An Introduction to Nonlinear Boundary Value Problems

An Introduction to Nonlinear Boundary Value Problems

This is Volume 109 in MATHEMATICS IN SCIENCE AND ENGINEERING A series of monographs and textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of books in this series is available from the Publisher upon request.

An Introduction to Nonlinear BoundaryValue Problems Stephen R. Bernfeld Department of Mathematics Memphis State University Memphis, Tennessee

V Lakshmtkantham Department of Mathematics University of Texas Arlington, Texas

Academic Press, Inc.

New York and London

I974

A Subsidiary of Harcourt Brace Jovanovich. Publishers

COPYRIGHT 0 1974, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

111 Fifth Avenw, New York. New York loo03

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON)LTD. 24/28 Oval Road, London NWl

Library of Congress Cataloging in Publication Data Bernfeld, Stephen R An introduction to nonlinear boundary value problems. (Mathematics in science and engineering, v. ) Bibliography: p. 1. Boundary value problems. 2. Nonlinear theories. I. Lakshmikantham,V., joint author. 11. Title. 111. Series. QA319.B41 515l.35 73-21996

ISBN 0-12-093150-8

Contents Preface . . . Acknowledgments

ChuprerI 1 .0. 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10. 1.11. 1.12. 1.13. 1.14. 1.15. Chapter 2

2.0. 2.1. 2.2. 2.3.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

MethodsInvolvingDifferentialInequalities . Introduction . . . . . . . . Existencein thesmall . . . . . Upper and Lower Solutions . . . TheModifiedFunction . . . . . Nagumo’s Condition . . . . . Existence in the Large . . . . . Lyapunov-Like Functions . . . Existence on Infinite Intervals . . Super-and Subfunctions . . . . Properties of Subfunctions . . . Perron’s Method . . . . . . . ModifiedVectorFunction . . . Nagumo’s Condition (Continued) . Existence in the Large for Systems. Further Results for Systems . . . Notesandcomments . . . . .

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ix

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. . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . Shooting Type Methods

Uniqueness Implies Existence . . . General Linear Boundary Conditions . Weaker Uniqueness Conditions . . . V

1 1 2 12 18 25 31 39 44 46 52 62 69 74 81 84 93 94 94

. . . . . . . . 94 . . . . . . . . 101 . . . . . . . . 109

CONTENTS

2.4. 2.5. 2.6. 2.7. 2.8. 2.9. 2.10. 2.1 1. 2.12.

Chapter 3 3.0. 3.1. 3.2. 3.3. 3.4. 3.5. 3.6.

Chapter 4 4.0. 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9. 4.10. 4.1 1. 4.12. 4.13. 4.14. 4.15. 4.16. 4.17.

Nonlinear Boundary Conditions . . . Angular Function Technique. . . . . Fundamental Lemmas . . . . . . . Existence . . . . . . . . . . . Uniqueness . . . . . . . . . . Estimation of Number of Solutions . . Existence of Infinite Number of Solutions Nonlinear Boundary Conditions . . . Notes and Comments . . . . . . . Topological Methods

. . . . . . . 113 . . . . . . . 116

. . . . . . . 117 . . . . . . . 121 . . . . . . . 127

. . . . . . . 136

. . . . . . . 142 . . . . . . . 145 . . . . . . . 152

. . . . . . . . . . . . . .

Introduction . . . . . . . . . . . Solution Funnels . . . . . . . . . Application to Second-Order Equations . . Wazewski Retract Method . . . . . . Generalized Differential Equations . . . Dependence ofSolutions on Boundary Data Notes and Comments . . . . . . . . Functional Analytic Methods

. . . . . . 153 . . . . . . 153

. . . . . . 160 . . . . . 168 . . . . . 175

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. . . . . 186 . . . . . 195

. . . . . . . . . . . 197

Introduction . . . . . . . . . . . . Linear Problems for Linear Systems . . . . Linear Problems for Nonlinear Systems . . . Interpolation Problems . . . . . . . . Further Nonlinear Problems . . . . . . . Generalized Spaces . . . . . . . . . . Integral Equations . . . . . . . . . . Application to Existence and Uniqueness . . Method of A Priori Estimates . . . . . . Bounds for Solutions in Admissible Subspaces . b r a y Schauder’s Alternative . . . . . . Application of LeraySchauder’s Alternative . Periodic Boundary Conditions . . . . . . Set-Valued Mappings and Functional Equations General Linear Problems . . . . . . . . General Results for Set-Valued Mappings . . Set-Valued Differential Equations . . . . . Notes and Comments . . . . . . . . . vi

153

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. . . . 198 . . . . 205

. . . . 208 . . . . 213 . . . . 225

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. . . . . 228

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. 232 . 248 . 256 . 263 . 265 . 269 . 278 . 282 . 289 . 295 . 302

CONTENTS

. . . . . 304 Introduction . . . . . . . . . . . . . . . . . 304 Existence in the Small . . . . . . . . . . . . . . 304

Chapter5

Extensions to Functional Differential Equations

5.0. 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8.

Existence in the Large . . . . . . . . . Shooting Method . . . . . . . . . . Nonhomogeneous Linear Boundary Conditions Linear Problems . . . . . . . . . . . Nonlinear Problems. . . . . . . . . . Degenerate Cases . . . . . . . . . . Notes and Comments . . . . . . . . .

Chapter 6 6.0. 6.1. 6.2. 6.3. 6.4. 6.5. 6.6.

. . . . . .

. . . . . .

. . . . . .

. 312 . 315 . 320 . 324 . 330 . 335

. . . . . . . . . . . . . . . . 337 Introduction . . . . . . . . . . . . . . . . . 337 Newton's Method . . . . . . . . . . . . . . . 337

Selected Topics

The Goodman-Lance Method . . . . . . . The Method of Quasilinearization . . . . . . Nonlinear Eigenvalue Problems . . . . . . . n-Parameter Families and Interpolation Problems . Notes and Comments . . . . . . . . . .

Bibliography . . . . Additional Bibliography Index

. . . . . 308

. . . . . .

. . . . 344 . . . . 349 . . . . 353

. . . . 358

. . . . 367

. . . . . . . . . . . . . . . 368 . . . . . . . . . . . . . . . 382

. . . . . . . . . . . . . . . . . . . . . .

vii

385

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'Ihe theory of nonlinear boundary value problems is an extremely important and interesting area of research in differential equations. Due to the entirely different nature of the underlying physical processes, its study is substantially more difficult than that of initial value problems and consequently belongs to a third course in differential equations. Although this sophisticated branch of research has, in recent years, developed significantly, the available books are either more elementary in nature, for example the book by Baily, Shampine, and Waltman, or directed to a particular method of importance, such as that by Bellman and Kalaba. Hence it is felt that a book on an advanced level that exposes the reader to this fascinating field of differential equations and provides a ready access to an up-to-date state of this art is of immense value. With this as motivation, we present in our book a variety of techniques that are employed in the theory of nonlinear boundary value problems. For example, we discuss the following: (i) methods that involve differentid inequalities; (ii) shooting and angular function techniques; (iii) functional analytic approaches; (iv) topological methods. We have also included a chapter on nonlinear boundary value problems for functional differential equations and a chapter covering special topics of interest. The main features of the book are (i) a coverage of a portion of the material from the contribution of Russian mathematics of which the English speaking world is not well aware; (ii) the use of several Lyapunov-like functions and differential inequalities in a fruitful way; (iii) the inclusion of many examples and problems to help the reader develop an expertise in the field. This book is an outgrowth of a seminar course given by the authors. We ix

PREFACE

have assumed the reader is familiar with the fundamental theory of ordinary differential equations, including the theory of differential inequalities, as well as the basic theory of real and functional analysis. It is designed to serve as a textbook for an advanced course and as a research monograph. It is therefore useful to the specialist and the nonspecialist alike. The reader who is familiar with the contents of the book, it is hoped, is fully equipped to contribute to the area.

X

Acknowledgments We wish to express our warmest thanks to Professor Richard Bellman whose interest and enthusiastic support made this work possible. We are immensely pleased that our book appears in his series. m e staff of Academic Press has been most helpfil. We thank our colleagues who participated in the seminar on boundary value problems at the University of Rhode Island in 1971-1972. In particular, we appreciate the comments and criticism of Professors E. Roxin, R. Driver, and M. Berman. Moreover, we gratefilly acknowledge several helpfil suggestions offered by Professor L. Jackson. We are very much indebted to Professors G.S. Lad& and S. Leela for their enthusiastic support in many stages of the development of this monograph and to Mr. T. K. Teng for his careful proofreading. Moreover, we wish to thank Mrs. Rosalind Shumate and Mr. Sreekantham for their excellent typing of the manuscript, and we wish to express our appreciation to Ms.Elaine Barth for her superb vping of the final copy. m e first-mentioned author wouM like to acknowledge some interesting helpfirl discussions on boundary value problems with the differential equation 3 group at the University of Missouri at Columbia. Finally, the final preparation of this book was facilitated by a National Science Foundation Gmnt GP-3 7838.

xi

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Chapter 1 METHODS INVOLVING DIFFERENTIAL INEQUALITIES

1.0

INTRODUCTION A variety of techniques a r e employed i n the theory of

nonlinear boundary value problems.

!Chis chapter i s primarily

concerned with the methods involving d i f f e r e n t i a l inequalities. The basic idea i s t o modify the given boundary value problem suitably, and then t o use the theory of d i f f e r e n t i a l inequalities and the existence theorems i n the small t o establish the desired existence r e s u l t s i n the large. After presenting needed existence theorems i n the small, we f i r s t concentrate on scalar second-order d i f f e r e n t i a l equations and associated boundary value problems.

We then

introduce upper and lower solutions, discuss the modification technique, and u t i l i z e Nagumo's condition t o obtain a p r i o r i bounds on solutions and t h e i r derivatives.

Once we have these

bounds a t our disposal, t o prove existence theorems on f i n i t e o r i n f i n i t e intervals i s r e l a t i v e l y simple and straightforward. Boundary value problems subjected t o nonlinear boundary conditions as well are treated i n t h i s framework.

We then

develop Q,rapunov-like theory f o r boundary value problems employing several w a p o v - l i k e functions and the theory of d i f f e r e n t i a l inequalities i n a f r u i t f u l way.

We also t r e a t

i n d e t a i l Perron's method of proving existence i n the large 1

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

by u t i l i z i n g the properties of sub-

and superf'unctions and

the existence r e s u l t s i n the small.

!Ibis technique works well

f o r scalar equations. We next extend the r e s u l t s considered f o r scalar equations t o 8 f i n i t e system of second-order d i f f e r e n t i a l equations. Here there are two directions t o follow, t h a t is, e i t h e r t r y t o obtain the required bounds componentwise o r i n terms of a convenient norm. We o f f e r r e s u l t s from both points of view indicating t h e i r r e l a t i v e merits and using I&-apunov-like theory, whenever possible, t o derive general r e s u l t s . 1.1 EXISTENCE IN l E 3 SMALL

denote t h e r e a l n-dimensional, Euclidean space

Rn

Let

and f o r x

E

Rn,

16(11

let

denote any convenient norm of

x.

be the i n t e r v a l

[a,b]. We s h a l l mean by C(n)[A,B] the class of n-times continuously differentiable functions Let

J

from a s e t A

into

8.

set

B.

We w i l l be concerned, i n t h i s section, w i t h the existence of solutions of the second-order d i f f e r e n t i a l equations of the form (1.1.1)

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

satisfying the boundary conditions (1.1.2) where

n n

f E C[JXRnXR ,R

3.

For the purposes of t h i s chapter,

we also need an existence r e s u l t under more general boundary conditions.

This we do consider f o r the s c a l a r case, l e a v i a

a thorough discussion of the general theory t o a l a t e r chapter. F i r s t of a l l , we observe t h a t the only solution of (1.1.3)

XI1 =

0,

subject t o the boundary conditions 2

1.1. EXISTENCE IN THE SMALL

i s the t r i v i a l solution.

This implies, from the theory of

linear d i f f e r e n t i a l equations, that there e x i s t s a unique solution of

(1.1.5)

h(t),

X" =

satisfying (1.1.4) f o r each h

E

C[J,Rn].

Moreover, since the

problem (1.1.3), (1.1.4) possesses the two l i n e a r l y independent solutions u ( t ) = (t - t l ) J v ( t ) = ( t 2 -t ) , the method of variation o f parameters readily gives the i n t e g r a l equation (1.1.6)

x(t) =

+

-

(t2- t ) (S tl)h(S)

jr2(t -

tl)(t2

- s ) h ( s ) ds

dS

I

for the solution x ( t ) of (1.1.5) subject t o (1.1.4). Relation (1.1.6) can be written i n the familiar form

"1 where

i s usually referred t o as the Green's f'unction f o r the boundary value problem i n question. Hence

%is function G(t,s)

the solution of (1.1.5) verifying conditions (1.1.2) takes the form

3

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

(1.1.8)

x(t) =

rt2

G(t,s)h(s) ds

+ w(t),

w(t2) = x where w”(t) = 0 and w(t,) = x 1’ 2‘ follows that if x ( t ) is a solution of (1.1.1), (1-1-9)x ( t )

=lt2

G(t,s)f(S,x(s),

~ ‘ ( 8 ) )ds

It therefore

(1.1.2),

then

+ w(t).

Conversely, i f x ( t ) i s a solution of (1.1.9), we can verify by differentiation of (1.1.6) that x ( t ) s a t i s f i e s ( l . l . l ) , (1.1.2). L e t us next r e c a l l some properties of the function G(t,s) f o r later use. For a fixed t, the maximum of IG(t,s)l i s attained a t s = t and IG(t,t)l has i t s maximum value at t = (t1+t2)/2, that i s ,

(1.1. lo)

Iw,~)I

5 ( t 2 - tl)/J+

Furthermore,

and consequently

Wreover,

-

ds = ( ( t tl)

2

+ (t2 - t ) 2 ) / 2 ( t 2

- tl) ,

the maximum of which is attained a t t = t l and t = t 2 . Hence, we obtain 4

1.1. EXISTENCE IN THE SMALL

We are now ready t o prove an existence and uniqueness r e s u l t by using t h e contraction mapping theorem,

where

K, L

>0

are constants such t h a t

Then the boundary value problem ( l . l . l ) ,

(1.1.2) has a unique

solution. Proof: u E CT[[tl,t2],

Let

B

Rn]

Define the operator Tu(t)

be the Banach space of functions with t h e norm

T: B + B

=Lt2

by

G ( t , s ) f ( s , u ( s ) , u t ( s ) ) ds

1

We then have, by (1.l.l.l)and (1.1.13),

Also, because of (l.l.l2), 5

+ w(t).

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

It then follows t h a t

ThisJ i n view of assumption (1.1.14)' shows t h a t

T

is a

contraction mapping and thus has a unique fixed point which i s t h e solution of t h e problem (l.l.l)J (1.1.2).

m e proof i s

complete.

An i n t e r e s t i n g problem i s t o f i n d the l a r g e s t possible i n t e r v a l i n which t h e preceding theorem i s v a l i d . f E C[JXRXR,R],

when

result.

I n t h e case

one can o f f e r such a best possible

We have i n t e n t i o n a l l y given such a r e s u l t i n t h e

following exercise with generous h i n t s . EXERCISE 1.1.1. Assume t h a t

s a t i s f i e s (1.1.13).

Let

(1.1.15)

U"

which vanishes a t number such t h a t

t

u(t)

and

be any solution of

+Lu' + K u = 0

t

and l e t a(L,K) be t h e f i r s t unique 1 u l ( t ) = O f o r t = t l + a ( L J K ) . Show t h a t t h e =

boundary value problem (1.1.1)' if

f E C[JxRXR,R]

(1.1.2) has a unique solution

t 2 - t l < 2a(LJK) and t h a t t h i s r e s u l t i s best possible. Hints:

Step 1. F i r s t show t h a t t h e r e i s a unique solution

t o t h e boundary value problem x 1 ( t 3 )=

3

if

XI'

-

(t3 tl) < a(L,K).

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

This can be shown by apply-

ing the contraction mapping theorem r e l a t i v e t o the M a c h space

E = C(1)[[tl,t3],R]

with t h e norm 6

1 . 1 . EXISTENCE IN THE SMALL I-

where u o ( t )

>0

i s a solution of

for

Lu' +Ku =

+

UI'

cy

s u f f i c i e n t l y close t o 1.

cy

Step 2.

Show t h a t the existence of unique solutions of

(l.l,l),(1.1.2) and of (1.1.1)with e i t h e r

x ( t l ) = x 1' ) = x 3 ' x ( t 2 ) = x 2 on any i n t e r v a l of

x'(t ) = x or x ( 3 3 ' length l e s s than

implies the existence of a unique solution

d

of (1.1.1)' (1.1.2) on any i n t e r v a l of length l e s s than Step 3. show t h a t ing

2d.

To prove t h a t the r e s u l t i s the best possible

u" + Llu'

I + Ku = 0

u(t,) = u ( t 2 )= 0,

where

has a n o n t r i v i a l solution verif'y-

t2- t l =2CY(L,K).

Since u ( t ) = O

also s a t i s f i e s the problem, argue t h a t the r e s u l t i s best possible.

Observe t h a t

EXERCISE 1.1.2.

i n Theorem 1.1.1f o r where N

m

a(L,K)

can be e x p l i c i t l y computed.

Show t h a t it i s s u f f i c i e n t t o define

t

E

/(XI( -< N,

[tl,t21y

IIx' 11 < 4N/(t2

f

- tl),

satisfies either

2 ( t 2 tl>

-a

-< N [ l - ( K

Hint: Apply hll05 N where

2 ( t 2 tl>

-a

t h e contraction mapping theorem on the b a l l

7

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

To obtain merely an existence result we employ Schauder's

fixed point theorem as is usual.

THEOREM 1.1.2. l e t , f o r t E J, IlxII where

f E

I& M > 0, N

ItYII ,< PJ,

5 m,

C[JFRnXRn ,Rn 1.

>0

be given numbers and

Ilf(t,x,Y)II

5

q,

Suppose that

6 = 1nin[(8M/q)~, 2N/ql. Then any problem (1.1,1), (1.1.2) such that [t1,t21 J, t2 tl I 6, Ilx1II 5 M, I ~ x, I 5 M, and Ilx, x211/(t2 t l ) I N, has a solution. Furthermore, given any E > 0, there i s a solution x ( t ) such that

-

-

-

llx(t) -W(t)ll < E, Ilx*(t) -wl(t)ll t2 tl i s sufficiently small.

-

is a closed, convex subset o f

bY

ICx(t)

=

< E on [tl,t21, provided

B. Define the mapping T: B+B

1"'

G(t,s) f (s,x(s),xl (s)) ds + w(t).

Using now estimates (l.l.ll), (l.l.E!),

8

we obtain

1.l.EXISTENCE IN

THE SMALL

and

t2- t l 5 6, T maps

Hence, f o r

II(W"(t)Il

5

into i t s e l f .

Bo

Ilf(t,x(t), x ' ( t ) ) l l

by Ascoli's theorem it follows t h a t

T

5

Also, since

q,

i s completely continuous.

Schauder's fixed point theorem therefore assures t h a t fixed point i n If

x(t)

which i s a solution of ( l . l . l ) ,

Bo,

x

IIx(t) - w ( t )

t

E

(1.1.2).

i s a solution of t h e boundary value problem

(l.l.l), (1.1.2), with

for

T has a

[tl,t2].

E

Bo,

then we have

-

II 5

( ( t 2 tl12/8) 9,

Consequently t h e last assertion follows and

the proof of Theorem 1.1.2 i s complete. COROLLARY 1.1.1. Assume t h a t

and i s bounded on

[tl,t2] XRn xRn.

f E C[[t,,t2]

x R n xRn,Rn]

Then every boundary value

problem (l.l.l), (1.1.2) has a solution. Proof:

where

q

Pick M

>0

s u f f i c i e n t l y large so t h a t

i s an upper bound of

f.

The conclusion then follows

from Theorem 1.1.2.

As was indicated e a r l i e r , we s h a l l next discuss the 9

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

s i t u a t i o n where f E C [J XR x R , R ] . Then a natural question is under what general boundary conditions can one construct a Green's m c t i o n f o r the scalar equation (1.1.3). L e t us look a t t h e boundary conditions

- a2x1 (t,)

(1.1.16)

alx(tl)

(1.1.17)

b x ( t 2 ) +b2X1( t 2 ) = 0 .

= 0,

1

A s long as there e x i s t two l i n e a r l y independent solutions of

(1.1.3) satisfying (1.1.16), ( l . l . l 7 ) , it is possible t o cons t r u c t the desired Green's f'unction. One sufficient condition i s t o f i n d two nonparallel l i n e s satisfying (1.1.16) and (1.1.17), t h a t i s , demanding t h a t alb2 + a2bl

(1.1.18)

#

0.

Another s u f f i c i e n t condition, for example, i s t o require t h a t a2b2 = 0 . I n view of these remarks, it is s u f f i c i e n t t o assume that a1+ b1 > 0,

2 0,

(1.1.19) al,a2,bl,b2

and

a2+b2

> 0.

L e t us now consider the nonhomogeneous boundary conditions (1.1.20)

alx(tl)

-

(1.1.21)

blx(t2)

+ bF'(t2)

a2x1(tl) = A, = B.

It i s not difficult; t o conclude the existence of a solution cp(t) satisfying (1.1.3) and (1.1.20), (1.1.21). Thus any

solution x ( t ) of (1.1.1)obeying the boundary conditions (1.1.20) and (1.1.21), takes the form (1.1.22)

x(t)

Go(t,s)f(s,x(s>,x(s>)

= Jt2

1 where 10

+ T'(t),

1.1. EXISTENCE IN THE SMALL

and

- v(t)u'(t).

c = u(t)v'(t)

Here we assume u ( t ) , v ( t )

are two l i n e a r l y independent solutions o f (1.1.3)which s a t i s f y

(1.1.16), (1.1.17). This discussion leads t o t h e following result. TEEOREM

1.1.3.

[tl,t2] X R X R . (1.1.20),

f E C [[t,,t2]

Let

X R xR,R],

bounded on

Then t h e boundary value problem (l.l.l),

(1.1.21) has a solution, whenever (1.1.18) o r (1.1.19)

holds. Proof:

Let

Define a mapping

M be the bound of T: E + E

where the Banach space

f

on

[tl,t2] x R X R .

by

E = C ( l ) [[tl,t2],R]

Letting

it follows t h a t 11

with t h e norm

1. METHODS INVOLVING DIFFERENTIAL INEQUALITiEs

Hence,

T maps the closed, bounded, and convex s e t

Bo = [X E E: ( x ( t )I 5 NM+L,

Ix'

(t) I

5 NIM+L1]

into i t s e l f . mreover, since I ( ~ ) " l < M, T i s completely continuous by Ascoli's theorem. The Schauder's fixed point theorem then yields the fixed point of T w h i c h i s a solution of (l.l.l), (1.1.20), (1.1.21), thus completing the proof of' the theorem. EXEXCISE 1.1.3.

-

Find the solution of the boundary value

problem x" = t, x(0) x' (0) = 1 and x(1) = 0. 1.2 UPPER AND lxlwER SOLUTIONS Let us consider the second-order equation

(1.2.1)

x" = f(t,X,X'),

where f E C[JxRxR,R], J being the i n t e r v a l [a,b] as before. The i n t e r i o r of J w i l l be denoted by J 0 L e t us define certain types of solutions of d i f f e r e n t i a l inequalities t h a t w i l l play a prominent part i n the subsequent work.

.

DEFINIITION 1.2.1. i s said t o be

A f'unction

cy E

C[J,Rl

(i) a lower solution of (1.2.1) on

-

D "'(t) =lim inf

h+ 0

(ii)

TI

a' ( t + h ) -a'( t - h ) 2h

J

h-t 0

-

C(l)[Jo,R]

if

,> f ( t , a ( t ) , a ' ( t ) ) ,

upper solution of (1.2.1) on J

D-a'(t) slim sup

n

if

a t ( t + h ) - a t ( t h) < f(tp(t),a'(t)), 2h

-

A f'undamental r e s u l t concerning the upper and lower 12

t cJo;

0

t EJ

1.2. UPPER AND LOWER SOLUTIONS

solutions i s t h e following.

THEOmM 1.2.1. (i) f

in x

C[JxRxR,R]

E

f o r each (ii)

Assume t h a t and

(t,y) E J x R ;

i s a lower solution and

cy

i s nondecreasing

f(t,x,y)

i s an upper solution

@

of (1.2.1) on J ; (iii)

@ ( a )5 @ ( a ) and cy(b)

5 B(b).

If one of t h e d i f f e r e n t i a l i n e q u a l i t i e s involved i s s t r i c t ,

then we have Proof: some t E J that

m(t)

sequently,

a(t)

0

< B(t) on J 0

.

Suppose, on t h e contrary, t h a t

.

@ ( t )2 B(t)

Then, s e t t i n g m(t) = a ( * ) - B ( t ) ,

m(to)

> 0,

m ' ( t o ) = 0,

we notice

to E J

has a nonnegative maximum q t some

0

D-mt(tO) 5 0.

and

for

.

ConHence

Thus

i n view of t h e monotonic character of

f.

!this i s a contra-

diction since one o f t h e d i f f e r e n t i a l i n e q u a l i t i e s i s assumed t o be s t r i c t .

The proof i s therefore complete.

We observe t h a t the proof of Theorem 1 . 2 . 1 breaks down i f one of t h e d i f f e r e n t i a l i n e q u a l i t i e s i s not assumed s t r i c t . Nonetheless, t h e following assertion i s t r u e , DIEOREM 1.2.2. 1 . 2 . 1 hold.

Assume -theses

Suppose f'urther t h a t 13

( i ) - ( i i i ) o f Theorem f(t,x,y)

obeys a one-sided

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

Lipschitz condition i n y

on each compact subset of

J X R xR,

that i s ,

Then the inequality a ( t ) _< p ( t ) on J, Proof:

Suppose t h a t

2& = m a x [ a ( t ) - p ( t ) ]

a(.)

- P ( c ) = E,

a ( % )> p ( t )

on J .

U(d)

- B(d)=

f o r some t

[c,d]

Let

a(t)

E,

i s true.

C

E

J

0

.

Let

be such t h a t

J

- p(t) 2

t

E,

E

[C,dI

-

to E (c,d) such t h a t a ( t o ) p ( t o ) = 2 E . Consider the compact subset given by

and there e x i s t s a

s

=

[(t,x,y): t

For any

6

E

[c,dl,

IPW-xl

5

1, I B ' W - Y I

_< 11.

> 0 s u f f i c i e n t l y small, l e t p ( t ) s a t i s f y the

conditions p" = ( L + l ) p ' ,

-min(6,E)

< p ' ( t ) _< 1,

0

Set m(t) = p ( t ) p'(t)

- p(t).

2 m'(t), t

-< p ( t ) _< 0 ,

E

Note t h a t

(c,d).

t

E

monotonicity of

E

[c,dl,

-> P ( t ) ,

t

E

(c,d).

m(t)

Moreover, f o r

using successively the assumptions on If we now l e t

t

v ( t ) = m ( t ) + E,

t

E

[c,d],

and

(c,d),

f.

we see, because of the

f,

D'v'(t) =D'm'(t) < f ( t , m ( t ) , m ' ( t ) ) _ < f ( t , v ( t ) , v ' ( t ) ) , Also, 14

t

E

(c,d).

1.2. UPPER AND LOWER SOLUTIONS

+

E

-< m(c) +

E =

v(c),

a ( d ) = @ ( d )+

E

< m(d) +

& =

v(d).

a(.)

= B(c)

An a p p l i c a t i o n of Theorem 1 . 2 . 1 y i e l d s t h a t a(t)

< v(t),

t

E

(c,d).

we o b t a i n

From t h e d e f i n i t i o n of m(t),

B(t) _ < m ( t ) and, as a r e s u l t , we o b t a i n a t

B(t) +

E

t=t

0'

This contradiction proves t h e theorem.

We can deduce a uniqueness r e s u l t f o r a c e r t a i n boundary value problem from Theorem 1.2.2. COROLLARY 1.2.1.

L e t t h e assumptions of Theorem 1.2.2

hold except ( i i ) and ( i i i ) . are s o l u t i o n s on x(t2) = y(t,).

[tl,t2]

with respect t o

y

i s no longer v a l i d .

x"=($)

C

Then x ( t )

REMARK 1.2.1.

This we s t a t e as a c o r o l l a r y .

Suppose t h a t

x,y

such t h a t

x(t,)

J

E

y(t)

on

E

C(2)[[t,,t2],R]

= y(tl),

[tl,t21.

If t h e Lipschitz condition on f ( t , x , y ) on compact sets i s omitted, Corollary 1 . 2 . 1 For example, t h e boundary value problem

(: 5

has s o l u t i o n s x ( t )

1 and

x ( t ) = lt15/2 on

[-1,11.

Instead o f the one-sided Lipschitz condition on f, one could assume t h a t s o l u t i o n s of i n i t i a l value problems XI'

= f(t,x,x')

Theorem 1.2.2.

a r e unique, t o conclude t h e v a l i d i t y of This i s p r e c i s e l y what t h e next theorem does.

THEOREM 1.2.3.

Suppose t h a t 15

I . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

(i) in

x

f

C[JXRXR,R]

E

f o r each (ii)

(tJy)

E

and

f(tJx,y)

i s a lower solution and

CY

i s nondecreasing

JxR; i s an upper solution

@

of (1.2.1) on (iii) are unique;

[c,d] C J; solutions of i n i t i a l value problems f o r (1.2.1)

(iv) a ( c >< B(c)

a(t)

Then we have

and CY(d-1 5 @ ( a ) .

5 p(t)

on

[CJd].

The proof of t h i s theorem r e s t s on a r e s u l t t h a t we are going t o consider i n t h e following section and therefore w i l l be given t h e r e . COROLLARY 1.2.2.

If

f(t,x,x')

s a t i s f i e s hypotheses ( i )

and ( i i i ) of Theorem 1.2.3, and i f solutions on

[tl,t2]

x ( t 2 ) = y ( t 2 ) J then

C

J

x,y

such t h a t

E

C(2)[[tl,t2]JR]

are

x(tl)=y(tl),

x ( t ) = y ( t ) on

[tl,t21.

Another i n t e r e s t i n g r e s u l t concerning lower and upper solutions which would y i e l d uniqueness of solutions with general l i n e a r boundary conditions i s t h e following. THEOREM 1.2.4.

(i)

f E C[JxRxR,R]

creasing i n (ii)

Assume t h a t

x f o r each CY

and

(t,y)

E

i s s t r i c t l y in-

f(t,x,y) JxR;

i s a lower solution and

@

i s an upper solution

of (1.2.1) on J; (iii)

f o r each

(t,x,yl),

If(tJx,yl)-f(tJxJy2)I (iv)

a,a(a>

(t,x,y2) E J X R X R j

5

LIY1-Y21J

- a2a' ( a ) 5 al@(a) - a2@'(81,

bl,(b)+b2ct'(b) 5 bl@(b)+b2@'(b), where b +b2 1

> 0,

>O;

a1,a2,blJb2

20,

and

16

a +bl 1

a 1+a2 > 0 ,

>o.

1.2. UPPER AND LOWER SOLUTIONS

Then a ( t ) Proof: -

5

p(t)

on J .

We s h a l l f i r s t show t h a t at the end points of

J

the desired inequality holds. and hence

that

> B(a) which implies from ( i v ) t h a t a2 # 0 a' (a) > p ' ( a ) . Then t h e r e e x i s t s a 6 > 0 such > p ( t ) on [a,6). Consequently, using t h e s t r i c t

.(a)

Let

a(t)

monotony of

and condition ( i i i ) , we obtain

f

D

-

0'

- D-p'

(t)

(t)

-

_> L [ a ' ( t ) B ' ( t ) ] ,

which, by t h e theory of d i f f e r e n t i a l inequalities, y i e l d s a' ( t )

- B ' ( t ) 2 [a'( a ) - B ' ( a ) le L(t-a)

This, together with a ' ( t ) U'(t)

p'(t)

on

[a,6].

- p ' ( t ) -< 0,

assures us t h a t

I n view of (1.2.2), we then a r r i v e

a t the contradiction

-

D-a' ( t ) D-B' ( t ) > 0 which establishes t h e claim

to E [a,s).

(to)

-

- Bf(t) > 0

argument may be extended t o show t h a t a(b)

>0

f o r some

Again, we appeal t o t h e d i f f e r e n t i a l inequality

(1.2.2) t o conclude a ' ( t ) that

(to)

> p(b).

on 6= b

[t0,6].

This

which implies

The l a s t conclusion, i n i t s turn, leads

t o the contradiction a l ( b )

-< p ' ( b ) 17

because of t h e second

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

inequality i n ( i v ) . lation

a(a)

>

p(a)

i s impossible.

a ( t ) 5 B(t)

a ( & ) > p(a)

Thus

a(b)

If we assume t h a t

>

i s impossible.

p(b), we a r e lead t o t h e re-

arguing as before, which we have shown

It therefore remains t o be shown t h a t

on

which i s exactly s i m i l a r t o t h e proof

(a,b),

of Theorem 1.2.1.

The proof of t h e theorem i s therefore

complete.

1.3 THE MODIFIED FUNCTION Employing the notion of upper and lower solutions and t h e existence theorem i n t h e small, it i s possible t o e s t a b l i s h t h e existence of solutions i n t h e large of some boundary value problems f o r a modified form of the d i f f e r e n t i a l equation XI' = f(t,X,X'),

(1.3.1) where

f E C[J xRxR,R].

Let us f i r s t define the modified

function. DEFINITION 1.3.1. on

J

and l e t

c

J.

Then define

>0

Let

a,p

C(l)[JJR]

E

be such t h a t

with

Icy'(t) I, I p l ( t )

I

a(t) < p(t)

< c on

2 c,

f(t,x,c)

for

XI

f(t,X,X')

for

IX'I

f(tJx,-c)

for

x'

5 CJ 5 -c,

and

F(t,x,x') =

F{ The function

F*( t ,B (t),XI) + [x-B(t) ] / 1 + x2

for

F*(tJx,x')

for a ( t ) < x < B ( t ) , for x < a ( t ) .

* (t , a ( t ) , x ' ) + [ x - a ( t ) ] / 1 + x 2

F(t,x,x')

J

w i l l be called the modification of

f ( t J x J x ' ) associated with t h e t r i p l e follows from the d e f i n i t i o n t h a t on J x R x R

x >p(t)

and t h a t 18

a ( t ) , p(t), c.

F(t,x,x')

It

i s continuous

1.3. THE MODIFIED FUNCTION

Let us consider now the modified boundary value problem

(1.3.2)

X"

= F(t,X,X'),

.(a)

=

y, x(b)

= 6,

r e l a t i v e t o which we have t h e following.

THEOREM 1.3.1.

a,f3 E C(l)[J,R]

Let

be, respectively,

lower and upper solutions of (1.3.1) on J a(t)

5

B(t)

on J .

such t h a t

Then t h e boundary value problem (1.3.2),

i s the modification of f associated with t h e t r i p l e cr(t)J B(t)J c, has a solution x E C(2)[J,R] s a t i s f y i n g where

F

a ( t >5 x ( t )

(1.3.3) provided t h a t

Proof:

5

.(a)

y

B(t)

on J,

-

p(a), a ( b ) < 6

5

B(b).

