, b E X , satisfying b(lI x 11)

Let u s choose S

< l’d4 x) < 4

ll),

( 4 ). fzJ x

so.

6 ( ~> ) 0 such that

:

b ( € ) > a(6).

Then, by Corollary 8.2.4, the uniform stability of the trivial solution of (7.1.1) results. Let us designate 6, = S(p). Assume that ]I +o ]lo < S o . T o prove the theorem, it is sufficient to show that there exists a T = T ( E )such that, , < S ( E ) . As in the proof of for some t , E [t, , to ?’],jl ~ ( t,+o)(t,)jl Theorem 4.2.1, this will be achieved in a number of steps:

+

8.6.

(1") If d[x(t,), x(t,)]

109

ASYMPTOTIC BEHAVIOR

> r > 0, t, > t, , then Y

< Mnl/2(t,

t,),

~


, 71/21. Suppose that x(t) satisfies, for t, t t,,, , the inequality 6 < 11 x(t)ll < p. Then, arguing as in the proof of Theorem 4.2.1, with obvious changes, we can show that

<
4 t k f 2 ) )

< Vdtk

> 4tk)) -

A,

*

We now choose an integer K* such that h,K* > a(S,) and let T = T ( E )= 4K*L/l. Assuming that, for to t to T , we have

< < +

/I 4t)lI 2

w

7

we arrive at the inequality, as in Theorem 4.2.1, I yi(to

+

7’9

X(tn

+ T ) ) < Vi(to, 40) ~

< a@,)

-

K*h,

K*hi

< 0,

which is absurd, since Vl(t, x) is positive definite. It therefore turns out TI such that (1 x(t*)ll < 6, and this proves that there exists a t* E [to , t, the uniform asymptotic stability of the trivial solution of (7.1 .I).

+

8.7. Notes T h e comparison theorems 8.1.1 and 8.1.3 are due to Lakshmikantham [ I , 61. See also Driver [3]. Theorem 8.1.4 is new. Theorems 8.2.1 and 8.2.2 are adapted from the work of Lakshmikantham [l, 61, whereas Theorem 8.2.3 is based on the result of Driver [3]. See also Krasovskii [2, 51. Theorems 8.2.4 and 8.2.5 are new. T h e examples in Sect. 8.2 are taken from Lakshmikantham [6] and Driver [3]. All the results of Sect. 8.3 are based on the work of Lakshmikantham [6], whereas Theorem 8.3.4 is new. Section 8.4 contains new results. See also Halanay [22] for particular cases. T h e results of Sects. 8.5 and 8.6 are new. For many similar results for delay-differential equations, see Oguztiireli [I]. For related work, see Driver [3], El’sgol’ts [4], Krasovskii [l-51, Lakshmikantham [I], Oguztoreli [l], and Razumikhin [2, 61. For the use of vector Lyapunov functions in studying the conditional stability criteria of invariant sets, see Lakshmikantham and Leela [2].

PARTIAL DIFFERENTIAL EQUATIONS

This page intentionally left blank

Chapter 9

9 .O. Introduction T hi s chapter is devoted to the study of partial differential inequalities of first order. We consider some basic theorems on partial differential inequalities, discuss a variety of comparison results, and obtain a priori bounds of solutions of partial differential equations of first order in terms of solutions of ordinary differential equations as well as solutions of auxiliary partial differential equations. We also treat the uniqueness problem, error estimation of approximate solutions, and simple stability criteria. We make use of Lyapunov-like functions to derive sufficient conditions for stability behavior. For systems of partial differential inequalities of more general type, we merely indicate certain analogous results.

9.1. Partial differential inequalities of first order We shall use the well-known notation am at

mi=-,

m, =

am

ax

-

wp a2m

mxg=

whenever convenient. It is, however, necessary to caution the reader not to confuse the symbol m l with the one used while considering functional differential systems. Let 01, ,l3 E C [ J ,R ] , and suppose that a ( t ) < P(t), t E J. Assume that a’(t), P ’ ( t ) exist and are continuous on J . For to E J , we define the following sets: 113

114

CHAPTER

9

E

=

[ ( t ,x): to

< t < co,.(t) < x < P ( t ) ] ,

E, aE,

=

[ ( t ,2 ) : t ,

< t < co,.(t) < x < P ( t ) ] ,

=

[ ( t ,x): t

aE,

== [ ( t ,x):

aE,

=

t o , &(to) < x < &to)], < t < 03,x = &(t)], /at.

D-m(t,)

Proof.

(9.2.1)

Choose a sequence {t,}, so that tk < to , t, + to as k + co, and (9.2.2)

For k sufficiently large, we have (9.2.3)

On the other hand, by the definition of M ( t ) and the fact that M(t,,) = u(t, , x,,), for k sufficiently large, it follows that

which, on account of the relations (9.2.2) and (9.2.3), yields (9.2.1).

LEMMA 9.2.2. Let G be a bounded open set such that G C R n and C[J x G, R].Let

uE

w ( t ) = max I u(t, x)I, XtG

M ( t ) = max u(t, x), XEG

and N ( t ) = maZ[-u(t, x)]. X€G

Let ( t o ,xo)E (0, m) x G. Then,

(1") w(t,) > 0 implies either w(t,) = M(to)or w(to) = N(t,,); (2") eu(to) > 0 and w(t,) = M(t,) implies D-w(t,) < D-M(t,,); (3") w(to) 0 and w(t,,) = N(t,,) implies D-w(t,) D-N(t,,).


0

Proof. Suppose that w(t,) the inequality U(to

, x)

< I u(to

9

.)I

and w(t,,)

=

u(t, , x,,). It follows from

< w(to) == ~ ( t,,xo),

x E G,

I20

9

CHAPTER

, If, on the other hand, we suppose that that M(to)= ~ ( t , xo). w(to) = -u(t0, xo) > 0, the inequality -u(t,

7

x)

< I u(t0

3

)I.


, 0 and

< z(t, x)

1 u(t, x)l

on

aE,

Under these assumptions, we have j u ( t , )I.

Pmof.


0

> z,(h , 4.

aE, , m ( t , , xl)

= 0,

9.3.

I33

UPPER BOUNDS

It therefore follows that

verifying condition (iv) of Lemma 9.1.1. T h e other assumptions of Lemma 9.1.1 may be checked similarly. Thus, by Lemma 9.1.1, we have

O n the other hand, defining

it is easy to show that n(t, x) satisfies the hypotheses of Lemma 9.1.1, and, consequently, -u(t, x)

< z(t,x) + y(t,

on

C)

E.

It therefore turns out that Iu(4 I).

since lim,,,y(t,

e)


, 0 is the solution of (9.3.18) such that

1 u(t, .Y) we have

-

1 u(t, x)

v(t, x)I

-

on

S z(t, X)

< z(t,x)

v(t, x)i

aE, ,

on

E.

9.4. Approximate solutions and uniqueness We shall consider the partial differential inequality

< 6.

I ut -f(4 x,u, u,)l

(9.4.1)

DEFINITION 9.4.1. A function u(t, x,6) is aid to be a &approximate solution of (9.3.1) if (i) u E C [ E ,K],and u(t, x, 6) possesses continuous partial derivatives LJ aE, ; /3(to); (ii) u(to, x,6) = (b(x), where +(x) is continuous on .i(to) x (iii) u ( t , x,6) satisfies the inequality (9.4.1) on E, .

on E, and total derivative on aE,

<
, 9,

on

aE,, (9.4.3)

4),

P 2 4,

on

aE,,

(9.4.4)

hold. I,et u(t, x), u(t, x,6) be a solution and a &approximate solution of (9.3.1). Then,

I u ( t , x)

-

u(t, x, S)l

< r(t, S)

on

B.

(9.4.5)

9.4. Pyoof.

APPROXIMATE SOLUTIONS AND UNIQUENESS

135

Consider the function m(t, x)

where y ( t , e )

=

4 4).

6,

c)

= y ( t , to , 0,

-

44 x, 8) - y ( 4 e ) ,

is a solution of

Y’ = g(4y)

+8 +

Y(td

6,

= e,

for small E > 0. Suppose that, for (tl ,xl) E aE, , m ( t , , xl) = 0 and , xl) 3 0. This shows that

m,(t,

> 4 t l , x1 , 8 )

4 t l 7 x1)

and

%it,

I

2 %it1

4

9

x1 , 8).

Thus, using (9.4.2) and (9.4.4), we have mt(t1

9

44

x1) = f(tl x1 7

-f(tl

f

7

+f@l 1

x1 9

P

Xl),

udt1 9 x1>>

U ( t l7 ’1

4% x1 9

UZ(tl

J

7

?

- g(t1

< -B’(t1)

,A t 1

9

%(tl

so that %(f,

I

-f@1 x1 4)- E - 6 8)

x1

Xl>

9

9

x1)

-

.1>)

61, %it, , .1>>

- f ( G , x1 > 4 t , > x1 , s>,%(tl -

9

, x1 , 8)) 9

4 t l I x1 7 8 ) , 4 1 , x1 9 8))l

€9

+ B’(tl>

,x1)

< 0.

This proves condition (iv) of Lemma 9.1.1. It is easy to show that the other assumptions of Lemma 9.1.1 also hold. Hence, by Lemma 9.1.1, ~ ( tx),

-

u(t, x,

6)

< y (t , e )

on

E.

on

E.

Proceeding similarly, we can show that ~ ( tX,, 8 )

-

u(t, x)

< y ( t ,E )

T h e estimate (9.4.5) results immediately, noting that limy(t, 6) €4

T h e proof is complete.

=

~ ( tto, 0,8).

136

9

CHAPTER

COROLLARY 9.4.1. If the function g ( t , u ) = Ku, K takes the form

1 u(t, x)

-

u(t, x,

s)i < (S/K)[exp K ( t - to) - 11

> 0,

then (9.4.5)

on

E.

We next state a uniqueness theorem of Perron type whose proof is an immediate consequence of Theorem 9.4.1 or Corollary 9.3.2.

THEOREM 9.4.2. Let f E C [ E x R x R,R] and the condition (9.4.2) hold. Assume further that the boundary conditions (9.4.3) and (9.4.4) are satisfied. If y ( t ) = 0 is the maximal solution of Y'

= g(4

Y),

Y(to) = 0,

for t >, t o , then the partial differential equation (9.3.1) admits atmost one solution.

THEOREM 9.4.3. Under the assumptions of Theorem 9.4.2, given E > 0, there exists a 8 ( ~ > ) 0 such that I d(x)

~~

$I).(

f " t ,

x,TI, U,Z)

9.5.

139

SYSTEMS OF INEQUALITIES

(v) u(t, x) < v(t, x) on aEl ; (vi) f ( t , x, u, p ) is quasi-monotone in u for each fixed ( t , x, p ) . These conditions imply that u(t, x)

Proof.

< v(t, x)

on

(9.5.4)

E.

Consider the function m(t, x)

=

u(t, x)

-

v(t,x).

Evidently, m ( t , x) verifies assumptions (i) and (v) of Lemma 9.5.1, on account of assumptions (i) and (v). Suppose that, for some index j , 1 <j N , ( t l , xl)E aE, , we have mj(t, , xl) = 0,mi(tl , xl) 0, i # j , and m,J(tl , xl) 3 0. This implies

5


az,

where g E C [ J x R, , R ] , and the maximal solution r(t, t o , 0) of (9.1.7) is identically zero. Then, whenever

u(t, .x)

< v(t,x)

on

E

u(t, x)

< v ( t ,x)

on

aE,.

We shall next prove a comparison theorem analogous to Theorem 9.2.3.

THEOREM 9.5.3. Let m E C [ E ,R+N], f that the following conditiQns hold:

E

C [ E x RN x Rn, RN] such

(i) m ( t , x) possesses continuous partial derivatives on E and total derivative on aE, LJ aE, ; I , 2,..., N , (ii) for ( t , x) E B and i (iii)

m t ( t , x ) 0. Since it is known

lim Y(4 .) t-0

=

that

r(t, t o 7 Yo),

it is sufficient to show that m ( t , x)

< y(t,

on

E)

(9.5.10)

B.

It is easy to verify that v(t,x) = m(t, x) - y ( t , E ) satisfies the hypotheses of Lemma 9.5.1. Clearly, assumptions (i) and (v) hold. Moreover, for an index j , 1 < j N , (tl , xl) E aE, , suppose that d ( t , , xl) = 0, v i ( t 1 ,xl) 0, i f j , and vE:3(tl, xl) 3 0. Then,


,O)

0,

=+ 87,

e Br(4 1, ,Yo>

Q(4 x, u(t, .))I

on

aH, ,

where(? E C[aH, x R,R],Q(t, x, 2) is increasing in x for each ( t ,x) E aH,, Q(t, x, -4 = -P(t, x,4, and P(t,t o 3 Yo) Q(4 x,44 t o ,Yo))-


z2 ; (10.3-9)

(iii) w E C [ H ,R,], v(t, x) possesses continuous partial derivatives 2 0, w t uzz, and T[v] >, 0; (iv) u(t, x) is a solution of the first initial-boundary-value problem satisfying 1 u(t, x)I < v(t, x) on Pi,, u 8H. u,

9

Then, we have

< v(t, x)

1 u(t, x)l Prooj.

on

H.

(10.3.10)

Let us consider the function

44 4 = -[u(t, 4

+ v(t, x) + Y ( t , 41,

where y(t, E) = y ( t , t o , 0, E ) is any solution of (10.2.9), for sufficiently small E > 0. Let (tl , xl)E Pt1, n(tl , xl) = 0, n,(tl , xl) = 0, and the quadratic form

for some vector A. This means, noting y(t, E ) -4t1 -U&1

> x1) f

> 0, that

> "(tl , 4,

4 = "&,

t

XI)

b 0,

168

CHAPTER

10

and

Because of the ellipticity off, it results that f(t1

, x1 , - 4 f 1 > X l ) , --%(f1 4,-u,,(t1, x1)) 5f ( t l , x 1 , - 4 t 1 , 4,-%c(t, , 4,%it1 , XI)). 9

Furthermore, using (10.3.8), (10.3.9), and the preceding inequality, we obtain "Ltl

7

x1)

= -

G

4 t l

-,f(t1

> x1)



x1)

, x1

,r ( t 1

?

?

XI),

1

I.'@,

c)) -

- Y'(t1

9

%(tl

4

9

4,%til1

-%(t,

, XI), %&I

'UAtl 9

4,%At,

XI>,

At1

7

> Xl), U,,(tl

Xl), %1 ti

1

x1 > v(t1 9 x1),

*2^1 1 - 4 t 1

Kfl

"dtl

4tl

x1

1

-f(t1

-

7

Y(tl

7

> x1))

, x1)) - Y'(t1 €1 , XI)) , x1)) YYtl 4 9

-

9

4) - e

which implies that n,(t, , xl) < 0. Clearly, n(t, 2) satisfies all the assumptions of Lemma 10.1.1, and hence

< v(t, x)

-u(t, x)

+y(t,

c)

on

H.

Proceeding similarly, we can show, on the basis of Lemma 10.1.1, that u ( t , x)

< v(t, x) + y ( t , c)

on

H.

T h e preceding two inequalities, together with the fact that lim,,,y(t, c) 0, yield the estimate (10.3.10). T h e theorem is proved. ~

COROLLARY 10.3.2. Let the hypotheses of Theorem 10.3.2 remain the same except that condition (iv) is replaced by (iv*) u ( t , x) is a solution of the mixed initial-boundary-value problem satisfying (a) 1 u(t, .)I v(t, x) on P t 0 and i3H - aH, ;


0

on

u(t, x) is said to be an over function.

satisfies

on

u(t, x) = +(t, x)

a(t, x)

If

H,

&/a7 and

Pt0

9 +

Q(t, x, u(t, x))

exists on aH, and u(t, x)

= +(t,x)

aH

-

aH,,

on

we shall say that u(t, x) is an under or over function with respect to the mixed initial-boundary-value problem according as T[u]< 0 or T[u]> 0, on H.

As a direct consequence of Theorems 10.1.1 and 10.1.2, we have the following.

THEOREM 10.3.3. Let f~ C [ H x R x Rn x Rnz,R] and the differential operator T be parabolic. Suppose that u(t, x) and v(t, x) are under and over functions with respect to the first initial-boundary-value problem. If x(t, x) is any solution of the same problem such that u(t, x)

< x ( t , x) < v(t, x)

then u(t, x)

< z(t, x) < v(t, x)

on

Pt0u a H , on

H.

(10.3.1 1)

T h e inequality (10.3.11) remains true, even when u(t, x) and v(t, x)

170

CHAPTER

10

are under and over functions with respect to the mixed initial-boundaryvalue problem, provided z(t, x) is any solution of the same problem and u(t, x)

< z ( t , x) < v(t,x)

on

and

Pt0

i3H - aH, ,

+ Q(t,x, u(t, x)) < a(t, x) !%a7?? + Q(t, x, z(t, x)) av(t,x) < "(4 x) ____ a7 + Q(4 x, 4 4 4).

a(t, x)

10.4. Approximate solutions and uniqueness We shall begin with the theorems that estimate the difference between a solution and an approximate solution of (10.3.1).

THEOREM 10.4.1. Assume that (i) f E C[H x R x Rn x Rnz7R],the operator T is parabolic, and .f(t, X, 2 1

,P, R ) - f ( t ,

X, ~

2

P,%R )

< g(t, z1 - zz),

~1

> zz ; (10-4-1)

(ii) n E C [ p ,R],v(t, x) possesses continuous partial derivatives u t , a,, v,, such that I 7-[74 G s(t>, (10.4.2) where 6 E C [ J ,R,]; (iii) g E C [ J x R, , R],g(t, 0)= 0, and r ( t , t o ,yo ) is the maximal solution of

y'

= g(t,y)

+ S(t),

(10.4.3)

Y ( t 3 = yo 2 0,

existing for t 2 to ; (iv) u(t, x) is any solution of the first initial-boundary-value problem such that on Pto u a f f . I u ( t , ). v(t, X)l r(4 t o ,Yo)

e

~

Then, the estimate

I u ( t , ).

-

v(t, .>I

< r ( t , to ,yo)

on

is valid.

Proof.

Define N t , '4

= u ( 4 x)

-

v(t, x)

-

y(t, E),

R

(10.4.4)

10.4. where y ( t , c )

= y ( t , to

Y’

,y o , E) is any solution of

= g(t,y)

+ S(t) +

+

YPO) = yo

6,

for sufficiently small E > 0. Suppose that (tl , xl) m,(t, , xl) = 0, and, for some nonzero vector A,

Since y(t, c )

171

APPROXIMATE SOLUTIONS AND UNIQUENESS

E

(10.4.5)

E,

Ptl , m(tl , xl)

= 0,

> 0, this implies that

and

T h e last inequality yields, because of the ellipticity off, At1

Y

x1

7

aft1 7

7 %(t, >

> x1

?

4tl

>

x1),

Uzz(f1

4,%it,

9

4)

7

X l ) , VU,U,(tl

, El)).

It follows, in view of the preceding inequality, that %(tl ? x1> -

vUt(t1 9 x1)

G f(tl x1 9

-f(t1 -

7

4tl

?

7

x1)7

x1 7 v(t1

9

uz(t1

>

Xl),

vJt,

Xl),

9

Trz(t1

Xl),

7

4)

vzz(t1 > x1))

T[v].

Hence, the relations (10.4.1) and (10.4.2) show that %(tl 7 5 1 )

-

4 t l

x1)

< g(t1

9

Y(tl

.>I

9

+ S(t).

We thus have mdt1 ? x1) =


R , ) -At, ,x, 2 2 p2 > 4) 2 ,G(t,2, z1 - 22 , Pl - P, , R, - RZ); 7

9

(10.4.6)

C [ f I ,R ] , v(t, x) possesses continuous partial derivatives

ZI~,

(10.4.7)

10.4.

173

APPROXIMATE SOLUTIONS AND UNIQUENESS

whereg E C [ J x R, , R],g(t, 0) = 0, and the maximal solution r ( t , to , 0) of (10.2.6) is identically zero; (iv) z E C [ R ,R+], z, 2 0, z t , z,, exist and are continuous in H , and

2 G(t, x, z,

zt

Then,

I 44).

- v(l,

+ s(t, 4.

zzz)

< z(t, x)

on

Pi0 u aH

41 e 4 4 4

on

H,

1 u(t, x) - v(t, x)I implies

232,

(10.4.9)

where u(t, x) is any solution of the first initial-boundary-value problem.

Proof. As usual, we shall reduce the theorem to Lemma 10.1.1. We shall first consider the function

where y ( t , E ) = y ( t , to , 0, e) is any solution of (10.2.9) for small E > 0. Suppose that (tl , xl) E Ptl , m ( t , , xl) = 0, m,(t, , xl) = 0, and, for some nonzero vector A,

Since z(t, x) >, 0, y ( t , E )

> 0, the preceding 4 t l , x1)

%(tl x1) 9

-

supposition implies that

> 4 t l 4,

%it, 1x1)

9

= zz(t1,

x1)

2 0,

T h e ellipticity of G shows that

In view of this and the relations (10.4.6), (10.4.7), we have

174

10

CHAPTER

Hence, using (10.4.8) and (10.4.9), we derive 4 t l

,

< G(t, ~


Y(tl

+

I

6))

< 0.