By Corollary 1.1.1, t h e boundary value problem

(1.3.2) has a solution show (1.3.3).

5

5

x c c ( 2 ) [ ~ , ~ ~~ h. u swe only need t o

We s h a l l only prove t h a t

x(t)

5

@ ( t ) on J .

The arguments are e s s e n t i a l l y t h e same f o r t h e case a(t)

x(t).

Assume, i f possible, t h a t

-

x(t)

> B(t)

f o r some

t E J . Then x ( t ) p ( t ) has a positive maximum a t a point 0 to E J Hence it follows t h a t x ' ( t , ) = @ ' ( t o ) , I x ' ( t O ) ( c c

.

and

x"(to) = F ( t O , x ( t O ) , x '( t o ) )

19

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

and therefore, we arrive at

-

which i s impossible a t a maximum of x ( t ) B(t). We conclude that x ( t ) 5 B(t) on J . The proof i s complete. Let us next consider the general linear boundary conditions

(1.3.4)

a,x(a)

-

(a) = A,

blx(b)

+ b2x1(b)

= B,

where a1 + a2 > 0, b 1 + b2 > O j a1ja2jbljb2 2 0, and a + b l > 0 . We w i s h t o prove a result analogous t o Theorem 1 1.3.1 w i t h respect t o conditions (1.3.4). F i r s t of a l l , we need the following lemma.

LFMMA 1 . 3 . 1 . Assume that ( i ) a,@ E C(l)[J,R] are, respectively, lower and upper solutions of (1.3.1) on J such that a ( t ) 5 B(t) on J; ( i i ) f E C[J XRXR,R] and for t E Jj a ( t ) 5 x 5 B ( t ) , f ( t , x , x ' ) s a t i s f i e s a Lipschitz condition i n X I f o r a constant L > 0. Then, there exists a f'unction f* E C [ J xRxR,R] which is bounded on J X R X R and i s Lipschitzian for the same constant L > 0 whenever t E J and a ( t ) 5 x ,< B(t).

Proof:

A s before, l e t

I a ' ( t ) l , lB'(t)l t E J and a ( t )

c > 0 be such that

< c on J. Then define f*(t,xJx') f o r 5 x 5 p ( t ) by setting

20

1.3. THE MODIFIED FUNCTION

f*(t,x,x*) =

{

x ' 2 c, ( x ' ) 5 c,

f(t,x,c) f(t,x,xl)

for for

f(t,x,-c)

for

We then extend the domain of definition of J X R X R by l e t t i n g f*(t,x,x')

XI

5 -c.

f*(t,x,x')

f*(t,B(t),x')

for x > B(t),

f*(t,a(t),x')

for x < a ( t ) .

to

=

It i s easy t o see that the function f* so defined possesses a l l the stated properties. Hence the proof i s complete. E-EOREM 1.3.2.

Let hypotheses ( i ) and ( i i ) of Lemma 1 . 3 . 1

Then there exists a solution x

hold.

(1.3.5)

E

C(2)[J,R]

of

XI' = f*(t,X,X')

which s a t i s f i e s the boundary conditions (1.3,4), provided that

(1.3.6)

.,.(a)

Proof: F(t,x,x') =

- a2a' (a) 5 A 5 alp(") - a 2 P (a),

Define the ftmction F(t,x,xl)

{

on J x R x R as

f*C(t,B(t),xl) + ~ ( [ x - ~ ( t ) ~ / l +for x ~ )x > ~(t), f*(t,x,x') for 4 t ) _ < X S B ( t ) , f*(t,a(t),x') + ( [ x - a ( t ) ] / 1 + x 2 ) f o r x < a ( t > ,

where f*(t,x,x') i s the f'unction obtained i n &ma 1.3.1. Since f* is bounded, F is also bounded. Hence, by Corollary 1.1.1, there exists a solution x E C ( 2 ) [ J , R ] of (1.3.8)

X"

= F(t,x,x')

satisfying the boundary conditions (1.3.4).

21

We now show that

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

t

for

E

J, a ( t )

-< x ( t ) -< @ ( t ) ,

which implies t h a t

a solution of the problem (1.3.4), (1.3.5). Assume t h a t x(a) > @(a) i n which case zero.

Then there exists a

[a,6).

By definition of

(1.3.9)

x"(t) - D - B ' ( t )

f*,

+

f*(t,B(t),x'(t))

-

cannot be

h(t) - B(t)l 2 l + x (t)

f*(tJ@(t)?B'( t ) )

-> -Llx' ( t ) - B ' ( t ) I since

Y

i s a l s o an upper solution of (1.3.8).

p(t)

is

> a such t h a t x ( t ) > p ( t ) on F, we have f o r a
6

2

a2

x(t)

Also,

r e l a t i o n (1.3.6) implies, because of the assumption x ( a ) _> p(a), t h a t x ' ( a ) 2 @ ' ( a ) . We claim that there i s a to E [a,6) such that

x'(t,)

on a

5

t

5

- @ ' ( t o )> 0 .

6.

If not, suppose t h a t

Then, by (1.3.9), x"(t)

- D-B'

x'(t)

- p'(t)IO

we would obtain

( t ) _> L[X' ( t )

- B' (t)1

which, by the theory of d i f f e r e n t i a l inequalities, yields x'(t)

- B'(t) 2

This, together with on

a
x'(t)

[x'(a)

- B'(a)le L(t-a)

- B ' ( t ) -4,

implies t h a t

x ' ( t ) Epl(t)

On the basis of (l.3.9), we then arrive a t the

contradict ion

-

x ' ( t o ) @ ' ( t o )> 0 f o r some to E [a,&). Again appealing t o the d i f f e r e n t i a l inequality (1.3.9), we conclude t h a t x ' ( t ) - B ' ( t ) > 0 on [t0,8]. I n

which establishes the claim,

fact, t h i s argument leads t o follows t h a t tradiction

x(b) > p(b).

6 = b and, consequently, it This, i n turn, leads t o the con-

< @'(b), i n view of the relations (1.3.4) x'(b) -

and (1.3.7). 22

1.3. THE MODIFIED FUNCTION

I f , on the other hand, we assume t h a t are lead t o the r e l a t i o n

x(b) > B(b), we arguing as before, which

x(a) > B(a)

has j u s t been shown t o be impossible. show that

x(t)

5

t

@(t) for

Assume then there e x i s t

E J

t

x(t)

- p(t),

It therefore remains t o

.

f o r which

E Jo

tl,t2

"his implies the existence of

The difference

0

E

J

> B(t).

x(t)

such t h a t

therefore, assmes a positive

maximum a t some to E (tl,t2). Clearly, x ' ( t o ) = @ ' ( t o ) and x"(to) D-B' ( t o ) _< 0 . A computation, however, leads t o the contradiction

-

noting ( x t ( t o l = l p T ( t o ) l< c and the definition of f". We < B(t) on J . One can similarly have thus established x ( t ) verify that

a(t) < x(t)

on J

completing the proof.

Let us now proceed t o supply the proof of Theorm 1.2.3.

For t h i s purpose we need the following r e s u l t . THEOREM

1.3.3. Assume t h a t

f E C[JxRxR,R]

and t h a t

,

solutions of i n i t i a l value problems f o r (1.3 .l)are unique. Let a,@ E C(l)[J,R] be lower and upper solutions o f (1.3.1) on J, respectively, such t h a t a ( t ) < p ( t ) on J . Then, i f a ( t o ) = @ ( t o ) and CY ( ) = p ( ) f o r some to E J, it follows ' ' that a ( t ) p ( t ) on J .

Proof:

Suppose t h a t

a ( t ) f p(t)

the case where there i s an i n t e r v a l

on J .

(to,tl]

C

We w i l l consider J

such t h a t

a ( t o ) = @ ( t o ) ,Q ' ( t o ) = B ' ( t o ) , and a ( t ) < B(t) on 23

[to,tlI-

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

Fl(t,x,x')

Let

be the modification jknction of

according t o Definition 1.3.1 f o r the i n t e r v a l the t r i p l e a,p,cl.

f(t,x,x') [to,t,]

and

a ( t l ) < S1 < @(tl), we have by

Then, i f

Theorem 1.3.1 t h a t the boundary value problem (BVP) =

X"

Fl(t,x,x'),

has a solution x1 a(t)

5 x,(t) _<

E

B(t)

x(to) = a ( t o ) C(2)[[to,t,],R] on

[tO,tll.

x ' ( t o ) = a l ( t o ) and since of

such that Therefore,

I x i ( t ) I < c1

to, it follows by the definition of

interval

[t0,t2]

(1.3.1).

If

C

t2 = t

[tO,tl]

x ( t 1) = 61

xl(tO)

=

u(to),

on some neighborhood

F1, there i s a maximal

on which x ( t ) 1

i s a solution of

1'

a ( t 2 ) = a@,) < S1 = Xl(t2) = x 1(t1) < @ ( t l ) .

to < t 2 < t

it i s s t i l l t r u e t h a t m(t2) < xl(t2) < p(t2) 1' for, if e i t h e r inequality were an equality, we would have

If

Ixi(t)l

< c1 and the i n t e r v a l [t0 , t 2 3 would not be maximal.

We can therefore construct another modification f'unction

[t ,t ] r e l a t i v e t o the t r i p l e 0 2 a , p, c2. A n application of meorem 1.3.1 again w i t h a ( t 2 ) < 62 < p ( t 2 ) , shows t h a t there i s a solution x2 E C(2) [[t0,t2],R] of the BVP F2(t,x,x')

of

f(t,x,x')

satisfying

a ( t ) < x2(t)

-

on

5 p(t)

on

[t

0'

t 2 1.

it follows t h a t there i s a maximal i n t e r v a l on which x 2 ( t )

A s before,

[to,t3] i s a solution of (1.3.1) and t h a t

C

[t0,t21

a ( t 3 ) < x2(t3) < p(t3). lnnis contradicts the assumption that solutions of i n i t i a l value problems f o r (1.3.1) are unique, since xl(tO) = x 2 ( t 0 ) and x l ( t ) = x;(to). conclude t h a t

a(t)

p(t)

1

on

24

0 J.

We therefore

1.4. NAGUMO'S CONDITION

We a r e now ready t o give t h e proof o f Theorem 1.2.3. Proof of meorem 1.2.3: some points o f

[c,d]

Assume t h a t

and l e t

Clearly t h e r e e x i s t s a

a(t)

-

> p ( t ) at

N = m a x [ a ( t ) @ ( t ) l on

-

[c~d].

to E [ c , d ] where a ( t O ) @ ( t o )= N

and a ' ( t o ) = B ' ( t o ) . The functions (Y and B , = p + N s a t i s f y t h e conditions o f Theorem 1.3.3 i n view of t h e nondecreasing nature f(t,x,xl)

of

@ ( t )+ N

a(t)

in

Consequently it follows t h a t

x.

on

[c,d],

a c o n t r a d i c t i o n which proves t h e

theorem.

1.4 NAGW'S CONDITION A s we have seen, t h e proofs f o r t h e existence theorems

for solutions o f boundary value problems depend on f i n d i n g a Here we

p r i o r i bounds f o r t h e s o l u t i o n and i t s d e r i v a t i v e .

present some s u f f i c i e n t conditions f o r obtaining such bounds. DEFINITION 1.4.1. C[J,R]

with

Let

E

t

J, w ( t ) _< x _< @ ( t ) and

(1.4.1) where h

If(tJXYX') E

C[JXRxR;R]

E

a(t) < @ ( t ) on

a,p E

f

x'

E

J.

and

Suppose t h a t f o r

R,

I 5 h ( Ix' 1 )

C [R+j ( o J m ) ] . If

THEOREM 1.4.1. condition on J

Assume t h a t

f

satisfies Nagumo's

with respect t o t h e p a i r 25

a , @ . m e n f o r any

1.

METHODS INVOLVING DIFFERENTIAL INEQUALITIES

solution x E C(')[J,R] of (1.3.1) with a ( t ) _< x ( t ) ,< @ ( t ) on J, there e x i s t s an N > 0 depending only on a,@,h such that Ix'(t) I

(1.4.4)

5N

on J .

Because o f (1.4.2), we can choose an N > h such

Proof: that

(b - a ) x l ( t o ) = x(b)

to E Jo is such t h a t

If

- x(a),

then by

I x l ( t O )I 5 h. Assume t h a t (1.4.4) i s not Then there e x i s t s an i n t e r v a l [tl,t21 C J such t h a t

(1.4.3), we have true.

the following cases hold: = N,

and

A < x'(t)

=

N, x l ( t , ) = A,

and

h

x ' ( t 1)

=

-A,

x'(t,)

= -N,

(i) x l ( t ) = A, x ' ( t , ) 1

< N,

t c (tl,t2),

( i i ) x'(t,)

t

E

(tl,t2), (iii)

t

E

(tl,t2), (iv)

t

E

< x ' ( t ) < N,

x'(t2)

= -N,

and

-N < x ' ( t )

< -A,

x

=

-h,

and

-N < x l ( t )

<

)

-A,

(tl,t2).

L e t us consider case ( i ) .

On

[tl,t21,

and as a r e s u l t

26

we obtain, by (l,b.l),

1.4. NAGUMO'S CONDITION

This leads t o t h e c o n t r a d i c t i o n

We can d e a l with t h e remaining p o s s i b i l i t i e s i n a s i m i l a r way

and therefore we conclude t h a t (1.4.4) i s v a l i d .

Then, f o r any s o l u t i o n exists an N lx'(t)l

5

>0 on

N

x

E

C(2)[J,R1 with

depending only on J.

Also,

N -10

M,h, (b

as

( x ( t ) ( 5 M,

- a)

there

such t h a t

M -0.

The conclusion of Corollary 1 . 4 . 1 i s n n C[JxRnxR , R ] and absolute values a r e re-

REMARK 1.4.1.

false, i f

f

E

placed by norms, as t h e following example shows. (cos n t , s i n n t ) llX"

( t ) 11 = n2 =

hold with

(Ix'

so t h a t ( t ) I/*.

Let

x(t)

=

Ilx(t)(I = 1, Ilxl(t)ll = n, Thus t h e assumptions of Corollary 1 . 4 . 1

M = 1, h ( s ) = s2

+

1. However, t h e r e does not e x i s t

an N > 0 such t h a t / / x t ( t ) )_<) N f o r a l l choices of n. Thus it is c l e a r t h a t Nagumo's condition i s not s u f f i c i e n t t o obtain the desired conclusion i n t h e case of systems. We s h a l l discuss t h i s f u r t h e r when we t r e a t systems of equations l a t e r i n t h i s chapter. 21

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

EXERCISE 1.4.1.

on

Suppose t h a t

J.

a(t)

Let

5

5

where

L

>

f E C[JxRxR,R]

p(t), Y 1 J Y 2

I 0

(

J xJ

C(l)[JJR] with

E

Y,)

and f o r

a(t)

t

E

-< p ( t )

JJ

RJ

-

(t J xJY2)

i s a constant.

Show

condition. THEOREM 1.4.2.

Assume t h a t such t h a t

( i ) a,p E C(l)[J,R] a(b) = p(b)J W

and a ( t )

5

p(t)

On

.(a)

= p(a),

J;

( i i ) (pJJI E C(')[uJR] with q ( t J x ) 5 Jl(t,x) on = [(t,X): a ( t ) <X < (u(t)J t E J ] and O n W,

(1.4.5)

q t ( t J x ) + qx(tJx)q(tJx)

< f(tJxJq(tJx)),

(1.4.6)

$ t ( t J x ) + qx(tJx)q(tJx)

> f(tJXJq(tJX)),

Proof:

a(t)

Let

x(t)

(u,

where

be any solution of (1.3.1) such t h a t

_< x ( t ) 5 p ( t ) on J . Then, by

(i)and ( i i i ) it follows

that ' ~ ( a , x ( a )5 )

X I(a)

Assume now t h a t there exists a

5

Jr(a,x(a)).

to E (a,b)

such t h a t

< ' p ( t o J x ( t o ) ) . Then there must e x i s t an i n t e r v a l [tl,t2]C J with tl < to < t2 such t h a t x ' ( t i ) = ' P ( t i > x ( t i ) ) Ji = 1,2 and x ' ( t ) < ' P ( t , x ( t ) ) , t i < t < t 2 J

x'(to)

28

1.4. NACUMO'S CONDITION

and f u r t h e r t h a t

This i s a c o n t r a d i c t i o n because of assumption ( i i ) .

f o r e conclude t h a t merit shows t h a t

x'(t)

x'(t)

-> ' p ( t , x ( t ) )

-< J r ( t , x ( t ) )

on

on J

We there-

A s i m i l a r argu-

J.

and t h e proof i s

complete. We remark t h a t t h e i n e q u a l i t y s i g n i n (1.4.5) could be reversed i n which case t h e c o n t r a d i c t i o n i s a r r i v e d a t because of

(1.4.6).

A s i m i l a r coment holds r e l a t i v e t o i n e q u a l i t y

Of

course, i n t h i s case, we have t o replace ( i i i ) by V(b,a@))

5 a'@>,

_< Jr(b,B(b))*

COROLLARY 1.4.2. Assume ( i ) of Theorem 1.4.2.

e x i s t constants B'(a),B'(b)

N

5N2,

N such t h a t 1 ' 2 and f(t,x,Ni)

a"(a), a' (b)

#

0, i = 1,2.

clusion of Theorem 1.4.2 i s t r u e w i t h $ { t A=

9.

EXERCISE 1.4.2. Assume t h a t

Ix'I

--fw

f

E

Let there

2 N1, Then t h e con-

cp(t,x) = N1

and

L e t hypothesis ( i ) of Theorem 1.4.2 hold.

C[JXRXR,R]

and

uniformly on compact

lf(t,x,x')I

(t,x)

sets.

-tm

as

Show t h a t t h e

conclusion of Corollary 1.4.2 i s t r u e . EXERCISE 1.4.3. 1 . 4 . 1 h o l d and flmctions

h(s)

Show t h a t i f t h e assumptions o f Corollary i s nondecreasing, t h e r e always e x i s t

'p(t,x), @ ( t , x ) (independent of

t ) satisfying the

hypotheses of Theorem 1.4.2. Consider t h e ' following example on J = [-1,11,' 29

1 , METHODS

= e-2(x+1)

XI'

where

n

INVOLVING DIFFERENTIAL INEQUALITIES

- (x')2nJ

x(-1) = 0 = x ( l ) ,

i s a p o s i t i v e integer.

B ( t ) = 0.

Choose N1 = - 2

Take

and

a ( t ) = t 2 - 1 and

N2 = 2.

Then we see a l l t h e

hypotheses of Corollary 1.4.2 a r e s a t i s f i e d . t h a t any solution x ( t ) Ixr(t)(5 2

on J.

such t h a t

Hence we conclude

t 2 -15 x ( t ) _< 0

However, we notice t h a t f o r n

satisfies

>

* J

Corollary 1 . 4 . 1 is not applicable. A variant of Theorem 1.4.2 which i s more useful i s t h e following r e s u l t .

Proof:

Let

f (t>XJX') such t h a t

o(t)

_< x ( t ) 5 B(t), t

to E (a,bl

be any solution o f

x E C(2)[J,R1

such t h a t

cp(a,x(a)) E

J.

5 x *( a ) 5

q(a,x(a))

x" = and

Suppose t h a t t h e r e e x i s t s a

x'(to)

> q ( t o , x ( t o ) ) . Define

30

15. EXISTENCE IN THE LARGE

Then,

i n some i n t e r v a l t o t h e l e f t of

to. This implies t h a t

V(t)

t increases and p o s i t i v e near to and hence on [a, toI. Thus we have V(a) > 0 o r equivalently x ' ( a ) > $ ( a , x ( a ) ) which i s a contradiction. Hence x ' ( t ) _< $ ( t J x ( t ) ) on J . S i m i l a r l y we can v e r i f y i s nonincreasing as

'p(tJx(t))

5 x'(t>

on J.

1.5 EXISTENCE I N THE LARGE We a r e now ready t o prove theorems on existence i n t h e large.

We begin with one of t h e b a s i c r e s u l t s i n t h i s d i r e c t i o n .

THEOREM 1.5.1.

Let

a,p

E

C(l)[J,R]

be, respectively,

lower and upper s o l u t i o n s of (1.3.1) on J

@(t) _< @ ( t ) on J.

Suppose f u r t h e r that

such t h a t f(t,x,x')

satisfies

Nagumo's condition on J any .(a)

(1.5.1)

5

c

5

r e l a t i v e t o t h e p a i r a,@. Then, f o r @ ( a ) , ~ ( b 5) d 5 B ( ~ ) J t h e BVP

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

has a s o l u t i o n x

E

x ( a ) = c,

c ( 2 ) [ J ~ R ] with

31

x ( b ) = d,

a ( t >_< x ( t > _< @ ( t )and

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

lx'(t)l

5N

on J,

where

depends only on a,@ and t h e

N

Nagumo's function h. By Theorem 1.4.1, t h e r e i s an N

Proof:

only on a,p,h with

a(t)

5

such t h a t

Ix'(t)l

x ( t ) _< @ ( t ) on J .

( c u ' ( t ) ( < cl,

(P'(t)(

<

5

on J

N

depending

f o r any solution

c > N so t h a t 1 Then, by Theorem 1.3.1, t h e

Choose a

on J.

c1

>0

BVP x" = F ( t , x , x ' ) , has a solution on J,

where

f(t,x,x')

x ( a ) = c,

x E C(2)[J,R]

a ( t ) 5 x ( t ) _< p ( t ) i s t h e modification f'unction of

F(t,x,x')

with respect t o

to

theorem t h e r e i s a

x ( b ) = d,

E Jo

such t h a t

a,p,

and

cl.

By t h e mean value

such t h a t

(b - a ) x ' ( t o ) = x ( b )

- x(a),

and as a r e s u l t using (1.4.3) it follows t h a t

Ix'(t,)l 5 h < N < cl. This implies t h a t t h e r e i s an i n t e r v a l containing toy where x ( t ) i s a solution of x" = f ( t , x , x ' ) . By Theorem

1.4.1, we have

Ix'(t)l

5

N

< c1 on t h i s i n t e r v a l .

However, I x ' ( t ) J < cl-

x ( t > i s a s o l u t i o n of

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

We conclude t h a t

i s a solution of (1.5.1) on J.

x(t)

as long as

The

proof i s complete.

COROLLARY 1.5.1.

Under t h e assumptions of Theorem 1.5.1,

any i n f i n i t e sequence of solutions of the relation

a(t)

,< x ( t ) _<

p(t)

XI'

on J,

= f(t,x,x')

obeying

has a uniformly con-

vergent subsequence converging t o a solution of

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

on J . A conclusion similar t o Theorem 1 . 5 . 1 may be drawn r e l a t i v e

t o t h e boundary value problem (1.3.1), Theorem 1.3.2,

EXercise 1.4.1,

(1.3.4), on the b a s i s of

and the proof of Theorem 1.5.1.

32

1.5. EXISTENCE IN THE LARGE

We state it in the following exercise. Let hypotheses (i) and (ii) of Lemma Show that for any A,B for which relations (1.3.6)

EXERCISE 1.5.1.

1.3.1 hold.

and (1.3.7) are valid, the BVP (1.3.1), (1.3.4) has a solution such that a(%) _< x(t) _< p(t) on J and x E C(*)[J,R] Ix'(t)l < N on J, where N > 0 depends o n l y on a,@,L. EXERCISE 1.5.2.

Let f E C [ J x R x R , R ] , f(t,x,x') be nondecreasing in x for each (t,x') and satisfy If(t,x,yl)f(t,x,y2)l5~ Iy1-y21 for (t,x> E J X R and yljy2 E R. Show that the BVP (1.3.1), (1.3.4) has a solution. If f is strictly increasing in x, then show that the solution is unique. Hint: -

Set M

max f(t,O,O), m = min f(t,O,O) and J J let p(t) > - 0, a(t) _< 0 be solutions of x" + L l x ' I - m = 0 , XI'- Llx'l - M = 0 , respectively, subject to (1.3.6), (1.3.7) which may be computed explicitly. Then, using monotony of f, show a,@ are lower, upper solutions. For uniqueness, use =

Theorem 1.2.4. 2

1.5.3. Verify that p(t) = t, a(t) = t 1 - 3t +2 are upper, lower solutions for the problem x'' = - 1x1' + t, x(1) = 0, x'(2) = 1, on J = [1,2]. Draw conclusions on the applicability of Exercises 1.5.1, 1.5.2 to this problem. Discuss the merits. EXERCISE

The foregoing discussion concerning the problem (1.3.1),

(1.3.4) imposes more restrictions on the function f compared This is apparently due to the techniques to the B W (1.5.1). involved in the process rather than the need, because the result that we present next involves nonlinear boundary conditions more general than (1.3.4) and yet the assumptions on f are just 33

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

those required f o r t h e BW (1.5.1). We s h a l l f i r s t prove t h e following.

THEOREM 1.5.2. (a) a,@

E

Assume t h a t

C(l)[J,R],

s o l u t i o n s of (1.3.1) on

respectively, a r e lower and upper such t h a t

J

a(t) 5 B(t)

on J

and

a(a) < B(a); (b)

s a t i s f i e s Nagumo's condition on

f(t,x,x')

respect t o t h e p a i r (c) y

g

E

f o r each

(1.5.2)

a(b) J

_<

d

g(x,y)

i s nondecreasing i n

and

g(a(a),a'(a))

2

g(B(a),B'(a))

0,

It i s enough t o show t h a t given

Proof:

with

a,B;

C[[a(a),B(a)] xR,R],

x,

J

_< f3(b),

there i s a solution

x(t,E)

5

0.

E

>

0

and

of (1.3.1)

on

that fulfills x(b,E> = d,

and

a ( t ) 5 x(t,E)

that that such

on

<E

J.

_< d _< B(b), l e t T(c) denote t h e s e t (1.5.1) such t h a t a ( a ) _ < c < B ( a ) a ( t ) _< x ( t ) _< p ( t ) on J. By Theorem 1.5.1, it i s c l e a r ~ ( c ) i s nonempty f o r a l l a(&)5 c _< @(a). Suppose now t h e theorem i s not t r u e . Then t h e r e exists an EO > 0 , t h a t f o r every a ( a ) _< c 5 p ( a ) and x ( t ) E ~ ( c ) , we have For a f i x e d

of s o l u t i o n s and

5 B(t)

Ig(X(a,E), x'(a,E))I

x(t)

a(b)

t o t h e BW

Ig(x(a),x'

(4)I 2 E O . 34

1.5. EXISTENCE IN THE LARGE

Define the set S by

and let co = sup[x(a) = c: x(t) E S]. We notice that x(t) E T(cy(a)) implies a'(a) _< x ' ( a ) which by (1.5.2) yields g(x(a>,xl(a>) E ~ . Similarly, x(t> E .rr(@(a)) implies

>

5 -EO. This observation x'(a) _< @'(a) so that g(x(a),x'(a)) proves that co < p(a). Let yo(t) be a solution of (1.3.1) which is obtained as a uniform limit of members of S so that (1-5.3) yO(a> =c0,

Yo@) = d ,

and g(Yo(a>, Y;)(a))

2

€0.

For n 2 N, Now let N 2 1 be such that co + (1/N) 5 @(a). satisfying, in addition, the inlet yn(t) E T(C~ f (l/n))

>

equality yn(t) yo(t) on J. This is clearly possible since yo(t) may be treated as a lower solution of (1.3.1). Then a subsequence of yn(t) converges to a solution xo(t) obeying xo(t) yo(t) on J. By definition of co,

>

E T ( C ~ )

we have

5 -cO and consequent- g(xo(a>, x;(a)) _< g(yn(a), y;(a)) Since x;(a) 2 y;)(a) and xo(a) = yo(a), this leads to g(y,(a), y;)(a)) 5 - E O which is a contradiction in view of (1.5.3).

-E~.

The proof is complete.

Employing similar arguments as in the proof of Theorem 1.5.2, which were based on a consequence of Theorem 1.5.1, namely Corollary 1.5.1, we may prove the following theorem. THEOREM 1.5.3.

Assume that

respectively, are lower and upper C(l)[J,R], solutions of (1.3.1) on J such that (a) a,@

E

a(t)

5

@(t)

on J and a(b) < @(b);

(b) f(t,x,x') satisfies Nagumo's condition on J relative to the pair a,@; 35

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

(c) h

C[[a(b),B(b)]

E

xR,R],

y for each x and h(a(b),

=

f(t,x,x'),

5

a'(b))

Then, for any .(a) 5 c 5 @(a) x C E C(2)[J,R] of the BVP x"

h(x,y)

2

h(B(b),B'(b))

0.

there is a solution

x(a) = c,

which satisfies a(t)

0,

is nondecreasing in

5 xc(t) 5

h(x(b),x'(b))

B(t)

= 0

on J.

Combining the proofs of Theorems 1.5.2 and 1.5.3, we may obtain our main result relative to nonlinear boundary conditions. THEOREM

1.5.4. Assume that

(a) a,@

E

C(l)[J,R],

respectively, are lower and upper

solutions of (1.3.1) on J .(a>

such that Q(t)

5 B(t)

on J and

4 b ) < Bb); (b) f(t,x,x') satisfies Nagumo's condition on J with

< B(a),

respect to the pair a,B; E C[[a(a),B(a)l xR,R], h E C[[o(b),B(b)] xR,R], (c) g(x,y), h(x,y) are nondecreasing in y for each x and

g(a(a),al(a)) h(a(b),a'(b))

2

0,

5

0,

Then there is a solution x XI' = f(t,x,x'),

g(B(a),B'(a)) h(B(b),B'(b)) E

which satisfies a(t)

= 0,

5 x(t) _<

@(t)

Proof: For each a(b) 5 d set of solutions of the BVP

x"

=

f(t,x,x'),

5

2

0-

h(x(b),

x'(b))

= 0,

on J.

@(b),

g(x(a>,x'(a))

0,

of the BVP

C(*)[J,R]

g(x(a),x'(a))

5

let T ( d ) denote the = 0,

x(b) = d,

which satisfy a(t) 5 x(t) 5 B(t) on J. By Theorem 1.5.2, is nonempty for all it is clear that the set T(d) 36

15. EXISTENCE IN THE LARGE

which contradicts the supposition that the theorem is not Now use y(t) in place of a(t) true. Hence y(a) < @(a). in Theorem 1.5.2 to conclude that the set D = [x(t): x(t)

E

T(d),

do

< d 5 B(b),

y(t)

5 x(t)

.< @(t>

on

JI

in nonempty. Moreover, x(t) E n(d) implies that h(x(b), x'(b)) > 0 . However, inf[x(b) = d: x(t) E D] = do, which leads to a contradiction by a convergence argument and the fact that h(x,y) is nondecreasing in y. Hence the proof is complete. On the basis of Theorem 1.4.3, it is possible to relax the condition of the basic Theorem 1.5.1 which is the content of the next result. THEOREM 1.5.5. Suppose that assumptions (i) and (ii) of Theorem 1.4.3 hold. Let a,@ be lower and upper solutions of (1.3.1), respectively, on J, such that a(&) = @(a) and

cp(t,y(t))

5 ~'(t)I +(t,r(t>),

for y(t) 37

=

a(t),B(t),

t

E

J.

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

Then, f o r any

5

a(b)

d

_< B(b) and u(a)

BVP (1.5.1) has a solution

x E C(2)[J,R]

(t,x(t),x'(t>) where

E

n,

t

= c = B(a),

the

such t h a t E J,

il i s t h e s e t given by

R = [(t,x,x'):

Proof:

(t,x>

E

(0

and

cp(t,x)

Define a modified function

5 x' 5

Jr(t,x)l.

F(t,x,x')

on

cuxR

by f(t,x,dt,x)) f(t,x,+(t,x)) and extend

F(t,x,x')

to

if x '

< cp(t,x>,

if

(t,x,x')

if

XI

>

E

n,

q(t,x),

J x R x R by s e t t i n g

F(t,a(t),x')

if

x < a(t),

F(t,B(t),x')

if

x > @(t).

F(t,x,x') = Observe t h a t

i s bounded on J x R x R

F(t,x,x')

s a t i s f i e s Nagumo's condition.

a',@'), w,B,

assumptions on

and therefore

Furthermore (because of t h e a r e a l s o lower and upper solutions

relative t o

Consequently, by Theorem 1.5.1 t h e r e exists a solution x E C(2)[J,R]

t o t h e modified BVP

x" = F(t,x,x'), such t h a t and

a(b)

a(t)

5

d

x ( a ) = c,

x(b) = d,

5 x(t) 5

_< B(b).

B(t), t E J f o r .(a) = c = B(a) We now apply Theorem 1.4.3 t o conclude

i s a c t u a l l y a s o l u t i o n of t h e BVP (1.5.1) s a t i s f y i n g ( t , x ( t ) , x ' ( t ) ) E R, t E J. Hence t h e proof i s

that this

x(t)

complete.

38

1.6. LYAPUNOV-LIKE FUNCTIONS

1.6.

LYAPUNOV-LIKE FUNCTIONS We wish t o employ, i n t h i s section, Wapunov-like functions

and the theory of d i f f e r e n t i a l i n e q u a l i t i e s t o e s t a b l i s h existence i n t h e large f o r t h e BVP (1.5.1).

+

We define t h e generalized derivative

x" = f ( t , x , x t )

the d i f f e r e n t i a l equation Dt,( t

J

X, X

V

Let

C[JxRxR,R].

E

D Vf(tJxJx') relative t o

by

')

- V(t,x,x')].

lim sup 1 [V(t + h , x + h x ' , x ' + h f ( t , x , x ' ) ) h+O+ h We s h a l l often use s e v e r a l f'unctions ViJ

i = 1,2,.

..,n,

which

we c a l l I(fapunov-like f'unctions and f o r n o t a t i o n a l consistency we write Vi

of

+

D V. ( t , x , x t )

t o denote t h e generalized derivative

If

with respect t o t h e d i f f e r e n t i a l equation

(1.6.1)

XI1 =

THEOREM 1.6.1.

(i) F(t,x,x') the t r i p l e (ii)

Assume t h a t

a , p c C(l)[J,R]

such t h a t

i s t h e modified f'unction of

a,p,c,

a(t)

-< p ( t )

f(t,x,x')

on

J

and

relative t o

as given i n t h e Definition 1.3.1;

t h e r e e x i s t two Wapunov functions

wi

x [x: x ,< a ( t ) ] xR, D2

= J

i = 1 J 2 J . where

B(t)]xR,

f(t,X,X').

D1

= J

E

C[D~,R],

x [x: x >

W ( t , x , x ' ) = 0 i f x = a ( t ) , Wl(t,x,xl)>O 1 x < a ( t ) , W2(t,x,xt) = 0 i f x = B(t), W2(t,x,x1) > O i f

if

such t h a t

x > B(t); (iii)

for

i = 1,2, g

locally Lipschitzian i n

+

D WP(t,x,xt)

+

D W2F(t,x,xt)

i

E

+

C[JxR , R ] , Wi(t,x,xl)

is

( x , x t ) and

_> gl(t,Wl(t,x,xl)) g2(t,W2(t,x,x1))

39

i n t h e i n t e r i o r of

D1j

i n t h e i n t e r i o r of

D2;

1.

to

E

METHODS INVOLVING DIFFERENTIAL INEQUALITIES

( i v ) the only solution of J , i s r ( t ) p 0 on J .