-€

+

, XI) Y(tl , E ) , %it, xl), &(tl , x1)) so1 ,)I. , X l ) , %it1 X l ) , Z , X ( t l ,x1)) - S(t1 , 4 9

Z(t1

9

7

--

This shows that m(t, x) verifies the assumptions of Lemma 10.1.1, and, therefore, on H. u(t, x) v(t, x) < z(t, x) + y ( t , E) -

Arguing similarly, we can prove w ( t , x)

-

u(t, x)

< z ( t , x) + y(t, 6)

on

H.

Since lim,,,y(t, e ) = 0, by assumption, it follows from the preceding two inequalities that

I u(t, .x)

-

v(t, .x)I

< z(t, x)

on

H,

proving the theorem. We shall next consider the uniqueness problem.

THEOREM 10.4.4. Suppose that (i) f~ C [ H x R x Rn x R’”, R ] , the operator T is parabolic, and

f(4 x, z1 , p,R ) -f(4

x,

2.21

(ii) g E C[/ x R, , R ] ,g(t, 0) of (10.2.6) is identically zero.

p, R) =

< f(t, z1 - 4,

z 1

>22 ;

0, and the maximal solution r ( t , to , 0)

Under these assumptions, there is at most one solution to either one of the initial-boundary-value problems. T h e proof is a direct consequence of Theorems 10.4.1 and 10.4.2.

10.5. Stability of steady-state solutions Let problem D represent the partial differential equation of the form ut = :

for x E [a, b], t

> 0, together

%(t, a )

= f,(u(t,

a)),

f(.x, u, u,

7

(10.5.1)

ux,),

with the boundary conditions u,(t,

b)

= f,(u(t,b)),

t

> 0,

10.5.

STABILITY OF STEADY-STATE SOLUTIONS

175

where fi ,fiare continuous functions with bounded derivatives. Let us assume that af/au exists and is continuous,f (x, u, P, R ) is nondecreasing in R for each (x, u, P ) . We use the notation u(t, x, $) to denote a solution of problem D such that

4 0 , x, 4 ) = 4 w ,

6

[a,

4,

where 4 E C [ [ a ,b], R]. DEFINITION10.5.1. Let u ( t , x, $) be a solution of problem D. We shall say that u(t, x, 4)is a steady-state solution if u(t, x, $) = $(x), t > 0.

DEFINITION 10.5.2. T h e steady-state solution u(t, x, 4) of problem is said to be stable if, given E > 0, there is a 6 > 0 such that

D

implies

DEFINITION10.5.3. A

Let

= [(x, u): x E [ a , b’J

< < #z(x)17

and

where z+hl , are arbitrary functions, twice continuously differentiable on [a, 61. Let B be the set of functions on [a, b] such that z+h E B implies

T h e steady-state solution u ( t , x, $) of problem D is said to be asymptotically stable if it is stable and lim[ max j u(t, x,+) - u(t, x, #)I]

t-m

x~[a,bl

=

0,

whenever I$ E B. T h e set A is called the domain of attraction. Sufficient conditions for a steady-state solution u(t, x, $) of problem D to be stable are given by the following theorem.

THEOREM 10.5.1. Assume that there exists a one-parameter family o(x, A), A E [A, , A,], of solutions of the equation f(X,

v, v z , vm)

=

0

(10.5.2)

176

10

CHAPTER

fulfilling the following conditions: (i) there is a A * E ( A , , A), such that v,(a, A*) =f,(v(a, A*)) v,(b, A * ) = J.(v(b, A")); (ii) zq,(x, A) > 0, x E [a, 61, A E [A, , A,];

(4a,(% 4 >f,(v(a, 4)and %(b, A)

f,(v(b, A)), A

(iv) a,(%

Then, if 4(x) lem D is stable.

Pyoof.

Let

E

=

given. Choose A, max

1 v(x, A*)

max

1 v ( x , A")

xt[a,b]

[A, , A*); E (A*, A,]. E

the steady-state solution u(t, x,4 ) of prob-

v(x, A*),

> 0 be

and

~

E

[A, , A*), Ao

E

(A*, A2] such that

v(x,A,,)\ < E

(10.5.3)

< €.

(10.5.4)

and xt[a,b]

-

W(X,

A*)(

Then, define the number 6 by 6

7-=

min[ min (v(x,A*)

-

v(x,An)),

xt[a,bl

min (v(x,An)

xt[u,bl

-

Since v,(x, A) > 0 for all x E [a, b ] , it is clear that 6 the inequality

and assume that $(x) a(x,A,)

5:

$A).(

= -

(10.5.5)

v(x,A*))].

> 0. Let $(x)

satisfy

n(x,A"). We then have, by (10.5.5),

s < #(x) < +(x)

+ < v(x, A,),

x E [a, b ] .

The fact that v(x,A) is a family of solutions of (10.5.2) and u(t, x, $) is a solution to the Droblem D imolies that

By a successive application of Theorem 10.2.4, we deduce that v(x, A") < u(t, x, I))

< v(x,A",

x E [a, b ] ,

t

2 0.

10.5.

177

STABILITY OF STEADY-STATE SOLUTIONS

This, because of the relations (10.5.3) and (10.5.4), yields u ( 4 x,

4)- E = v(x, A*)

and u(4 x, 4)

+

=

€

v(x, A*)

-E

< v ( x , A,) < u(t, x, $)

+

> v(x, An) > u(t, x, #).

€

It is evident from the preceding inequalities that

< 6.

whenever max,.[,,,l I $(x) - +(.)I

T h e proof is complete.

I n the situation in which it is difficult or impossible to find a oneparameter family u(x,A), satisfying the conditions of Theorem 10.5.1, it may still be possible to find an upper bound. This we state as a corollary.

COROLLARY 10.5.1. Suppose that there exists a solution u(x)of (10.5.2) satisfying %(a) < fl(44) and vdb) > f,(v(b))* Then, if u(t, x, $) is a solution to problem D such that #).(I

we have

< v(4,

u(4 x,4) < +),

x E [a, bl,

x E [a,bl,

t b 0.

A similar corollary may be stated establishing a lower bound.

THEOREM 10.5.2. Let the hypotheses of Theorem 10.5.1 hold. Suppose further that, for x E [a, b] and A E [A, , A,], fv(x,

v(x,

3,%(X,

A),

%,(X,

Then, if +(x) = u(x,A*), the steady-state solution u(t, x, totically stable, and the set A

=

[(x, u ) : x E [a, b]

and

(10.5.6)

A)) f 0-

v(x, A,)

4) is

< u < v(x, A,)]

asymp(10.5.7)

is a region of attraction.

Proof. T h e stability of steady-state solution u(t, x, #J) follows by Theorem 10.5.1. Let A be the set defined by (10.5.7), and let B be the set of functions such that E B implies [(x, +(x)) : x E [a, b]] C A. We

178

10

CHAPTER

shall first show that, for any E > 0 and any 4 E B, there exists a T I > 0 such that max [@, x, $) - 44 x, 411 < E . (10.5.8) x~Ca,bI t>T,

Let

E

> 0 be given, and let ha E (A*, v(x, AO)

u(x,A*)

-

A,] be such that

< E,

(10.5.9)

x E [a, b].

Without loss of generality, we may assume that f, > 0, in view of ,u2, p3 by

(10.5.6). We then define three positive numbers p l ,

v(x,

A)

~

'u(x,

< p1/2,

A*)

x E [a, b].

Define a positive number p4 by

Let H(A) be a function defined for A E [Ao ,A,], such that, for h

fZ('u(b,

4 - h ) -fz('u(b, 4)< P4/2.

< H(A), (10.5.12)

Consider the function w

=

w(x, A) = v ( x , A)

-

where 6 > 0 will be specified later. Since small 6 > 0, we have f(X,

w, w, > w,,) = f(x.

'u -

< f(X, =

0.

'u, u ', 'up

u ',

f, exists, for sufficiently

8 , u' , ,' u 3

= f(X, 71%'us > u ),',

fi(+, 4) - (P4/2) > v,(a, 4

= f*(fl(a,

==

w,(a, A).

~

(10.5.18)

180

CHAPTER

10

Similarly, from (10.5.10), (10.5.12), (10.5.14), and (10.5.15), we derive $dw(b,

4) = fz(v(b, A) < v,(b, A)

9+ ( d 2 )

8) < f2(7@, wc(b, A).

-

=

Clearly, (10.5.18) and (10.5.19) hold for all X E [A*, A,]. Theorem 10.2.4, it follows that m(t, ).

2

u(t, x,

x E [ a , 61,

$17

t

E

[O,

(10.5.19)

On the basis of 7-11>

since m(0, x) > $(x), x E [a, b ] . From the definition of m and Tl , we get

> u(T1 , x, $),

v(x, AO)

x E [a,4 .

Thus, by Corollary 10.5.1, v(x, Ao)

> u(t, x, $),

x E [a, 61,

t

> T, ,

which, together with (10.5.9), gives us (10.5.8). T h e next step of the proof is to show that there exists a T, > 0 such that (10.5.20) max [u(4 x, 4)- 4 4 x, $11 < xe[a.bl t>T,

T h e proof of this consists of showing that there is a lower bound for $) which can be increased with time until it is within E of u ( t , x, 4) at some time T, . T h e proof of this fact is similar to the first part and differs only in minor details. Let T = max[Tl , TJ. Then, from (10.5.8) and (10.5.20), we obtain u(t, x,

max

xtla.61

I u(t, x, 4)

~

u(t, x,

$)I < E .

t>T

This completes the proof of the theorem. COROLLARY 10.5.2. T h e conclusion of Theorem 10.5.2 remains valid if (10.5.6) is replaced by either (i) f,, = 0 andf,&, v(x, 4, %(X, 4, % r ( X , (ii) f r = 0, f U Z = 0, and fv,,(x, 4x, A), x E [a, 4 , A E [A, A,].

4) f %(X,

0 ; or A), %,(x,

A)) f 0 for

9

We now give an example to illustrate Theorems 10.5.1 and 10.5.2. Consider the partial differential equation u.

-

(1

+

242)

u,. - uu,z,

xE

[I, 21,

10.6.

181

SYSTEMS OF PARABOLIC INEQUALITIES

subject to the boundary conditions I)

T h e equation (1

+

uz(4 2) = fz(u(4 2)).

and

= f&(4 1))

242)

u,,

-

uu,2 = 0

has a one-parameter family of solutions v ( x , A ) given by v(x, A ) Notice that v,(x, A) = x-l[l + &(x, A)]1/2 sinh-l v ( x , A).

=

sinh Ax.

Suppose that

+ u2)1/2sinh-l u, fl(u) < (1 + u2)1/2sinh-l u, ti(.) < +(l + u2)1/2sinh-l u, f2(u) > t(l + u2)1/2sinh-' u,

fl(u) > (1

u u u u

> 0, < 0, > 0, < 0.

fl(0)

=

0,

f2(0)= 0,

Then, by Theorem 10.5.1, u(t, x, 4) = 0 is stable. T o apply Theorem 10.5.2, we observe that fv(x, v, 0,

,v,,)

=

2vv,, - vz2 2A2 sinh2Ax - X2 cosh2Ax

=

A2(sinh2Ax- 1).

=

Thus, fv < 0 if sinh2 Ax < 1 or u < 1. Hence, u(t, x, 4) = 0 is asymptotically stable, by Theorem 10.5.2.

10.6. Systems of parabolic differential inequalities in bounded domains Let us consider now a partial differential system of the type where

u: = f i ( t , x, u,u,i, u;,),

u,i =

, d2,-*.,

i

=

1, 2 ,...)N ,

u&),

For convenience, we shall write the preceding system in the form

182

CHAPTER

10

where f E C [ w x R N x Rn x Rn', RN]and each function f is elliptic,

so that the system is parabolic.

We shall first state the following lemmas, which are extensions of Lemmas 10.1.1 and 10.1.2.

LEMMA 10.6.1. Suppose that (i) m E C [ H ,RN],m(t, x) possesses continuous partial derivatives m ,, m, mzz in H ; (ii) m(t, x) < 0 on Ptou aH; 7


0 and to E J be given. Assume that the trivial solution of (10.8.2) is equistable. Then, given b ( e ) > 0 and to E J , there exists a 6 = 8 ( t o ,E) > 0 that is continuous in to for each E such that Y ( t , to ,Yo)


t o , we

have

36

Then, by condition (b), there exists an xo E int P,, such that

II +I

7

x0)Il

= E.

E.

192

CHAPTER

10

It now follows from relations (10.8.1), (10.8.3), and (10.8.5) that

< i;(tl

44

, xo "(t,, %))
0 and to E J be given. It then follows, on account of (S3*), that, given b(c) > 0 and to E J , there exist positive numbers 6, = so(to) and T = T ( t o ,E) such that

+

(10.8.7) 2 to T , provided yo < 6, . Choosing yo = a(t, , 1) $(to , -)I\ plo),as before, we can Y ( 4 t o 3 Yo) < b ( 4 ,

t

show the existence of a positive number

/I C(to -)llp'o ,< &I I

and

8,

=

8,(t,) such that

4 t o 7 I1 +(to

9

.)llPJ

< so

hold at the same time Suppose now that n ; (a) I/ +(to * ) l / P t o (b) I1 +(t, *>/laIf < ' t 3 to

+ T.

+

Let there exist a sequence { t k } , t, 3 to T and t,+ co as k + co, such that / / u(t, , *)llPtk 2 E for some solution u(t, x). Then, there exist x, t int P t k satisfying (1 u(t, , x,)[[ = E in view of condition (b). Thus, using relations (10.8.1), (10.8.5), and (10.8.7), we arrive at the contradiction b(c) < V(t, , X k , U ( t , , X k ) ) < Y ( t , , to ,Yo) < 4).

10.8.

193

STABILITY AND BOUNDEDNESS

Thus, the quasi-equi-asymptotic stability holds, and, as a result, the trivial solution of ( 10.6.1) is equi-asymptotically stable.

THEOREM 10.8.4. Under the assumptions of Theorem 10.8.2, the uniform asymptotic stability of the trivial solution of (10.8.2) implies the uniform asymptotic stability of the trivial solution of (10.6.1). Proof. Assume that the trivial solution of (10.8.2) is uniformly asymptotically stable. Then, we have (S,*) and (S,*). By Theorem 10.8.2, the uniform stability of the trivial solution of (10.6.1) follows. T o prove the quasi-uniform asymptotic stability of the trivial solution, we proceed as in Theorem 10.8.3 and choose 8, = u-l(S0), observing that So and T are both independent of t o . T h e proof is complete.

THEOREM 10.8.5. Assume that there exist functions V(t,x, u), g(t, y ) , and A(t) satisfying the following conditions: (i) A(t) > 0 is continuously differentiable for t E J , and A(t)+ co as t + co; (ii) g E C[J x R , , R],and g(t, 0) = 0; (iii) V E C[H x R N ,R,], V(t,x, u ) possesses continuous partial derivatives V , , V, , V,, , V, in H, and (10.8.1) holds; (iv) f E C [ H x R N x Rn x Rn', R N ] ,G E C[H x R, x Rn x Rnz,R], G is elliptic, and

(v)

< G(t,x,4 t ) w, x,4, A ( t ) vz , A ( t ) V z z ) ; G(t,x, z, 0 , O ) < g(t, 4,27 3 0.

Then, the equistability of the trivial solution of (10.8.2) guarantees the equi-asymptotic stability of the trivial solution of (10.6.1). Proof. Let E > 0 and to E f be given, and let u = min,,, A(t). By assumption (i), G > 0. Set 7 = o ~ ( E ) . Assume that (S,*) holds. Then, given 7 > 0, to E J , there exists a 6 = 6(to , E ) such that ~ ( ttn, yo) 9

whenever y o

t

(10.8.8)

to,

< 6. Let u ( t , x) be any solution of (10.6.1) such that A(t0) V(t0 P x,+(to

and

< 7,

>

4)G Yo

7

A ( t ) v(t,x,+ ( t , x)) G r(t, t o >Yo)

x E pto

9

on

aN.

194

CHAPTER

10

Then, by Theorem 10.7.2, it follows that




, to T , and t, co as k + c o such that 11 u(t,, .)I1 3 E for Ptk 6, and II+(t, -)[IaH < E , some solution u ( t , x) satisfying ll+(to, -)/Ip t 0 t >, to + T . Also, there exist xlj E int Ptk such that 11 u(t, , xk)lI = E . T h e relations (10.8.8) and (10.8.9) yield

+

---f


0; (iii) G(t, x,x, 0, 0 ) g(t, 0 ) 3 0 ; (iv) llF(t,x,u)ll

< g(t, z),

z

>, 0, where g E C [ J x R , , R],

< vV(t,x,u), and a 3 Kv.

Then, one of the stability notions of the trivial solution of (10.8.2) implies the corresponding one of the stability results of the trivial solution of the perturbed system (10.8.17).

I98

Proof.

CHAPTER

10

Using the respective assumptions in (i), (ii), and (iv), we find

I t is evident, from this inequality, that we can directly apply Theorems 10.8.1-10.8.4 to obtain the desired result. T h e proof is complete. Although we can prove a number of results by the techniques just used, that is, by reducing the study of partial differential system to the study of ordinary differential equations, in certain situations, this method does not yield all the information about the given system. For instance, consider again the example (10.8.11). Suppose we now assume that the ui,k(t, x) hihk is positive definite instead of positive quadratic form semidefinite, as demanded in (10.8.12). This stronger hypothesis has no effect. I n other words, we do not get more information because of this assumption. T o be more specific, suppose F = 0 so that g = 0. Then, we can conclude by Theorem 10.8.1 that the trivial solution of (10.8.1 1) is stable. This conclusion remains the same even when the preceding quadratic form is assumed to be positive definite. I n such situations, the following theorem is more fruitful.

xF,k=l

THEOREM 10.8.9. Assume that (i) V E C[H x R N ,R,], V ( t ,x, u ) possesses continuous partial derivatives V , , V , , V,, , Vt,in N,and b(ll u II)

< v(t,x, u ) e 44 II u Ill,

where b E Z, a E C [ ] x R, , R,], and u E X for each t E f; (ii) f E C[H x RN x Rn x Rn2,RN], G E C[f7 x R, x Rn x Rn2,R ] , G is elliptic, G(t,x, 0, 0, 0) = 0, and

10.8.

where g E C[J x I?, r(t, t o , 0) of

199

STABILITY AND BOUNDEDNESS

,R ] , g(t, 0) E 0, and the maximal soiution Y’

= &Y),

Y(t3

0

=

is identically zero. Then, the equistability of the trivial solution of zt =

(10.8.18)

G(t,x, z, z, %), 9

implies the equistability of the trivial solution of (10.6.1). Proof. Let u(t, x) be any solution of (10.6.1) such that

and

V(to,x, +(to

,.I)

,< z(t0

9

4

v(t,x, +(4 x)) < z(t, x)

on

Pto

on

aff,

where x ( t , x) >, 0 is the solution of (10.8.18). Define m(t, x)

V ( t ,x, u(t, x)).

=

Then, we get

< G(t,x, 4 4 4, m d t , 4,mm&, 4). If we write T[v]= v t - G(t, x,v, v,, vZJ, then it is clear that T[m]< T [ z ] . Furthermore, m(t, x) < z(t, x) on PL0u aH. All the assumptions of Theorem 10.2.2 being verified, we deduce that V ( t ,x, u(t, x))

< z(t, x)

on

R.

Let E > 0 and to E J be given. Assume that the trivial solution(l0.8.18) is equistable. Then, given b ( ~ > ) 0 and to E J , there exists a 6 = S(to , E) such that


x) 8, (ii) maxZEaHx(t, x) < E , t b to , implies

max z(t, x) XSP,

< E,

t 3 to

Let maxZSp ~ ( t ,, x) = .(to, 11 # ( t o , -)[!pto),and let 6, = &(to, E) be the 50 same number chosen according to the inequalities (10.8.6) in the proof of Theorem 10.8.1. Suppose that I I + ( t o , ,< 6, and II+(t, - ) / I aH < E , to

200

CHAPTER

10

t 3 t o . Assume that there exists a solution u(t, x) of (10.6.1) such that, for some t , > t o , I/ u(t, , -)I/ 3 E . It then follows that there is an xo E int Pil satisfying / j u(t, , xo)lI = E . From this, we deduce the inequality VE)

< L’(t1 , xo , 4 t , , xo)) < Z(t1 , xo) < b ( E ) .

This contradiction proves the equistability of the trivial solution of (10.6.1). On the basis of this theorem, we can formulate other stability results in this setup. We notice, however, that we now have the problem of knowing the stability behavior of partial differential equation (10.8.18). I n the cases where the function G(t,x,x, x, , x,,) is simple enough to know the behavior of its solutions by other methods, this technique is useful.

10.9. Conditional stability and boundedness I n this section, we shall consider the partial differential system of the type (10.9.1) U f = f(4 x, % u, , u,,), where U, =

(F

7-7

au2 __

au,

--

ax,

7

ax,

,*--,

au, __

ax, ’-*’

au, ax, ,-*., -)ax,

au, __

and

azu,

azu,

a2u,

ax, ax, ’...’ ax,z

’*..’

a2u, ax,2

**..’

w). a2uN

I t will be assumed that the first initial-boundary-value problem with respect to (10.9.1) admits the trivial solution and that all solutions exist on H . I n the sequel, a solution of (10.9.1) will always mean a solution of the first initial-boundary-value problem. Let k < N and M(N--k)denote a manifold of ( N - K ) dimensions containing the origin. Let S(a) and represent the sets, as before,

s(a)

S ( a ) = [ u : (1 u (1

< a]

and

S(a) = [u : 1) u (1

< a],

respectively. Parallel to the conditional stability and conditional boundedness definitions (C,) to (C16)of Sect. 4.4, we can formulate the definitions of conditional stability and boundedness of the trivial solution of (10.9.1.). Corresponding to (C,), we have

10.9.