5

Then, if .(a) (1.6.2)

c

5

has a solution x

d

5

r ( t o ) = 0,

the BVP

B(b),

x ( a ) = c,

C(2)[J,R]

E

5

B(a), a ( b )

x" = F ( t , x , x ' ) ,

r ' = gi(t,r),

x(b) = d,

satisfying

a(t)

-< x ( t ) 5 p ( t )

on J . Proof: x

E

By Corollary 1.1.1, the BVP (1.6.2) has a solution

C(2)[J,R].

on J .

It s u f f i c e s t o show t h a t

We w i l l only prove t h a t

proof f o r

x(t)

5

[tl,t21C (a,b)

E

such t h a t

< a ( t ) for t

E

a(t)

5 x(t)

-

< x(t) < @(t) on J since the

i s e s s e n t i a l l y the same.

< a ( t ) f o r some t

x(t) x(t)

p(t)

a(t)

(a,b).

Then t h e r e e x i s t s an i n t e r v a l

x ( t l ) = a(t,),

(tl,t2).

Assume t h a t

x(t2) = a ( t 2 ) ,

and

We thus have by (ii)

From ( i i i ) and the theory of d i f f e r e n t i a l i n e q u a l i t i e s , we obtain (1.6.4) where

Wl(t,x(t),x'(t)) r(t,t2,ro)

r ( t2 ) = r 0 clear that

5

r(t,t2,ro),

a

5

t

i s the l e f t maximal solution of

5 t2, r ' = gl(t,r),

~y ( i v ) and (1.6.3), it i s 1( t2, x ( t 2 ) , x t ( t 2 ) ) . we r ( t , t 2 , 0 ) 1 0 . However, f o r any t E (tl,t2),

= W

obtain, because of ( i i ) and (1.6.4), the contradiction 0

< W1(t,x(t),x'(t)) 5

0.

The proof i s therefore complete. We notice t h a t t h i s theorem generalizes Theorem 1 . 3 . 1 . In case

a(t), p(t)

respectively, on J,

a r e lower and upper solutions of (1.6.2),

it is enough t o choose W l ( t , x , x r ) = a ( t ) - x 40

1.6. LYAPUNOV-LIKE FUNCTIONS

It i s easy t o check that

and W2(t,x,x') = x - p(t).

Wl

s a t i s f i e s a l l the assumptions of Theorem 1.6.1 w i t h g1 = 0 . The only condition t h a t requires a l i t t l e explanation i s ( i i i ) , + For this, namely D WD(t,x,x') 2 0 i n the interior o f D

1' does not a t t a i n a it is sufficient t o prove that Wl(t,x,x') localmaxhum i n the interior of Arguing as i n the proof D1. of Theorem 1.3.1, we see that it i s necessary t h a t a ( t l ) = x ( t ) 1 a t some tl E (a,b) i n order for W1 t o a t t a i n a local maxjmum.

However, for t = t we obtain DWb(tl,x(tl), x ' ( t l ) ) > 0 as 1' i n the proof o f Theorem 1.3.1 which i s impossible a t a local maximum of W1. A similar reasoning holds for W2 and thus the conditions of Theorem 1.6.1 are verified. The next theorem offers a general s e t of conditions t o ensure that / x ' ( t ) I is bounded.

-

Assume that a,p E C(l)[J,R] and a ( t ) < p(t) on J . Define the s e t s El = J x [x: a ( t ) 5 x 5 p ( t ) ] x [x': X ' ,> 01 and E2 = J X [x: a ( t ) ,< x @ ( t ) ]X [x'; X I 5 01.

THEOREM 1.6.2.

Suppose that there exist Iyapunov f b c t i o n s Vi, i = 1,2,3,4, that are locally Lipschitzian i n (x,xl) such that ( i ) Vi E CIE1,R] and Vi(t,x,x') + m as uniformly on J x [x: a ( t ) < x 5 B(t)], i = 1,2; and ( i i ) D+Vlf(t,x,xl) 5 gl(t,Vl(t,x,x'))

-

XI

+m

-+

D V2f(t,x,x') 2 g2(t,V2(t,x,x')) on El; ( i i i ) Vi c C[E2,R] and Vi(t,x,x') + m as x' +-uniformly on J X [x: a ( t ) x 5 B(t)], i = 3,4; + (iv) on E2, D V3f(t,x,xt) -c g3 (t,V3(t,x,x')) and -i.

D V4f(t,X,X-) ,> gq(t,Vq(t,x,x')); (v) Vi(t,x,x') 5 Jri(lxII), i = 1,2,3,4, for r > 0 are continuous f b c t i o n s . Suppose f i r t h e r that

41

where

Jli(r)>O

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

(Vi)

gi

C[JXR,R], i = 1’2’3’4

f

and f o r

to f J, ro

E

R,

a l l solutions of

e x i s t on rt = g i ( t J r ) , r ( t o ) = roJ i = 1 , 3 [ t O , b l J and a l l solutions of rt = g i ( t J r ) ’ r ( t ) = r

i = 2,4

e x i s t on

Then there e x i s t s an N with

a(t)

Define f o r

>0

-< x ( t ) -< @ ( t )

t

0

[aJto].

0’

such t h a t every solution of

on

J

satisfies

Ixt(t)l

-<

N

(1.6.1) on

J.

E J

and L.1 =

max

to E J

lri(tJtoJqi(A))

IJ

i =

2’4,

where ri(tJtOJjri(h)), i = 1’3 (i = 2’4) i s the r i g h t ( l e f t ) maximal solution of rt = g i ( t J r ) ’ r ( t o ) = qi(A). Then cond i t i o n ( v i ) guarantees t h a t t h e L.’s a r e f i n i t e . By ( i ) and ( i i i ) , there e x i s t

(1.6.5)

Ni

>0

1

i = 1,2

3 4 such t h a t

> Lij

i = 1’2’

v.1(tjxj-Ni) > Lij

i = 3’4.

Vi(tjXjNi)

and

kt N >maX[Nijh]y i = 1J2j3y4. We Claim t h a t IX’(t)l 5 N on J where x ( t ) i s any solution of (1.6.1) with a ( t ) 5 x(t)

5

@ ( t ) on J .

There e x i s t s a

to E (a,b)

42

such t h a t

x(b)

- x ( a ) = x ’ (tO)(b-a)

1.6. LYAPUNOV-LIKE FUNCTIONS

and therefore it follows t h a t Ix' ( t o ) I 5 h < N. There are four cases t o be considered depending upon whether there e x i s t s

t2 such t h a t x ' ( t 2 ) = N or x ' ( t 2 ) = -N and whether t2 > to o r t2 < to. Assume, f o r example, x ' ( t 2 ) = N and a

t 2 > to. Then there e x i s t s a tl with that

xl(t,) = A

h < xl(t)

and

t

< N for t

(v) we obtain Vl(tl>x(tl)Jxl(tl)) arrive a t the estimate

5

< t 1 < t2 such

0 -

E

(t,,t,).

From

$1(~) and from ( i i ) we

using the theory of d i f f e r e n t i a l inequalities. Relations (1.6.5) and (1.6.6) lead us t o the contradiction L1 < v1(t2JX(t2)JX' ( t 2 ) )

because of the definitions of

5

L1

L1 and N.

On the basis of the preceding argument, we can arrive a t a similar contradiction i n the remaining three cases. the d e t a i l s .

We omit

The proof i s therefore complete.

The r e s u l t we have j u s t proved offers a general s e t of sufficient conditions which imply that

Ixl ( t ) I

i s bounded

and thus relaxes the more stringent Nagumols condition assumed i n Theorem 1.4.1.

In f a c t , the intent of the following

exercise i s t o c l a r i f y t h i s advantage f u r t h e r . EXERCISE 1.6.1.

Let

hi

E

C[R+,(O,m)], f

E

C[JxRxR,R],

and

Suppose t h a t

t

E

J

and

aJf3E C[J,R]

a(t)

5x5

with w ( t ) < B(t)

@ ( t ) J assume that

43

on J.

For

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

-

-h2(xf) c f ( t , x , x ' ) -h (-XI)

3

5 f ( t , x , x ' ) 5 hq(-xl)

Show t h a t for any solution x a ( t ) 5 x ( t ) 5 B(t) on J, only on cu,p,hi such t h a t Hint: -

hl(xl)

E

C(2)[J,R]

if

x'

if

x'

-> 0, -< 0 .

of (1.6.1) w i t h

there e x i s t s an N > 0 I x l ( t ) l L N on J .

depending

Under the assumptions show one can construct four

Qfapunov functions satisfying the conditions of Theorem 1.6.2. For example s e t V1 = exp[t - x + ./:' s ds/hl(s)] on El show the assumptions are verified w i t h g l ( t , r ) = r and and V4. similarly construct V2, V

and

3'

we may ccanbine Theorems 1.6.1 and 1.6.2 t o obtain the existence of solutions (1.5.1). "he proof i s similar t o Theorem1.5.1. Wemerely statethefollawingtheoremwhichisagener~izationofTheorem1.5.1.

THEOREM 1.6.3.

-

a,@ E C(l)[J,R] w i t h a ( t ) c B(t) Assume t h a t there e x i s t two l$fapunov f'unctions W1, W2 Let

on J . s a t i s Q i n g the hypotheses of Theorem 1.6.1. Suppose further t h a t there exist four Qrapunov functions Vi, i = 1,2,3,4 such t h a t they obey the hypotheses of Theorem 1.6.2. Then f o r any .(a) c c c p(a), u(b) 5 d 5 p(b) t h e BVP (1.5.1) has a solution x E C(*)[J,R] with a ( t ) c x ( t ) C p ( t ) and Ix'(t)l S N on J .

- -

-

-

1.7 EXISTENCE ON INFINITE INTERVALS On the basis of Theorems 1.4.1 and 1.5.1 it i s possible t o obtain the existence of solutions on i n f i n i t e intervals.

THEOREM 1.7.1. Assume t h a t f o r each b > a , f ( t , x , x ' ) s a t i s f i e s Nagumofs condition on [a,b] r e l a t i v e t o the p a i r a,B E C(l)[[a,m),R] with u ( t ) 5 B(t) on [a,-). Suppose 44

1.7. EXISTENCE ON INFINITE INTERVALS

also that a , @ are lower and upper solutions of (1.6.1) on [a,m), respectively. Then f o r any a ( a ) 5 c 5 @(a) the B W (1-7.1)

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

he8 a solution

x

on

E

x(a) = c,

C(2)[[a,m),R]

-

such t h a t a ( t ) c x ( t ) < p(t)

[a,m). Proof: -

By Theorem 1.5.1, it follows that f o r each n 1 1 there i s a solution xn E C(2)[[a,a+n],R] such t h a t xn(a) = c,

xn(a+n) = p ( a + n ) ,

and a ( t ) 5 x n ( t )

5 p ( t ) on [ a , a + n l .

-

Furthermore, there i s an Nn > 0 such t h a t I x t ( t ) l < Nn on [ a , a + n l f o r any solution satisfying a ( t ) 5 x ( t ) 5 p ( t ) on > 1, x,(t) is a solution on [a,a+n]. Thus f o r any fixed n [a,a+nI verifying a ( t ) 5 x,(t) 5 p ( t ) and Ix;(t)l 5 N~ on [ a , a + n ] f o r a l l m > n. Consequently, f o r m i n the sequences {xm(t)], {x;(t)] are both uniformly bounded and equicontinuous on [a, a + n ] Then, employing the standard diagonalization arguments, we obtain a subsequence which converges uniformly on a l l compact subintervals of [ a , ~ ) t o a solution x ( t ) . The desired solution of (1.7.1) is precisely this x ( t ) . Hence the proof is complete.

.

Essentially a similar proof may be given t o the following, s a t i s f i e s Nagumo's condition on [-a,a] f o r each a > 0 w i t h respect t o the pair @,p E C(l)[R,RI w i t h a ( t ) 5 p ( t ) on R. Suppose also t h a t (Y,B are lower and upper solutions of (1.6.1) on R . Then there i s a solution of (1.6.1) on R such t h a t a ( % ) 5 x ( t ) 5 B(t) on R. !MEOREM 1.7.2.

Assume t h a t

f(t,x,xt)

We shall merely s t a t e below a r e s u l t analogous t o Theorem 1.7.1 i n terms of Qfapunov-like functions.

45

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

THEOREM

on

[a,m)

1.7.3. L e t a,@

and l e t

f

f

c C(')[[a,-),R]

C[[a,-)xRxR,R].

with a ( t ) 5 p ( t ) Assume there e x i s t

two Wapunov functions Wl(t,x,xl), W2(t,x,x1) obeying the hypotheses of Theorem 1.6.1 where we now replace

[a, b 1 with

[a,m).

Assume also t h a t there e x i s t two Iiyapunov functions V2, V4 s a t i s w i n g the corresponding properties of !Theorem 1.6.2 w i t h [a,bl replaced by [a,-). Wre precisely suppose that ( i ) V2(t,x,x') subsets of

-

[a,m) x [x: a ( t )

+

(ii) D V2f(t,x,xl)

a ( t ) < x 5 B ( t ) and x' ( i i i ) V4(t,x,xt) 4 subsets of

as

-t-

uniformly on compact

+Q)

5 x 5 @(t)];

3 g2(t,v2(t,x,x1)) for t

2

E

[a,-),

0;

m

[arm) x [x: a ( t )

+

XI

as

x'

5x5

uniformly on compact

--f-c=~

p(t)l;

( i v ) D V4f(t>x,x1)2 g4(t,v4(t9x,xt)) f o r t E [a,m), a(t) 5 x B(t) and x ' 5 0; (v) there e x i s t qi, i = 2,4 where qi e C([a,m) xR,R] and Qi(t,u) i s increasing i n t for each u such t h a t f o r each

2 a, Vi(t,x,x') e qi(T, Ix' I ) 5x5 T

for

t

E

[a,T]

and

(vi) gi E C[[a,-) XR,R], i = 2,4 such t h a t f o r each t 0 e [a,-) and ro 2 0 a l l solutions of r' = gi(t,r), r ( t o ) = ro e x i s t on [a,tol. !Then for any

c

satisf'ying

a(.)

6 c 5 B(a) there e x i s t s

-

a solution of (1.7.1) such that a ( t ) c x ( t )

5 B(t)

on

[a,-).

Clearly a theorem analogous t o Theorem 1.7.2 may be formulated, which we leave as an exercise.

1.8 SUPER- AND SUBFUNCTIONS Here we define subf'unctions and superfunctions r e l a t i v e t o the solutions of XI' = f ( t , x , x ' ) and discuss necessary and 46

1.8. SUPER- AND SUBFUNCTIONS

sufficient conditions I o r such functions t o be, respectively, lower and upper solutions.

To avoid repetition, most of the

results w i l l be stated only i n terms of lower solutions and When it becomes necessary l a t e r t o r e f e r t o a

subfunctions.

result concerning superfunctions and upper solutions, we s h a l l simply r e f e r t o the subfunction statement of the r e s u l t .

DEFINITION 1.8.1.

A function

function r e l a t i v e t o solutions of val J

[tl,t2]

i f f o r any

C

Cp(t)

is said t o be a

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

2-

on an inter-

and f o r any solution

J

.)> cp(t.), i = 1,2 implies x ( t ) 2 cp(t) C(2) [[tl,t21,R] , x (t 1 1 on [t t 1. A superfunction may be defined similarly by re1 ' 2 versing the respective inequalities.

x

E

We have immediately the following r e s u l t . THEOREN

1.8.1.

Assume t h a t

cp c C[J,R] I I C(l)[J0,R]

is a

subfunction on J w i t h respect t o the solution of

(1.8.1)

XI1

where f E C[J X R xR,R]. (1.8.1) on J . Proof:

Let

Then

to 6 Jo

sufficiently small.

= f(t,X,X') Cp

and h > 0, k > 0, h

-

E

C(2)[[t0-k,t0+h],R].

subfunction, we readily obtain Cp'

-

be

x ( t O+ h ) = q ( t o + h ) ,

x ( t O k) = g ( t 0 k),

has a solution x

+k >0

Then by Theorem 1.1.2 t h e BVP

-

X" = f(t,X,X'),

i s a lower solution of

-

(to+ h ) Cp' ( t o k) h+k

->

X' (to + h )

X'

cp

is a

-

(to k)

h+k =

for some t o- k < s < to + h.

-

Since

x'l(s) = f ( s , x ( s ) , x ' ( s ) )

!the continuity of 47

f

together

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

with "heorem 1.1.2, implies that f ( s , x ( s ) , x t ( s ) ) --3 f ( t O , x ( t O ) , x t ( t O ) )

as h + k

+ 0.

Thus

and, i n particular,

from which it follows t h a t (1.8.1) on J.

i s a lower solution r e l a t i v e t o

cp

Assuming only the continuity of f , it w i l l not be possible i n general t o show t h a t a lower function is a subf'unction. Since a solution i s a lower solution, if lower solutions are subf'unctions, then solutions are subf'unctions From the definition of subfunctions it would then follow t h a t , if a BVP on an i n t e r v a l has a solution, thet solution is unique. Hence stronger assmptions other than continuity of f are required t o conclude that a lower solution i s a subfunction. A s e t of sufficient conditions i s given i n the next theorem.

.

THEOREM 1.8.2.

Assume t h a t

( i ) f E C[JxRxR,R] and i n x f o r each (t,g); ( i i ) either f satisfies a i n y on each compact subset of i n i t i a l value problems f o r (1.8.1) Then a lower solution on J t o solutions of (1.8.1).

f(t,x,y)

i s nondecreasing

one-sided Lipschitz condition J x R x R o r solutions of are unique.

is a subfunction on J r e l a t i v e

The proof i s a direct consequence of Theorems 1.2.2 and 48

1.8. SUPER- AND SUBFUNCTIONS

1.2.3.

I n t h e preceding r e s u l t conditions a r e imposed on f ( t , x , y ) which a r e s u f f i c i e n t t o imply t h a t lower solutions a r e subf'unctions and, therefore, t h a t solutions of BVP's, when they

I n t h e next r e s u l t we take t h e uniqueness

exist, a r e unique.

of BVP's as one of t h e hypotheses. THEOREM 1.8.3. Assume t h a t each i n i t i a l value problem f o r (1.8.1) has a s o l u t i o n which extends throughout [a,b]. Suppose

further t h a t solutions of boundary value problems X"

= f(t,X,X'),

x ( t l ) = xl,

~ ( 6 =~x2, )

[tl,t2] C J,

when they e x i s t , a r e unique.

Then, i f

i s a lower solution on

i s a subfunction on I.

Proof:

Assume t h a t

there i s an i n t e r v a l such t h a t (c,d).

CY E

is a

x,(c)

C(l)[[c,d],R],

>0

cy

[c,d]

and

i s not a subfunction on C

I

and solution

F(t,x,x')

on

C(l)[I,R]

cy E

I.

Then

xo E C(2)[[c,d],R]

and x,(t) [c,d] X R x R

< a(t)

on

by

i s continuous on [c,d] X R X R and it follows from Theorem 1.1.2 t h a t there

F(t,x,x') 6

cy

= cy(c), xo(d) = cy(d),

We now define

Since

I,

IC J

such t h a t

[tl,t2]

C

[c,d]

and

t2-tl 5

6

implies the BVP X"

= F(t,x,x'),

has a solution

x

x(t,)

= Q(tl>,

E C(2)[[tl,t2],R].

x ( t 2 > = a(t2),

Using t h e f a c t t h a t

a ( t ) 5 x ( t ) on with t h e same type of argument as used i n t h e proof

i s a lower solution, we can show t h a t [tl,t2]

Ly

of Theorem

1.3.1.

As a r e s u l t , f o r 49

[t1,t2] C [c,d]

and

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

t2- tl

5 XI1

t h e BVP

6, =

f(t,X,X'),

has a s o l u t i o n x

x(tl) = a(tl), C(2)[[t,,t,],R]

E

x ( t 2 ) = a(t,),

with

a(t) < x(t)

on

[tl,t2]. Thus a i s a subfunction " i n t h e small." Clearly d c > 6; f o r otherwise t h e r e would be a s o l u t i o n

-

x(t)

with

[c,d].

x(c) = a ( c ) , x(d) = a(d),

an% x ( t )

-> a ( t )

This s o l u t i o n would be d i s t i n c t from x ( t ) 0

on

contra-

d i e t i n g t h e assumption concerning t h e uniqueness of s o l u t i o n s of BVP's.

Now f o r each p o s i t i v e i n t e g e r

n,

let

P(n)

t h e proposition t h a t t h e r e e x i s t s an i n t e r v a l [cn,dn] with 0 < dn - c < d - c - (n 1)s and a s o l u t i o n n-

-

x n on

E

C(2)[[cn,dn],R]

such t h a t

d i s t i n c t s o l u t i o n with boundary values

X"

C

[c,d]

x (c ) = a ( c n ) , xn(d ) = a ( t ) n n n i s t r u e with [cl,dl] = [ c , d ]

( c , d ). Evidently P(1) n n and x 1( t ) = x o ( t ) . Assume P(k) i s t r u e . Then otherwise we would o b t a i n a c o n t r a d i c t i o n o f x , ( t )

z,(t)

be

a(ck)

4,- Ck >

6,

being t h e

and a ( % ) .

Let

be t h e s o l u t i o n of t h e BVP =

f(t,X,X'),

x(c,)

=

X ( Ck + 6) = a ( c k + 6 ) .

CU(C,),

Since each i n i t i a l value problem has a s o l u t i o n extending throughout t h a t z,(t)

J,

i s not t r u e .

t h e r e i s a s o l u t i o n z 2 ( t ) on [ck,%I such z,(t) on [ c k , c k + 6 ] . Suppose t h a t P ( k + l ) Then we must have

Also we must have

z2(%)

z2(t)

> "(4,). If

2 a ( t ) on [ c k + 6 , 4 , ] . 4, - ck - 6 5 6, t h e

BVP

E C ( 2 ) [ [ ~ k + 6 , 4 , ] , R ] with a ( t ) < z ( t ) . z3 - 3 Then, s i n c e z ( ) < z2(4,) and s o l u t i o n s of BVPls a r e unique 3 % t E cCk+6,%1. a(t) 5 z 3 w 5 z , ~ ,

has a s o l u t i o n

50

1.8 SUPER- AND SUBFUNCTIONS

This implies z (Ck+6)

3

= z2(Ck+6)

Consequently,

u(t),

and

Z'(C

3

k

+ S ) = Z'(C +6).

2

k

defined by

i s of c l a s s C(2)[[ck,%],R]

and i s a s o l u t i o n on

[ck,%]

However, u ( t ) # \ ( t ) with u ( ck ) = %(ck), u($) = \(\). on rek,%] and t h i s c o n t r a d i c t s t h e uniqueness of s o l u t i o n s of BVP's.

We conclude t h a t

% - ck >

This being t h e case,

6.

the BVP X"

= f(t,X,x'),

has a s o l u t i o n

x(ck+ S ) = a(ck+ S),

z4

E

X(C +26) k

C(2)[[c + 6,ck+26],R]

k

+ 6) 4 k

= z;(ck+ 6).

v(t) =

!Phis s o l u t i o n P(k+1)

on

v(t)

z4(ck + 6 ) = z2(ck + 6) Hence

z,(t)

on

[ck,ck+61,

z4(t)

on

(ck+ 6,ck+281,

has an extension

z (t)

5

a(ck+28),

with

This again assures us t h a t z'(c

=

and

on J .

Since

i s assumed t o be f a l s e , we must have

[ck+2S,%]

and

z,(%)

> a(\).

z5(t) 2 a ( t ) I n t h i s way t h e fore-

going arguments can be repeated and t h e assumption t h a t 51

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

P(k+l)

i s f a l s e permits us t o work our way across the

interval

[ckJ%] by subintervals of length

6 u n t i l we

> a ( t > on a(%) = \(%).

obtain a solution w c c(*) [ [ c k J % l , ~ 1 with w ( t >

[C,J%]J w(ck) = a ( ck = s ( c k ) and w(%) = xJt) on [c,J%] which again i s a contradiction Then w(t)

+

t o uniqueness of solutions of BVP’s. We therefore conclude t h a t , i f i s true.

P(k+l)

leads t o the contradiction 0 Hence a

P(k)

i s true, then

i s true f o r a l l n

Thus P(n)

i s a subfunction on

2

1, which

< d - c - (n - 1)s f o r a l l n > I

1.

and the proof i s complete.

Show t h a t i n Theorem 1.8.3 the hypothesis

EXERCISE 1.8.1.

t h a t each i n i t i a l value problem f o r (1.8.1) has a solution which extends throughout

can be replaced by Na@;umo’s condition.

J

1.9 PROPERTIES OF SUBFUNCTIONS Before proceeding t o outline the Perron’s method, it w i l l be necessary t o make a more detailed examination of the propert i e s of subt h i s section.

and superf’unctions.

This w i l l be undertaken i n

Again most r e s u l t s w i l l be stated i n terms of

subf’unctions and the obvious analogs f o r superf’unctions w i l l be omitted. IMEOREM I C J,

then

1.9.1. cp

If

cp

has r i g h t and l e f t limits i n the extended

r e a l s a t each point i n

Io and has appropriate one-sided

limits a t the end points of

Let

i s a subfunction on an i n t e r v a l

I.

Proof: Clearly it i s sufficient t o consider one case. to E Io and suppose t h a t cp(t-) 0 = limt+t_, cp(t) does

not e x i s t i n the extended r e a l s . numbers a,@ such t h a t

52

Then there e x i s t r e a l

1.9. PROPERTIES OF SUBFUNCTIONS

cp(t) < a < p < l i m sup cp(t)

lim inf t4t; It,},

Let

{s,]

t4;

be s t r i c t l y increasing sequences i n

I

t n < sn < tn+l f o r n 2 1, l i m t n = lim sn = to,

that

iim cp(tn) Taking

iim sup cp(t) t-tt;

=

t ( B -a),

E =

6> 0

is a

x" = f(t,X,X'),

has a solution x $(a+B)I

it follows from Theorem 1.1.2 t h a t there

<

[t,,t2]

C

[tl,tO]

with

the BVP

t2 tl

(1.9.1)

lim cp(sn) = lim inf cp(t). t+t;

and

such t h a t f o r any

- -< 6

such

E

E

on

x ( t 1) = x ( t 2 ) = $ ( a + p ) ,

C ( ~ ) [ [ ~ , , ~ ~ I satisfying ,R]

[tl,t2].

Let

Ix(t)

-

n be a fixed positive integer

-

sufficiently large so t h a t

sn+l sn -< 6, rp(sn) < a, q ~ ( s ~ <+a~,) and cp(tn+l) > p. This implies t h a t there i s a solution of (1.9.1) with [ t , t 1 = [S , S 1 and 1 2 n n +l Since ]. cp i s a subfunction lx(t) -$(a + B) I < E on [ S ~ , S ~ + ~ on 1 and cp(sn) < X(Sn)' cp(sn+l) < X(sn+l), it must be the case t h a t

cp(tn+l)

5 x(tn+l).

X(tn+l> <

3. + B )

However, + E

< B < cp(t,+,>,

a contradiction from which we conclude extended r e a l s .

cp(t',)

e x i s t s i n the

The proof i s complete.

COROLIARY 1.9.1. If cp i s a bounded subfunction on IcJ, Cp has a t most a countable number of discontinuities on A t each to E I0 cP(tO) 5 max[T(ti),Cp(t,)l.

then I.

Proof:

The f i r s t assertion i s a c l a s s i c a l r e s u l t t h a t

follows from the f a c t t h a t

+

cp(to) 53

and

cp(ti)

e x i s t a t each

to

E

0

I

.

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

By Theorem 1.1.2 and t h e f a c t t h a t

i s a sub-

cp

function, t h e second a s s e r t i o n r e a d i l y follows. We consider next t h e d i f f e r e n t i a b i l i t y of subfunctions For a f'unction

with a f i n i t e r i g h t l i m i t

h(t)

h(t:)

.

at

to, we define

-

+

Dh(tO) = lim

+

h ( t ) h(t;) t-t 0

t+tO provided t h e l i m i t e x i s t s .

THEOREM 1 . 9 . 2 . then

to

E

Dcp(t:), 0

I

.

cp

Proof:

i s a bounded subfunction on

I c J,

e x i s t i n t h e extended r e a l s f o r each

I. A s i n Theorem l . g . l w e s h a l l consider only one

Suppose t h a t

lim i n f

+

t-tt

to

t

E

- to

Io and t h a t

< l i m sup t-tt

0

Let

Dh(ti).

The appropriate one-sided derivatives e x i s t a t f i n i t e

end points of

case.

If

Dcp(ti)

Analogously we define

+

-

cp(t> cp(to>

t

- to

0

m be a r e a l number s t r i c t l y between these two limits.

Then t h e i n i t i a l value problem x" = f ( t , x , x ' ) , has a solution

+

x(t,) = cp(tO), x ' ( t o ) = m

x E C ( ~ ) [ [ ~ ~ , ~ ~ f+o r~ some I , R I6

54

>

0.

AS

1.9. PROPERTIES OF SUBFUNCTIONS

there e x i s t

t 1, t2, t3 such t h a t

It follows t h a t

x(t2).

Since

cp

to < t < t2 < t < t 3

1

0

cp(t,) < x(t,), cp(t,) < x ( t 3 ) , and cp(t2) i s a subf’unction t h i s i s a contradiction

+

6,

>

and therefore we are done.

The c l a s s i c a l r e s u l t s i n the theory of r e a l variables give as a consequence the following corollary.

COROLLARY 1 . 9 . 2 . If cp i s a bounded subfunction on I c J, then cp has a f i n i t e derivative almost everywhere on I. LEMMA 1.9.1. If

cp

i s a subf’unction on

I

bounded above on each compact subinterval of

I,

i s a subfunction on

I.

y(t) = l i m

cp(s)

Proof:

Let

[tlyt21

C

J

and i s

then

I a n d l e t x ( t ) b e a s o l u t i o n o f (1.8.1)

with y(t,) 5 x(t,) and y ( t 2 ) 5 x ( t 2 ) . Then cp(ti) 5 y ( t i ) 5 x(ti), i = 1,2, and, because cp(t) i s a subf’unction,

5 x ( t ) on [t,,t,]. We therefore have f o r each tl < t < t2, y ( t ) = l i m sups+t cp(s) 5 x ( t ) . Hence y ( t ) 5

p(t) x(t)

on

[tl,t2]

and y ( t )

i s a s u b f b c t i o n on

I.

We now proceed t o discuss some l a t t i c e properties of subf’unctions.

THEOREM 1.9.3. Assume t h a t the s e t of subfunctions 55

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

on the i n t e r v a l I C J i s bounded above a t each I. Then cp,(t) = supCYEA cpCY ( t ) i s a subf’unction on

{cpCY: CY E A]

point of

I. I and x E C(2)[[tl,t2],R] Suppose t h a t [tl,t2] i s a solution w i t h cp,(t) < x ( t ) a t t = tl,t2. By the definition of q0, we then have cpCY(t)5 x ( t ) a t t = t t 1’ 2 f o r each CY E A . Since each cpCY i s a subfunction on I, it follows t h a t cpCY(t)5 x ( t ) on [tl,t2] f o r each CY E A . This implies cpo(t) < x ( t ) on [tl,t,] and cpo(t) i s a subfunction on I.

Proof:

IIHEOREM 1.9.4.

I&

cp be a subfunction on an i n t e r v a l

I1 w i t h I1= i1n I. Suppose further that cp,(t) 5 cp(t) a t f i n i t e end points of I1 which are contained i n I. Then cp2 defined by IC J

and cpl

a subfunction on

cp,(t> =

{

~“P1(t)>cp(t)l>

I1J

t

cp(t>,

E

I - 11,

i s a subf’unction on I. Proof: By h y p t h e s i s cp,(t) = cp(t) i s a s u b m c t i o n on I I1 and, by Theorem 1.9.3, cp2(t) i s a subfunction on I1. Consequently, we need only t o show t h a t we have the correct behavior on intervals [tl,t,] C I which are not contained i n e i t h e r I1 o r I-I1. Again l e t us consider j u s t one case since the arguments are similar. Assume tl E 11, t2 E I 11,

-

-

56

1.9. PROPERTIES OF SUBFUNCTIONS

i s a subf'unction on

In particular,

-

cp(t) < x ( t )

we obtain

I,

5

cp2(t) = cp(t)

x(t)

on

on

(t3,t2].

[t,,t,].

Also

'~l(t35 ) c ~ ( t 3 )_< x(t3) and cpl(t1) 5 ~ 2 ( t l 5 > x ( t l > , hence cp,(t) 5 x ( t ) on [ t t I because cpl(t) i s a subfunction 1'

3

cp2(t) = max[cp(t), cpl(t)] < x(t)

We thus i n f e r t h a t

on

I1.

on

[tl,t3].

Putting these things together we see t h a t

cp2(t) 5 x ( t ) on [tl,t2]. The other p o s s i b i l i t i e s a r e dealt with i n a similar way and we conclude t h a t cp2(t) i s a subfunction on

I.

MEOREM

(i) x

f

for each

1.9.5. E

Suppose t h a t

C[J xR x R , R ] , f ( t , x , x I )

i s nondecreasing i n

(t,xl)

and ( i i ) t h a t solutions of i n i t i a l value problems are unique.

Then, i f

$(t)

i s an upper solution on

cp(t> i s a bounded subfunction on and cp(t;> 5 $(t,>, we have

Proof:

Suppose t h a t

[tlJt2]

[tlJt2]

cp(t1.5 $ ( t )

cp(t) > $ ( t )

C

J

and

with cp(t1) on

5 $(tl>

(tl't2).

a t some points of

(tlJt2). Observe t h a t it i s enough t o consider the case where cp i s upper semicontinuous on (tl,t2). To see t h i s , l e t

cp(t) = l i m sup cp(s) on (t,,t,). Then by Lermna 1.9.1, s+t -cp(t) i s a subfunction on (t t ). By Corollary 1.9.1, 1' 2 - + + cp(tl>= cp(t,> 5 $(tl> and F(t; cp(t;> 5 J'(t2). mreover, rp(t) > @ ( t ) a t some points i n

(t t ).

at some points i n

rp(t;)

5

(tl,t2).

a t some points i n

$(t,)

implies

cp(t)

[to

- 6, to+ 61

6

>0

(tlJt2),

and 0

57

This being the case,

(tl,t2)

- $(t)

which i s assumed on a compact s e t Then there e x i s t s a

implies T ( t ) > q ( t )

As a r e s u l t we may assume cp(t)

1' 2 i s upper semicontinuous on

cp(t>> + ( t )

(tlJt2)

<

and

cp(tf)

5

$(tl>,

has a positive maximum M E C (tl,t2).

E

>0

&

< $(to+ E ) +

Let

to= lub E.

such t h a t M - 9(t0 + 6 ) ~

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

and such t h a t the BVP

has a solution x

C(2) [[to

E

Since

by Theorem 1.1.2.

- 6,

$(t)

+

to+ 6],R]. This i s assured M i s an upper solution, we

+

x(t) < - $(t)

obtain by Theorem 1.2.3,

However ,

V(t0

- 6) 5 $(to - 6)

cp(t

+ 6) < +(to + 6) +

+

on

M

( t o - 6, to+ 6).

-

M = X(tO 6)

and

which yields

0

5 x(to),

cp(to)

subfunction on cp(to) = + ( t o )

+

M-

E =

X(tO + S ) ,

because of the f a c t

t0 We therefore conclude t h a t

(tlJt2). M.

This contradicts

E

cp(t) i s a E

-

q ( t ) < $(t)

(tlJt2).

On

and

EXERCISE 1.9.1. Prove t h a t the assertion of Theorem

1.9.5 remains valid

i f i n place of the assumption "that

solutions of i n i t i a l value problems are unique" we suppose that to

f (t,x,x')

XI

s a t i s f i e s a Lipschitz condition with respect

on each compact subset o f

J X R xR.

We s h a l l now discuss properties of bounded functions t h a t are subfunctions and superfunctions simultaneously.

We w i l l

need a well-known r e s u l t concerning solutions of i n i t i a l value problems which we merely s t a t e . LEMMA 1.9.2.

6

> 0, % >

0,

If

and

%>0

i n i t i a l value problem i s defined on

I

6

=

(tOJXOJxi)E J X R X R ,

such t h a t every solution of the

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

[to

- 6,

there exist

to+ 61 58

n

J.

x ( t o ) = xo, x f ( t o ) = xd Moreover,

1.9. PROPERTIES OF SUBFUNCTIONS

THEOREM 1.9.6.

Assume t h a t

f(t,x,xt)

i s such t h a t

C(2)

solutions of boundary value problems, when they exist, are unique. I

and t h a t

J

C

Suppose t h a t z(t)

i s bounded on each compact subset

z(t)

i s simultaneously a subfunction and a

superfunction on

I. Then z ( t ) i s a solution of x " = f ( t , x , x t ) on an open subset of I the complement of which has measure zero.

Furthermore, i f

z(t)

a t which

z(t)

to E Io i s a p i n t of continuity of does not have a f i n i t e derivative, then

either

+

Dz(tO) = Dz(t-) = + m 0

+

+

If z ( t o ) > z(t,)J D z ( t 0 )

+

DZ(tO) = Dz(t')

=

0

Proof:

-

+

or =

D z ( t ) = Dz(t-) = 0

Dz(t-) 0

=+m

0

and

z(ti)

-

W.

< z(t,),

W.

By Corollary 1.9.2,

z(t)

has a f i n i t e derivative

to E Io i s a point a t which z ( t ) has a f i n i t e derivative, there i s a 6 > 0 such t h a t almost everywhere on

[ t o -6, t 0 + 6 1 and

I.

I J Iz(t)l

If

5

Iz(to)l

1 On

[ t o - 6, t 0 + 6 1 ,

It then follows from Theorem 1.1.2 t h a t there i s a 0

<

45

6, X"

61

= f(t,x,x'),

has a solution x(t)

a solution of

0,

x

E

C(2)[[to

~ ( t , + 6 ~= ) ~ ( t ~ + 6 ~ ) ,

- 61,t0+61]JR].

Since

z(t)

a t the same time a subf'unction and a superfunction on z(t)

>

such t h a t the BVP

on

[to

- 61Jt0 + S1].

xtt = f ( t , x , x t ) 59

We conclude t h a t

on an open subset of

is

I, z(t) I

the

is

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

complement of which has measure zero.

to f 'I be a point of continuity of z ( t ) at which ~ ( t )does not have a f i n i t e derivative. By Theorem + 1.9.2 Dz(tO) and Dz(ti) both exist i n extended r e a l s . If Next l e t

both are f i n i t e , then by the same argument as used above there

i s an i n t e r v a l around to i n which This contradicts the assumption t h a t

Dz(t0)

Dz(t:)

=

+m

z(t)

i s infinite.

and Dz(ti)

# + m.

Then there e x i s t numbers

> w(t)

>0

f(t,X,X'),

x(tO

has a solution x1 there i s a

b2

solution on

>0

-

E

- 4)=

C(2)[[to

= f(t,X,X'),

41

W(tO

such that the x(to) =

Z(to),

1.9.2, can be extended t o be a

x,(t)

By Lemma

and such t h a t a l l solutions of

x ( t o ) = Xl(tO)'

[ t O , t O+ 6*1

- 61)y

- 61,tO],R].

such t h a t

[to 61 ,t 0 + the i n i t i a l value problem

e x i s t on

-

4, 0 < 61 5 6,

BVP

XI1

6

on to 6 5 t < to.

= z(to) + N ( t -to)

By Theorem 1.1.2, there i s a

=

does not have a

such t h a t

and N

XI1

z(t)

to. Consequently, a t l e a s t one of To be specific, suppose

f i n i t e derivative a t Dz(t:),

i s a solution.

z(t)

and s a t i s f y

x'(to) =

lx(t)

X 1' ( t

0 ),

- x(to) I 5 M ( t - t o )

[ t 0 , t 0 + 6 2 ] y where M = I x ' ( t ) I + 1. Again applying 1 0 Theorem 1.1.2 we i n f e r t h a t there i s an q, 0 < 7 5 min[fj ,6 1, 1 2 such t h a t f o r 0 < 6 < q the BVP

on

3-

X" = f(t,X,X'),

X(tO

- 63 ) = Xl(tO - 63),

x ( t O+ 6 ) = Xl(tO)

3

60

+

(M+ &)6

3

1.9. PROPERTIES OF SUBFUNCTIONS

has a solution E

> 0.

that

As 0

<

x2

- 63 , t0 + 63 ],R]

f o r some fixed

E C ( 2 ) [[to

4

Dz(ti) = + m, we can assume t h a t i s chosen s o 5 q and xl(tO) + ( M + & ) 6 < z ( t 0 + 6 ). Then

4

3

3

xl(tO + 6 ) < x 2 ( t 0 + 6 )

3

3

< z ( t O +6 ) 3

and

- 9'

Xl(tO

=

X2(t0 - 6+

5

- 4,.

z(tO

The l a s t inequality follows from the f a c t t h a t

- 9)

~ ( t )i s a

- 4).

superfunction and xl(tO) = z ( t o ) , xl(tO < z(tO Since ~ ( t )i s a superfunction, we deduce from the above inequalities t h a t

-< z ( t )

x2(t)

on

[to

- 63 ,t0 + 63 ].

From the

same inequalities and the f a c t t h a t the solutions of BVPIs, when they exist, are unique, we also i n f e r t h a t on

[to - 6 ,t + 6 3

3I.

0

x i ( t o ) = x;(to),

Thus

hence,

xl(t) 5 x 2 ( t ) x (t ) = z ( t o ) = x2(t0) and 1 0 x 2 ( t ) i s a solution of t h e i n i t i a l

value problem with the i n i t i a l conditions x l ( t o ) = x ' ( t ). 1 0

However,

Ix2(to +

4,- X2(t0) I

=

x(t,) = xl(tO),

(M+ €183

which contradicts the f a c t t h a t a l l solutions of t h i s i n i t i a l Ix(t) - xl(tO) I 5 M ( t - t o ) on We are forced t o conclude t h a t 'Dz(t;) = + m.

value problem s a t i s f y

[ t O , t O+ 4 1 . By similar arguments using t h e f a c t t h a t

z(t)

i s also a

subfunction, the other statements regarding the behavior of a t a point of continuity can be established.

z(t)

We consider now the behavior of If

z ( t ) i s bounded on each compact inter+ Furthermore, z ( t o ) # z ( t 0 ) since, by Corollary

and are f i n i t e since Val of

a t p i n t s of

to E Io i s a p i n t of discontinuity of and z ( t i ) both e x i s t then by Theorem 1.9.1, z(t:)

discontinuity. z(t),

z(t)

I.

1.9.1, 61

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

+ > .(ti)

Assume t h a t

is a z(t)

6

z(to)

>0

< w(t)

and an N =

= f(t,x,xf),

has a solution Xl(t0) = z(t:) and

z(t)

x

2

Dz(ti)

6

Then t h e r e

[tO,tO + 61 C I and on to < t 5 to + 6. By

-to)

4,

0

<

45

such t h a t t h e BVP

6,

x(tO+fjl) = w ( t O + 61),.

x(to) = z(ti), 1

# + m.

such t h a t

+ z(to) + N ( t

Theorem 1.1.2 t h e r e i s a XI'

and t h a t

C ( 2 ) [ [ t o , t o + 611,R].

x1 (t0 '61)

z(to),

i s a subfunction,

x,(t)

= w(to+61)

1 z(t)

on

>

z(to+61),

[to,to+61].

Now proceeding as i n t h e previous paragraph and using the f a c t that

z(t)

i s a subfunction, we can obtain a solution of t h e

i n i t i a l value problem XIt = f ( t , X , X ' ) ,

x ( t o ) = Xl(tO),

X'(t0) = x;(to),

t h e graph of which i s not contained i n t h e sector t o t h e l e f t of

to in which such solutions must be.

diction, we conclude t h a t

+ Dz(tO) = +=J.

From t h i s contraThe other assertions

concerning derivatives a t points of discontinuity of are d e a l t with i n a similar w a y .

z(t) The proof of t h e theorem i s

now complete. 1.10

PERRON'S METHOD mploying t h e properties of sub-

and superfunctions and

the existence " i n the small" theorem, we consider the existence

i n the large f o r t h e boundary value problems by Perron's method. DEFIXITION 1.10.1.

A bounded real-valued function

62

1 .lo. PERRON'S METHOD

defined on J

i s said t o be an underfunction with respect t o

the BVP (1.10.1)

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

where

C[J X R xR,R],

f E

i s a subfunction on J

.(a)

= A,

i n case

v(a)

x(b) = B,

5 A,

cp(b)

5

and cp

B,

r e l a t i v e t o (1.8.1). An overfunction

i s defined similarly i n an obvious way. THEOREM 1.10.1.

x"

=

f(t,x,x')

unique.

Assume t h a t solutions of BVP's f o r

on subintervals of

when they e x i s t , are

J,

Suppose t h a t there e x i s t both an underfunction cpo

and an overfunction $ r e l a t i v e t o BVP (1.10.1) such t h a t 0 cpo(t) 1. q0(t) on J . Let Q be the s e t of a l l underfunctions cp

such t h a t

cp(t) < $o(t)

i s simultaneously a sub-

function on J.

a solution x

5

E

z(t,),

W e define

Then

z ( t ) = sup

and superfunction on J.

It follows from Theorem 1.9.3 t h a t

Proof:

subfhnction on J .

x(t2)

on J .

zl(t)

Suppose now t h a t

z(t)

but

x(t)

>

such t h a t

z(t)

[tl,t2]

is a C J

x(tl) < z(tl),

a t some points of

LtlJt21

on J Then by Theorem 1.9.4, zl(a) = z(a)

zl(t)

-< A,

- Ctl,t21.

i s a subfunction on z (b) = z(b) 1

-< B.

and

(tl,t2).

on J by

Zl(t) = { ~ ~ ~ ( t ) J z ( t ) l on

kreover

z(t)

i s hot a super-

Then there i s a subinterval C(2)[[tlyt2],R]

cp(t) v=Q

J

and

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

and

jro

[tl,t2].

a superfunction implies t h a t Consequently, we have

we i n f e r t h a t x(t)

> z(t)

which proves

x(t)

5

z,(t)

5

jro(t)

jro(t)

on

on Hence,

J.

E 0 and z,(t) < z ( t ) . However, z1( t ) = 1 a t some points i n (t,,t2), a contradiction

z

i s a superfunction on

z(t)

J.

This proves

t h e theorem. The function

DEFINITION 1.10.2.

obtained i n t h e

z(t)

preceding theorem depends on t h e BVP (1.10.1) function

We s h a l l say t h a t

jr0.

z(t)

and on t h e over-

i s a generalized

solution of the BVP (1.10.1) and we s h a l l designate it by z(t;JIO). Notice t h a t since

z(t;jr ) 0

i s a subfunction and a super-

function a t t h e same time, the assertions made i n Theorem

1.9.6 apply t o z(t;Jro). We therefore have t o consider the behavior of

a t t h e end points of

z(t;jro)

J = [a,b].

We

discuss t h i s i n t h e following theorem. THEOREM 1.10.2.

Assume t h a t the hypotheses of Theorem

1.10.1 are s a t i s f i e d and l e t

z ( t ) = z(t;Jro) be the corresponding generalized solution of (1.10.1). Then .(a) = A. If

Dz(a+)

Hence, i f

z(a+)

#+m,

Dz(a+)

-< .(a).

i s finite,

t

assertions are t r u e a t Proof: cp*(a) = cp(a)

cp*(t)

+

c, c

cp(t)

Assume now t h a t z(t)

6,0

Dz(a+)

< 6 < b - a,

< w(t)

= z(a+)

-m.

Similar

i s a subfunction on J,

cp*(t) = cp(t)

on

i s also a subfunction.

gether with the d e f i n i t i o n of there i s a

=

= b.

defined by

> 0,

< A, Dz(a+)

z(a+) = z ( a ) = A.

Observe t h a t i f

the function

z(a+)

If

z(t),

# +m

and

and an

+ N(t 64

yields

-a),

N

z(a+)

(a,bl

This, t o -

z ( a ) = A.

>

~ ( a ) . Then

such t h a t a

and

< t 5 a + 6.

1.10. PERRON'S METHOD

By Theorem 1.1.2 it follows t h a t f o r

0

<

E

< z(a+) -.(a)

and 0 < 6 < 6 s u f f i c i e n t l y small, t h e BVP 1-

has a solution x

E

C(2)[[a,a+61],R]. z ( t ) _< x ( t )

f'unction, we have z(a+)

5

on

x(a+) = .(a)

This contradiction proves t h a t Finally, suppose t h a t

[&,a+ = z(a+)

Dz (a+)

Z(a>* z(a+)

Since

41

z(t)

i s a sub-

which yields

- E.

# +m

implies

< A and Dz(a+)

# - m.

using an analogous argument as above and the f a c t t h a t

i s also a superfunction on J, implies

-

Dz(a+) =

If Dz(a+)

z (a+) 5

we can conclude t h a t

Then

~ ( t ) z(a+) < A

m.

i s f i n i t e , then combining the assertions of

t h i s theorem we arrive a t

A = .(a)

i s complete.

> A. -> z(a+) -

The proof

From the preceding r e s u l t s it i s clear t h a t Perron's method of studying t h e BVP (1.10.1) canbe separated i n t o twoparts. The f i r s t p a r t dealswiththeproblemofestablishingtheexistenceof an overfunction qo and an underfunction cP,(t) 5 q o ( t )

on J.

rp 0

such t h a t

The second part consists of finding

conditions under which the generalized solution of class

C(2)[J,R]

and i s a solution on J .

z (t;qo)

is

I n view of

Theorems 1.9.6, 1.10.1, and 1.10.2, accomplishing t h i s i s equivalent t o showing t h a t Dz(t-)

i s f i n i t e on

Dz(t+)

i s f i n i t e on

[a,b)

and

(a,b].

Let us f i r s t give s u f f i c i e n t conditions f o r the existence

of under-

and overf'unctions

LEMMA 1.10.1.

nondecreasing i n x

Let

.

f E C[JxRxR,R]

f o r fixed 65

(t,x').

and f ( t , x , x ' ) be Assume t h a t f ( t , x , x ' )

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

i s such t h a t lower and upper solutions of the d i f f e r e n t i a l equation x'' = f ( t , x , x 7 ) are subrespectively.

and superfunctions,

Then,

i s a solution of XI' = f ( t , x , x I ) (a) i f u E C ( 2 ) [ J , R ] on J, there e x i s t overf'unctions and underf'unctions r e l a t i v e t o any BVP on J; (b)

i f there i s a

(1.10.2)

k

>0

such t h a t

5

If(t,O,x') -f(t,O,O)l

t

klx'),

E J,

XI

E

R,

there e x i s t overfunctions and underfunctions r e l a t i v e t o any BVP on

J.

For any s u f f i c i e n t l y large M > 0, q o ( t ) = u ( t ) + M i s an overfunction, rpo ( t ) = u ( t ) M i s an underfunction, and

Proof:

cpo(t)

5

q 0 ( t ) on J.

-

This proves ( a ) .

To prove (b), we l e t

sider the solution w(t)

M = maxlf(t,O,O)l

5

M

>0

q 0 ( t ) = w(t)

v" = - k v ' v(t)

5

0

As a

on J

+

and we a s s e r t t h a t ,

M w i n be an overfunction

v ( t ) be the solution of BVP

+ My

and v ' ( t )

that dt = klv'

-< f(t,w,w')

i s s u f f i c i e n t l y large.

Similarly, l e t

on J .

wf(b) = 0.

and w t ( t )

f(t,O,wt)

f o r a given BVP, provided

Then

w(a) = 0,

-> 0 on J. Hence, by (1.10.2), = - kw' - M = - klw' I - M < f (t,O,w') - f (t,O,O) - M. 410

Then w(t)

Thus w"

and con-

of BY!?

w" = - k w * - M y

W"

on J

v(a) = 0,

-< 0

on J,

v'(b) = 0. from which it follows

I +M> f(t,O,v') -f(t,O,O) + M > f(t,O,v') r e s u l t , vl' > - f ( t , v , v * ) on J and, f o r M > 0

s u f f i c i e n t l y large,

cp,(t) = v ( t ) 66

-M

i s an underfunction

1.10. PERRON'S METHOD

relative t o a given BVP. Obviously, we have q 0 ( t ) on J. !Chis completes the proof of the lemma.

5

qo(t)

We are now ready t o s t a t e our main r e s u l t . THEOREM 1.10.3.

Assume t h a t

i s nondecreasing i n x

f o r each

f

Q

(t,x'),

of i n i t i a l value problems are unique. that there i s a

k

C[JxRxR,R], f ( t , x , x ' )

> 0 Such t h a t

and t h a t solutions

In addition, suppose

I f ( t J 0 J x ' ) -f(tJOjO)l < klx'

t E J and x t e R . Then f o r any BVP on J with an associated overflrnction q0 ( t ) , the generalized solution 0 Z(t) = Z(t;qo) E C(2)[Jo,R] and Z" = f ( t J ? d J Z ' ) On J for

.

Proof:

By Lemma l . l O J I J we readily obtain the existence

rpo

Jr0 with respect - JIo(t) on J . Consequently, t o BVP (1.10.1) such t h a t cpo(t) < the generalized solution z ( t ) = z(t;q0) i s defined. b r e over, the hypotheses imply t h a t solutions of BVP's when they exist, a r e unique and, therefore, the conclusions of Theorem 1.9.6 apply t o z ( t ) . Thus it suffices t o show t h a t Dz(t+), of an underfunction

and an overfunction

0

0

.

Let to E Jo and Dz(t0) a r e f i n i t e f o r every to E J + suppose t h a t z ( t O ) 2 z ( t i ) . The alternative case can be dealt with i n a similar way and w i l l not be t r e a t e d . We break the discussion i n t o two p a r t s .

F i r s t assume

that z ( t i ) 2 0. Let + ( t ) be a solution of Jr" = - kJr' -M, where M = max If(t,O,O)l, with Jr(to) = 0 , + ' ( t ) _> 0 on [tOJb], and

J

Jr(b) _>z(b-), which can be computed.

in Lemma 1.10.1, +,(t)

=

q(t)

+ + z(to)

on [t0'b1 w i t h z(tA) _< ql(tO) and follows from Theorem 1.9.5 t h a t z ( t )

$ ( t ) be a solution of

i s an upper solution

z(b-) Jr,(t)

_< Jrl(b). on

It (tOJb),

q ' ( t ) < m. Applying Theorem 1 0 z ( t ) i s continuous a t to. Now l e t

which implies t h a t Dz(tt)

1.9.6, we a s s e r t t h a t

Then as

Jr"

5

= kJr'

67

-M

on

[a,to]

such t h a t

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

$ ( t o ) = 0 , $(a)

2

z(a+),

$'(t) <0

and

again by Theorem 1.9.5, we have Dz(t0) 2 $ ' ( t o ) > - m. z ( t ) has a f i n i t e derivative a t implies

Finally, suppose t h a t solution of

cpl'

z(a+), cp(to) = 0, Theorem 1.9.5,

kcp'

=

+

and

on

2

<

0

0

on

cp,(t) be a

and l e t such t h a t [a,tol.

cp(a)

Theorem 1.9.6 t h a t we can show t h a t

z(t)

i s continuous a t

+ > - m.

5

Then, by

,> cp,(t) - z ( t 0 ) on (a,to), Dz(t0) 5 cp;(to) < m. It follows

z(t)

assures us t h a t

Then

to.

[a,to]

cp'(t)

[a,tol.

z ( t ) 5 $ ( t ) + z ( t o ) , which We i n f e r t h a t i n t h i s case

.(ti)

M

on

which from

to. Similarly

Thus we again conclude t h a t

Dz ( t o )

to. z ( t ) has a f i n i t e derivative a t each point of

z ( t ) has a f i n i t e derivative a t J

0

,

Since

it follows from !Theorem 1.9.6 t h a t z E C(2)[Jo,R] and 0 z" = f ( t , z , z ' ) on J This completes the

.

i s a solution of proof.

The following exercise i s instructive. EXERCISE 1.10.1.

Under the hypotheses of Theorem 1.10.3,

show t h a t the BVP x" = f ( t , x , x ' ) , x(a) = x(b) = 0 has a unique solution x ( t )

where M,k

on J,

Hints:

- k$, - M)

If

such t h a t

as i n Theorem 1.10.3.

$,(t),$2(t)

are the solutions of BIT'S

9; = k$, -M, q2(b) = 0, = 0, show t h a t Jrg(t) = min[$,(t),$,(t)] i s an overfunction. Similarly obtain an underfunction cpo(t) 5 $ o ( t ) $'I

=

1 $;(a)

on J . cpo(t)

5

Jrl(a) = 0, $;(b)

= 0,

Using Theorems 1.10.2 and 1.10.3, proceed t o show that z(t;q0)

5

q 0 ( t ) on J 68

and

z(t;q0)

i s a solution of

1.1 1 . MODIFIED VECTOR FUNCTION

t h e desired BVP which i s unique.

Compute

qoJq0 and est a b l i s h t h e bound on I x ( t ) l ; show t h a t m e x Jr ( t ) = J O -minJ cpo(t>. If to E JO and x ( t o > > 0 , l e t q1(t;t0)J q 2 ( t' - 0t ) be the respective solutions of BVP's

- kJr' -M,

$1

=

+"

= kJr'

- M,

q(t,) = 0,

$'(b) = 0;

+ ( t o ) = 0, $ ' ( a ) = 0 .

prove t h a t

q$(to;tO) ,< x ' ( t O ) 5 Jri(t,;t,),

x ( t o ) < 0,

obtain analogous relations.

Similarly, i f

Since the functions

involved and t h e i r derivatives can be computed, deduce the desired bound on

Ix' ( t ) I.

1.11 KlDIFIED VECTOR FUNCTION

Some of the r e s u l t s considered so f a r f o r scalar secondorder d i f f e r e n t i a l equations w i l l now be extended, i n the sections t h a t follow, t o f i n i t e systems of second-order difTo avoid repetition, l e t us agree on

f e r e n t i a l equations. the following:

..,n;

1,2,.

the subscripts

range over the integers

i,j

vectorial inequalities w i l l be used freely with

the understanding t h a t the same inequalities hold between t h e i r respective components. L e t us consider a system of second-order d i f f e r e n t i a l

equations written i n the vector form (1.11.1)

where

XI1 =

n

f E C [J xRn X R

,Rn 1.

f(t,X,X'), We s h a l l next define modified

vector functions i n different ways. DEFINITION 1.11.1. Let a(t) that

5

p(t)

on J

IcY'(t)l,

cr,B

E

C(l)[J,Rn] with

and l e t a constant vector

\pl(t)( < c F*(t,X,X')

on J . =

69

Then define

f(tJx,z')

c

>0

be such

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

("s

where

-x!J

=

if

J if -c < x! j - J if x!

-C

J

Fr((t,f;,x')

,x,

,x,x') =

XI

+ (xi

-

>c 3' 5 cj , < -c 3'

2 B i ( t ) ) / l + x i if xi

>

Bi(t),

Fy ( t x ) if a i ( t ) 5 ~ i ( ~ i ( t ) , 2 F y ( ( t , x , x ' ) + (xi - a i ( t ) ) / l + x i if xi < a i ( t ) ,

where

The f'unction f(t,x,x')

will be c a l l e d t h e modification of

F(t,x,x')

relative t o the t r i p l e

the definition t h a t J xRnxRn.

a,@,c.

It i s c l e a r from

i s continuous and bounded on

F(t,x,x')

IF'I ,< c and a ( t ) ,< 5 5 @ ( t )

Also note t h a t

on J . DEFINITION 1.11.2. on

J

A,

where, as before,

and l e t

cp,$

E

Let

a,@

E

C[J,Rn]

with

F ( t , x , x ' ) = f(t,x,f;'),

-(.

Bj(t)

x3

j

aj(t)

< B(t)

such t h a t cp(t,x) 5 q ( t , x ) on A = [(t,x): a ( t ) <x< @ ( t ) , t E J].

C[A,Rn]

men define

where

a(*)

if x.J

>

Bj(t), if a . ( t ) < x . < B . ( t ) ,

if

and

70

J - J - J x. < a . ( t ) , J J

1.11. MODIFIED VECTOR FUNCTION

x! J

-

=

> J;i(t,x),

Jrg(t,X)

if xj'

x'

if cp.(t,x)

{ scpj(t,x)

5 x!J 5 Jlj(t,X), J if x! < cpj(t,x). J

We shall c a l l the f'unction F(t,x,x'), the modified f'unction o f f ( t , x , x ' ) associated w i t h a,@,cp, and $. fgain, it i s easily seen t h a t F(t,x,x') i s continuous and bounded on JxRnxRn.

Furthermore,

a(t) <

x -< @ ( t ) and

cp(t,x)

5

5

$(t,x>* DEFINITION 1 . 1 1 . 3 .

Let

E = [(t,x,x'): t

+ +

E

J,

C[R ,R 1 be such that 6(s) = 0 accordingly as 0 5 s respectively. Then define Let

6

E

N > 0 be given and l e t

p,

(Ix((

C P, X'

E

Rnl.

c

6(s)

< 1, and 5 N, N < s < 2N, and s > 2N 6(s) = 1, 0

We s h a l l sey that F ( t , x , x ' ) i s a modified function of f ( t , x , x ' ) r e l a t i v e t o p,6. Clearly the f'unction F ( t , x , x ' ) n n i s continuous and bounded on J xR xR

.

A r e s u l t analogous t o Theorem 1.3.1 w i l l be proved next.

where

71

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

and I a ' ( t ) l J I p ' ( t ) l < c. Suppose f u r t h e r t h a t n n f E [ J X R ~ X ,R R I and f ( t J x , y ) i s quasimonotone nonincreasing i n

t h a t is, f o r fixed

x,y,

increasing i n y

for j is nonincreasing i n x t h e modification of

j

f

and f o r fixed

i

for

(t,x), f i ( t , x , y )

j

#

f(t,x,x')

t h e BW

(1.11.2)

x" = F ( t , x , x ' ) , x

has a solution

E

5y5

.(a)

x ( a ) = y,

C(*)[J,Rn]

is

F(t,x,x')

associated with

cording t o Definition 1.11.1, and 6 5 p(b),

(t,y), fi(t,x,y)

Then, i f

i.

i s non-

a,p,c

ac-

p(a), a ( b ) 5

x ( b ) = 6,

satisfying

a(t)

-< x ( t )

_< p ( t )

on J . Proof:

By Corollary 1.1.1, t h e BVP (1.11.2) has a solution.

Hence we need only t o show t h a t

w i l l show t h a t x(t)

5

p(t)

a(t)

on

5

x(t)

5

a(t)

on J,

x(t)

5

p(t)

on J .

since t h e proof of

follows s i m i l a r l y .

J

Suppose it i s not t r u e t h a t t h e r e e x i s t s an index k

a(t)

_< x ( t ) on J.

and an i n t e r v a l

Then

[ t l J t 2 ] C (a,b)

such t h a t 5 ( t 2 > = ak(t2)J

%(tl> = ak(tl), and 5(t) Thus

ak(t) -\(t)

%(to) = a i ( t o ) .

< ak(t>J

(tlJt2)'

has a maximum at some Hence

I

/%(to)< ck

have

72

to

E

(tl,t2)

and

and consequently, we

We

1.1 1 . MODIFIED VECTOR FUNCTION

i n view of t h e d e f i n i t i o n of nonincreasing character of possible at a maximum of

F(t,x,x') f(t,x,x')

cik(t)

and t h e quasimonotone

x,x'.

in

- %(t)

and hence t h e proof.

The quasimonotone nonincreasing nature of x,x'

This i s im-

in

f (t,x,x')

assumed i n Theorem 1 . l l . l b e c o m e s superfluous i f t h e

assumptions concerning

a,p

are made stronger.

"his we s t a t e

i n the following exericse. EXERCISE 1.U.1. Suppose t h a t with

a(t)

5

p(t)

on

J

and f o r

fi( t,A(t,i),At ( t , i ) ) , D-p;(t) A(t,i)

,

=

A ( t i) =

5

n 2

a,p E CIJ,Rnl

t

E

J

0

, D-a;(t)

C(l)[Jo,Rn]

fi( t , B ( t , i ) , B ' ( t , i ) ) ,

( X1' .. ., x i - l > q t ) , x i + l , ( xi, .. .,x i - 1,ai ( t ) ,xi+1,

*

where

.,xn),

...,xn ),

...7xi-19Pi(t),xi+l,. . .,xn), B 1( t , i ) = ( x i , .. . p i (t),xi+l, .. n ) ,

B(t,i)

= (xl,

.,XI

provided a . ( t ) < x . < p . ( t ) and - c . < x! 5 c f o r j # i, J - J - J J J j c being any vector s a t i s f y i n g I a ' ( t ) l , l p t ( t ) l < c . Show .(a)

-< y < p(a),

p(b) has a solution x E C(2)[J,Rnl

satisfying

that t h e BVP (1.11.2) such t h a t

73

a(b)

-<

6 _<

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

a(t)

-< x ( t ) 5

B(t)

on J.

REMARK 1.ll.1.

From t h e foregoing discussion, a n a t u r a l

question t h a t a r i s e s i s whether a r e s u l t analogous t o Theorem 1.11.1remains t r u e i f

a,B

ww,

solutions, i n a n a t u r a l

equation x" = f ( t , x , x ' ) .

a r e defined as lower and upper r e l a t i v e t o the vector d i f f e r e n t i a l

This can be done by defining a

modified function which has a jump discontinuity along t h e upper and lower solutions.

EXERCISE 1.U.2.

Let

x E C(2)[J,Rn]

be a solution of

xff = F ( t , x , x ' ) ,

where F i s the modified function of n n f E C[JxRnxR ,R ] r e l a t i v e t o p,6 as i n Definition 1.11.3,

Ilx(a) 1 , Ilx(b) 11 5 p. Suppose t h a t there e x i s t s a V E C[D,R+] such t h a t V(t,x,y) i s l o c a l l y Lipschitzian i n such t h a t

( x , ~ ) ,V(t,x,y) = 0 if llxll = P, V(t,x,y) > 0 if IIxll > P + and D VF(t,x,x') 2 0 in t h e i n t e r i o r of D, D being t h e set on

D = [[a,bl

x [x: llxll > P I xRn].

Show t h a t

Ilx(t)II

-< P

[a,b].

1.12 NAGUMO1s CONDITION

(CONTINUED)

I n remark 1.4.1, we noted t h a t Nagumo's condition i s not suff i c i e n t t o conclude

IIx' ( t ) 11 i s bounded, i f

n

f E C[J xRn xR ,R

n

Nonetheless, we can make conclusions on t h e a r c length of t h e curve

Ilx(t)II,

that i s

/ / / x ' ( s ) l / ds,

LEMMA 1.12.1. Assume t h a t

h E C[R+, (0,m) (1.12.1) where

.few

Ilf

1.

as the next lemma shows.

n n f E C[JxRnXR ,R ]

and

Suppose t h a t

(t,x,x')

11 5 h ( IIx' [I),

s ds/h(s) =

m.

(t,x,x')

E

J xRn xRn,

Then, t h e r e e x i s t s a 74

y: R' - t R +

1.

1.12. NAGUMO'S CONDITION (CONTINUED)

Proof: Since lorn s ds/h(s) y: R + Z satisfying

=

there e x i s t s a f'unction

rn,

From assumption (l.I2.1), we obtain

Hence i f

-

a < v, w < b,

by the change o f variables are equal (and zero) f o r respect t o w

((XI

v=w,

(s)II

--f

s;

the two integrals

and the derivatives with

e x i s t almost everywhere and are equal.

follows t h a t the integrals are equal since

Ilx'(w)II

It

is

absolutely continuous. Choose v and w such t h a t 11x' (v) 1) = el/ (b-a) (which i s possible by the mean value theorem) llxt(w)(I = max Ilx'(s)II on a 5 s 5 b . The desired r e s u l t then follows from (1.l2.2) and ( 1 . 2 . 3 ) and t h e proof i s and

complete.

It i s , however, possible t o extend Theorem 1.4.1 t o deal with the vector s i t u a t i o n . DLEORFM 1.12.1.

Suppose t h a t the following assumptions

hold: 75

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

n n ( i ) f E C[JxRnxR ,R 1, a , p E C[J,Rn] such t h a t a ( t > 5 p ( t ) on J and f o r t c J, u ( t > 5 x 5 p ( t ) , x f E R",

(1.l2.4)

5 hi(lX11),

Ifi(t,X,X')l

+

h, E C[R ,(O,m)]

(ii)

...,n,

i = 1,

and s a t i s f i e s

) - min a, ( t ) , where

Then, f o r any solution a(t) N

2

5

5

x(t)

@ ( t ) on

(1.u .6)

J

Ixf(t)l

N

E

C(2)[J,Rn]

of (1.11.1) with

t h e r e e x i s t s a constant vector u,p,h,

-< N

such t h a t on J .

I n view of ( l e a . ? ) , t h e r e e x i s t s a constant

Proof: vector

x

depending only on

0,

I.

- a ) = max[Ia(a) - @ ( a )

h(b

>

0

such t h a t

If (1.12.6) i s not t r u e , we m q r assume, without loss of

generality, t h a t t h e r e i s an index

t

J, I s ( t ) have E

I > Nk.

to = t O ( k )

\

E

such t h q t f o r some

However, by t h e mean value theorem, we

- s ( a >= where

k

(a,b)

x$t0)(b

- a),

and consequently, it follows t h a t

I%(tO) I 5 < Nk. The remainder of t h e proof i s almost identical t o the proof of Theorem 1 . 4 . 1 and we therefore leave it t o the reader.