CONDITIONAL STABILITY AND BOUNDEDNESS

20 1

DEFINITION 10.9.1. T h e trivial solution of the partial differential system (10.9.1) is said to be conditionally equistable if, for each E > 0 and to E J , there exists a positive function S ( t , , E ) , which is continuous in t n for each E, such that, if (i) ). M(N--k) (ii) $(t,4 c S ( E ) , ( t , 4 E aH, 9

9

then u(t, x)

“lo

c S(E),

>

( t , x) E H.

Sufficient conditions for the conditional stability of the trivial solution of (10.9.1) are given by the following result.

THEOREM 10.9.1. Assume that (i) g E C [ J x R+N,R N ] , g ( t , 0 ) = 0, and g ( t , y ) is quasi-monotone nondecreasing in y for each t E J ; (ii) V EC[H x R N ,R + N ] , V(t,x, u ) possesses continuous partial derivatives V t , V , , V,, , V , in H , and b(lI u 11)

where b E X , a (iii)

E

N

< 1 vik x,4 < 4 4 II u Ill, i=l

C [J x R+ , R+], and a E X for each t E J ;

f E C[H x RN x RNnx RNn2,R N ] ,G E C[H x R,N x RN x RN2,R N ] , G(t,2,

2, z:,

ZLJ

is elliptic, and

av

-

at

av + au .f(t, x,

u, u,

, uzz) < G(t,X, v(t,X, u ) , I/aci, VL);

(iv) G(t,x,x, 090) < g(t, 4, z 3 0; (v) Vi(t,x, u ) = 0 (i = 1, 2 ,..., k), k < N , if u E , where M(N--k) is an ( N - k)-dimensional manifold containing the origin. Then, if the trivial solution of the ordinary differential system y‘

= g(t,y),

= yo

>, 0,

to

2 0,

(10.9.2)

is conditionally equistable (in the sense of Definition 4.4.2), the trivial solution of the partial differential system (10.9.1) is conditionally equistable.

202

Proof.

10

CHAPTER

For any

> 0,

E

if

11 u 11

=

we have from assumption (ii) that

E,

(10.9.3) Suppose that the trivial solution of (10.9.2) is conditionally equistable. ) 0 and to E J , there exists a 6 = 6(to , c) > 0 such that Then, given b ( ~ >

c Yz(4 to ,Yo) < 44, N

t

2 to

(10.9.4)

9

2=1

provided v

C

y20



9

and

c N

V 2 ( 4

x , 4(t, .))

2=1

E

pto>

2=1

2=1


0 such that 7

Ild.(ro

9

9

7

.)/IPt0

< 61

and

4to

>

II d(to

7

.)llPJ

9

(a) (b) +(4 4 c S(C--),( 4 x) E aH, 9

pto

9

10.9.

CONDITIONAL STABILITY AND BOUNDEDNESS

203

and has the property that u(t, x) $ S(E) for some t, > to and x E Ptl. Because of relation (b), there exists an xo E int Ptl such that 11 u(t, , x,,)lI = E. Hence, by (10.9.3), (10.9.4), and (10.9.6), we are led to the following absurdity:

< c Vi(t1 , xo N

b(c)

i=l

< i=l c Ti(t1 , to ,Yo) < b(4. N

9

, .a>>

U( t 1

Consequently, the trivial solution of (10.9.1) is conditionally equistable, and the proof is complete. On the strength of Theorem 10.9.1 and the parallel theorems on conditional stability and boundedness (in Sect. 4.4), we have the following

THEOREM 10.9.2. Assume that the hypotheses of Theorem 10.9.1 hold, and suppose that a(t, r ) = a(r), a E X . Then, one of the notions (C,*) to (C&) relative to the ordinary differential system (10.9.2) implies the corresponding one of the conditional stability concepts (C,) to (C16). T h e following example, in addition to demonstrating the conditional stability, serves to show that the system (10.9.1) need not be parabolic. Consider the system

au, at

au, at

where F, F, F,

+ cos t ) u , + (1

=

(1

=

(1 - ePt)ul

=

(cos t

~

cos t)u,

+ (cos t

-

l)u, ;

+ (1 + e+)u, + (ct- l)u, ;

&)ul

~

+ (ect - cos t)u, + (e+ + cos t)u, .

204

CHAPTER

10

Assume that the quadratic forms C;j=, aiih,hj , Crj=1bi& , and C:,i=, cijhihi are all nonnegative for arbitrary vector A. Choosing the vector Lyapunov function V = ( V l , V , , V,) such that

we observe that the functions b(r) and a(t, r ) reduce to b(r) = [(u12

+ u: + u1;3)1/2]2

Furthermore, the function G

and =

n(t, r )

=

5[(u,2

+ u: + u32)1/2]2.

(G, , G, , G,) takes the form

T h e differential system (10.9.2) can be reduced to

We find that g = (g, ,g, ,8,) fulfills the monotonic requirements. Choose k = 1. T h e n the solution ~ ( tto, ,yo) of (10.9.2) is given by rl(4 t o Yo)

= YlO

r2(t, to ,yo)

T

7

exp[4(t

-

yzoexp[sin t

to)],

-

sin to],

y3(t, to ,yo) = y30exp[4eKt0 - 4.81.

Here we have A!l(N-,i) = M , , the set of points u such that u2 - u3)2 = 0. I t is clear that the condition (C,") holds, which,

(ul

+

10.10 UNBOUNDED DOMAINS

205

in its turn, implies, from Theorem 10.9.1, the conditional equistability of the trivial solution of the system considered previously.

10,lO. Parabolic differential inequalities in unbounded domains Let D be a region in Rnfl of (t, x) space, satisfying the following conditions: (i) D is open and contained in the zone to < t < m; (ii) for any t, E [ t o , m], the intersection Stl of D with the plane t = t, is nonempty and unbounded; (iii) for any t , , Stl is identical with the intersection of the plane t = t, with the closure of that part of D which is contained in the zone to t < t, .


KO. We shall retain the meaning of the symbols D, , r, , Sk, aD, , C, and Dh , DRh,aDk, aDRh, CRilas defined in the proof of Theorem 10.10.1. Set z

Then, we have m observe that az

at H

=

=

z(t, x) = G(t, x)

-

v(t,x).

x H defined in Dh, where h < p-l. Moreover, we aH

+zat = ut - vt

GH + G H , , R"') f ( t , x , iiH, u>H + u i H x ,R 2 ) ) +f ( t , X , GH, uTH + U"H,, R ( 2 ) ) - f ( t , x , c H , z H + GH,, R ( 3 ) ) ,

= f ( t , X , GH,

~

(10.11.2)

212

CHAPTER

10

Let {R,} be an increasing sequence, R, 4 GO as we consider the domain D:, . Let us denote A,

=

01

-+

00.

For a fixed a ,

max[ sup - I xi ( t , x)i]. (t,x)eDh Re

-

Then, there exists an index i, and a point ( t , , x,) E D i , such that A , = 1 zi.(t,, x,l). We shall show that p, y , and h may be conveniently chosen such that (t, , x,) E u Sta implies I zi=(t,, x,)l = 0. Let us suppose the contrary. Then, there are two cases to be considered: (a) zi"(t, , x,) (b) ziu(t, , x,)

> 0, and < 0.

I n case (a) holds, we have, for an arbitrary vector A,

and

Since f ie is elliptic, it follows that

CH + Z H x , I?"') , u(t, , x,), G H + .".Hx

f " ( t , , xu , ~ ( t ,,x,),

E $0 ( 4 ( t , , x,) E D? (iii) ( t , , x,) E aha, (iv) ( t , , x,) E C ~ O . 7

9

Evidently, in case (i), zi-(t,, x,) holds, we have either or

(iia) zi=(t,, x,)

=

(iib) zi,(t,, x,)

>0

(iic) zi-(t,,x,)

< 0.

or

=

0, and therefore A,

= 0.

I n case (ii)

0

Clearly, case (iia) implies A, = 0. If case (iib) is true, then, using a similar argument as in the proof of Theorem 10.10.1, we arrive at the inequality 0


,

means weak convergence.

REMARK12.1.1. Suppose now that E is a Hilbert space and that f ( t , u)satisfies the monotonicity condition, that is, Re(f(t,

u ) -f(t,

v), u - v)

<M I u

-

1)'

( 12.1.2)

12.2. NONLOCAL

24 1

EXISTENCE

where we denote the scalar product by ( x , y ) and the norm by1 x I = (x, 3 0. Then V(t, u, v) = e-2Mtl u - v l2 satisfies all the hypotheses of Theorem 12.1.1, and, consequently, the conclusion of Theorem 12.1.1 is true. We shall now give an example to show that, even whenf(t, u ) does not satisfy the monotonicity condition, there does exist a V(t, u, v) satisfying the hypotheses of Theorem 12.1.1. Consider the example du - = f ( t , u) =

dt

u

< 0,

assuming that E = R. Clearly (12.1.2) is not fulfilled. However, there exists a V ( t ,u, v) defined as follows:

I(";

V(t,u, n) =

I

-

dV

-

log(1

+ dU) + log(1 + dVy,

(G- log(1 + 4;)- &v)2, (4. - dV + log(1 + dV))Z,

\a, 0,

< 0,

n 3 0,

v < 0.

12.2. Nonlocal existence We shall use the functional method of Leray-Schauder to prove the nonlocal existence of solutions of the differential equation (12.1.1).

DEFINITION 12.2.1. Let A and B be completely continuous operators defined for u E S , , where S , = [u E E : I u I p ] , with values in E and Au E E, Bu E E for u E S, . Then we say that A and B are homotopic if there exists an operator T(u,A), which is completely continuous on E x [0, 11 such that T(u,0) = Au and T(u, 1) = Bu for u E S, and T(u, A) # u for I u I = p. We need the following lemma of Leray-Schauder.




+

wheref(t, u ) is completely continuous for t E [to , to a], u E E ; (ii) g E C[to, to u] x R+n,Rn], g ( t , y ) is quasi-monotone nondecreasing in y for each t , and the maximal solution r(t, to ,yo ) of

+

+

Y’

= g(t,y),

Y@o)= Yo

exists on [to , to a ] ; (iii) @(u) = maxi V,(u) and @(u)-+ co as I u

I -+ co.

Then, for every uo E E, there exists at least one solution u(t) of the differential equation (12.1 .l) defined on [to , to u] such that u(to)= uo .

+

Proof. Let us consider the space B of all continuous functions u(t) with values lying in E, continuous on the interval [ t o ,to u]. Define

1u1

=

max,oGIGl,+a I u ( t ) / .Also, consider the operator

+

which maps B into itself. Denote this operator by T(u,A). Sincef(t, u ) is completely continuous, it follows that T(u,A) is completely continuous on B x [0, I]. Suppose now that u(t, A) is a solution of the equation ~ ( t= ) XU,

+X

f ( ~u(s)) , ds. t0

The last inequality implies that

12.3.

243

UNIQUENESS

choosing yo = V(Au,). Write K

max ri(t, t o , Vl(Au,), ..., Vn(Auo)).

=

i.d,t

Then, it follows that Vi(u(t,A))

< K,

t

+ a ] , 0 < X < 1. (12.2.2) < M for 0 < X < 1 and to < t < to + a.

E

[ t o , to

There is such an M that I u(t, A)l Indeed, if it were not so, then there would exist a sequence A, E [0, I] and a sequence t, E [ t o ,to u] such that 1 u(tn , A,)[ + co. Hence, @ ( u ( t nA,)) , -+a.This contradicts (12.2.2). Thus, 1 u(t, A)l M, 0 A 1 , t E [ t o , t, a]. We conclude that T(u,A) # u for 0 X 1 and 1 u 1 = M E, f > 0. It is easily seen that T(u, I) is homotopic to zero in the region I u 1 M E. By Lemma 12.2.1, we find that, in the space B , there exists at least one solution of the equation

+


t o , we have

4 ( c tl)

-

(6(t'; t l )

=

f-tcs, +(s; 5 ) )ds,

to

i

< tl < t ,

and hence, by letting t, -+ to , we get 4(t)

-

442)

=

Jt.f(.,

&)> ds-

This means that +(t)is a solution of (12.1.1) in D.Uniqueness of solutions is obvious, since, if +*(t)is such a solution, then

c-(4 4(t; tl), C * ( t ) )

< V(tl

-

= V(tl

as t, + t,, shows that

4(c tl) This completes the proof.

7

-

4*(tl)> 4*(tl)) 0

4(tl ; tl),

> +(tl)>

4*(9

Let ff be d I-lilbert space and f ( t , u ) be a continuous function on Q -- [ ( t ,zi): t,, ,t t, a , j u - uo 1 c] taking values in H . Suppose that


0, a constant 6 = S ( E ) > 0 such that, whenever v(t) is a step function in [0, To] with v(0) = uo(0) and I v(t) u,,(t)I ,-: 6 in [0, To],there is a neighborhood F = r ( E ) C A of A, for which h E r implies ~

12.5.

249

EXISTENCE

t E [0, To].Thus, we have

which will be used in the main theorem that follows, as in Theorem 3.20.1.

THEOREM 12.4.1. Suppose that (i) V E C[J x E , R+], V(t,0 ) = 0, V ( t ,u ) is positive definite and satisfies a Lipschitz condition in u for a constant 111 > 0; (ii) g E C[J x R, , R], g(t, 0 ) = 0, and r ( t ) -=0 is the maximal solution of Y’ = Y) passing through (0, 0); (iii) for any step function v(t) on J , with values in H and for every t E J, u ~ H , y e A , D+U(t,u, 4

< g(t, V ( t ,v ( t ) 4); -

(iv) the relation (12.4.2) holds. Then, given any compact interval [0, To]C J and any E > 0, there exists a neighborhood T ( E )of ho such that, for every h E T(E),(12.4.1) admits a unique solution u ( t ) with u(0) = uo(0),which is defined on [0, To] and satisfies

I 141) - u,(t)l < E ,

t

E

[O, To].

12.5. Existence (continued) Hereafter, we shall be concerned with the nonlinear evolution equation u’

=

A ( t ) u +f(4 u),

(12.5.1)

where A(t) is a family of densely defined closed linear operators on a Banach space E andf(t, u ) is a function on [ t o ,to a] x E taking values in E. First of all, we shall summarize some of the known results for the linear equation u’ = A(t)u + F ( t ) , (12.5.2)

+

where F ( t ) is a function on [to, to is unbounded.

+ u] taking values in E. Usually A(t)

250

12

CHAPTER

Let us make the standing assumptions that there exists an evolution operator U ( t ,s) associated with A(t).This means that ( U ( t ,s)} is a family of bounded linear operators from E into E defined fort, s t to a, strongly continuous in the two variables jointly and satisfying the conditions

< < < +

U(S,s)

U ( t ,s) U(s,r ) = U ( t ,r ) ,

au(t,s) u = A ( t ) U ( t ,s)u, 3s

I,

for u in a subset of E , specified in each case.

at

___ au(t9 s, 1L =

=

U(t,s) A(s)u

A function u(t) defined on [to,to

DEFINITION 12.5.1.

+ a] is said to be

a strict solution of (12.5.2) with the initial value uo , if u(t) is strongly continuous on [ t o ,to $- a ] , u(to) = uo , strongly continuously differentiable, and satisfies (12.5.2) on ( t o ,to a).

+

IfF(t) is continuous, any solution of (12.5.2) is of the form

DEFINITION 12.5.2. A function u ( t )is said to be a mildsolution of (12.5.2) with the initial value u,, if u(t) is continuous on [ t o ,to a] and satisfies (1 2.5.3).

+

12.5.3. T h e family (A(t)} of operators is said to be DEFINITION uniformly parabolic if (i) the spectrum of A ( t )is in a sector S,, /(A

= ~

[ z : 1 arg(z A(W’

I

- T)\

< w < n/2],

< Mil A I,

and I(A(t))-’

I

$ s, ,

< M,

where u and M are independent o f t ; (ii) for some h = n-l where n is a positive integer, the domain of A(t)”is independent of t , that is, D[A(t)”]= D and

I A(t)hA(s)-hI < M , I A(t)hA(s)-h- I I < M I t for t, s E [to, to

+ a], 1

-

h

0, there exists a positive number M such that If(4 41
0,

existing on [to , 00). Assume that, for each t E J and x E E, lim R(h, A(t))x = x

h-O+

and

I R(h7 4 t N x + h ? ( t ,41 2 I x I - hwz(t, I x I) for all sufficiently small h > 0 depending on t and x. Let x(t) be any solution of (12.6.1) such that 1 x(to)\ 3 po . Then, for all t for which p ( t ) 2 0, we have IX(t>l

3P(9

Proof. Defining m(t) = I x(t)I as before, it is easy to obtain the inequality lim inf h-l[m(t h-O+

+ h)

-

m ( t ) ] 2 -w(t, m(t)).

This is enough to prove the stated result using an argument essentially similar to that of Theorem 1.4.1. For various choices of w1and w 2 ,Theorems 12.6.1 and 12.6.2 extend many known results in ordinary differential equations to abstract differential equations. Suppose that w 1 = E and that x ( t ) is an E-approximate solution of (12.6.2). Let w 2 = Kzi, K > 0. Then, Theorem 12.6.1 gives an estimate of the norm of +approximate solution, namely,

I x(t)l '2 1 x(t,)/ eK-)

+ (c/K)(eK(t-Q

--

l),

t

2 to,

whereas Theorem 12.6.2 yields a lower estimate,

Again, suppose that w1Y 0 and that x(t) is a solution of (12.6.2) existing on [t, , 00). Let w2= A(t)g(u), where g(u) > 0 for u > 0 and

12.6.

APPROXIMATE SOLUTIONS AND UNIQUENESS

257

X E C [ J ,R ] . Then, we obtain the following upper and lower bounds of the norm of a solution, namely,

Jz,

where G(u) = [g(s)]-lds, u,, 3 0. If we suppose that wl = v(t)u, v(t) >, 0 is continuous on J , we have a variant of Theorem 12.6.1 which offers a sharper estimate.

THEOREM 12.6.3. Let the assumptions of Theorem 12.6.1 hold except that the condition (12.6.6) is replaced by

+

< I x I (1 - ah) + hw2(t, I x 1

I R(h, A(t))x hf(t, .)I where

01

> 0. Then (12.6.7) is replaced by &to)

Proof,

e--u(t-to), (12.6.10)

I x(t)l

e r(t),

t

3 to.

Let R(t) be the maximal solution of R'

= -&

+ w(t, Rp(t-to)), - E ( t - f o ) ,


B(t)

=0

t+o+

be satisfied, where the function B(t) is positive and continuous on

258

CHAPTER

12

< t < CO, with B(0) = 0. Let g ( t , u ) 3 0 be continuous on J x R+ . Suppose that the only solution u(t) of

0

on 0

u'

< t < co such that

= g ( t , 24)

is the trivial solution. Assume that, for each t E J , lim R(h, A(t))x = x

h-O+

for every x E E and that

1 R(h, A ( W A ( t ) ) y+ h [ f ( 44 -f(t,Y)lI 5: I x - y I -1hg(4 I x - y I) ~

wt

for each t E (0, GO), each x, y E E, and for all sufficiently small h > 0, depending on t and x. Then, there exists at most one solution of (12.6.2) on J .

Proof. Suppose that there are two solutions x(t) and y ( t ) of (12.6.2) on J , with the initial condition x(0) = y(0) = 0. Let m ( t ) = 1 x ( t ) - y(t)i. Then, m(0) = 0. Now, using an argument similar to that of Theorem 12.6.1, we obtain D+m(t) < g(t, W ) . From now on, we follow the proof of Theorem 2.2.8 with appropriate changes to complete the proof. 12.7. Chaplygin's method

By the one-parameter contraction semigroup of operators, we mean a one-parameter family { T(t)),t 3 0, of bounded operators acting from E to E , such that

+

(i) T ( t , t z ) = T(t,) T(t,) for t , , t, >, 0; (ii) 1irnjb->,,T(h)x = x for x E E ; (iii) I T(t)I 1 for t E J.


0, I R(h, A ) ] 1. It is well known that, if A is 1, then there exists a closed and densely defined and if I R(h, A)l unique contraction semigroup {T(t)}such that A is its infinitesimal generator. For x E D [ A ] ,the function x(t) = T(t)x satisfies the equation


l,

that is, the norm of the solution x(t) is a decreasing function. We observe that limb+, R(h, A ) x = x for every x E E, if A is closed, D ( A ) is dense in E, and limb,, sup/ R(h, A)I < 00. I n view of this fact and on the basis of Theorem 12.6.1, we can prove the following

THEOREM 12.7.1. Assume that


O; ( i i ) g E C [ J x R + , R + l , f ~ c [ JE ,xE l , a n d l f ( t , x ) l < g ( t , I x l ) f o r t E J and x E E ; (iii) r(t) is the maximal solution of u’ = g(t, u),u(t,) = u,, existing on J. Then, if x(t) is any solution of x’ =

such that I x(t,)l

Ax + f ( t , x)

< u,, existing on J,

we have

I x(t)l

t

< +>

(12.7.1)

>to.

We shall now prove a result that generates the Newtonian method of approximations in a version given by Chaplygin.