76

1.12. NAGUMO'S CONDITION (CONTINUED)

The next r e s u l t i s an extension of Theorem 1.4.3 t o the present situation. TE30REM 1.12.2. (i) (ii) (t,x)

E

C(l)[J,Rn]

with

a(t)

'p,+

E

C(')[A,Rn]

with

cp(t,x)

+ 'px(t,x)

'pt

+t (iii)

-< p ( t )

a,@

where A = [ ( t , x ) : a ( t )

A,

E

Suppose t h a t

+

5x5

5

on J;

+(t,x)

p(t), t

E

and for J],

cp(t,x> 5 f(t,x,'p(t,x)),

*

Jrx(t,x> * + ( t , x )

f E C[JxRnxRn ,Rn 3

2

and

f(t,x,+(t,x)); cp(a,x(a))

< x ' ( a ) _<

-

+(a,x(a)) * Then, f o r any solution x a(t)

5 x(t) 5

(1.12.7)

p(t)

on J,

cp(t,x(t))

Proof:

Let

C(2)[J,Rn]

E

we have

5 x'(t>5

x(t)

of (l.ll.1) such t h a t

Jr(t,x(t)),

t

E J.

be any solution of (l.U.1) satisfying

'p(a,x(a)) _< x ' (a) 5 +(a,x(a)). We s h a l l only prove t h a t p ( t , x ( t ) )

-< x ' ( t )

on J,

since the proof of the other case i n ( l . X . 7 ) i s similar. Assume t h a t t h i s i s not t r u e . and a

on

to

E

(a,b]

such t h a t

Then there e x i s t s an index k X;c(to) < cpk(to,x(to)). Set

(tl,to]. Such an i n t e r v a l

'pk(t,x(t)) -X;c(t)

>

0.

(tl,to] e x i s t s where

It then follows t h a t

77

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

on

(tl,to].

5

zi(t)

t

0

This together with assumption ( i i ) yields on

(tl,to]

increases.

Thus

and hence zk(tl)

tended t o conclude t h a t

>

0.

z,(t)

This argument can be ex-

>0

z,(a)

i s nonincreasing as

which implies t h a t

cp,(a,x(a)) > %(a), a contradiction t o assumption ( i i i ) . This proves cp(t,x(t)) 5 x l ( t ) on J and the proof i s complete. Another r e s u l t t h a t may be u s e h l i n some s i t u a t i o n s i s t h e following. THEOREM 1.12.3.

(i) .(a)

a,p E C(l)[J,Rn]

(ii)

#

with

a(t) < p(t)

h E C[JxRy,R~], h ( t , u )

i

where

t h a t is,

u,

t

f o r each

( 1.12.8)

E J,

and

u ( a ) = A,

A = max[la'(a)), l p l ( a ) l ] ,

for

t

E

J, a ( t )

5x5

I

e x i s t s on

> 0,

x(t)

5

B(t)

on J,

[a,b];

p(t),

5 h(t,

Then f o r any solution x E C(*)[J,Rn]

5

i s nondecreasing i n u

and t h e maximal solution of

If(t,X,X')

N

J

i s quasimonotone non-

hi(t,u)

u 1 = h(t,u),

(iii)

a(t)

on

= B(a);

decreasing i n j

Suppose t h a t

Ix'

I). of ( l . l l . 1 ) with

t h e r e e x i s t s a constant vector

depending only on a,@,h such t h a t I x l ( t ) l IN,

t

E J.

j'

1.12. NAGUMOS CONDlTlON (CONTINUED)

Proof:

a(t)

5

that

Let

5

x(t)

.(a)

x(t)

be any solution of (1.11.1)with

@ ( t ) on

= p(a),

This, i n view of t h e assumption

J.

implies t h a t

fine m(t) = l x ' ( t ) l .

5

al(a)

Then m(a)

-< A

D+m(t) < Ix"(t)I = I f ( t , x ( t ) , x ' ( t ) ) l

xl(a)

5

p'(a).

De-

and, by ( i i i ) ,

5

t

h(t,m(t)),

E

J.

Consequently, by t h e theory of d i f f e r e n t i a l i n e q u a l i t i e s , we have m(t) where

[a,b].

5 N,

r(t,a,h)

t

r(t,a,A),

J,

E

i s t h e maximal solution of (1.12.8) which

r(t,a,A)

exists on

5

t

Let

E J.

N

>0

be a vector such t h a t

Then t h e s t a t e d conclusion follows

immediately. One can a l s o deduce a bound on Wapunov-like method.

IIx'(t))I by employing a This i s t h e content of t h e next two

theorems. THEOREM 1.12.4. (i)

in x,y, bl(u)

V

CID1,R+],

E

as

V(t,x,y)

D1 = J X[x:

where

+ w

Assume t h a t

u

+ w

( ( ~ 1 1-<

i s l o c a l l y Lipschitzian

5 b2( IIYII 1,

+

( i i ) D V(t,x,x') = lim sup h+O

(t,X,Y) +

+

D1;

(t,x,x')

and t h e m a x i m a l solution

of exists on

E

- V(t,x,x')]

-< g ( t , V ( t , x , x ' ) ) f o r

C [ J x R ,R]

u'

+ + 1,

C[R ,R

( l / h ) [V(t + h , x + h x ' ,

x' + h f ( t , x , x ' ) )

E

E

and

b l ( I h l l ) ~V(t,X,Y)

(iii) g

bl,b2

p] XRn,

u ( a ) = A ? 0,

= g(t,u),

[a,b].

79

E Int D ~ ;

r(t,a,h)

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

Then, f o r any solution

x E C(2)[J,Rn] of (1.11.1)such t h a t

llx'(a)II

5

pending only on

p, po,g

such t h a t

Proof:

x(t)

Ilx(t)((_< p

Ilx(t)II

5

and

Let and

P

t h e r e e x i s t s an

po,

N

> 0 de-

be any solution of (1.11.1)such t h a t

5

llx'(a)II

Define

p0.

m(t) = V ( t , x ( t ) , x ' ( t ) ) .

Then by ( i ) , we obtain m(a) 5 b2( IIx'(a)lI) 5 b2(po). h = b2(po). Also, because o f t h e assumptions on V,

Choose

it

follows t h a t ~+m(t5 > @;(t,m(t>),

t

E

(a,bl

and consequently, by t h e theory of d i f f e r e n t i a l i n e q u a l i t i e s , we obtain m(t)

-< r ( t , a , h ) ,

t

E J.

By ( i ) , t h i s i n t u r n yields

< r(t,a,h), bl( I I x l ( t > I I ) 5 V ( t J x ( t ) , x ' ( t ) ) = m(t> Since that

r(t,a,h) r(t,a,A)

e x i s t s on

5

M on

[a,b],

[a,b].

there is a

t

M

Furthermore, as

E

J.

> 0 such

bl(u)

--3m

> 0 such t h a t M < b(N). These considerations imply t h a t IIx' ( t ) 11 5 N on J . Clearly

as N

u

--3 m,

t h e r e e x i s t s an

depends only on

pJp0,

THEOREM 1.12.5.

where

and

E = [ [ a ~ b X] [x: llxll

5

V(t,XJX') _> ( t - a ) b ( [ l x ' I I ) ,

r

+m,

+

and

that ing

5

p] XRn],

on

[a,b],

Ilxl(t)ll _< N

on

[a,bl.

such t h a t

V

6

C[E,R+],

V(t,x,y)

< L i n the i n t e r i o r of

x E C(2'[[a,b],Rn]

p

Ilx(t)ll

This completes t h e proof.

(x,y), V ( a , x ( a ) , x ' ( a ) ) = 0, + + where b E C[R ,R ] with b ( r ) + m

D Vf(t,x,x')

Then f o r any solution

g.

Suppose t h a t t h e r e e x i s t s a

i s l o c a l l y Lipschitzian i n as

N

of (1.11.1)such

there e x i s t s an

80

E.

N >0

satisfy-

1 . 1 3 . EXISTENCE IN THE LARGE FOR SYSTEMS

Proof: such t h a t

By assumption on

b(r)

>L

if

r

b(r),

> N. -

t h e r e e x i s t s an

N

>0

A l s o , we have by t h e theory

of d i f f e r e n t i a l i n e q u a l i t i e s

v ( t , x ( t > J x ' ( t > )5 V ( a , x ( a ) , x ' ( a ) ) + L(t I f we now suppose t h a t f o r some

t2

E

- a),

t

(a,bl, Ilx(t2

then we a r e lead t o t h e contradiction

< (t2-a)[b(llx'(t2)11)-L] 5 V ( a , x ( a ) , x ' ( a ) )

0

= 0.

Hence t h e conclusion of the theorem i s t r u e . EXISTENCE IN THE LARGE FOR S

1.13

Y

S

~

We s h a l l f i r s t prove an existence r e s u l t analogous t o Theorem 1 . 5 . 1 f o r t h e BVP (1.13.1) where

f

XI'

E

= f(tJXJX')J

X(a) = A,

x(b) = B,

C[JxRnxRn,Rn], A,B E Rn.

THEOREM 1.13.1. Suppose t h a t t h e following conditions

hold: on J D-ai(t)

( i ) a , p E C[J,Rn] II C(l)[JoJRn] 0 and f o r t E J ,

such t h a t

a(t)

5

@(t)

_> f i ( t ~ a ( t ) ~ a ( t ~ i ) D-B'(t) )~ 5 fi(tJp(t)Jb(t>i))>

where

81

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

(iii)

hi

E

-I.

C[R ,(O,-)]

s a t i s f i e s (1.12.5).

and hi

Then f o r any .(a) 5 A 5 B(a), a(b) 5 B (1.13.1) has a solution x E C(2)[J,Rn] a(t)

5

vector

x(t)

No

p(b),

t h e BVP

such t h a t

5 p ( t ) on J . k r e o v e r t h e r e e x i s t s a constant

>0

Proof: -

5

such t h a t

Ix'(t)

I 5 No

on

J.

Ey hypothesis ( i i ) , we can choose a vector

depending only on c?,p,h,

as i n Theorem 1.12.1. L e t

N

> 0,

No be

such t h a t

Choose

c

> No.

Let

F(t,x,x')

be t h e modification function

as defined i n Definition 1.11.1r e l a t i v e t o the t r i p l e

a,p,c.

Then by Theorem 1.11.1it follows t h a t t h e BVP XI'

= F(t,x,x'),

has a solution

J.

x

x(a) = A,

x(b) = B,

with a ( t )

E C(2)[J,Rn]

-< x ( t ) -< B(t)

on

Hence we have X"

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

i n view of t h e d e f i n i t i o n of by ( i i )

,

5x 5

B(t), t

F.

Consequently, we obtain,

Recalling t h a t 1Z;l < ci and proceeding as i n the proof of meorern 1.12.1, it i s e a s i l y

whenever

shown t h a t

a(t)

( x '( t ) I

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

E

J.

No on Jo.

This then implies t h a t

a d therefore

solution of t h e desired BVP

x(t)

is actually a

(1.13.1) completing t h e proof.

Based on Theorem 1.12.2, it i s possible t o exhibit.

82

1.13. EXISTENCE IN THE LARGE FOR SYSTEMS

another existence theorem which i s an extension of Theorem

1.5.5.

on JoJ where

83

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

SZ being the s e t

n = [(t,x,x): (t,x)

E

n

and cp(t,x)

,< x' ,< Jr(t,x)l.

-

Proof: Let F(t,x,xt) be a modified f'unction of as in Definition 1.11.2 f ( t , x , x l ) associated w i t h cu,@,cp,@ which i s continuous and bounded on J xRn xRn and therefore s a t i s f i e s -theses ( i ) and ( i i i ) of Theorem 1.13.1. Furthermore, in view of the assumptions relative t o a l , p l i n ( i i i ) , it i s easily verified that hypothesis (i) is true w i t h respect t o F ( t , x , x l ) i n place of f ( t , x , x ' ) . This implies that

condition (i) of Theorem 1.13.1 holds w i t h cp,@ instead of -c,c respectively. As a result, it follows, by Theorem 1.13.1 that there exists a solution x E: C(2)[J,Rn] t o the modified

BW x" = F(t,x,x'),

x(a) = A,

x(b) = B,

such that a ( t ) 5 x ( t ) 5 @ ( t ) on J, where @(a)= A = @(a), a ( b ) 5 B 5 @(b). We now apply Theorem 1.12.2 t o assert that x ( t ) is actually a solution of the BVP (1.13.1) satisfying ( t , x ( t ) , x ' ( t ) ) E 12, t E J. For t h i s , it is necessary t o check that a l l the hypotheses of Theorem 1.12.2 are s a t i s f i e d which we leave t o the reader since it i s similar t o the proof of Theorem 1.5.5. This completes the proof. EI(ERC1SE 1.13.1.

Under the assumptions of Exercises show that the BVP XI' = f ( t , x , x t ) , x(a) = A, x(b) '= B has a solution. 1.11.2 and Theorem 1.12.5,

1.14 FUR'MER RESULTS FOR SYSTEM Let us consider the d i f f e r e n t i a l system

(1.14.1) where f

X'l

E

= f(t,X,X'),

n n C [ [ O , ~ ] X R ~ X,R R

3,

84

subject t o the boundary

1.14. FURTHER RESULTS FOR SYSTEMS

conditions (1.14.2)

= 0, ~ ( 0 -AOx'(0) )

(1.14.3)

x(l)+A1x'(l) = 0 ,

AOjA1

being

matrices.

dxd

Here we s h a l l study t h e existence of solutions of t h e BVP (1.14.1)

- (1.14.3)

earlier results.

i n a more general s e t up than t h e

We employ mapunov-like fbnctions and t h e

theory of d i f f e r e n t i a l i n e q u a l i t i e s i n a s l i g h t l y d i f f e r e n t way which throws much l i g h t on t h e underlying ideas.

The

technique i s , of course, the modified function approach.

The

following lemma i s very u s e f u l i n our discussion. LEMMA 1.14.1.

(i)

U E

Assume t h a t

C ( * ) [ [ O J ~ ] J R + ] Jg

i s nonincreasing i n u

g(t,u,v)

c [ [ o J 1 X] R + X R J R - ] J

f o r each

-> g ( t J u J u ' ) ; ~ ' ( 1< ) 0, and

(1.14.4)

(t,v)

and

U''

(ii) some ct

E

u'(0)

2

0,

u(0)

-< m ' ( 0 )

for

-> 0 ;

(iii)

G E C [ [ O J ~ ] X R + ~ R ]and t h e r e exists an

such t h a t f o r u

2 L,

t

s a t i s f i e s t h e estimate

E

L>

[0,1],

r(t,TJO)

< aoJ t

E [oJT]J

where

a0 = min(+, 1/a); (iv)

t h e l e f t maximal solution

right minimal solution

(1.14.7)

p(t,O,O)

of

v ' = g(t,2L,v) 85

r(tJ1,O)

and t h e

o

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

e x i s t s on

[0,13.

Then there e x i s t s a

(1.14.8)

u(t)

5

Bo

>0

such that

and

Bo

5

lu'(t)l

Bo,

Proof: Assume t h a t the maximum of

0

u(t)

-< t -< 1.

occurs a t a

-

tl. From conditions u ' ( 0 ) > 0 and ~ ' ( 1 <) 0, it follows t h a t u'(t,) = 0 . Clearly tl > 0 , f o r otherwise

point

we would have

u(tl)

5 mt(tl)

>

Define

2L.

-

u ( t ) 10.

u ( t ) < 2L, 0 L for t E [O,tll.

W e s h a l l show t h a t

u(tl)

and consequently

= 0

If not, define

to = sup[t Since u(tl)

>

E

[O,t11:

U'(t)

to is well

by the mean value theorem,

2L,

defined.

It i s then e a s i l y seen t h a t

(1.14-9)

u'(to)

Setting

-> +u(t)].

2

L ,< u ( t ) ,

cuou(tO),

z(t) = u'(t)/u(t)

for t

t

[tO,tll

E

E

[tO,tll.

and using

assumption (1.14.5) we readily obtain ' ~ ' ( t 2) G ( t , z ( t ) ) , Notice t h a t

z(to) = 0

and

t

z(to)

E

[to,t,l.

1 cyo > 0 .

By the theory

of d i f f e r e n t i a l inequalities, we then i n f e r t h a t z(t) where with on

5

r(t,tl,z(tl)),

r(t,tl,z(tl)) T =

t

[tO,tl],

1'

t

E

[tO,tlI,

is t h e lef't maximal solution of (1.14.6)

Since z(tl) = 0, we see t h a t r(t,tl,O) < CY 0 and as a r e s u l t , we are lead t o t h e contradiction

This proves t h a t

u(t)

5

2L on

86

[0,11.

1.14. FURTHER RESULTS FOR SYSTEMS

U s i n g t h i s i n e q u a l i t y and t h e nonincreasing nature of g(tJuJv) i n

u,

we obtain

u"

-> g ( t , 2 L , u ' ) .

-> 0, ~'(1)5 0, d i f f e r e n t i a l i n e q u a l i t i e s , we have 0 p(tJ0JO)J

Again, using t h e f a c t t h a t theory of

where

for

and t h e

r ( t J 1 , 0 ) , p ( t J O J 0 ) are, respectively, l e f t maximal and

r i g h t minimal s o l u t i o n s of

on

u'(0)

[O,L]. 0

(1.14.7)which

Thus, we can f i n d a

-< t -< 1, B = max

[I

B

>0

a r e assumed t o e x i s t

such t h a t

Iu'(t)I
where max O< t <-l

r(tJIJo)lJ

I

min

- -

O < t < l

p(t,O,O)l].

The conclusion of t h e lemma now follows by choosing B = max[2LJB]. 0

The proof i s complete. 1

The functions g ( t , u , v ) = - k [ l + (2u)'+ 2 0, G(tjZ) = - ( a + k \ z I + Z ) J V(tjX) *11X112 are

COROLLARY 1.14.1. k

IVl]j

>

admissible i n Lemma 1.14.1, where h = $ a0e -(k+l),

provided

a

La

=

k(1

< aOke- (k+l)

+

(1/2h)

+ hk),

Our aim i s t o prove t h e following r e s u l t . IREOREM 1.14.1.

(a)

Assume t h a t

f E c[[OJ1]xRnxRnJRn]

and

A ~ , A a~r e p o s i t i v e

d e f i n i t e o r i d e n t i c a l l y zero; (b)

v

E

c ( 2 ) [ [ 0 , 1 ] XRnrR+],

g E C [ [ O J ~ ] X R + X R ~ R -g](~t , u , v )

each

( t J v ) ; and f o r

(t,X,X')

E

V(t,x)

i s positive definite,

i s nonincreasing i n [0,1] X R n X R n ,

(1.14.10) Vg(tJx) > g ( t J V ( t J x ) J V ' ( t J x ) )+ v / l f ( t J X J X ' )( 1 )

87

u for

u>oJ

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

where V'(t,x) = Vt(t,x) VF(t,x) = Vtt(t,x)

+ vx(t,x) x ' , + 2Vtx(t,x)

+ vx(t,x)

x ' + Vxx(t,X)X'

x'

f(t,x,x');

( c ) The boundary conditions (1.14.2), (1.14.3) imply, for

some a 2 0,

that

(1.14.12)

2 0, V'(l,x(l)) 5 V(O,X(O>) 5 (Yv'(O,x(O)); V'(O,x(O))

(d) G that f o r

u

E

and there e x i s t s an L

C[[O,l]R',R]

2 L,

t

E

7 E

and

>

0

such

[O,11

(l/u)g(t,u,v) and f o r any

0

-

(V/UI2

2 G(t,v/u)

(0,1], the l e f t maximal solution

r(t,7,0)

of Z'

= G(t,z),

Z(2)

= 0

s a t i s f i e s the inequality r(t,T,O) < ao, t E [O,T], where = min(+,l/a), (e) the l e f t m a x i m a l solution r ( t , l , O ) and the r i g h t minimal solution p(t,O,O) of a.

v ' = g(t,2L,v) exist on

[0,11.

Then there exists a solution x boundary value problem (1.14.1) Proof:

E

C(2)[[0,1],R

- (1.14.3).

Define the function

88

6(u,v)

dl

of the

on R+xR+

as

1.14. FURTHER RESULTS FOR SYSTEMS

osu, (1.14.1'3)

vLB,

( 1 i B - v),

O l u l B l v l B + 1,

(l+B-u)(l+B-v),

Blu,

0 5 v_
~ ( u , v )=

vlB+1,

otherwise. Next define t h e modified function

F

of

f

on

d d [0,1]X R X R

by

(1.14.14)

F(t,x,x'

Clearly t h e function [0,1] xRdxRd.

F

= S(llxll, IIX' Il)f(t,XjX'

1.

i s continuous and bounded on

Hence t h e r e e x i s t s a solution

x

E

C(*)[[O,l],R

d

]

of the boundary value problem

(1.14.15)

X"

= F(t,x,x'),

Set m(t) = V ( t , x ( t ) )

x(O)-Ax'(O) = 0 ,

X(1) + A I X ' ( l ) = O *

so t h a t , because of assumption ( c ) , we have

the r e l a t i o n s

which leads t o t h e f u r t h e r inequality

(1.14.18)

m"(t)

_> .g(t,m(t),m'(t)) + allx"(t)ll.

Hence, by Lemma 1.14.1,

it follows t h a t t h e r e exists a Bo > 0

such t h a t

89

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

As a result, s e t t i n g

-N = [min g(t,u,v): 0 ,< t 5 1, u ,< Bo, Ivl ,< Bol, we obtain from (1.14.18) m"(t) Thus, for 0

5

s

5

t

2 -N+ullx"(t)ll.

5 1,

2 IIx'(t)ll- IIx(l>ll- IIx(O)ll. Since V(t,x) i s assumed t o be positive definite, it follows, from the estimate V ( t , x ( t ) ) = m(t) 5 Bo, 0 5 t 5 1, t h a t (lx(t)II 5 B*, 0 5 t ,< 1. Consequently, we deduce that Ilx(t)ll

5 a*+

(2Bo +N)

O < t l l .

"B,

Eivdently, t h i s implies t h a t (1.14-19)

IIX(t)ll

€3

IIX'(t)ll

and

This, i n view of the definition of

5

B,

0

5 t 5 1,

assures us t h a t x ( t ) is actually a solution of t h e boundary value problem (1.14.)(1.14.3). The proof i s cmplete.

90

F,

1.14. FURTHER RESULTS FOR SYSTEMS

If f satisfies Nagumo's condition, assumption (1.14.10) may be weakened as the next theorem shows.

THEOREM 1.14.2. Let the hypotheses of Theorem 1.14.1 hold except that inequalities (1.14.10) and (1.14.U) are replaced by

2 g(t,V(t,x),V'(t,x))

(1.14.20) V;(t,x)

(1.14.21)

U(t,X,X')

+ u ~ ~ x ' ~ ~u,> 0,

+ 1 ,> IIX'II.

Suppose that Ilf(t,x,x')ll 5 h JJx'/Ifor (t,x,x') E [0,1] xRnxRn, and s ds/h(s) = 00. Then there where h E C[R+,(O,.o)] exists a solution x E C(2)[[0,1],Rn] of the boundary value problem (1.14.1) (1.14.3).

-

Proof:

We proceed exactly as in the proof of Theorem 1.14.1 until we arrive at inequalities (1.14.16). Consider first the case when u 11. Then, in view of (1.14.17), relations (1.14.20) and (1.14.21) yield the inequality (1.14.22)

Vi(t,x)

2 g(t,V(t,x),V'(t,x)) + (1-S,[IIx'II

-> g(t,V(t,x),V'(t,x))

z g(t,V(t,x),V'

+ u6I]x'II

- 13 -1+

IIx'~~[~(u1)+ 1 I

-

I(x'11,

(t,x>) 1

+

using the facts g 5 0, 0 5 S 5 1. Here we have used 6 for 6(u,v). If, on the other hand, 0 < u < 1, noting that (1.14.21) implies uIIx'(I 5 1 + U(t,x,x'), we obtain

(1.14.23)

Vi(t,x)

g(t,V(t,x),V' +

(t,x))

-t

u6IIx'11

(1-E)[ullx'II -13

,> g(t,V(t,x),V'(t,x))

-> g(t,V(t,x),V'(t,x)) 91

- 1 + IIx'II[uS + (1-s)al

- 1 + u(Jx')(.

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

Since by L a m a 1.14.1 we have

inequalities (1.14.22), (1.14.23) lead to m"(t)

2- ( N + 1 ) +

m"(t)

2- ( N + 1 ) + ollx'(t)ll,

IIx'(t)ll

and respectively, where, as before,

It then follows that 0 ( l ) From Lemma 1.12.1, we have IIX'(t>jl

5 (2B0 + N +l)/o

L r ( e W ) 5 ?(MI,

Letting B = m[B*,y(M)], the proof as before.

0

M

L

t

5

in any case. 1.

we obtain (1.14.19)which concludes

Finally, for later use, we shall state a uniqueness result leaving the proof as an exercise. THEOREM

1.14.3. Assume that f

V 6 C(2)[[0,1]xRnXRn,Rf], X = Y and

V(t,x,y)

E

C[[O,1] xRnxRn,Rn], 0 if and Only if

1.15, NOTES AND COMMENTS

Then the BVP (1.14.1), one solution.

x(0) = xo and x(1)

=

x has at most 1

1.15 NOTES AND COMMENTS For the existence results in the small contained in Section 1.1, see Hartman [31, Jackson [2], and Bailey, et al.

[31. The results of Sections 1.2 and 1.3 are taken from Jackson [2] except Theorems 1.2.4 and 1.3.2 which are based on the work of Schmitt [5]. Theorem 1.4.1, the Nagumo's condition, is taken from Jackson [2], while Theorems 1.4.2 and 1.4.3 are adapted from Schmitt [2] and Ako [I.], respectively (see also Hartman 133). Theorem 1.5.1 is taken from Jackson [21. Theorems 1.5.2, 1.5.4 are due to Erbe [l] and Theorem 1.5.5 is due to Schmitt [2]. The results of Section 1.6 are due to Bernfeld et al. [l]. Fxercise 1.6.1 contains the work of Schrader [2]. See George and Sutton [l] for the use of Wapunov-like functions. The contents of Sections 1.7-1.10 are taken from Jackson [2] where other related references may be found. The work contained in Sections 1.11-1.13 is due to Bernfeld et al. [41 except Lemma 1.12.1which is due to Lasota and Yorke [13]. The definition of the modified function as given in Definition 1.11.3 may be found in Hartman [31 where a number of results for second-order systems are given. The contents of Section 1.14 are due to Bernfeld et al. [j]. For further results in this direction, see Hartman [?I. For related results, see Knobloch [1,21, Schrader [5,61, Jackson and Schrader 111, Schmitt [31, Moyer [l], Gaines [3,41, Halikov [l], Gudkov and Lepin [l], Gudkov [21, and Mamedov [l].

93

Chapter 2 SHOOTING TYPE METHODS

2.0

INTRODUCTION

This chapter is essentially divided into two parts in which the shooting method serves as the underlying technique. The first part is concerned with the question as to whether uniqueness of solutions of boundary value problems implies the existence of solutions. Although the interdependence between uniqueness and existence is complicated for nonlinear equations, it can be formulated in simple terms for second-order differential equations. We examine this problem under linear and nonlinear boundary conditions and show that the results obtained are the best possible. The second part is devoted to an hportant method known as the "angular function technique." A number of results concerning existence, uniqueness, and criteria for existence of a finite OF infinite number of solutions are studied using this method as a tool. Einploying Lyapunov-like functions and the theory of differential inequalities, this technique is extended to cover nonlinear boundary conditions and systems of differential equations. 2.1

UNIQUENESS IMPLIES EXISTENCE

For linear differential equations, it is well known that the uniqueness of solutions implies the existence of solutions. For nonlinear equations the interdependence between uniqueness and existence is much more complicated. However, in the case of second-order differential equations this relationship can surprisingly be formulated in simple terms.

94

2.1. UNIQUENESS IMPLIES EXISTENCE

An extremely useful technique in handling these questions for second-order boundary value problems is the so-called shooting method. The idea of this method is to fix one initial value and to allow the slope at the initial point to vary through the real numbers. From the connectedness of the solution funnel and the uniqueness of BVP‘s, it is then possible to show that the values of the solutions at the final point cover the real line. In what follows we plan to illustrate this important technique in a detailed manner. Our discussion depends on the following variation of Kneser’s theorem. LEMMA 2.1.1. Consider the initial value problem X’ =

F(t,x),

x(to>

= X0’

where F E CCRXRn,Rn 1. Let S be any compact, connected set. Assume all solutions x(t,tO,xO) exist on [tO,tl]. Then

is a compact, connected set. Our first result will be concerned with a system of two first-order equations subject to simple boundary conditions. Consider the differential system x ’ = f(t,x)

(2.1.1)

with the boundary conditions Xl(tl)

(2.1.2)

where f result.

E

C [R x R ,R

Xl(t2)

=

2 2

= C2’

1. For this problem we have the following

95

2 , SHOOTING TYPE METHODS

0

J

THEOREM 2.1.1. =

(a,b).

Let J

=

(a,b],

-m

5 a < b < m, and

Assume

(i) fl(t,xl,x2) is an increasing function of x2 for fixed (t,x,) satisfying fl(t,xl,x2) + & m as x2 + k m uniformly on compact sets in (a,b)xR; (ii) all solutions of (2.1.1) exist on J; (iii) there exists at most one solution of (2.1.1), 0 (2.1.2) for all tl,t2 E J and all cl, c2. Then every BVP (2.1.1), (2.1.2) has exactly one solution 0 if tl E J , t2 E J. Proof: We may assume without loss of generality t2 = b. Let tl, c1 be given where a < tl < b. Let x(t) = be a solution of (2.1.1) satisfying the initial (x,(t),x,(t>) condition

Then define

S = [x(b): x(t) is a solution of (2.1.1) k satisfying (2.1.3) for each Irl 5 k]. By Lemma 2.1.1,

N

is connected. Let Sk be the projection of Sk on the x axis. Hence S is an interval which is 1 $ 0 nondecreasing in k. Define S = - Sk; then is an interval. To prove the stated result, it is sufficient to show S = (-m,m). If this is not true, suppose that S is bounded above. A similar proof holds if is bounded below. Then c (-m,M) for some M > 0 . Let xn(t) be a sequence of solutions of (2.1.1) satisfying Sk

-

N

z

&

N

N

s

(2.1.4) If s1,s2,

xn(tl)

...,sn

=

(cl,n),

n

=

,... .

1,2

is a sequence of t-values on a compact 96

2.1. UNIQUENESS IMPLIES EXISTENCE

f o r otherwise, from the well-known convergence theorems a subsequence of

[x ( t ) ]

n

converges uniformly t o a solution of

(2.1.1)J on a l l compact subsets of

J

0

,

a contradiction t o

(2.1.4). Let

x ( t ) be the f i r s t component of x n ( t ) . Observe n, 1 t h a t ~ ~ + ~ ,> x~n J( l (t t )) and x (s) < x (s) f o r n+l, 1 n, 1 t E (t,,6] and s E [ t l - 6 , t 1) f o r s u f f i c i e n t l y small 6. However, from ( i i i ) we immediately obtain

for

a < s < t l < t < b .

W e claim t h a t the sequence x

bounded on any subinterval of there e x i s t s a

C

J

0

.

n,l

(t) is not uniformly

If we assume not, then

> 0 and an i n t e r v a l [al,bll

such t h a t

From the mean value theorem

Ix ( t ) l 5 C f o r t E [a,,b,I. n, 1 there e x i s t s rn E [a J b ] such t h a t /xh,l(rn)( 52C/(bl- al). 1 1 Then condition ( i ) implies llxn(rn)ll i s bounded f o r all n. However, t h i s contradicts (2.1.5). of

[ t l - 6m,tl],[tl,tl+6m]

uniformly bounded. vn, wn

Letting

Hence, on any subintervals [x ( t ) l i s not n, 1 we see there e x i s t points

the sequence 6m + 0

where vn < tl,

v +t n 1’

wn>

w +t

n

1

such that, i n view of (2.1.6),

(2.1.7) as

n

x

n,l

(v ) n

+-m

and

xn,l(wn>

++a0

+m.

Let x*(t)

be any solution of (2.1.1) satisfying

91

2. SHOOTING TYPE METHODS

x*(b) = M; hence, since S E (-m,M),xT(b) 1 n. From (2.l.7), f o r l a r g e n

x ( t ) and xT(t) n, 1

This implies

d i c t i r q assumption ( i i i ) .

> xn,l(b)

for a l l

i n t e r s e c t twice, contra-

This completes the proof of Theorem

2.1.1. COROLLARY 2.1.1.

where

Consider the BVP

h E C[JxRxR,R], J = (a,b]. (i)

Assume t h a t

a l l solutions of (2.1.8) e x i s t on

(a,b];

( i i ) t h e r e e x i s t s a t most one solution of (2.1.8), (2.1.9) f o r tl,t2 E (a,b], c1,c2 E R. Then there e x i s t s exactly one solution of (2.1.8) and (2.1.9). Proof:

The BVP (2.1.8),

xi

= x2,

(2.1.9) can be w r i t t e n as = h ( t , x ,X

X'

1 21,

2

x (t ) = C2' 1 2

x ( t ) = c 1 1 1'

We see immediately t h a t t h e conditions i n Theorem 2.1.1 a r e satisfied.

Hence Corollary 2.1.1 follows.

EXERCISE 2.1.1.

suppose t h a t

Consider t h e BVP (2.1.8),

h(t,x,x')

is s t r i c t l y increasing i n x.

Assume that a l l solutions of (2.1.8) exist on there e x i s t s exactly one solution of (2.1.8),

tl,t2

E

(a,b]

Hint:

and a l l

(2.1.9) and

c1,c2

E

(a,b].

Then

(2.1.9) f o r a l l

R.

Show t h a t there exists a t most one solution by 98

2.1. UNIQUENESS IMPLIES EXISTENCE

showing t h a t t h e difference of any two solutions cannot a t t a i n a positive maximum o r negative minimum.

Then use Corollary

2.1.1. EXERCISE 2.1.2.

Assume

h(t,x,x')

satisfies

where

Then (2.1.8),

Hint:

(2.1.9) has a unique solution. Show t h a t t h e inequality

q(t)ixl + p(t)lx'I

lx"I

with boundary conditions x ( t l ) = 0, x(t,) = 0 has only the zero solution, where I t2- t I C b a. Then use Corollary 1 2.1.1.

-

Observe t h a t i n Theorem 2.1.1we obtained t h e existence of a solution on

(a,b].

t h a t we cannot s u b s t i t u t e EXAMPLE 2.1.1.

I n t h e following example we show [a,bI

for

(a,b].

The implicit equation

P > -2,

cp + (P/2) t a n - l cp = 9,

The family of a l l solutions of

has a unique solution cp(p,q). (2.1.10)

x'1 = - x +

1

3

tan-'

cp(sin t , x sin t

+

x ' cos t )

can be represented by x ( t ) = A cos t where A

and

B

+

B sin t

+ 3 tan-1 B,

a r e a r b i t r a r y constants.

Consider the

boundary conditions ( 2 . 1 . ~ ) x ( t l ) =xl

and

x ( t2 ) = x2, 99

tl,t2

E

[o,~].