THEOREM 12.7.2. Suppose that (i) A is an infinitesimal generator of contraction semigroup;

260

CHAPTER

12

(ii) f(t, x) is FrCchet differentiable in x t o j , ( t , x) and

I f Z ( 4 Y)-

fZ(4

41 e gl(4 I Y

-2

0,

where g, E C [ J x R, , R,] and gl(t, u ) is nondecreasing in u for each t s J; (iii) the sequence of functions {xn(t)} such that I xn(t)1 M, t E [O, a ] , n = 0, I , 2,... satisfies


l -
+F ( t ) + [gi(t, r ( t ) ) + G(t)]u,

existing on [0, a ] . Setting g ( 4 u, 4 = 2g,(t, v)v

+ 2G(t)v + F ( t ) + [g,(t, v) + G(t)lu,

we see that all the assumptions of Theorem 1.4.4 are satisfied, and, as a result, R(t) = r ( t ) on [0, a]. T h e assertion of the theorem then follows from (12.7.8).

REMARK12.7.1. I t follows from the preceding theorem that, for a sequence of solutions xn(t) of xA+,(t) = Axn+1(t)

+f&

xn-l(t))[xn+l(t)

-

4 0 )=%EaAI,

the estimate

I x,(t)

-

holds, provided z(t) = xl(t).

< r(t),

t

E

[O,aI,

+fk x n w ,

264

CHAPTER

12

12.8. Asymptotic behavior

We shall now suppose that the norm in E is differentiable in the sense of Gateaux, namely, lim 1 A

1

+

4

h

-

1

'

=

(rx,h),

where r x = grad I x 1 and (1, x) is the value of the linear functional 1 E B*,the conjugate space of E, at an element of E. It is easy to check that acts from E into E* and that

r

DEFINITION 12.8.1.

Consider a function y ( t ) for which the estimate (%

holds for all t

E

J and x E D

494 < A t ) I x I =

D [ A ( t ) ] .Introduce the notation

Q~ = lim sup

t-1

t-m

and

Q

=

(12.8.1)

f

y(s) ds,

0

inf Q, ,

where the inf is taken over all functions y ( t ) . T h e number Q is called the central characteristic exponent.

THEOREM 12.8.1. Assume that (i) Q is the central characteristic exponent; (ii) f ( t , x) allows the estimate ( T x , f ( t ,x))

< 8 I x 1,

Then, given an E > 0, there exists a 6 of (12.6.2) admits an estimate lx(t)l

> 0.

(12.8.2)

> 0 such that

< I 4 ) l c exp[(Q + 2EY1,

where the constant C depends on

Proof.

6

t

any solution x(t)

2 0,

(12.8.3)

E.

Let m ( t ) = I x ( t ) i . Then, using (12.8.1) and (12.8.2), we have

12.8.

265

ASYMPTOTIC BEHAVIOR

so that

By Theorem 1.4.1, we get

From the definition of SZ, given E > 0 there exists a function y ( t ) such that (12.8.1) is satisfied and, at the same time, Qy < Q

I n other words,

+

€.

where C i s a constant depending on E . 'Then, taking 6 < E and considering the last inequality, we obtain from (12.8.4) the desired inequality (12.8.3), and the theorem is proved. As an application of Theorem 12.8.1, we prove the following theorem, which gives sufficient conditions for the asymptotic stability of the null solution of (12.6.2).

THEOREM 12.8.2. Suppose that (i) the central characteristic exponent 52 is negative; (ii) the functionf(t, x) satisfies

4) < P I x

(&f(t,

01

I1+Or,

> 0.

(12.8.5)

Then, the trivial solution of (12.6.2) is asymptotically stable. Proof. Choose a X function

> 0 such that 9, = 52 + X < 0, and consider the x ( t ) = exp( -At) y ( t ) -

Then where

Y"t>

=

[4t)

+ A11 Y ( t ) + A t , Y ( t > ) ,

(12.8.6)

g(t, Y ) = e x p ( A t ) f ( t , e.p(--ht)y).

It then follows from the properties of Y = Y(t),

r and

(123.5) that, setting

(TY, g(4 Y ) ) = eAt(rY,f(4 = e A t ( T ( e c n t y ) , f ( ecAty)) t,

< eAt/3 1 ecAty

J1+a,

266

CHAPTER

12

(12.8.7)

+

Hence, the central characteristic exponent of the operator A(t) h l is equal to Q, = 9 A, and, therefore, choosing an E > 0 such that Q, 2~ < 0, we can find a function y ( t ) that satisfies (12.8,l) and Q < 9, E < 0, because of the definition of 9. Let 6 > 0 be such that 6 0 so large that, for t to and small I y I, we get, by (12.8.7),

+

+

+

(T Y ,g(4 Y ) )

e 8 I Y I.

Thus, the operator g ( t , y ) verifies the hypotheses of Theorem 12.8.1, and hence

+

I Y(t)l < I Y(t0)l c exp[(Q,

+ 2+19

t

2 to .

Since Q, 2~ < 0, the null solution of (12.8.6) is asymptotically stable, and, as a result, the trivial solution of (12.6.2) is also asymptotically stable. T h e theorem is proved. Another set of conditions for the asymptotic stability is given by the following

THEOREM 12.8.3. Assume that (i) g E C[J x R, , R ] , and the solutions u(t) of the scalar differential equation u’

=

g(t, u),

u(to) = uo 3 0,

(12.8.8)

are bounded on [ t o ,a]; (ii) for each t E J , x E E, lim R(h, A(t))x = x

and

h-Of

I ~ ( h~ ,( t ) ) + x hf(t, x ) ~,< 1 x j ( I

where

O(

~

ah)

+ hg(t, 1 x I ea(t-tO))e--a(*--fJ,

> 0, for all sufficiently small h > 0 depending on

Then, the trivial solution of (12.6.2) is asymptotically stable.

t and x.

12.9.

Proof.

LYAPUNOV FUNCTION AND COMPARISON THEOREMS

267

Following the proof of Theorem 12.6.3, we obtain

I x(t)l

< r ( t ) exp[--or(t

-

to)],

(12.8.9)

t >, t o ,

where r ( t ) is the maximal solution of (12.8.8) and x(t) is any solution of (12.6.2). By assumption, r(t) is bounded on [ t o , 001. Hence, the asymptotic stability of the trivial solution of (12.6.2) is immediate from the estimate (12.8.9). T h e proof is complete.

12.9. Lyapunov function and comparison theorems We shall continue to consider the differential equation (12.6.2) under the same assumptions on the family of operators {A(t)}as in Sect. 12.6. Let us prove the following comparison theorems.

THEOREM 12.9.1. Assume that (i) V E C [ J x E, R,] and

I q t , .I>

-

q t ,4 1 < c ( t ) I 2 1 --

x2

(12.9.1)

I>

for t E J , x1 , x2 E E, c ( t ) 3 0 being a continuous function on J ; (ii) g E C[J x R, , R], r ( t ) is the maximal solution of the scalar differential equation u‘

= g(t, u),

existing on J , and, for t L)+l’(t, x)

=

u(tn) = zco

t J,

>, 0,

to

(12.9.2)

0,

x E E,

lim sup /z-l[V(t + h, R(h, A ( t ) ) x + /zf(f, x)) h-O+

s g ( t , r -(t,~y));

-

l’(t, x)]

(12.9.3)

(iii) for each t E J , lim,,,, R(h, A(t))x = x, x E E, and x ( t ) is any solution of (12.6.2) existing on [to, co) such that V ( t , , .%,(to))< ug .

Under these assumptions, we have V(t, x ( t ) )

< r(t),

t

2 to.

(12.9.4)

satisfying Proof. Let x(t) be any solution of (12.6.2) existing on [to, a), V ( t o ,x(to)) u,,. Consider the function


0,

m(t)

< c(t)[l

x(t

+ h ) - R(h, 4 t ) )x ( t )

-

hf(t, x(t))l]

+ v(t + h, R(h, 4 t ) )4 t ) + hf(4 x ( t ) ) )

~

because of (12.9.1). Since, for every x E D [ A (t)] ,

V(t,x ( t ) ) , ( 12.9.5)

R(h, A(t))[l- h A ( t ) ] x= x ,

it follows that

which, together with (12.93, implies that m(f

+4

-

m(t)

< c(t)[l x ( t + 4 - x ( t ) - h [ A ( t )x ( t ) +f(4 x(t))lll

+

h[l R(h, 4 t ) ) 4f) x(t)

-

A(t)4t)Il

+ L'(t + h, R(h, A ( t ) ) + hf(4 x ( t ) ) ) x(t)

-

V(t,x ( t ) ) .

Using the relations (12.6.2), (12.9.3), and assumption (iii), we obtain the inequality D+m(t)

< g(t, 40).

An application of Theorem 1.4.1 now yields the stated inequality (1 2.9.4), and the proof is complete.

THEOREM 12.9.2. Let the assumptions of Theorem 12.9.1 be satisfied except that the condition (12.9.3) be replaced by p ( t ) ntq44

+ "(t, ).

D+p(t)

< g(t, V ( t ,x ) p ( t ) ) ,

(12.9.6)

where p ( t ) > 0 is continuous on J . Then, whenever

~ ( t dV t n 1 4 t n ) ) G uo the inequality (12.9.4) takes the form p ( t ) qt,X ( t ) )

P ~ o o f . DefineL(t, x)

< r(t),

t

2 to *

p ( t ) V ( t ,x). Then, using (12.9.Q we have

1

12.10.

where

E +

269

STABILITY AND BOUNDEDNESS

0 as h + 0; a rearrangement of the right-hand side gives

It then follows that D+L(t,x)

=

lim sup h-l[L(t h a +

+ h, R(h, A(t))x + hf(t, x))

-

L(t, x)]

which implies that Theorem 12.9.2 can be reduced to Theorem 12.9.1 with L(t, x) in place of V ( t ,x). Hence we have the proof.

12.10. Stability and boundedness Let M be a nonempty subset of E containing {0}, and let d(x,M ) denote the distance between an element x E E and the set M . Denote the sets [x: d(x, M )< q] and [x: d(x, M ) 771 by S ( M , 7)and s ( M , q), respectively. Suppose that x ( t ) is any solution of (12.6.2) existing in the future. Then, we may formulate the various definitions of stability and boundedness with respect to the set M and the differential system (12.6.2) corresponding to the definitions (S,) to (Slo)and (B,) to (Bl,,) given in Chapter 3. As an example, (S,) would run as follows.


0, to E J , there exists a S = S ( t o ,E ) that is continuous in to for each E , such that x(t)

c S ( M , 4,

t

>, t o >

provided that x(to)E s ( M , 6). T h e following theorem gives sufficient conditions for stability.