2 . SHOOTING TYPE METHODS

Then x i

=

A cos ti + B sin ti +

Eliminating A, (2.1.12)

1

2

tan-1 B,

i

=

1,2.

we obtain

-

B sin(t2 t ) + ~ ( C O Stl - cos t2) tan-1 B 1 = x2 cos tl- x cos t2' 1

We now claim that there exists at most one solution of (2.1.10), (2.1.11). If 0 5 t l < t2 < T or 0 < t < t2 5 T , then 1

sin(t2- tl) > 0

cos t2- cos t < 0 . 1

and

Thus from (2.1.12), we see that, for any xl, x2, tl, t2, B and A are uniquely determined. Thus solutions of (2.1.10), (2.1.11) are uniquely determined. For the case t = 0, t2 = 7, (2.1.12) reduces to 1 (2.1.13)

tan-'

B

=

x1 + x2'

Once again uniqueness of solutions foUows. Thus on [ O , T ] the conditions of Corollary 2.1.1 hold. However, observe (2.1.13) implies there exists no solution, if )x1+x21 = T/2EXERCISE 2.1.3.

(2.1.14)

Show that solutions of y"

=

-y + arctan y,

(2.1.15)

when they exist, are unique for t ,t E [ 0 , 7 ] . Then show 1 2 there exists no solution of (2.1.14) satisfying (2.1.16)

Hint:

Y(0)

= 0,

Y(T)

=

3T.

Assume there exist two solutions y,(t),

100

y,(t)

of

2.2, GENERAL LINEAR BOUNDARY CONDITIONS

(2.1.14), y,(t)

tlyt2 E

(2.1.15) f o r

- y2(t)

satisfies

i s a subfunction of subinterval of W(T) > u(T),

w"(t)

y" = -y. ul' = -u

u ' ( ~ / 2 ) . Prove that

u(t)

leads t o a contradiction. of (2.1.15), (2.1.16), vl' = -v

and

Argue .that y(T) y(O) = 0 , ~ ' ( 0 )= m.

m.

and

x'.

> u(O),

has two zeros on

[O,T]

which

To show t h e r e exists no solution

~ ' ( 0 )= m

+ 1,

of

[O,T]

with respect t o (2.1.14) f o r any

< v(T)

= 2T,

where

y

s a t i s f i e s (2.1.14),

I n t e r e s t i n g l y enough we can, under c e r t a i n derive t h e r e s u l t s of Corollary

2.1.1 f o r t h e case when J = [a,b]. h

i s not a proper

w(0)

v(t)

conditions on h ( t , x , x ' ) , example, t h a t

w(t)

w ( ~ / 2 )= u ( T / ~ ) , w ' ( T / ~ )=

v(0) = 0 ,

i s a subfunction on

and prove that

[tlyt2]

If

=

w(t)

prove t h a t t h e solution

+ T,

REMARK 2.1.1.

2 -w(t)

then show that

[O,T], where

Show t h a t

(0,T).

It i s s u f f i c i e n t , f o r

s a t i s f y a Lipschitz condition i n both x

This follows by considering t h e l i n e a r equation

derived from t h e Lipschitz condition, and using t h e f a c t t h a t t h e uniqueness of t h e BVP f o r t h e l i n e a r equation on a l l subi n t e r v a l s of

implies t h e existence of unique solutions

[a,bl

on a l l subintervals of

of BVP's f o r x" = h ( t , x , x ' ) Moreover, an estimate of

[a,b]

[a,b].

can be derived i n terms of

hyperbolic f'unctions. 2.2

GENERAL LINEAR BOUNDARY CONDITIONS "he question as t o whether uniqueness of solutions of

(2.1.8) with t h e general l i n e a r boundary conditions (2.2.1)

alx(tl)

+

ap'(tl)

= cl,

blx(t2)

+ b p ' ( t 2 ) = c2

implies existence i s n a t u r a l l y more d i f f i c u l t since it depends upon t h e c o e f f i c i e n t s

al,

a2, bly b2.

101

2. SHOOTING TYPE METHODS

We shall show, as a consequence of our next result, that Corollary 2.1.1 holds for (2.1.8) and (2.2.1) whenever a2b2 = 0 . THEOREM 2.2.1. Let J, x, f be as in Theorem 2.1.1 except that condition (iii) is replaced by the following:

(iv) there is at most one solution of (2.1.1) satisfying (2.2.2)

x1(t1)

for any tl,t2

E

=

J

0

c1

and

x2(t2)

and c1,c2

E

=

c2'

tl

#

t2'

R.

Then (a) every boundary value problem (2.1.1), (2.1.2) has a unique solution; (b) every boundary value problem (2.1.1), (2.2.2) has a unique solution. Proof: In order to show (a), we must show condition (iii) of Theorem 2.1.1 holds. Let x(t), y(t) be distinct solutions 0 of (2.1.1) satisfying xl(tl) = y (t ) for some tl E J It 1 1 0 suffices to show that xl(t) # yl(t) for tl f t E J We show this for tl < t < b as a similar argument holds for Without loss of generality, we may assume from t E (a,t,). (iv) that y,(t) > x2(t) for t E (tl,b). Then either

.

.

Y2(t1> = X2(tl) or Y2(t1> > X2(tl)' If Y (t 1 = X2(tl)' 01 then, by (iv), yl(t) f x,(t) for tl # t E J If then yi(t,) > xi(tl) by (i), so that there y2(tl) > x2(tl), exists a 6 > 0 such that yl(t) > x,(t) for t E (tl,tl+6). Assume that there exists a least T,tl< 7 < b such that xl(T) = yl(T). However, then ~'(7) 2 y'1(T), which implies 1 x2(T) 2 y2(T). This contradiction shows that xl(t) # yl(t) on (tl,b) and thus completes the proof of part (a).

.

102

2.2. GENERAL LINEAR BOUNDARY CONDITIONS

The proof of part (b) i s similar t o t h a t of Theorem 2.1.1.

t2 =

Assume without l o s s of generality t h a t Define

Sk a s i n Theorem 2.1.1 and l e t

projection of

3

on t h e

Sk

i s an i n t e r v a l . A

bounded below).

3

axis.

Let

It suffices t o show

s^

3 =U,

m

be t h e r

z

but

$

(--,MI

2.

then

S,;

i s the r e a l l i n e .

We may assume t h a t t h e r e e x i s t s

(-m,M)

C

x2

8,

i s bounded above ( a similar proof holds i f

Suppose S that

b.

M

A s before, l e t

^s

>0

is such

xn(t)

be

a sequence of solutions of (2.1.1) s a t i s f y i n g

n

xn(tl) = (c,,n),

(2.2.3)

= 1,2

,... .

Then there e x i s t sequences

such t h a t (2.1.7) holds vn, "n [ t h i s follows since ( a ) i s the same as ( i i i ) i n Theorem 2.1.11.

From (2.2.3) there e x i s t s a sequence {zn] such t h a t t
+ t l and xn2(zn) 1 n solution of (2.1.1) s a t i s f y i n g

as

n

403.

x*,(b) = M.

Let

x*(t)

be any

From t h e f a c t t h a t

^s c (-m,M), xz(b) > xn2(b) f o r a l l n, but f o r l a r g e x:(zn) < xn2(zn). Thus there exists zk such t h a t

n

(2.2.4)

From (2.1.7)

< x,(wn),

x;(wn)

so t h a t t h e r e e x i s t s

zk

1

#

d',

> xnl(vn)

n

v n

n

and

n

< znt t < wn.

then (2.2.4) and (2.2.5) contradict ( i v ) .

Suppose z ' = z t t - then define x ( t ) = x n ( t )

n

for large

such t h a t

z"

X*(Zt') = Xd(ZB),

(2-2-5) If

xT(vn)

n'

x ( t ) = x*(t)

for

of (2.1.1)

such t h a t

x*,(b) = M,

so t h a t

z'

n

Gl(tl)

(--,MI

< t 5 b. = xnl(tl) E

3,

for

Then x ( t ) = c1

a


zk,

i s a solution

and x2(b) =

a contradiction.

This

completes t h e proof of Theorem 2.2.2. COROLLARY 2.2.1.

Consider t h e BVP (2.1.8),

103

(2.2.1) and

2 . SHOOTING TYPE METHODS

assume

a2b2 = 0.

Suppose a l l s o l u t i o n s of (2.1.8)

e x i s t on

( a , b ] and t h e r e e x i s t s a t most one s o l u t i o n o f (2.1.8), (2.2.1) f o r every tl,t2 E ( a , b ) . Then t h e r e e x i s t s

b

(a)

a unique s o l u t i o n o f (2.1.8),

(2.l.g),

(b)

a unique s o l u t i o n o f (2.1.8),

(2.2.1).

Proof: -

2

= 1 and

Without loss of g e n e r a l i t y , assume a1 = 1. Then (2.1.8),

a s t h e f i r s t - o r d e r BVP f o r X'

1

L-

x2-b x 1 1'

X$ =

x

=

(2.2.1)

(xl,x;

h(t,Xl,x2

+

a2 = 0,

can be w r i t t e n

blxl)

- b 1x 1)

and

i n which

+ bl(X2

- blxl),

We t h e r e f o r e see t h a t t h e hypotheses o f Theorem 2 . 2 . 1 h o l d and t h e r e s u l t follows. Assume i n (2.1.8) t h a t t h e f u n c t i o n

EXERCISE 2.2.1. h(t,x,y) lhxI 5 M, for

E

2

C(l) [ (a,b) xR , R ] s a t i s f i e s t h e i n e q u a l i t i e s IhyI 5 K. If I b - a l < l y / b ((u 2 +Klul +M)-')du

al = 1, a2 = 0, bl = 1, b2

#

0,

2

then t h e BVP (2.1,8),

(2.2.1) has a unique s o l u t i o n .

Hint: -

Prove (2.1.8), (2.2.1) has a t most one s o l u t i o n .

!Phis can be done by assuming t h a t t h e r e e x i s t s o l u t i o n s x2

a n d b y s e t t i n g x = x1-x2

obtain

xl,

lx"I < Mix1 + Klx'I,

x ( t ) = 0, x ( t 2 ) + b 2 x T ( t 2 )= 0. Then l e t t i n g w = x ' / x , 1 < w2 + KIWI + M, w ( t l - 0 ) = + m and w(t 2 ) = -l/b2. deduce ( w ' 1 By t h e theory of d i f f e r e n t i a l i n e q u a l i t i e s conclude t h a t

w(t) < u ( t ) ,

where

-u' = u2

u(t)

+

i s the solution of

Klul + M,

F i n a l l y o b t a i n a contradiction. 104

u(t2) =

- l/b2.

2.2. GENERAL LINEAR BOUNDARY CONDITIONS

I n t h e next two examples, we show t h a t Corollary 2.2.1 does not necessarily hold without t h e assumption t h a t a2b2 = 0 . EXAMPLFi 2.2.1.

(2.2.6)

The equation

2t l + t2

=

2 - x, 1+t2

X'

has the general s o l u t i o n x ( t ) = A(t2

-m


- 1) + B t .

+m

Hence there

e x i s t s one solution s a t i s f y i n g t h e boundary conditions x l ( t , ) = c1,x1(t2) = c2.

Observe

x 1( t )

are two d i f f e r e n t solutions s a t i s f y i n g

and

0

x2 ( t )

p

x(-1) = 0, x(1) = 0.

Moreover, t h e r e e x i s t s no solution s a t i s f y i n g

x(-1) = 0,

x(1) = 1. Thus (a) i n Corollary 2.2.1 i s violated.

EXAMPLE 2.2.2.

Let

be defined as i n Example 2.1.1.

Cp

Consider t h e d i f f e r e n t i a l equation

with t h e boundary conditions xt(t,)

- x(tl)

x ' ( t 2 ) - x ( t 2 ) = r 2'

= r 1'

The family of solutions of (*) i s given by x(t) = A where

A

and

B

+

Bet + t tan-'

B,

a r e a r b i t r a r y constants.

conditions lead t o ( l - t i ) tan Eliminating A

-1

B-A=

we obtain (tl

- t2)

tan

-1

105

B = r

2-rl

The boundary

-

t2 1

2 . SHOOTING TYPE METHODS

and thus t h e r e e x i s t s a t most one s o l u t i o n of t h e BVP.

How-

ever, i f

-

Ir2 rll _> (7J/2) It2-t1I, t h e r e e x i s t s no s o l u t i o n .

Hence (b) i n Corollary 2.2.1 does

not hold. When (2.2.7)

a2b2

#

rx(tl)

we can r e w r i t e (2.2.1) i n t h e form

0

+

x l ( t ) = c 1’ 1

s x ( t 2 ) + x l ( t2 ) = c

2‘

We now present a r e s u l t i n which uniqueness of (2.1.8) and (2.2.7)

implies existence under c e r t a i n added conditions.

-

Consider t h e ordering on

(al,bl)

Write

I: (a0,b0) (al,bl)

defined a s

R2

a. < bo

and

a1

5

a.

and

bl

3 bo,

> bo

and

a

2

a.

and

b

5

a.

1

1

bo.

< (aO,bo) when (al,bl) 5 (aO,bo) and

(al,bl) # (aO,bo). noted by ( h ; r , s ) . THEOREM 2.2.2.

Problem (2.1.8) and (2.2.7) w i l l be de-

Assume a l l s o l u t i o n s of (2.1.8) e x i s t on

(a,b) and a r e uniquely determined by t h e i r i n i t i a l values. Assume problem ( h ; r o , s O ) i s unique f o r any tl,t2 E (a,b), then t h e r e e x i s t s a unique s o l u t i o n o f

(h;rl,sl)

whenever

(rl,sl) < (r0,so)* Proof:

Assume t h a t

The o t h e r cases have s i m i l a r arguments and t h e r e f o r e we s h a l l consider t h i s case only. x(t,v)

Fix

tl, t2, and

c1

be t h e s o l u t i o n of (2.1.8) s a t i s f y i n g

106

and l e t

2.2. GENERAL LINEAR BOUNDARY CONDITIONS

x ( t 1, v ) = v,

x'(t,,v)

= cl-r

+

soX(t,v).

0

v.

Define T(t,v) = x ' ( t , v )

(h;ro,sO), v + T (t , v )

From t h e uniqueness o f mapping f o r

t

1 tl.

Define

( T(t,v2)

AT/Av Then

AT/&

t'tl f o r a l l t 2 tl.

=

i s a one-to-one

-ro + so

>

- T(t,vl))/(v2 - vl). 0.

Thus

+

so h / A v

AT/Av

i s positive

'ken, s i n c e

AT/Av = h ' / A v

>

0,

we see t h a t

Setting S(t,V) = x ' ( t , V ) + SIX(t,V) = T(t,V)

Hence

v +S(t2,v)

+

(sl

- S,)X(t,V),

i s a one-to-one mapping of

This i s equivalent t o

(h;rl,sl)

R

onto i t s e l f .

having exactly one s o l u t i o n

and thus t h e proof i s complete.

i s s t r i c t l y increasing and t h a t s o l u t i o n s of (2.1.8) e x i s t on (a,b) and a r e

COROLLARY 2.2.2. in

x

Assume

h(t,x,y)

uniquely determined by i n i t i a l conditions.

s

> 0,

problem (2.1.8),

Then, f o r

(2.2.7) has a unique solution. 107

r < 0,

2 . SHOOTING TYPE METHODS

Proof:

ro, so

To prove t h i s , choose

r < ro < 0 , 0 < so < s .

Hence

(r0,so) > (r,s) and by it s u f f i c e s t o show (h;ro,sO) i s unique.

Theorem 2.2.2,

Assume t h e r e e x i s t two s o l u t i o n s x = x1-x2.

and x2 and l e t 1 attains either its

x ( t ) f 0, x ( t )

Since

x

p o s i t i v e maximum o r negative minhum on x

such t h a t

Assume t h a t

[tl,t2].

a t t a i n s i t s p o s i t i v e maximum (the case i n which

i t s negative minimum uses a similar argument).

tl,

occurs a t

then

maximum occurs at

t h e second condition

s x(t ) 0

to

t h e maximum occurs at

>

x'(tl)

5

If t h e maximum

and t h e f i r s t

0,

r o x ( t ) + x ( ) = 0 cannot hold. If t h e 1 ' t2, then x ( t 2 ) > 0, x ' ( t 2 ) 2 0, and

boundary condition

x(t,)

> 0,

x(tl)

x attains

2

E

+

xt(t,) = 0

(tl,t2),

x ' ( t o ) = 0,

0,

however, by t h e monotonicity o f

cannot hold.

If

then x"(to)

h(t,x,x'),

5

0;

we have

a contradiction.

I n t h e next e x e r c i s e , we show t h a t t h e s t r i c t i n e q u a l i t y (rl,sl)

< (ro,so) cannot be weakened t o t h e i n e q u a l i t y

(rlYs1) 5 (ro,so). This w i l l provide another example i n which a2b2 # 0 and uniqueness o f t h e boundary value problem does not imply existence. EXERCISE 2.2.2.

Construct an example i n which t h e

hypotheses of Theorem 2.2.2 hold with replaced by t h e weaker condition

( y l ) < (ro'so) (rl,sl) 5 (ro,so) such

t h a t t h e r e e x i s t s no s o l u t i o n of t h e problem (2.1.8), (2.2.7).

Hint:

Let

Cp(p,q)

CP + p a r c t a n CP = q.

be t h e s o l u t i o n of t h e equation

Consider t h e d i f f e r e n t i a l equation 108

2.3. WEAKER UNIQUENESS CONDITIONS XI'

= F"cp(l/F',

x t / F t ) on t h e i n t e r v a l

w i l l be determined.

Let

w = x';

(-1,2),

w

then

where

F

s a t i s f i e s the

general solution

+

w ' ( t ) = BF'(t)

Solve f o r

x(t)

i n terms of

w(t)

conditions, where we may assume conditions on

F(tl),

This leads t o

1 where

F2

+

-r ( t + l )

1. Then obtain

t

F'(t)

>0

E

[-LO],

6

[0,11,

E

[1,21,

on

satisfies

tl

=

I (rc2 - c1)/(2r

-

F2(0) = 1 e'r, g(0) = l / r t g(1) = 1, g ' ( 0 ) = g'(1) = 0. Show

such t h a t

-4,t2 = 3/2 - 1)I 2 T / 2 ,

-< 1,

and

and f o r cl, c2 chosen so t h a t t h e r e e x i s t s no solution o f

(Use ideas i n B m p l e 2.1.1,)

(2.1.8) and (2.2.7). 2.3

>

t t

( ~ ~ (- 1l)el-t, )

i n which g 6 C ( l ) [ [ O , l ] , R ] l/r < g ( t ) < 1 for 0
tha t for

s = 1, r

and such t h a t

F2 + g(t)F; = 1,

-

and s u b s t i t u t e t h e boundary

F f ( t l ) , F ( t 2 ) , F 1 ( t 2 ) such t h a t unique-

ness holds (see Example 2.1.1) (-1,2).

a r c t a n B.

WEAKER UNIQUENESS CONDITIONS

It i s n a t u r a l t o ask whether it i s possible t o weaken t h e uniqueness condition of t h e BVP (2.1.8), (2.1.2) by not demanding t h a t t h i s condition hold f o r a l l tl,t2 [a,b+&) (or

tl,t2

E

(a-&,b]).

6

I n t h i s section, we plan

t o discuss how some of t h e previous r e s u l t s remain t r u e even when t h e uniqueness assumptions a r e weakened f o r and f o r

t2

E (b

- &,b + E )

THMlREM 2.3.1.

Let

f o r some h

6

E

>

0.

C[[a,b+E) xR2,R]

109

t1 = a f o r some

2 . SHOOTING TYPE METHODS

>

E

and suppose t h a t a l l s o l u t i o n s of i n i t i a l value problems

0

[a,b + E).

f o r (2.1.1) e x i s t on

Assume t h a t f o r a f i x e d

t h e r e e x i s t s a t most one s o l u t i o n on

[a,t2]

c

o f t h e BVP

1

(2.1.1) and (2.3.1)

.(a)

for a l l

= c1 '

and t2

c2 E R

x ( t 2 ) = C2'

(b

E

- E,

b + E).

Then t h e r e e x i s t s

e x a c t l y one s o l u t i o n of (2.1.1) and (2.3 - 2 )

.(a)

f o r each

= cl,

x ( b ) = c2

c2 E R. For each m E R

Proof:

define

Am = [x(t,m) E C ( 2 ) [[a,b

+E)

x R , R ] : x(t,m)

i s a s o l u t i o n of (2.1.1) with

x(a) = c

x ' ( a ) = m].

and Define t h e mapping

r(m) = [x(b,m)]; I?

maps p o i n t s i n t o s e t s

and an a p p l i c a t i o n of Lemma 2.1.1 y i e l d s t h a t connected s e t s i n t o compact, connected s e t s . then

i

5

r(m)

m <m2 < m 1

r(mi).

m2 < m

t h e BVP.

3

Since

there exists

m*

ml

#

m2,

5 max r(m2).

We now show E

maps compact,

fl r(m2) = $, because of t h e uniqueness of s o l u t i o n s We can thus define r(m,) 5 r(m2) i f and only i f

max r(m,)

x

r

If

r(ml)

of BVP.

that

1

m*

3'

E

i s monotone.

Assume, f o r example,

r(m,) < r(m3) < r(m2). Let < x < x2 and F ( [m ,m 1 ) i s connected, 1 3 1 2 [m,,m2] such t h a t x(b,m+) = x Since 3' and

x

we have a c o n t r a d i c t i o n t o t h e uniqueness of

The o t h e r cases may be t r e a t e d by a similar argument.

I n order t o complete t h e proof of Theorem 2.3.1, enough t o show

r ( R ) = R.

If not, t h e r e exists a

9

We f i r s t prove t h a t E

r(R) 110

r(R)

it i s i s open.

t h a t i s not i n t h e i n t e r i o r .

2.3. WEAKER UNIQUENESS CONDITIONS

Hence there i s an

r

t o n i c i t y of

r ((;,-GI)

m

such t h a t

x(b,m) = JI.

we can find m > m N

contains

-

< m such t h a t

or

as an i n t e r i o r point.

JI

contradict ion.

r(R)

It thus suffices t o prove t h a t

above nor below,

By the mono-

r(R)

Assume

This i s a

i s neither bounded

i s bounded above (a similar

argument i s t r u e f o r the other case) and l e t '1 = sup r(R). L e t xn(t,m ) denote a sequence of solutions of (2.1.1), n (xn(a,mn)) = cl,(x;(a,mn)) = mn such t h a t xn(b,mn) + q . For c E ( b , b + & ) and from the uniqueness of BW,

-

xn(c,mn> xn(b,mn) C-b

where

K = min {0, (x1(c,m,)

> -

- xn(b,mn)

xl(c,ml)

-> K

C-b

- q)/(c - b)].

We m a y assume t h a t , f o r i n f i n i t e l y many nls, we have x;(b,mn) 5 0, because a similar argument can be made f o r c E (b &,b) i n case x;(b,mn) _> 0 f o r i n f i n i t e l y m a n y n ' s .

-

We r e s t r i c t our arguments t o those Let

sn

=

[t:

o> x;(t,mn) 2

n

such t h a t

K, t E [ b , c ~ ] .

This s e t i s nonempty by the mean value theorem.

-

s = min S then b < s n < c and n n' without loss o f generality, sn + s

x;(b,mn)

If

x;(b,mn) x;(t*,mn)

t*

E

<

= 0,

0,

= 0

then

b = s

n then xn(t,mn) < q

x;(b,mn)
0 and.

kt

- n n n 5

K < x t ( s ,m )

0.

Thus,

(s m ) +?,. n n' n xn(sn,mn) < q. If

and

XI

t E [b,t*l, where and t* i s the f i r s t such point. men

Sn and therefore

for

sn E [b,t*].

Thus again xn(sn,mn) < q . [b,c]] (from the

Since xn(sn,mn) _> min [xl(t,mn): t E uniqueness of BVP), there e x i s t s a subsequence of

{xn(sn,mn)] which we choose t o c a l l by the same labeling, such t h a t

-

x (s ,m ) +xo. Thus, sn + s o , x (s ,m ) + x o , x;(sn,mn) + x o . n n n n n n Let S(6) be a 6 - b a l l i n R3 about ( S ~ , X ~ , ~ ~ ) , 111

2. SHOOTING TYPE METHODS

where

6


- so.

E

For

n

s u f f i c i e n t l y large

(snJxn(~nJmn),x~(sn,~n)) E ~(6). Since a l l solutions of (2.1.8) e x i s t on Ix$(t,m,)I

[a,b

there exists

+&),

< Q for t

E

[ a , s O + 6 ] . Let

be solutions o f (2.1.8) with montonicity of X(b9-Q)

2

q

or

x(b,*), x(b,Q)

Q

x(b,-Q)

> Xn(bjmn)

such t h a t

x(t,Q)

x(a,+Q) = cl;

either

>0

and

x(t,-Q)

then by the

> xn(b,mn)

which means

so that

x(b,Q)

2

q.

This i s a contradiction and t h e proof of Theorem 2 . 3 . 1 i s

complete. COROLLARY 2.3.1.

the BVP (2.1.8),

Under the conditions of Theorem 2.3.1,

(2.2.1) has a unique solution, i f

We leave the proof as an exercise.

m a y be helpful.

For a fixed

a2b2 = 0.

The following h i n t s

Consider the boundary conditions

clJ

define

r(m) = [x(b,m): where

x(t,m)

of (2.1.8) such t h a t

x' (.a,m)

= c1

- rlml.

i s a solution

x(a,m) = m

Then using the same techniques as i n Theorem 2.3.1,

and

show

r(R) = R.

By combining the techniques employed i n Theorems 2.1.1 and 2.3.1,

one can deduce t h e following generalization, the

proof of which we leave t o the reader. THEOREM 2.3.2.

Consider t h e BVP (2,1.1),

(2.1.2)

assuming conditions ( i ) and ( i i ) of Theorem 2.1.1.

Suppose

t h a t t h e condition r e l a t i v e t o t h e uniqueness of BVP, i n the sense of Theorem 2.3.1 i s t r u e . solution of (2.1.1),

(2.1.2).

112

Then t h e r e e x i s t s a unique

2.4. NONLINEAR BOUNDARY CONDITIONS

2.4

NONLINEAR BOUNDARY CONDITIONS We now present a r e s u l t i n which t h e boundary conditions

are. nonlinear.

Under some additional hypotheses on h ( t , x , x ‘ ) ,

it i s possible t o deduce existence r e s u l t s f o r general l i n e a r boundary conditions f o r which corollary t o our main r e s u l t . THEOREM 2.4.1.

S1, let

as

x2

-L

g2 (c)

+ y2

--tm

i n e i t h e r direction.

R2

be nonempty.

Assume f o r

t

uniformly f o r

E

(b

- E,b + E)

and a l l

one solution of (2.4.1)

Proof:

f = (fl,f2)

E [a,b+&),

and t h a t a l l

(2.4.2) c E R,

- (2.4.3).

and

g,(x(s),y(s))

= c

there e x i s t s exactly

The idea of the proof i s similar t o t h a t of the

proof of Theorem 2.3.1. gl(x,y) = c1

For every

[ a , b + & ) . men i f there e x i s t s

a t most one solution of (2.4.1),

s

This we s t a t e as a

Y ’ = f2(t,X,Y),

solutions of (2.4.1) e x i s t on for

0.

when parameterized i n terms of a r c length, leaves

every bounded region i n C,

#

Consider t h e boundary value problem

x’ = fl(t,X,Y),

(2.4.1)

then

a2b2

Let

Po

and parameterize 113

be an a r b i t r a r y point of

S1 by a r c length with

Po

2 . SHOOTING TYPE METHODS

a s t h e i n i t i a l point.

The points on

s

For any f i x e d

t h e parameter.

E

S1 w i l l be denoted by

[a,b + E ) ,

define f o r

s1

P E

Ts(P) =

[ (x(S,P),Y(s,P)),

where

(x(t,P),y(t~P))

denotes a s o l u t i o n of (2.4.1) s a t i s f y i n g ( x ( a , p > , y ( a , p > ) = PI, and l e t

rs(p) =

[g2(x(s,p),y(s,p)):

( X ( S , P > ~ Y ( S , P >E ) T ~ ( P > ] .

It i s s u f f i c i e n t t o show rb(R) = R, t o prove t h e theorem. From Lemma 2.1.1, Ts(A) i s compact and connected, i f A

i s compact and connected.

g (x,y) i s continuous, 2 It follows from a simple

Since

i s compact and connected.

rs(A)

argument t h a t

rs(R)

i s connected. rs(P1) n rs(P2) = @ f o r i s an i n t e r v a l o r p o i n t . We

By uniqueness o f t h e BVP, s E (b

- E,b + E ) .

rs

now show t h a t

Thus

r S (P)

i s monotone f o r

s

(b - E,b

E

+ E).

Let

< P2 < P (ordered by a r c l e n g t h ) and assume rs(P2) < 3 rs(P 1) < r s ( P 3 ) ( t h e o t h e r cases follow s i m i l a r l y ) . Let

P1 x

E rs(Pi), i = 1,2,3. Since i t h e r e e x i s t s P*, P2 5 P* < P

Since

P

1

< P2 5 P*,

i s connected,

rs([P2,P3])

E rs(P*). 1 t h i s c o n t r a d i c t s uniqueness o f s o l u t i o n s

- 3

such t h a t

x

of boundary value problems. Assume now

rb(R)

i s not a l l of

R

and t h a t

bounded above [ a s i m i l a r argument holds i f below], that

q

Let E

q = sup rb(R).

rb(R).

(x(t),y(t))

We claim

rb(R)

q d, rb(R).

is

rb(R)

i s bounded Suppose

Then t h e r e e x i s t s a s o l u t i o n of (2.4.1),

s a t i s f y i n g (2.4.2) and g2(x(b),y(b)) = q.

Let

and choose P2, P3 such t h a t P2 < P1 < P 3' Then from t h e monotonicity of rb, e i t h e r rb(P2) > rb(P1)

P1 = ( x ( a ) , y ( a ) )

or

rb(P3)

>

rb(Pl),

Since

q E rb(Pl)

114

t h i s contradicts the

2.4. NONLINEAR BOUNDARY CONDITIONS

maximality of

q k, rb(R).

and thus

q

There e x i s t s

zn

rb(R)

E

such t h a t

zn t q.

To each

t h e r e e x i s t s a unique point P E S1 and a unique s o l u t i o n n n of (2.4.1), ( x n ( t ) , y n ( t ) ) through Pn a t t = a such t h a t z

g2(xn(b),yn(b)) = zn.

Also

qn(t) = g2(xn(t),~,(t)), t

Pn E

i s monotonic i n

n.

Let

[a,b +&I. Then

A s i n t h e proof of Theorem 2.3.1, we want t o o b t a i n a

convergent subsequgnce of i n i t i a l p o i n t s

t

E (b

- &,b+ & ) ,

monotone i n

or

$A(b)

$A(b)

5

-< 0

t

E

(b

i s monotone i n n since {Pn] i s n and BVP's a r e unique. Also, e i t h e r $;(b) $ (t)

n

f o r i n f i n i t e l y many n ' s .

[a similar argument i s v a l i d i f

monotone decreasing i n

b

-< sn 5 b +A(b)

+ s

<

Let

E

n

+so,

monotonicity of Jln(sn), +A(",)

t*

E

$

n

[b,b

E

Let

+ E).

> -

0

Assume

qn(t)

is

Then

From t h e mean value

sn = min S

n'

Then

K < $;(sn) < 0. Thus, without l o s s o f $A(",) +qo. If $;(b) = 0, t h e n b = s

and

0, q n ( t )

However, then

t*

_> $A(t) 2 K].

i s not empty.

Sn

generality If

n].

>b, 0 Sn = [ t : t -

theorem,

We may assume

$'(b) > 0. n i s monotone increasing i n n f o r

qn(t)

- &,b + & )

Define

For f i x e d

A similar argument holds when

0.

also that

(xn,yn).

<

Sn

q

until and thus

we have

$'(t*)= 0 n qn(sn)

JIl(sn)

< q.

< JI,(sn).

a r e bounded condition (2.4.4)

t* > b.

f o r some From t h e Since implies

l i e i n a bounded region.

{xn(sn),yn(sn)] Choose a convergent subsequence

115

sn + so, x (s ) n n

--f

x0'

n

.

2 . SHOOTING TYPE METHODS

yn(sn) + y o . where

6

Let


be a b a l l of radius

Ss

- so.

6 about

For large n, (sn,xn,yn)

E

S6.

(so,xo,yo) Since a l l

[a,so + 61, we have ( x n ( a ) , y n ( a ) ) E Sk s u f f i c i e n t l y large. Let Pn = ( x n ( a ) , y n ( a ) ) , Pick

solutions e x i s t on for

k

Ql,%

E

S1

llqll > k, Ilgll > k,

such t h a t

rb

From the montonicity of rb(Q2)

> rb(Pn)

f o r a l l n.

either

rb(%)

Thus e i t h e r

Q1 > Pn > %. > rb(Pn) o r rb(%) 2 q o r and

rb(&2) 2 q. This i s a contradiction t o t h e maximality of and the proof follows. COROLLARY 2.4.1.

q

Consider t h e BVP

(2.4.5)

X"

,

= h ( t X, x ' ),

(2.4.6)

r x(a) 1

+

r2x1(a) = c 1'

(2.4.7)

s x(b) 1

+

s 2 x ' ( b ) = c2,

2

where h E C[[a,b+E)xR ,R]

and assume t h e BVP (2.4.5),

(2.4.6) and s 1x ( t ) + s 2 x ' ( t ) = c has a t most one solution on [ a , t ] f o r each t E ( b - E , b + & ) and a l l c E R. I f

[s x + s 2 x 1 I 2 + [s x ' + s 2 h ( t , x , x ' ) ] 2 1 1 uniformly f o r

t

E

solution of (2.4.5) 2.5

[b

- E , b + E],

- (2.4.7).

+m

as

x2

+ xf2 +m

then there e x i s t s exactly one

ANGULAR FUNCTION TECHNIQUE

I n the preceding sections of t h i s chapter, we used a shooting type method t o prove t h e existence of solutions from uniqueness assumptions.

I n what follows, we wish t o employ

the angular f'unction technique t o study a number of r e s u l t s concerning existence and uniqueness i n a unified way.

In

p a r t i c u l a r , we are i n t e r e s t e d i n discussing the solutions of the d i f f e r e n t i a l equations

116

2.6. FUNDAMENTAL LEMMAS

(2.5.1)

Y' = g ( t , x , y )

x ' = f(t,X,Y),

s a t i s e i n g the boundary conditions (2.5.2)

.(a)

(2.5.3)

x(b) s i n p - y ( b ) cos f3 = 0.

Here we assume

s i n a - y ( a ) cos CY = 0 ,

2

f,g E C[[a,b] xR ,R],

0

-< CY < T

and

0

< f3 5

T.

We may geometrically i n t e r p r e t t h e problem as finding a solution z ( t ) = ( x ( t ) , y ( t ) ) of (2.5.1) which l i e s on the s t r a i g h t l i n e x s i n . a , y cos CY = 0 a t t = a, and on the l i n e

-

x s i n B - y cos B = O

at

t=b.

With respect t o t h e solution polar angle cp(t)

z ( t ) = ( x ( t ) , y ( t ) ) the

i s defined i n the

q(t)

xy plane.

i s called the angular function of

defined as long as formulas r e l a t e

cp(t)

and

z(t)

x ( t ) = Ilz(t)ll cos d t ) ,

=

and i s well

z(t)

~ ( t )does not vanish.

The function

Thus the following

(x(t),y(t)):

Y(t) = llz(t)II sin cp(t),

and

cp'(t>= Observe a l s o t h a t (2.5.3)

x(t)y' ( t )

- y(t)x' ( t )

x 2 ( t ) +Y2(t) z(t)

i s a solution of the BVP (2.5.1)-

i f and only i f i t s angular function

cp(a) = a, f o r some integer

q(t)

satisfies

cp(b) = B + k-rr

k.

The following lemma i s a version of Lemma 2.1.1.

117

2. SHOOTING TYPE METHODS

LEbW 2.6.1.

[a,b].

s

Let

Assume a l l s o l u t i o n s of (2.5.1) e x i s t on 2 be m y compact connected s e t i n [a,b] X R

.

Then t h e s e t o f a l l s o l u t i o n s passing through 2 connected s e t i n R

.

form a compact,

S

Consider now t h e i n i t i a l conditions (2.6.1)

x(to) =

y ( t o ) = YoJ

xoJ

and denote t h e s o l u t i o n s of (2.5.1) and (2.6.1) a s

z(t,z0) =

( x ( t J x 0 J ~ o ) JY ( ~ J X ~ J Y ~where ))J 0 = (xoJy0)* ~ e t~ ( t , z o ) denote t h e angular function of any s o l u t i o n of (2.5.1) s a t i s % ing (2.6.1).

Observe t h a t

cp(t,zo),

f o r each

t

E

[aJbIJ is

multivalued s i n c e s o l u t i o n s a r e not i n g e n e r a l wniquely defined We now present a lemma s i m i l a r t o

by t h e i n i t i a l conditions.

2.6.1 f o r angular functions.

-ma

LEMMA 2.6.2.

Let

be any compact connected s e t o f

S

R2

such t h a t t h e r e e x i s t s a l i n e through t h e o r i g i n which does not intersect all

t

E

S.

Assume t h a t f o r a l l

[a,b].

#

0

for

Then 2

u

0

forms a segment f o r each EXERCISE 2.6.1.

Hint: -

zo E S, z ( t , z o )

dtJzo)

ES

t

E

[a,b].

Prove Lemma 2.6.2.

Assume t h a t t h e r e s u l t i s not t r u e and use Lemma

2.6.1 t o o b t a i n a contradiction. LEMMA 2.6.3.

There e x i s t functions

m(u)

and

that m(IIzoII) _< IIZ(tJZO)ll _< ~ ( l l z ~ l l ) ~

118

M(u)

such

2.6. FUNDAMENTAL LEMMAS

where

lim m(u) =

u+ Proof:

+m,

+m

Define

and m(u) =

inf IIz(t,zO)Il. llzo II 2 u

t c [a,b I By Leinma 2.6.1,

M(u)

lhu++w m(U) =

show

and +m.

a r e w e l l defined.

m(u) Let

ro

>0

be given.

t h e s o l u t i o n s of (2.5.1) passing through t h e s e t

We now Consider

S =[(t,z):

E [ a , b l , I/zII 5 ro]. By Lemma 2.6.1 t h e r e e x i s t s a u > 0 0 such t h a t llz(t)l/ 5 uo where ~ ( t )passes through S. Hence

t

if

I/zoII > uo,

then

> r0

for

thus

m(u)

-

llz(t,ZO)ll > ro f o r a l l t E [a,b] u

> u 0'

and

This proves t h e lemma.

We now s t a t e t h e following known r e s u l t on t h e semicontinuity behavior of s o l u t i o n funnels. IEMMA 2.6.4.

Assume t h a t s o l u t i o n s of

r' where

F

E

n n C[[a,b] X R ,R

there e x i st s

6

>

0

6(t)

e x i s t on

= F(t,r)

i s continuous and

solution r 2 ( t )

of

[a,b].

For each

such t h a t f o r each s o l u t i o n rl

where

1,

= F(t,r),

r ' = F(t,r),

satisfying 119

+

rl(t)

6(t),

116(t)II < 6, where

there i s a

r ( a ) = r,(a), 1

E

of

>0

2. SHOOTING TYPE METHODS

- r 2 ( t ) II < E

Ilr,(t) f o r a l l t E [a,b]. We s h a l l now compare (2.6.2)

x' =

(2.5.1) with

N

Y ' = &,X,Y),

f(t,X,Y),

2 f , g E C[[a,b]xR ,R].

N

where

N

side of (2.6.2)

Assume also t h a t t h e right-hand

are p o s i t i v e l y homogeneous, t h a t is, N

N

(2.6.3)

f ( t , c x , c y ) = c?(t,x,y),

for a l l

c

2 0.

We write

xg(t,X,Y)

[f,g]

@;(t,cx,cy)= cg(t,x,Y)

2 {F,;i?

- Yf(t,X,Y) 2 &(t,X,Y)

if N

-Yf(t,x,Y).

We now present a comparison r e s u l t . LEMMI1 2.6.5.

L e t solutions

(2.5.1) and (2.6.2), never vanish.

Let

z(t)

and 2 ( t )

of systems

respectively, be defined on cp(t)

and

2 t )

[a,b]

and

be the respective angular

f'unctions of t h e solutions and assume

v(a> 2 5(4,

(2.6.4)

Cf,d ,> C G I .

Then

V ( t > 2 6m(t) where (2.6.5)

for

a l l t E [a,b],

N

(pm(t) i s the minimal solution of $ ' ( t ) = z ( t , c o s $,sin

$)

120

N

N

N

cos cp-f(t,cos cp,sin

5) s i n $.

2.7. EXISTENCE

The theory of d i f f e r e n t i a l i n e q u a l i t i e s then gives

The proof i s complete. If solutions of (2.5.1)

and (2.6.2)

a r e uniquely determined

by i n i t i a l conditions, then we may conclude t h a t Observe a l s o t h a t

?(t,O,O)

i s a solution of (2.6.2).

g(t,O,O)

E

E

0

cp(t) 1 ;(t); so t h a t z ( t ) E 0

Moreover, we notice, from the

homogeneity t h a t Il?(t,x,y)II 5 A (x2 + y2)*, llz(t,x,y)/l 5 I 2 A (x + y2)2 f o r some A > 0. This implies t h a t a l l solutions of (2.6.2) e x i s t on

[a,b].

From the calculations used i n t h e

previous l m a , we see t h a t the angular function s a t i s f i e s a d i f f e r e n t i a l equation independent of t h e solution, so it i s consistent t o discuss the angular function of a system with a given i n i t i a l angle. 2.7

EXISTENCE Before proving existence r e s u l t s f o r (2.5.1)

- (2.5.3),

it

i s necessary t h a t c e r t a i n r e s t r i c t i o n s be placed on the solutions of t h e comparison equation (2.6.2) which s a t i s f y t h e boundary conditions (2.5.2)

and (2.5.3).

When solutions of (2.5.1)

and

(2.6.2) are uniquely determined by i n i t i a l conditions, these r e s t r i c t i o n s reduce t o t h e following:

l e t ?+(t)

denote the angular functions of (2.6.2) conditions

-4-

cp (a) = a, ?-(a) = CY

121

+

T

and ? ( t )

s a t i s f y i n g the i n i t i a l

and suppose t h a t

2 . SHOOTING TYPE METHODS

f3

+ kT<

y ( b ) < f3

+ (k+l)T

and

f3

+

( k + l ) T < $-(b) < p

f o r a certain integer

k.

+ (k+2)~

This condition, i n f a c t , implies

t h a t t h e only s o l u t i o n o f t h e BVP (2.6.2), Z(t)

=

(2.5.2),

(2.5.3) i s

0.

Since we do not assume t h e uniqueness of s o l u t i o n s , t h e previous conditions w i l l be i n terms of minimal and maximal solutions. Consider t h e a u x i l i a r y systems

(2.7 1)

x'

= :(t,x,Y),

Y ' = b(t,X,Y)

x'

= C(t,X,Y),

Y ' = d(t,X,Y),

and

(2.7.2) where in

a,b,c,d

x and y.

2 C[[a,b] X R ,R]

E

Let

8(t)

of (2.7.1) and (2.7.2).

and

and a r e p o s i t i v e l y homogeneous $(t)

be t h e angular flmctions

Then a computation similar t o one i n

Lemma 2.6.5 y i e l d s

(2.7.3)

€ I t ( t ) = b ( t , c o s @ , s i n 8 ) cos e - a ( t , c o s B,sin 8) s i n 0,

and

(2.7.4)

$ ' ( t ) = d ( t , c o s $ , s i n 9) cos $ - c ( t , c o s $ , s i n $) s i n $.

I n order t o avoid r e p e t i t i o n , l e t us l i s t below c e r t a i n hypotheses t h a t w i l l be needed subsequently, (H ) The m a x i m a l s o l u t i o n s $ i ( t ) and $ i ( t ) 1 . satisfying and +-(a) = CY + T obey M (a) = M

$i(b) < p

+ (k+l)T

f o r some i n t e g e r

and

k;

122

o f (2.7.4)

$ i ( b ) < f3 + ( k + 2 ) 1 ~

2.7. EXISTENCE

(H2) The minimal solutions satisfying

0+(a) = m

$(b)

CY

and e-(a) m

+ krr

>p

f o r t h e same i n t e g e r

k

@:(t)

+

=

IT obey

0 i ( b ) > f3

and

+

(k+l)r

a s i n (Hl);

(3)The following i n e q u a l i t y holds between

(2.5.1),

(2.7.1),

b(tJxJy)x

- a(tJx,y)y +i(tJxJy)

t

1. g(tJxJy)x-

-

E

(t,x>y)y

- c ( ~ J x J Y ) +Y% ( ~ J X J Y ) ,

[a,bl.

Observe t h a t when solutions a r e unique (H1) t h a t t h e only s o l u t i o n of (2.7.l),

(2.5.2),

systems

and (2.7.2):

< d(t,x,Y)x

uniformly f o r

of (2.7.3)

and e i ( t )

(2.5.2),

and

(2.5.3)

( 5 )imply

and (2.7.2),

(2.5.3) i s t h e t r i v i a l s o l u t i o n .

We now s t a t e and prove our main existence r e s u l t . THEOREM 2.7.1.

(2.5.1)

- (2.5.3)

Proof:

Under t h e hypotheses (H1)

-

(3), t h e BVP

admits a solution.

Let

z(t,c) = (x(t,c),y(t,c))

be any solution of

(2.5.1) sat i s fying t h e i n i t i a l condition (2.7.5) Let

cp(t,c)

x ( a ) = c cos a,

y ( a ) = c s i n a.

be the angular function of cp(a,c) =

("

IT

123

z(t,c)

if

c 7 0,

if

c
such t h a t

2. SHOOTING TYPE METHODS

From (€$), we o b t a i n 'pf(t,c)

(2.7.6)

5 b(t,cos

(&sin

cos v - a ( t , c o s

'p)

'p,sin

s i n cp

'p)

+ B0(t),

2 2 _s(t,x(t,c),Y(t,c))/(x ( t f c ) + Y ( t j c ) ) . be t h e minimal s o l u t i o n s of

where 60(t) -+ e m ( t ) , g-(t) m

rv

;l(t)

(2.7.7)

Let

g)

= b ( t , c o s 0 , s i n 0 ) cos z - a ( t , c o s 5 , s i n

s i n ;i'

+ 60(t),

-+ 0 (a)

where

=

m

a, g-(a) m

= cf

+ T, it follows t h a t

From (2.7.6) and t h e proof o f Lemma 2.6.5,

(%),

In view of

>

0:(b)

(2.7.9)

f3

we may p i c k

+ krr +

@-(a) =

cf

(2.7.10) whenever

Oi(b) > B + ( k + l ) T +

E,

6>0

so t h a t t h e r e

e+(t), 0-(t)

o f (2.7.3),

where

0 (a) = a,

p

>0

E

satisfying

- e-(t>I

le;(t)

and

< 6 for a l l t

I6,(t)I

+

that

- e+(t>l < E

Iz:(t)

[a,b]. Bo(t)

<

IcI

llz(a,c)I(

> ro,

We thus obtain, using (2.7.8) (2.7.11)

'p(b,c)

> 0:(b)

then

6 if

124

b(t,c)

- (2.7.10),

-> e+(b) - E > 0:(b)

<

E,

7I

Because o f (

MOreover, Lemma 2.6.3 implies t h a t t h e r e exists an such t h a t , i f

E.

we may p i c k

+ TT,such

there exists a

> 0 so small t h a t

(%),

By Lemma 2.6.4 and exist solutions

E

llz(t,c)

> P.

ro > 0

11 > P.

t h a t for

-E

),

>B+

c

krr,

2 ro

2.7. EXISTENCE

By applying a s i m i l a r analysis t o t h e maximal solutions of

(2.7.4) and using (H1)’ we i n f e r the existence of that, for

c

2 rl

(2 7.13

q(b,C)

r1

>0

)

l i e on

such

< B + (k+l)Tf~

and f o r c

(2.7.14)

5 -r 1 (k 4-2)T.

q(bJC) < @

Hence, f o r

c

2 r3

max(r J r ), 0

1

B + ki-r < cp(b,c) <: B + (k + l ) ~ , and f o r

5 -r3

c

B + ( k + l ) n < q(b,c) < This a s s e r t s t h a t the points

@

+ (k+2)7r.

z(b,r )

3

d i f f e r e n t sides of the s t r a i g h t l i n e Consider the s e t of solutions

2.6.1, is

x sin

3 3

such t h a t

@ - ycos @ = 0.

Thus

z(b,-r

x sin @

-y

uIlz I1 <: r3 [ z ( t , c ) 1.

t h i s s e t i s connected i n

c* E (-r , r )

and

R2-for each

z(b,c*) z(t,c*)

3

cos f3 = 0.

t.

BY

Lemma

Hence there

l i e s on the s t r a i g h t l i n e i s a solution of the BW.

This concludes the proof of Theorem 2.7.1.

We leave the proof of t h e following theorem as an exercise since i t s proof i s very similar t o t h e proof of Theorem 2.7.1. 13IEOREM 2.7.2.

(H2) holds. and

ei(t)

Consider the system (2.7.1) and assume

Moreover suppose t h e maximal solutions of (2.7.3)

satisfying

125

+ OM(&)=

cy

and

ei(t) @;(a) = a +7r

2 . SHOOTING TYPE METHODS

e+(b) < f3 + ( k + l ) n and M e x i s t s a s o l u t i o n of

obey

EXERCISE 2.7.1. COROLLARY 2.7.1. (Theorem 2.7.2)

ei(b)

< f3

f

(k+2)T.

Then t h e r e

Prove Theorem 2.7.2. Under t h e hypotheses of Theorem 2.7.1

t h e r e e x i s t s s o l u t i o n s of (2.5.1)

[(2.7.15)]

s a t i s f y i n g t h e boundary conditions x(a) s i n a

(2.7.16)

- y(a)

cos CY = c

x ( b ) s i n p - y ( b ) cos

f o r any

c1,c2

E

Hint: -

1' c2

R.

EXERCISE 2.7.2.

bounded by

f3 =

Prove Corollary 2.7.1.

U s e t h e f a c t t h a t a homogeneous expression i s 2 A(x + y2)' and make a change of v a r i a b l e s by

t r a n s l a t i n g t h e p o i n t of i n t e r s e c t i o n of t h e two l i n e s i n (2.7.16) t o t h e o r i g i n .

Then apply Theorem 2.7.1 (Theorem

2.7.2). EXAMPLE 2.7.1.

The equation x"(t)

+a

sin x + h ( t )

= 0

has a s o l u t i o n which s a t i s f i e s t h e condition

x(a)

= x(b) = 0 .

To show t h i s consider t h e s o l u t i o n s of XI'

= 0,

x ( a ) = x ( b ) = 0.

It i s clear t h a t t h e only s o l u t i o n of t h i s BVP i s t h e t r i v i a l

126

2.8. UNIQUENESS

solution x ( t )

as

1x1 + I x ’ (

=

0.

bbreover

(a s i n x + h ( t ) ( / ( ( x l+

(XI

I)

+O

An immediate application of Theorem 2.7.2

+m.

y i e l d s the desired r e s u l t . EXERCISE 2.7.3.

Show t h a t t h e boundary value problem

x” + g ( t , x , x ’ ) = 0,

x ( a ) = x(b) = 0,

has a solution, i f Ig(t,x,x’)1 2.8

1. Alxl +

B,

where

E

C[[a,b] xR+,R]

A(b-a)

2

<

2

IT

.

UNIQUUENESS

Assume t h a t

Li,

Mi,

Ki,

Ni

homogeneous i n t h e i r second variable. conclude t h a t

127

and a r e positively

From (2.8.1), we may

2. SHOOTING TYPE METHODS

Define now the Functions

128

2.8. UNIQUENESS

uv

-> 0,

uv

<

uv

> 0, -

uv

<

UV?

uv

<

UV?

0,

0, 0, 0, 0,

uv < 0. Consider t h e comparison systems (2.8.2)

u'

= P1(t,u,v),

v' = P2(t,u,v),

(2.8.3)

U'

= Ql(t,U,v),

V'

Observe t h a t u

and

v.

= &(t,u,v).

Pi, Qi, i = 1 , 2 a r e p o s i t i v e l y homogeneous i n + Letting $ ( t ) , $ - ( t ) be t h e angular f'unctions

associated with (2.8.2) and

+

8 (t), e-(t)

be t h e angular

functions associated with (2.8.3), we have (2.8.4)

$ ' ( t ) = Pl(t,cos Fl(t,

(2.8.5)

$1,

e i ( t ) = Ql(t,cos @ , s i n e ) cos e - & ( t , c o s @ , s i n e ) s i n 3

Since

$,sin +) cos + - P 2 ( t , c o s $ , s i n JI) s i n JI

Pi,

Q.

1

F2(t,8).

a r e continuous everywhere except along t h e co-

o r d i n a t e axes, we agree t h a t s o l u t i o n s of (2.8.2), d i f f e r e n t i a b l e almost everywhere.

(2.8.3) are

A s i m i l a r statement holds f o r

t h e angular functions which a r e not d i f f e r e n t i a b l e along t h e angular rays

n~r/2, n = 0 , +I, 2 2 ,

... .

We a r e now a b l e t o s t a t e and prove a uniqueness r e s u l t .

129

e

2 . SHOOTING TYPE METHODS

!EBOREN 2.8.1.

Assume

t h a t conditions (H1) and

(2.8.9, respectively.

f

and g

(5)hold

s a t i s f y (2.8.1) and

r e l a t i v e t o (2.8.4) and

Then i f solutions of (2.5.1) a r e

uniquely defined by the i n i t i a l conditions, there e x i s t s one solution of t h e BVP (2.5.1)- (2.5.3).

then z ( t ) = (;(t),y(t)) (2.5.2),

(2.5.3).

satisf'y the boundary conditions

Also l e t

angular function of -~ ( t ) . For convenience, we s h$atl)l l ebet the G(t), = y(t), -z = T ( t ) , and = $ t ) . We claim t h a t -x%(t,z,F) -T€$(t,x,y) 5 FIG-GIT (2.8.6) -< 5;P2(t,<,7) - y P p , Z , T ) . 3

From our previous estimates

-x(%(t,;) -

-

x ( % ( t , x ) +N2(t,y)) if i;

= xP2(t,Z,T). Similarly,

130

+N2(t,y)) if 2

y 2 0, ? < 0,

2.8. UNIQUENESS

The lower estimates i n (2.8.6) a r e obtained i n t h e same manner. Hence we obtain from (2.8.6), (2.8.7) where

F2(t,p)

5 7' =

6 y1-: z')/(s;2+y2)5 Fl(t,$),

F1, F2 correspond t o (2.8.4) and (2.8.5), respectively.

#

Notice t h a t ( z ( a ) , y ( a ) )

(0,O)

by t h e uniqueness of

i n i t i a l value problems and t h a t t h e r e e x i s t s no y ( t o ) = ( x ( t o ) , y ( t o ) ) = (0,O).

such t h a t

to

E

[a,b]

This implies

-

cp(t)

i s w e l l defined. From (H1),

(H2), and (2.8.6) and the theory of d i f f e r e n t i a l

inequalities, which i s applicable since continuous everywhere except a t

Fl(t,%),F2(t,5)

0 = nn/2, n = 0, 51,2 2 ,

are

...,

we obtain, using exactly t h e same methods as i n the proof of that either

"heorem 2.7.1, f3

+

krr < F(b) < p

+

(k+ l)a

if

cp(a) =

CY

+

( k + 2)n

if

q(a) =

CY

or

p

+

( k + 1)n < p ( b )

I n e i t h e r case

<

f3

x(b),y(b)

+ n.

cannot s a t i s f y

-

x(b) s i n p - y ( b ) cos p = 0 ,

unless ( z ( b ) , y ( b ) ) E 0,

a contradiction.

The existence of a

solution follows from t h e next corollary.

COROLLARY 2.8.1.

Assume t h a t t h e hypotheses of Theorem

2.8.1hold with t h e exception of the uniqueness of solutions Then there e x i s t s a solution of

of i n i t i a l value problems. the BVP (2.5.1) Proof:

(3) holds.

- (2.5.3).

I n view of Theorem 2.7.1, Let

x2 = y2

E

0

and

131

it s u f f i c e s t o show

x 1 = x, y1 = y.

Then we

2 . SHOOTING TYPE METHODS

o b t a i n from (2.8.6)

Letting

we see t h a t

G(t,x,y) = xg(t,O,O) -yf(t,O,O), G(t,x,y)/(x

lim

2 2 x +y + m

t

uniformly f o r

E

[a,b].

2

+y

2

1=

0

Thus (H ) holds and t h e proof of

3

Corollary 2 . 8 . 1 i s done.

A s before, our r e s u l t s immediately apply t o t h e more

g e n e r a l boundary conditions (2.7.16).

THEOREM 2.8.2. (Corollary 2.8.1)

Under t h e conditions of Theorem 2.8.1 t h e BVP (2.5.1),

(2.7.16)

has e x a c t l y one

s o l u t i o n ( a t l e a s t one s o l u t i o n ) . EXERCISE 2.8.1.

Prove Theorem 2.8.2.

We now apply our r e s u l t s t o t h e case i n which 2

C(l)[[a,b] xR ,R].

f,g E

I n o r d e r t o do so we must consider f i r s t

t h e case of l i n e a r systems; namely, t h e system x ' = all(t)x

(2.8.8) where

a

ij

E

+ a12(t)y,

C[[a,b],R].

(2.8.9)

e l ( t ) = a21 cos

8 = e(t)

Let

a s s o c i a t e d with (2.8.8).

Y ' = agl(t)x

a22(t)~,

be t h e p o l a r angle

Then 2

F3(t,e).

8

+ (a22 - a 11)

cos 0 s i n 8

Assume now t h e r e e x i s t two l i n e a r systems (2.8.10 )

+

z' = A(t)z,

z ' = A(t)z,

132

- a 12 s i n 2 8

2.8. UNIQUENESS

where

ma

Letting

Cp,

z t = x(t)z

JI be the angular functions associated with and

z7 = A(t)z,

V(t)

F3(t,cp)

respectively, we obtain, as i n

(2.8.91, (2.8.11) Since and

f

and g

and

J17(t) = F3(t,q).

a r e d i f f e r e n t i a b l e we have, f o r

(x2,Y2),

where

133

(xl,yl)

2 . SHOOTING TYPE METHODS

f o r a l l t E [a,bl and -m < x,y,u,v uniqueness statement i s true.

<

Then t h e following

+m.

COROLLARY 2.8.2. Assume i n (2.5.1) t h a t f , g E 2 C(l)[[a,b]xR ,R] and t h a t there e x i s t systems (2.8.10)

satisfying (2.8.14).

Assume a l s o t h a t t h e solutions

cp(t),

of (2.8.11) s a t i s f y (H1) and (H2), respectively.

Jr(t)

the B W (2.5.1), Proof:

(2.7.16) has exactly one solution.

Replacing

x

by

x2

+

s(yl-y2),

and integrating (2.8.14)

respect t o

s,

Hyptheses (H1),

Then

s(x1-x2)' from

0

y to

by

y2

1 with

we obtain

(%),

and (2.8.15)

imply t h a t t h e r e e x i s t s no In

n o n t r i v i a l solution of the BVP (2.8.8) and (2.7.16). (2.8.12) and (2.8.13) then (2.5.1)

where

+

let

x2 = y2 = 0

and

x1 = x, y1 = y;

reduces t o

6 ( t ) = f(t,O,O)

3

s3(t)/(x2+y2)

-10,

and

s 4 ( t ) = g(t,O,O).

s4(t)/(x2+y2)

134

-10

as

Hence x2

+

y2

-+ma

2.8. UNIQUENESS

Because of Theorem 2.7.1, (2.7.16)

t h i s assures us t h a t (2.8.16),

o r t h e equivalent problem (2.5.1),

(2.7.16)

has a

solution. Assume now t h a t t h e r e a r e two s o l u t i o n s (X1W,Yl(t)),

Z,(t)

z (t) =

1

= ( X 2 ( t 1 , Y 2 ( t ) ) of (2*5*1), ( 2 * 7 * 1 6 ) *

m e n x ( t ) = x,(t) - x 2 ( t ) , y ( t ) = y,(t) - y 2 ( t ) i s a s o l u t i o n As j u s t mentioned, t h e r e e x i s t s of (2.8.8), (2.5.2), (2.5.3). no n o n t r i v i a l s o l u t i o n of (2.8.8),

-

x(t) = y(t) complete.

(2.5.2),

(2.5.3).

Hence

"hus s o l u t i o n s a r e unique and t h e proof i s

0.

EXAMPLE 2.8.1.

We show t h a t t h e equation

x"(t)

+

6 sin x

+ h(t)

= 0

has a unique s o l u t i o n s a t i s f y i n g t h e boundary conditions .(a)

= x(b) = 0

if

I n Example 2.7.1 we have shown t h e r e e x i s t s a t least one solution.

We now apply Corollary 2.8.2 t o prove t h e r e exists

a t most one s o l u t i o n .

Observe t h a t t h e system x " ( t ) k 161x = 0

s a t i s f i e s (2.8.14)

with

.(a)

= x ( b ) = 0.

of t h i s problem a r e unique, if

according t o Corollary 2.8.2,

x(a)

= x(b) = 0

.

-

t h e equation with conditions

has a unique s o l u t i o n .

EXERCISE 2.8.2. XI'

(61<

We know t h a t s o l u t i o n s 2 Hence, T2/(b a)

+

Show t h a t t h e boundary value problem x ( a ) = x ( b ) = 0,

g ( t , x ) = 0,

has a unique s o l u t i o n i f E

5

gx(t,x)

5

2 /(b - a ) 2 )

(T

135

-

E,

E

> 0.

2 . SHOOTING TYPE METHODS

2.9

ESTIMATION O F

NUMBER OF SOLUTIONS

I n some cases, it i s of i n t e r e s t t o determine i f a BW has s e v e r a l s o l u t i o n s and t o estimate t h e number of t h e s e s o l u t i o n s . Before s t a t i n g our main r e s u l t we need some preliminary developments.

Assume t h a t t h e r e e x i s t two systems

(2.9.11

x ' = r(t,x,y),

Y ' = S(t,X,Y),

(2.9.2)

x'

Y ' = W(t,X,Y),

where

Let

8

r,s,v,w and JI

= V(t,X,Y),

E

C [(a,b] xR2,R]

and p o s i t i v e l y homogeneous.

be t h e p o l a r angles a s s o c i a t e d with (2.9.1) and

(2.9.2),

respectively.

Then

(2.9.3)

0 ' = s ( t , c o s @ , s i n 8 ) cos 8 - r ( t , c o s 0 , s i n 8 ) s i n 8

and (2.9.4)

$I

= w(t,cos + , s i n

$) cos $ - v ( t , c o s $ , s i n J I ) s i n 9.

We now assume conditions similar t o (H1) t h a t i s , assume t h a t :

( 5 )of

Theorem 2.7.1;

(HI) t h e maximal s o l u t i o n s & ( t ) and $;(-ti) of (2.9.4) 1 . s a t i s f y i n g $ ( a ) = a and *-(a) = a + 7r have t h e property M

JIL(b)

(H;) satisfying

M

< p + (a+l)n

and

t h e minimal s o l u t i o n s

+ 8 (a) = m

a

and

$&b) 0:(t)

@'(a) = a m

136

< B + (a+2)n;

and

+

T

0i(t)

of (2.9.3)

have t h e property

2.9. ESTIMATION OF NUMBER OF SOLUTIONS

where

a r e continuous an d

Ej2

,ljt

B1 (t,x, Y 1

lim

x2+ y2 4 0

x

= o

and

lim 2 x +y24 0

+ Y

,X,Y 1

62 (t

2

x + Y

2

= 0.

We a r e now able t o estimate t h e number of solutions of the BVP (2.5.1)-

(2.5.3).

Assume t h a t conditions (H1) - (?), ( H i ) THEOREM 2.9.1. (5)a r e s a t i s f i e d . Then t h e BVP (2.5.1) - (2.5.3) has a t l e a s t 2 ( k- P, I nonzero solutions, whenever k # a .

Proof:

z(t,c) = (x(t,c),y(t,c))

Let

be any solution of

(2.5.1) s a t i s f y i n g t h e i n i t i a l conditions = c cos

.(a) and l e t

y(a) = c s i n

a,

(Y

be t h e angular function o f t h i s solution such

cp(t,c)

that

Assume now Theorem 2.7.1, an

rl

>0

c

>

Using the same pmof as t h a t of

0.

we conclude from (H1)

such t h a t f o r

p

+ kn<

c

2

cp(b,c)

0

such t h a t f o r

0

there e x i s t s

rl


Similarly, we obtain from (Hi)

r

- (3) that

< c -< r0

(k+l)n.

- (3) the

existence of an

ro < rl, we can apply Lemma 2.1.1, q(b,[ro,rl]) i s connected, The previous

Since we may assume and conclude t h a t

two i n e q u a l i t i e s on

cp(b,c)

then imply the existence of 137

2. SHOOTING TYPE METHODS

p o s i t i v e numbers

+

cp(b,c.) = p J

m

where

+

c

E

3

+

[ro,rl]

such t h a t

( j = m+l,m+2

jT

,...,m + ( k - J l ) ,

= min(k,J).

S i m i l a r l y t h e existence of negative numbers

c-

3

found f o r which cp(b,cT) = @ +

J

T +

jn

(j = m + l , m + 2

Hence t h e r e e x i s t s o l u t i o n s

+

z(t,cj)

,...,m +

and

2 ( k-

implying t h a t t h e r e a r e a t l e a s t

can be

Ik-Jl).

z(t,cT)

J

of t h e BVP

s o l u t i o n s , completing

t h e proof o f Theorem 2.9.1.

We now present conditions on system (Hj) a r e s a t i s f i e d . and s a t i s f y

Assume t h a t

f(t,O,O)

2.5.1)

so t h a t ( H i ) 2 ,R]

f , g E C'')[[a,blxR

g(t,O,O)

0.

Let

and consider t h e l i n e a r system (2.9- 5) The angular f'unction (2.9.6)

p(t)

then s a t i s f i e s

P ' ( t ) = gx(t,O,O) cos

x cos

p

2

IJ. + [gy(t,O,O) -fx(t,O,O)l

sin p - f

2

Y

(t,O,O) s i n

+

p.

+

% ( t ) = pm(t) = + pi(t) p - ( t ) where & ( t ) , p i ( t ) a r e , p ( t ) and G ( t ) Let r e s p e c t i v e l y , maximal and minimal s o l u t i o n s of (2.9.6). The s o l u t i o n s of (2.9.6) a r e unique and thus

us assume t h a t

138

-

2.9. ESTIMATION OF NUMBER OF SOLUTIONS

a.

f o r some i n t e g e r Now l e t

cp(t,c)

be t h e angular function of any s o l u t i o n

z ( t , c ) = ( x ( t , c ) J y ( t J c ) ) of (2.5.1). x(t) c x(t,c),y(t)

E

we o b t a i n

y(t,c),

T ' ( t ) = gx(tJoJo) cos

2

Consequently, l e t t i n g

cp + ~ ~ y ~ t J o J o ) ~ f x ( t J o , O ) ~

2 x cos cp s i n cp-fy(t,OJ0) s i n cp

+

6(t),

where

Since

- gy(t,oJo)y(t)

[@;(tJxJy)-gx(tJo,o)x(t)

6(t)

g

fx(tJOJO)x(t)

- fy(tJO>O)y(t)1/(x2(t) + Y 2 ( t ) ) *

is differentiable,

6(t) + O

as

x

Hence (HI) i s s a t i s f i e d , i f we choose 6(t)(x2

+

y 2 )3 and

systems (2.9.1)

-f(t,xJy)

+

y2 - 0 .

Gl(t,x,y) Our comparison f

Gl(t,xJy).

b2(t,x,y)

2

and (2.9.2) a r e p r e c i s e l y system (2.9.5) and

the conditions (HI) and (H;) follows immediately from (2.9.7). 1 We thus deduce t h e following r e s u l t . COROLLARY 2.9.1.

f J g E C ( l ) [ [ a J b ] XSP,R] and s a t i s f y

f(t,O,O)

Assume t h a t (H1) J

where

sp

= g(t,O,O) = 0 .

- (H

) hold.

c (X,Y)3 IX2

y2

Let

5

P23 ,

If t h e s o l u t i o n s of

(2.9.6) s a t i s f y (2.9.7), then t h e r e exist a t least s o l u t i o n s of t h e BVP (2.5.1) (2.5.3).

-

21k- 41

It i s i n t e r e s t i n g t o f i n d conditions such t h a t t h e 2 I k - R 1 nonzero s o l u t i o n s .

boundary value problem has e x a c t l y

Clearly, t h i s w i l l always occur when monotonic i n

c,

both f o r

a s u f f i c i e n t condition f o r

c

>0

cp(t,c)

139

cp(t,c)

and

c

<

is strictly 0.

We now present

t o be s t r i c t l y monotonic.

2 . SHOOTING TYPE METHODS

Assume system (2.5.1) has the following

LENMA 2.9.1.

properties : 2 xg(t,X,Y) -Yf(t,x,Y) 2 0 , ( t , X , Y > E [a,bI xR 2 the paths of two d i f f e r e n t solutions z,(t) and

(a) (b)

z 2 ( t ) which s a t i s f y the conditions cp (a) = cp (a) = t a n 1 2

tan where and

-1

a

cp ( t )

1 z2(t), (c)

5

cp,(t),

and

-1 a,

cp,(t> 5

cp (b) = rp2(b) = t a n 1

B

tan-'

t

for

E

-1 p,

[a,bl,

cp2(t) a r e the angular functions of

zl(t)

respectively, do not i n t e r s e c t ; ( l / r ) [ g ( t , r cos cp,r s i n cp) cos cp

F(t,cp,r)

i s s t r i c t l y increasing i n

f ( t , r cos cp,r s i n cp) s i n cp] Then f o r each solution

z(t,c)

the angular function

i s s t r i c t l y increasing function i n

r.

cp(t,c)

c.