THEOREM 12.10.1. Assume that (i) g E C [ J x R,

, R] and g(t, 0) = 0 ;

270

12

CHAPTER

(ii) V E C [ J x S ( M , p), I?+], V ( t ,x) is locally Lipschitzian in x, and, for ( t ,x) E J x S ( M , p),

)< V t , 4 d 4, 4 x 9 MI),

b(d(.x, W

where b E 2,a E C [ J x [0, p), R+],a E X for each t E J ; (iii) for ( t , x) E J x S ( M , p), D'Vt,

(iv) limh+O+ R(h, A(t))x

=

4 d g(4 vt,4);

x for t E J and x E E.

Then, the equistability of the. null solution of (10.2.1) implies the equistability of the set M with respect to the system (12.6.2).

P Y O O ~Let . 0 < E < p, to E J be given. Assume that the trivial solution of (10.2.1) is equistable. Then, given b ( ~ > ) 0, to E J , there exists a 6 = S ( t o , E ) that is continuous in to for each E such that u(t, t o , uo)

< b(4,

t

2 to,

(12.10.1)


, to . Suppose that this is not true. Then, there would exist a solution x(t) with .(to) E s ( M , S,) and a t, > to such that and

d(x(tl), M ) = E

so that b(6)

d(x(t),

<W

l

M ) < E,

t

E

, "W).

[to , tl],

(12.10.2)

Since this implies that x(t) E S ( M , p), t E [to , tl], the choice uo = V ( t o x(to)) , and condition (iii) yield, by Theorem 12.9.1, the estimate V ( t ,4 9

< r(t, to , uo),

t

E

[ t o , tll,

(12.10.3)

where r(t, t o , uo) is the maximal solution of (10.2.1). I t is easy to see that the relations (12.10.1), (12.10.2), and (12.10.3) lead us to the following contradiction: b(E)

d

qt1,

X(tl)>

d

9

t o , .o>

< b(4.

Hence, the stated result is true, proving the theorem.

12.10.

STABILI'PZ AND BOUNDEDNESS

27 1

THEOREM 12.10.2. Let the assumptions of Theorem 12.10.1 hold except that the function a(t, u ) in condition ii is independent of t, that is, a(t, u ) = a(u), where a E 37. Then, one of the stability conditions of the trivial solution of (10.2.1) implies the corresponding one of the stability conditions of the set M with respect to the system (12.6.2). On the basis of the proof of Theorem 12.10.1 and that of the parallel theorems in Chapter 3, the proof of Theorem 12.10.2 may be constructed easily. We therefore omit its proof. As an application of Theorem 12.9.2, we shall give a result that offers sufficient conditions for equi-asymptotic stability.

THEOREM 12.10.3. Let the hypotheses of Theorem 12.10.1 hold except that assumption (iii) is replaced by p ( t ) D + W ,).

+ v(4).

D+P(t) d g(t, V

t , 4P ( t ) ) ,

where p ( t ) > 0 is continuous on J and p ( t ) -+00 as t + CO. Then, the equistability of the trivial solution of (10.2.1) assures the equi-asymptotic stability of the set M with respect to the system (12.6.2). Proof. The proof of this theorem is analogous to the proof of Theorem 3.4.7, if we introduce the necessary changes. Finally, we state a theorem giving conditions for various boundedness notions, the proof of which may be constructed on the basis of the proof of Theorem 12.10.1 and the corresponding boundedness results of Chapter 3.

THEOREM 12.10.4. Assume that (i) Y ECCJ x E, R,], Y(t,x) is locally Lipschitzian in x, and, for (t, x) E J x E, &(x,

W )
, 0,

2 0,

u(0) = a:

and that zt"(t)

0

>, G(t, u(t)),

012

0,

< a,

(13.2.3)

then any solution of (13.2.1) for which

is regular for J z

1 < a.

Proof. Suppose that arg z integral equation

= 6' = const.

Then, from (13.24, we have the

where the integration is carried out along the ray 6'

=

const. Then,

(13.2.4) where s

=

I 5 1, t

1 z 1. Also, since u(t) satisfies (13.2.3), it

=

follows that

(13.2.5)

Setting m ( t ) 13.1.2, that

=

1 y(teis)Jfor fixed

8, it is easily checked, as in Theorem

Hence, in particular,

I i

1 m+'(O)\ < dY(0) dz

=

/3

0.

We consider the function u(t, 8) that satisfies (13.2.2) and u(0, S)

= a:,

u'(0, 6 ) = (6

+ s.

13.2. SINGULARITY-FREE REGIONS

It is easy to show that m(t) < u(t, 6 ) for 0 contrary, there exists a t, > 0 such that

28 1

< t < a.

Supposing the (13.2.6)

and (13.2.7)


1,

(1 3.2.9)

I < 1 if (Y-2,

E

> 0.

(13.2.10)

Proof. I n view of Theorem 13.2.1, it is sufficient to show that there exists a solution of

(13.2.11)

which is continuous on [0, 1). We consider whether (13.2.11) can have

282

CHAPTER

a solution of the form /?(I we must have

- t)-p,

p,

p > 0. If

(1

__ qz+nu--e

+

P b 1) -~

(1

qu+2

~

13

u p 1

such a solution exists,

'

that is to say, p(p

Since n

+ 1)

and

= u p 1

p = np

- c.

> 1, we obtain >0

E

p =

n-1

E(2+ 1) n-1

'

12-1

= up-1.

If /? is determined by the last equation, u(t) = p(1 - t)-* will be a solution of (13.2.11) for which u(0) and u'(0) are positive. T h e proof is complete. We shall next consider the complex differential system (13.1. I), wherefis regular in z , 0 1 z I .< a and entire in y E Cn.T h e following theorem gives an upper bound of the norm of solutions of (13.1.1) along each ray z : tei8.


=

t

< uo satisfies

3 0,

< 8 < 2n.

I Y(teze>l,

where y(x) is any solution of (13.1 . l ) such that I y(0)I

Define the vector

< uo . Proceeding

as in the proof of Theorem 13.1.2 with obvious modifications, it is

easy to obtain the differential inequality Dfnz(t)

< g(t, m ( t ) ) .

Corollary 1.7. I now assures the stated componentwise bounds.

13.3.

COMPONENTWISE BOUNDS

285

Analogous to Theorem 13.2.3, we can state a theorem for componentwise bounds which yields sharper bounds in some situations.

THEOREM 13.3.2. Let the condition (13.3.1) in Theorem 13.3.1 be

replaced by

IY

+ hf(z,Y>l < I Y I + hg(l

I, I Y I) -1 W )

for all small h > 0, where g E C[[O, a ) x R+" , Rn], and g ( t , u ) is quasimonotone nondecreasing in u for each t E [0,a), other assumptions remaining the same. Then, the conclusion of Theorem 13.3.1 is true. Instead of the complex differential inequality (13.1.2),we shall consider the system of inequalities (13.3.2)

where E is a positive vector. Definition 13.1.1 has to be slightly modified in an obvious way. Corresponding to Theorem 13.1.2, we have the following

THEOREM 13.3.3. Let g E C[[O, a ) x R+", R f n ] g, ( t , u ) be quasi-monotone nondecreasing in u for each t E [0, a), and r(t) be the maximal solution of the system u'

= g(t,

u)

+

E,

>0

u(0) = uo

existing on [0, u). Suppose further thatf(z, y ) is regular-analytic in D and

If yl(z, el) and y z ( z ,c2) are such that

el-

I YdO, €1) then we have, on each ray z

and Y ( Z 0 ) ) P(.o>l

then

I J+,

Y(Z))P(Z)l

< r(to>,

< r(t>,

z

I z o I = to

E D,

I2 I

( 13.4.8)

?

=

4

(13.4.9)

for all t 3 t o . We shall prove below Theorem 13.4.2, since Theorem 13.4.1 can be deduced from Theorem 13.4.2 by taking p(2:) = 1. We have stated Theorem 13.4.1 separately, as it is a basic comparison theorem by itself.

Proof of Theorem 13.4.2. Define

wherey(2) is any solution of (13.4.1) verifying (13.4.8). For each fixed 8, set nz(t) = I L(teie,y(te"))l.

Then, if h m(t

> 0 is sufficiently small,

+ h)

~

m(t)

.< I L((t + h)eie,y ( ( t + h)eiO))

-

L(teis,y(teie))l.

We can easily verify that

Also, dL(teie,y(teie))

I=)

dL(z,y(z))eie dz

288

CHAPTER

13

It therefore follows from the foregoing considerations that

< g ( 4 %(q,...,m,(l)). Now a straightforward application of Corollary 1.7.1 yields the desired inequality (13.4.9).

13.5. Notes T h e results of Sect. 13.1 are due to Deo and Lakshmikantham [l]. Theorem 13.2.1 and Corollary 13.2.1 are taken from the work of Das [l]. See also Das [4]. Theorem 13.2.2 is due to Wend [2], whereas Theorem 13.2.3 is new. T h e results of Sects. 13.3 and 13.4 are adapted from the work of Kayande and Lakshmikantham [l]. For further results, see Deo and Lakshmikantham [2] and Kayande and Lakshmikantham El], where stability and boundedness criteria are discussed.

Bibliography

AGMON,S., AND NIRENBERG, L. [l] Properties of solutions of ordinary differential equations in Banach spaces, Comm. Pure Appl. Math. 16 (1963), 121-239. ALEXIEWICZ, A., AND ORLICZ, W. [I] Some remarks on the existence and uniqueness of solutions of the hyperbolic equation a2z/ax ay = f (x, y , z, a z / a x , &lay), Studia Math. 15 (1956), 201-215. ANTOSIEWICZ, H. A. [l] Continuous parameter dependence and the method of averaging, Proc. Int. Symp. Nonlinear Oscillations, 2nd, Izd. Akad. Nauk. Ukrain, SSR, Kiev, 1963, pp. 51-58. ARONSON, D. G. [l] On the initial value problem for parabolic systems of differential equations, Bull. Amer. Math. SOC.65 (1959), 310-318. [2] Uniqueness of solutions of the initial value problem for parabolic systems of differential equations, /. Math. Mech. 11 (1962), 403-420. [3] Uniqueness of positive weak solutions of second order parabolic equations, Ann. Polon. Math. 16 (1965), 285-303. ARONSON, D. G., AND BESALA, P. [I] Uniqueness of solutions of the Cauchy problem for parabolic equations, Anal. Appl. 13 (1966), 5 16-526.

/. Math.

AZIZ,A. K. [l] A functional integral equation with applications to hyperbolic partial differential equations, Duke Math. J. 32 (1965), 579-592. [2] Periodic solutions of hyperbolic partial differential equations, Proc. Amer. Math. SOC.17 (1966), 557-566. AZIZ,A. K., AND DIAZ,J. [l] On a mixed boundary value problem for linear hyperbolic partial differential equations in two independent variables, Arch. Rational Mech. Anal. 10 (1962), 1-28. 289

290

BIBLIOGRAPHY

Aziz, A. I