If the inequality i n (a) i s reversed and i f

s t r i c t l y decreasing, then EXERCISE 2.9.1.

Hint:

Let

cp(t,c)

Prove Lemma 2.9.1.

cpl(t,cl)

and

f o r two solutions

zl(t,cl)

and show

> cp;(a,c2).

t

E

cpi(a,c,)

(a,a+6)

if

F(t,cp,r) i s i s s t r i c t l y decreasing i n c.

6

cp2(t,c2)

and

be angular functions

z2(t,c2).

Assume

c1

>

c2

This inequality holds f o r

i s s u f f i c i e n t l y small.

Show t h a t the

(a,b] by using (b) and (c) and the theory of d i f f e r e n t i a l i n e q u a l i t i e s . inequality holds on

We have thus proved the following r e s u l t . COROLLARY 2.9.2.

Assume t h a t the conditions of Theorem

2.9.1 and Lemma 2.9.1 a r e s a t i s f i e d . 21k-

Then there e x i s t exactly

a ( nonzero solutions of the BVP (2.5.1)

140

- (2.5.3).

2.9. ESTIMATION OF NUMBER OF SOLUTIONS

Observe t h a t Corollary 2.9.2 remains v a l i d when i s s t r i c t l y decreasing i n

EXAMPLE 2.9.1.

cp(t,c)

c.

We o b t a i n an estimate, depending on

6,

of the number of s o l u t i o n s of t h e equation

+

X"

6 sin x

=

0,

s a t i s fying x ( 0 ) = 0,

x(1) = 0 .

W e apply Corollary 2.9.2 t o o b t a i n t h e r e s u l t . 2 Then system (2.9.5) becomes 161 < 7r

.

(2.9.8)

x ' = y,

Assume

y' = -&

with t h e boundary conditions x(1) = 0 .

x ( 0 ) = 0,

Since t h e r e e x i s t no nonzero s o l u t i o n s , it follows e a s i l y t h a t

a

= -1.

I n order t o find

k,

we observe t h a t t h e r e e x i s t s no

s o l u t i o n of x" = 0,

x ( 0 ) = 0,

x ( 1 ) = 0.

It follows e a s i l y t h a t k =-1. Thus f o r 161 <

2

7r

, we s e e (even

though k = a ) from Corollary 2.9.2 t h a t t h e r e e x i s t no n o n t r i v i a l 2 2 solution; hence t h e r e e x i s t s o n l y t h e t r i v i a l s o l u t i o n . If n T < I 61 < (n + 1)2T2, n = +1,+2, , t h e n an easy c a l c u l a t i o n y i e l d s

...

= n - 1. Hence t h e r e e x i s t s We l e a v e t h e case EXERCISE 2.9.2.

x"

+

2 [ ( n - 1)+1] 2n

1 61 = n2T2

solutions.

f o r t h e following exercise.

Determine t h e number of s o l u t i o n s of x(0) = 0 = x ( l ) ,

6 sin x = 0 ,

s a t i s fying

141

2 . SHOOTING TYPE METHODS

x ( 0 ) = x(1) = 0

when 2.10

161 =

n

2 2

T ,

,... .

n = 1,2

EXISTENCE OF IJYFINITE NUMBER OF SOLUTIONS I n Section 2.7,

we discussed t h e existence of s o l u t i o n s

of t h e BVP by showing

l i e s i n two s e c t o r s located on

z(b,c)

x s i n @ - y cos @ = 0

d i f f e r e n t s i d e s of t h e s t r a i g h t l i n e when

Ic I

i s sufficiently large.

It i s geometrically c l e a r t h a t t h e boundary value problem a l s o has a s o l u t i o n i f

z(b,c)

has a s p i r a l character when

c

tends t o i n f i n i t y . THEOREM 2.10.1.

Consider t h e BVP (2.5.1)

- (2.5.3).

Let

the inequality F(t,cp,r)

=

(l/r) [ g ( t , r cos cp,r s i n cp] cos cp f ( t , r cos cp,r s i n cp) s i n cp]

-

> E hold f o r

r

2

p0.

o > O

Suppose t h a t t h e

i n t o a f i n i t e number of s e c t o r s

such t h a t

-

t

E

[a,bl,

(t,cp), qj +

where

E

.

S1,. ,,Sm;

F(t,(P,r) =

lim r+ uniformly i n

xy plane can be divided

EL

cps ~

+a

P ~ + ~( j- = E

,...,n),

1

i s an a r b i t r a r i l y small constant.

Then

t h e BVP has an i n f i n i t e number o f s o l u t i o n s . Proof:

By Lemma 2.6.3,

there exists

142

co

>

0

such t h a t

2.10. EXISTENCE OF INFINITE NUMBER OF SOLUTIONS

z(t, c) function and a

p1

lo

1)

2

E

and t h e r e f o r e t h e angular

of t h i s s o l u t i o n i s w e l l defined.

cp(t,c)

Pick any

(2.

1 c 1 > co

does not vanish f o r

>

M > 0.

and choose

0

There e x i s t s

{'pj];=l

so t h a t

p0

2

F(t,cp,r)

r 2

holds f o r a l l

I5

,pl

I C P - CP.

M

where

-< t -< b,

a

M,

E.

J Define t h e discontinuous function H(CP) =

{

for

(61 5

c p = cp. -t 6 ,

J

E,

all o t h e r values of

cp.

\

We s e e immediately F(t,cP,r)

(2.10.2)

for

t

E

[a,b]

2

r

and

> H(V)

pl.

Consider t h e s o l u t i o n

$(t)

of +(a) = a-

+ ' ( t ) = H(+), Observe t h a t t h e time assume t h e value

ty

z =

Hence

$(b)

By Lemma

lcl

2

cp(t,c)

then

p2

is

2T

- $(a) 2 2nk,

z -10;

++a,

+

( ~ E / E ~n ) +

i n t e r v a l s of length M

z which i s required f o r $ ( t ) t o

where

k

i s t h e l a r g e s t number of

z contained i n

and thus $(b)

2.6.3,

(27r-2~n).

(l/M)

4 + m

there exists

Ilz(t,c)ll

[cp(a,c) = a, c

2

2

p,

E -0

As and

E

and

--3

0

M

-++a.

> 0 such t h a t when

Hence, t h e angular f'unction

pl.

of

p,]

[a,b]. as

satisfies t h e

z(t,c)

d i f f e r e n t i a l inequality lb(t,c)ll

CP'(t,c) = F ( t , T ( t , c ) ) ,

2

By t h e theory of d i f f e r e n t i a l i n e q u a l i t i e s

t

E

[a,b].

Hence

cp(b,c)

4 + m

143

as

c

++m.

H(cP(t,c)). cp(t,c) _> $ ( t )

for

Also, i n a s i m i l a r

2 . SHOOTING TYPE METHODS

manner, we conclude

cp(b,c)

By Lema 2.6.2,

of

c,

as

++m

c

+-a.

t h e r e e x i s t s an i n f i n i t e number of values

both p o s i t i v e and negative f o r which cp(b,c) =

B

(mod

TI-).

Hence our BVP has an i n f i n i t e number of s o l u t i o n s .

The proof

i s complete.

EXAMPI8 2.10.1. x' where

gJ h

=

-h(y),

Consider t h e BVP y ' = g(x),

a r e continuous and odd.

= x(b) = 0

.(a) Assume

xg(x)

2,

2

Yh(Y) _> EoY2 and 1bx+,g(X)/X = +ooj limy+m h(y)/y = +a. To show t h a t t h i s problem has an i n f i n i t e number of s o l u t i o n s , we merely check t h e hypotheses o f Theorem 2.10.1.

We n o t i c e

inmediately

Consider t h e four segments

SiJ i = l J 2 , 3 , k J i n t h e

xy plane

defined by

where

E

i s any small constant less than a,b]xSi,

where

M(r)

T/4.

For

we have

has t h e property t h a t

144

g(x)/x

> M(r)

for

1x1

>r

2.1 1 . NONLINEAR BOUNDARY CONDITIONS

and h(y)/y SO

that

> M(r)

M(r)

for

lyl

r

as

-++a

> r.

Since

M(r)

can be chosen

we see t h e hypotheses of

-++a,

Theorem 2.10.1 hold. EXERCISE 2.10.1. x"

Show t h e BW

+ x ( ~ + x ' ~ ) ~+ /h' ( t )

x(a) = x(b) = 0

= 0,

has an i n f i n i t e number of s o l u t i o n s . 2.11

NONLINEAR BOUNDARY CONDITIONS

Using techniques s i m i l a r t o those used i n Section 2.7, we i n v e s t i g a t e t h e existence of s o l u t i o n s of a two-dimensional d i f f e r e n t i a l system s a t i s f y i n g nonlinear boundary conditions. Results on t h e uniqueness and number of s o l u t i o n s may a l s o be deduced under conditions s i m i l a r t o those i n t h e previous s e c t i o n s , but we s h a l l omit them here. Before s t a t i n g our existence theorem c e r t a i n preliminaries need some mentioning.

I n p a r t i c u l a r , consider system (2.5.1).

Assume t h e r e e x i s t two functions, C(')[[a,b] W

such t h a t f o r each

xR2,R]

i s a l l of

W(t,x,y),V(t,x,y)

t

E

t h e range o f

V

and

and t h e s e t

R

(2.11.1) f o r a l l r e a l numbers (2.ll.2)

as

c1

and

c2;

IV(t,X,Y)l + IW(t,X,Y)l

llxll + llyll

--)a

uniformly f o r

The d e r i v a t i v e of V'(t,x,y)

=

av/at +

V

E

[a,bl.

along s o l u t i o n s of (2.5.1)

(t,x,y) +

av/+

t

-tm

av/:X

is

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

(t,x,y)*g(t,x,y)-

A s i m i l a r expression holds f o r Let

x(t)

and

y(t)

W'(t,x,y). be any s o l u t i o n of (2.5.1). 145

Now

2 , SHOOTING TYPE METHODS

T(t)

consider t h e angular function defined i n t h e

where V

VW

Fi,Gi

E

W;

and

and

(2.11.4)

plane.

Gi(t,V,W)/(IV(

uniformly f o r

t

E

p(t)

!%us

C [[a,bl XR2,R] Gi(t,V,W)

and polar radius

E

are p o s i t i v e l y homogeneous i n 2 such t h a t C[[a,b] xR ,R]

+ IW() - + 0

( V ( + (WI

as

[a,b].

Define

146

-> 0,

if

V

if

V < 0,

if

W

-> 0,

if

W

< 0,

if

V

-> 0,

if

V

<

0,

-+m

2.1 1 . NONLINEAR BOUNDARY CONDITIONS

Ql(t,V,W) =

Fl(t,V,W)

if W

-> 0,

F2(t,V,W)

if

W

< 0.

where

v_>o, w_>oJ v10,

w
v
w10,

v
t h e comparison systems a r e

(2.11.6)

A' = &l(t,A,B),

B' = &2(t,A,B),

(2.11.7)

s'

T'

= Pl(t,S,T),

= P2(tJS,T).

As before we see t h a t systems (2.11.6)

have d i s c o n t i n u i t i e s along t h e l i n e s well as

S = 0, T = 0.

A = 0

and (2.11.7)

and

B = 0

may

as

Assume t h a t a l l s o l u t i o n s of (2.ll.6)

and (2.11.7) e x i s t on [a,b] and are continuously d i f f e r e n t i a b l e everywhere except a t t h e previously mentioned l i n e s . Observe that

P1,

P2,

%, %

a r e p o s i t i v e l y homogeneous i n t h e i r last 147

2. SHOOTING TYPE METHODS

two v a r i a b l e s . 0(t)

Let

and

ing t o (2.11.6) B(t)/A(t),

+(t)

be t h e angular functions correspond-

and (2.11.7),

where

B(t)

respectively.

Since

tan e ( t ) =

are s o l u t i o n s of (2.U.6),

and A ( t )

we have @ ' ( t ) = Q2(t,cos 8 , s i n 0 ) cos 8

(2.11.8)

-

%(t,cos @,sin e ) s i n

e.

S i m i l a r l y,

+

+ ' ( t ) = P2(t,cos + , s i n +) cos

(2.11.9)

-

Pl(t,cos

+ , s i n +) s i n

+.

Consider t h e BVP (2.11.10)

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

y ' = g(t,x,y),

with t h e boundary conditions V(a,x(a),y(a)) s i n a - w ( a , x ( a ) , y ( a ) ) cos

(2.11.11) where

0

CY

= 0,

V(b,x(b),y(b)) s i n B - W ( b , x ( b ) , ~ ( b ) ) cos B = 0 ,

-< CY < T

and

0


We now assume e s s e n t i a l l y t h e same hypotheses (H1)-

as i n Section 2.7.1.

Since t h e n o t a t i o n i s t h e same we w i l l

not repeat t h e d e f i n i t i o n s .

Therefore, assume t h e following:

( G ~ ) The maximal s o l u t i o n s

++ ( a )

(2.11.9) such t h a t +;(b)

<

f3

f o r some i n t e g e r (G2) such t h a t

M

+

=

(k+l)n

m

+L(t)

CY, +'(a) M and

= a

+i(b)

of

and

+

Tr

satisfy

< p + (k+2)i'f

k;

The minimal s o l u t i o n s

+ 8 (a)

(3)

=

a, @'(a) m

=

a +

148

ei(t) T

and

satisfy

@:(t)

of (2.11.8)

2.1 1 . NONLINEAR BOUNDARY CONDITIONS

0i(b)

>p +

(k+l)T

f o r the same integer

and

>@ +

e:(b)

kT

k as i n (G1).

We a r e now able t o s t a t e our main existence r e s u l t .

Since

the proof i s similar t o t h a t of Theorem 2.7.1 we leave it as an exercise. For any

'MEOFEM 2.11.1.

- (2.11.3)

ing (2.11.1) (2.11.10),

(2.11.11)

R~

and W(t,x,y)

satisf'y-

t h e r e e x i s t s a solution of t h e BVP provided (G1) and ( G g ) hold. Prove Theorem 2.11.1.

EXERCISE 2.11.1.

Hint:

V(t,x,y)

For any

c

let

(x(a,c),y(a,c))

be a point i n

satisfying V(a,x(a,c),y(a,c)) = c cos a, W(a,x(a,c),y(a,c)) = c s i n CY. Letting

through cp(t,c)

(x(t,c),y(t,c))

(x(a,c),y(a,c)),

be any solution of (2.11.10)

we obtain the angular flmction

where t a n q ( t , c ) = ~(t,x(t,c),~(t,c))/~(t,x(t,c),~(t,c))

Proceed now as i n the proof of !theorem 2.7.1 and obtain f o r s u f f i c i e n t l y large

N

the two points

and

l i e on d i f f e r e n t s i d e s of t h e s t r a i g h t l i n e V s i n p - W cos B = 0

149

2 . SHOOTING TYPE METHODS

VW

i n the

plane.

Show then t h a t t h e r e e x i s t s a s e t V(a,S) = [-N,N].

S

C

R2

such t h a t

Thus t h e s e t

i s connected i n

R

2

.

The r e s u l t then follows.

L e t t h e hypotheses of Theorem 2.11.1

COROLLARY 2.11.1.

Then t h e r e e x i s t s a s o l u t i o n of (2.11.10) s a t i s f y i n g the

hold.

boundary conditions RIV(a,x($),y(a))

+ R2W(a,x(a),y(a)) = clY

R3V(b,x(b),y(b)) + R&W(b,x(b),y(b)) = c2, f o r any r e a l numbers

Ri,

i = 1,2,3,4

and

c., i 1

= 1,2.

Under t h e conditions of Theorem 2.11.1 and Corollary 2.11.1,

we n o t i c e t h a t when s o l u t i o n s of systems (2.11.8)

(2.11.9)

and

a r e uniquely determined by i n i t i a l conditions, then

and (G2) t o g e t h e r a r e equivalent t o t h e fact t h a t t h e r e

(G1)

e x i s t no s o l u t i o n s of (2.11.6)

and (2.11.7)

satisfying the

boundary conditions (2.11.11).

EXAMPLF 2.11.1. x’

(2.11.12)

=

2

x where

f,g

lf(Y)l

E

Consider t h e boundary value problem

x

(8)

+

f(y),

-Y(a) =

y ’ = 2y C1’

+o

Define

x,

w= 150

g(x);

x ( b ) = C2’

C[

v=

+

2

x -y.

2.1 1 . NONLINEAR BOUNDARY CONDITIONS

We a r e thus i n t e r e s t e d i n finding s o l u t i o n s of (2.11.12) s a t i s f y ing

V(b)

w(a) = c 1'

V

Observe t h a t curves

and k2.

V = kl

and W

and W

c

=

2'

s a t i s f y (2.11.1) intersect i n

= k2

since t h e l e v e l

R2 f o r every kl

It i s not d i f f i c u l t t o v e r i f y (2.11.2). V'

=

x'

=

x

+

W' = 2 x x ' - y '

f(y) = 2 2x

=

= 2y

v+ +

Now

f(y),

2xf(y)-*-g(x)

- 2w + 2xf(y) - 2y - g ( x ) +

= -2w

- g(x).

2xf(y)

F = F = V , 6 = 6 = f ( y ) , G = G = -2w, and 1 2 1 3 1 2 S~ = 62 = 2 x f ( y ) - g ( x ) . Since 1 2 x f ( y ) - g ( x ) I / ( 1 x 2 - y ~ + I ~ I ) - ~ o as 1x2-yl + 1x1 -+a, we see t h a t (2.11.4) holds. Systems Let

(2.11.6)

and p2 (2.11.9)

and. (2.ll.7) are then t h e same, with

=%

I

-2w.

P1

=

Ql

Because t h e s o l u t i o n s of (2.11.8)

=V and

a r e uniquely determined by i n i t i a l conditions due t o

the l i n e a r i t y of

Pi

%,

and

it i s s u f f i c i e n t t o

i = 1,2,

show t h e r e e x i s t no s o l u t i o n s of

v,

V' =

W ' = -2w,

satisfying W(a) = 0,

V(b) = 0 ;

t h a t t h i s i s t r u e i s c l e a r f o r any

a

and b.

Show t h e r e e x i s t s a s o l u t i o n of t h e BVP

EXERCISE 2.11.2. XI

=

x

+

2y s i n

y ' = (y/2) y ( a ) = cl,

+

2

x,

2 s i n x;

2 x ( b ) - Y (b) = c2.

151

2. SHOOTING TYPE METHODS

EXERCISE 2.11.3. 2.8 and 2.11,

Keeping i n mind t h e r e s u l t s o f Sections

s t a t e and prove a uniqueness theorem f o r a BVP

with nonlinear boundary conditions. 2.12

NOTES AND COMMENTS Example 2.1.1 i s

Theorem 2.1.1 i s due t o Hartman [41. taken from Lasota and Opial [81. Lasota and Opial [7] and Jackson

For Corollary 2.1.1,

[el.

r e s u l t o f Lees [l] while Exercise 2.1.2 2.1.3 may be found i n Jackson [2].

see

Exercise 2.1.1 i s a

i s of Levin [l]. Exercise

Theorem 2.2.1 i s due t o

Hartman [41.

Exercise 2.2.1 i s a c l a s s i c a l r e s u l t of De La Vallge

Poussin [ I ] .

meorem 2.2.2

i s due t o Tutaj [ l l .

i s taken from Keller [l] and Exercise 2.2.4

Corollary 2.2.2

i s based on Heidel

[l]. Section 2.3 i s adapted from Schrader and Waltman [4], while Section 2.4 is t h e work of Waltman [3]. For r e l a t e d discussion on uniqueness questions, see Bailey e t a l . [3], Shampine [ l ] and Sherman [ l ] . The contents o f Sections 2.5

- 2.10

a r e based on t h e work

of Perov [ l ] and Krasnoselskii e t a l . [2]. on existence and uniqueness

, s e e Waltman

due t o Bernfeld and Lakshmikanthan [2].

152

For r e l a t e d r e s u l t s

[3 1.

Section 2. ll i s

Chapter 3 TOPOLOGICAL METHODS

3.0

INTRODUCTION

I n t h i s chapter, various t o p o l o g i c a l p r i n c i p l e s a r e u t i l i z e d i n solving boundary value problems.

For example, t h e

Wazewski's t o p o l o g i c a l method t o g e t h e r with t h e connectedness properties of s o l u t i o n funnels i s used t o prove t h e existence of s o l u t i o n s .

I n t h i s s e t up, t h e boundary conditions consist

of s e t s which, i n p a r t i c u l a r cases, can be constructed from the knowledge o f upper and lower s o l u t i o n s and Nagumo's condition. We a l s o develop t h e Wazewski-like method f o r boundary value problems associated with contingent equations and s e t valued d i f f e r e n t i a l equations.

F i n a l l y , t h e continuous dependence of

solutions on boundary d a t a i s discussed.

3.1 SOLUTION FUNNELS We s h a l l consider t h e existence of s o l u t i o n s of boundary value problems a s s o c i a t e d with t h e two-dimensional system

153

3. TOPOLOGICAL METHODS

Let

f(t,x,y)

and g ( t , x , y )

be continuous i n a s e t

E

which

i s open r e l a t i v e t o

Q(to) where Q(to) I 7 R C E. tl,t2 2 0 and l e t S be a subset of C(tl). Denote by IE(S,tlJt2), the s e t of a l l points (x,Y) such t h a t there Let

is a solution

(x(t),y(t))

of (3.1.1) on

[tl,t2]

i n which

( x ( t l ) , Y ( t l ) ) E S, (x(t,),Y(t,)) = (%Y> and ( t , x ( t ) , Y ( t ) ) E E This s e t , as we have seen i s called the f o r a l l t E [t,,t2].

t = t 2' Before s t a t i n g our main r e s u l t s , we s h a l l need the following hypotheses w h i c h r e s t r i c t the behavior of solutions as they

solution funnel cross section a t

cross the

cp

and

surfaces.

$

We s h a l l always assume:

'

(Hi) f o r a n t i E [OJtoIJ (Xl'Y1) I ~ ( S l J 0 , t l ) sq(t1) implies t h a t there e x i s t s a solution of (3.1.1) emanating from w i t h a trajectory which i s on or above the $-surface (tlJxl,yl)

on some right neighborhood of

t

1'

(H2) f o r a l l tl E [O,tOl, (xl,Y1) E IE(Sl~O,tl) n)lt(,,.s , implies t h a t there e x i s t s a solution of ( 3 . 1 . 1 ) which emanates w i t h a trajectory which is on or below the from (tl,xlJy1) cp-surface on some right neighborhood of tl.

Our f i r s t r e s u l t describes qualitatively the behavior of solutions

of (3.1.1) assuming (H1) and (H2) hold. THEOREM 3.1.1. C(0)

Let

which intersects (i)

S1 be a compact connected s e t i n S+(O)

and

Sp(0).

Then e i t h e r

IE(SIJO>t) contains a compact connected component

C(t) which intersects both S ( t ) and S ( t ) f o r a l l Jr cp t E [OJto], or ( i i ) there i s a solution of (3.1.1) w i t h (x(O),y(O)) E S1 having a maximal right i n t e r v a l of [O,t+] C [O,tO] such t h a t in

154

3.1. SOLUTION FUNNELS

E: (p(t,y) - 1 < x < Jr(t,y) + 11. Assume t h e r e i s no number K such t h a t f o r a l l t E [O,tO], Proof:

Let

E'

(x,y) E IEl(Sl,O,t)

i s some i n t e r v a l

[(t,x,y)

implies [O,t+]

> K for t

C

(xn,yn) E IEl(Sl,O,tn).

lynl

+a.

IyI

5

K.

Then t h e r e

Ixn I + m,

1x1

>K

or

tn + t Ixnl + m o r

Then, e i t h e r +(t,y)

and

-k

and

it

(P(t,y),

Iyn I -+ m.

then

>0

K

Pick any sequence

By t h e continuity o f

case we have

and

such t h a t e i t h e r

let

follows t h a t i f

-< K

1x1

+ + [ t - &,t).

E

E

[O,tO] such t h a t f o r any

&(K) > 0

there e x i s t s an IyI

E

Hence i n e i t h e r

lynl + m.

By a standard diagonalization process, using t h e s o l u t i o n s

{ (xn,yn)},

associated with

of (3.1.1) with

(x(t),y(t)) on

[O,t]

[O,t+)

C

Moreover,

we may construct a s o l u t i o n which e x i s t s

(x(O),y(O)) E S1

and such t h a t

ly(t)l

+.

t +t

as

+m

This i s case ( i i ) .

( t , x ( t ) , y ( t ) ) E E' c E.

Hence, t o complete t h e proof we may assume t h e r e e x i s t s a

K

number implies

>

1x1

Let

such t h a t f o r a l l

0

-< K

t

T be the s e t of a l l p o i n t s the set 0 E T

[O,tO] such t h a t ,

E

IEl(Sl,O,t)

C ( t ) which i n t e r s e c t s both

i s nonempty s i n c e

contains a component

S J l ( t ) and

Sq(t).

Then

T

to. We

and i s bounded above by

i s closed and thus i f we l e t

s h a l l show T

(x,y) E I E l ( S l , O , t )

E (O,tO],

J y J5 K.

and

f o r a l l t E [O,T], in

t

s = sup T,

then

s = to implies t h e conclusion of Theorem 3.1.1. Let

be a sequence of p i n t s i n

{s.}

s with Ci

1

C

C(si)

S (si)

intersects both

l i m i t set of

a component of

{Ci}

Jr

and

and l e t

Then there e x i s t s a sequence where

(x;,yf)

E

C;

and

I E (S l

Let

S (s.).

' p 1 (a,b) E

L

E(x;,y;)I

converging t o

T

1'

0 si)

L

since

'

which

denote t h e L

i s nonempty.

converging t o

i s a subsequence of

{C;]

155

(a,b)

{Ci}.

3. TOPOLOGICAL METHODS

Let

be the s e t o f l i m i t points of

L'

compact and since for a l l If

intersects both

C;

then

i

{C;};

hence

S (s!) c p 1

and

is

L'

S (s;) Jr

L' intersects both S (s) and S (s).

cp Jr contains no component which intersects both

L'

S (s) and ST("), then since L1 is compact and intersects Jr both S (s) and S (s), L' i s the union of two nonempty cp compact s e t s

that

A

n L'

Assume {C!].

There e x i s t s a point

s; +s,

C

in

C(s)

closed.

a subsequence of

c (x;,~;)]

t o be the subsequence of

Thus

(p,q)

(p,q)

(p,q) E L 1 and

and we conclude

and

For i s u f f i c i e n t l y large the connected i n t e r s e c t s A and l e t (pi,qi) be the

C;.

IE,(Ci,si,s)

then

be any point i n N

(z',?') E C i ,

where

points of intersection. and since

(c,d)

Let

E M.

+ (c,d),

contained i n

such

= cp.

Choose { (x;,yy)]

1

set

J,

and N which are separated by an arc A

(a,b)

(?',Ti)

let

M

(pi,qi)

E

n S.

IEl(C;,s;,s)

which i s a l i m i t point of

i s a limit point of

(p,q) E A fl LI.

{C;].

{(pi,qi)] However,

This i s a contradiction

IEl(Sl,O,s) contains a component intersecting both S (s) and S (s). Thus T i s cp JI L'

and thus

By assumptions (H1)

and (H2) there e x i s t s a

6>0

such

that

and

IEl(C,s,t)

contains a component which ir,tersects both

S (t)

and S ( t ) f o r a l l t E ( s , s + 6 ) . However, t h i s JI cp contradicts the f a c t t h a t s = sup T. Thus s = to and ( i )

i s proven.

This concludes the proof of Theorem 3 . 1 . 1 .

Theorem 3.1.1 w i t h conditions t h a t r e s t r i c t the possibility of ( i i ) occurring can be used t o deduce existence theorems. have seen i n Chapter 1t h a t one such condition is Nagumo's condition.

The following condition w i l l thus be imposed:

156

We

3.1. SOLUTION FUNNELS

(H3) Given any number N(tO,n)

n

>0

and

to > 0 t h e r e e x i s t s a

such t h a t f o r any s o l u t i o n ( x ( t ) , y ( t ) )

(3.1.1) with

ly(O)[ < n and ( t , x ( t ) , y ( t ) ) E F, f o r l y ( t ) l < N(tOJn) f o r a l l t E [O,tO).

we have

of

t

[O,tO)

E

We now p r e s e n t an existence theorem. Assume conditions (H1)- (H ) hold.

THEOREM 3.1.2.

S1 be a compact connected s e t i n Sq(0)

and

C(to)

such t h a t

S (0). cp

Let

(x(t),y(t))

t h a t ( x(O),Y(O) )’ E S1’ for a l l t E [O,tOl. Proof: (x,y)

and l e t

S1,

E

s1

Since

l y ( t ) l < N(tO,n) (x(O),y(O)) E S1 IE(SIJO,tO)

which i n t e r s e c t s both

be a closed connected subset of

S2

S2 f l [(x,y): y

e x i s t s a s o l u t i o n of

C(0)

# $,

arbitrary]

of (3.1.1) on

( x(to)tY(t0))

E

then t h e r e [O,to]

n = sup (yI f o r

5 ).

N = N(tO,n) be a s i n (

f o r ally ( x ( t ) J y ( t ) )

Of

C ( t o ) which i n t e r s e c t s both S2

Then

(3.1.1) with

t E [OJtO]. By Theorem 3.1.1,

contains a compact connected component

conditions imposed on

such

S2 with ( t , x ( t ) , ~ ( t ) )E E

i s a compact s e t , l e t

and a l l

Let

3

S (t ) $ 0

and

insure t h a t

S2

S (t ).

cpo

n

C

The

IE(SIJO,tO)

in

# 9.

This concludes t h e proof of t h e theorem. Another approach t o t h i s problem i s t h e a p p l i c a t i o n of t h e Wazewski’s method.

We s h a l l introduce t h i s method here.

Consider t h e d i f f e r e n t i a l system (3.1.2) where

f

X’

E

= f(tJx),

n C[nJR 3, R

be an open set of closure of

x ( t 0 ) = xo

2

J

being any open set i n

OJ

Rn+l.

Let

n, anoJ t h e boundary, and E0 t h e

Ro

no.

DEFINITION 3.1.1.

A point

157

( t O J x O )E

fi

n ano

is said

3 . TOPOLOGICAL METHODS

t o be an egress point of

with respect t o t h e system

Ro

(3.1.2) i f , f o r every solution E

>0

such t h a t

x(t)

of (3.1.2),

to- E 5 t < to. An egress

for

( t , x ( t ) ) E Ro

there i s an

no i s called a s t r i c t egress point of no, Tio f o r to < t 5 to + E f o r a small E > 0.

point

( t O , x O ) of

if ( t , x ( t ) ) Denote t h e s e t of a l l points of egress ( s t r i c t egress) as

S

(S") DEFIXITION 3.1.2. topological space and B

into A

such t h a t

A

If T:

C

~ ( p =) p

i s s a i d t o be a r e t r a c t i o n of a r e t r a c t i o n of

B

onto

B

are any two s e t s of a

is a continuous mapping from

B +A

f o r every

B onto

A, A

p E A,

then

T

When t h e r e exists

A.

i s called a r e t r a c t of

B.

The following theorem of Wazewski i s quite useful. THEOREM 3.1.3.

Let

f E C[R,Rn), R

Assume t h a t through every point of solution of (3.1.2).

Let

Ro

be a nonempty subset of

r e t r a c t of

S,

there passes a unique

be an open subset of

Qo U S

but not a r e t r a c t of

at least one point

$+l R.

Suppose

no are s t r i c t egress points.

t h a t a l l egress points of Z

R

open i n

such t h a t 2.

Z fl S

Let

is a

Then there e x i s t s

no such t h a t the solution ( t , x ( t ) ) of (3.1.2) remains i n no on i t s maximal i n t e r v a l (tO,xO) E Z fl

of existence t o the right of

to.

W e now apply Theorem 3.1.3 t o the BVP prescribed i n Theorem 3.1.2.

Although we assume uniqueness of solutions of (3.1.1)

t h i s i s not e s s e n t i a l a s a Wazewski-like theorem f o r nonuniqueness has been developed.

Let

R = [(t,x,y): t

2 03 -S2;

a r e l a t i v e l y open subset of the h a l f space

no = [(t,x,y): 0 5 t < tl,

then

[(t,x,y): t

2

is 01. Let R

q ( t ) < x < q ( t ) , IyI < m1 where now a r e independent of y. Let Z f S From 1' hypotheses (H1) and (%), it i s not d i f f i c u l t t o see t h a t S = S", $(t,y), q ( t , y )

158

3.1. SOLUTION FUNNELS

and

consists of t h e union of t h e s e t s

S

We see from t h e properties of

i s a r e t r a c t of

S.

i s not a r e t r a c t of

and t h e s e t

that

S1n S

S

of (3.1.1) such t h a t

solution ( x ( t ) , y ( t ) )

S2

n S i s not connected S1 n S 1 Hence from Theorem 3.1.2 there i s a

Since

S1.

S

( tl,x(tl),y(tl))

E

and such t h a t ( t , x ( t ) , y ( t ) ) remains i n no on i t s S1 n C(t,) r i g h t m a x i m a l i n t e r v a l of existence. This implies ( t 2 , x ( t 2 ) J

due t o (H ).

Hence t h e r e e x i s t s a solution s a t i s f y -

y ( t 2 ) ) E S2 3 ing t h e BVP prescribed i n Theorem 3.1.2.

We now can obtain under t h e same hypotheses the existence of a solution remaining i n THEOREM

3.1.2, that

n

&(to).

3.1.4. Under the same hypotheses as i n Theorem

t h e r e e x i s t s a solution ( x ( t ) , y ( t ) ) of (3.1.1) such (x(O),Y(O)) E S1' ( x ( t , ) , y ( t O ) )

n

~ ( t ) E) R

Hint: -

=E

6

S ~ Jand

(t,x(t>,

Q(t,).

EXERCISE 3.1.1.

E~

R

Prove Theorem 3.1.4.

Choose a sequence of open s e t s

n

~ R+ ~~ ( t ,I ),

E~ = E,

and

n,,

W

Show t h e r e e x i s t solutions which l i e i n

where E

En boundary conditions by applying Theorem 3.1.2.

n Q(t,). and s a t i s f y t h e

= R

Then apply

Ascoli's theorem t o obtain t h e desired solution. We can now prove a r e s u l t f o r a semiinfinite BVP.

THEOREM 3.1.5.

Let

f , g E C [ [ O , ~ ) ~ R * , Rand ] assume 159

3. TOPOLOGICAL METHODS

-

kt

(H1) (H3) hold. intersecting both solution

S1 be a compact connected s e t i n C(0)

(x(t),y(t))

and ( t , x ( t ) , y ( t ) )

E

Sq(0).

e x i s t i n g on

R

n

Then (3.1.1) has a with

[O,m)

2

Q(to)for a l l t

(x(O),y(O)) cS1

0.

Prove Theorem 3.1.5.

EXERCISE 3.1.2.

Hint: -

and

S (0) $

Use Theorem 3.1.4 and a standard diagonalization

argument. 3.2

APPLICATION TO SECOND-ORDER E a T I O N S

In t h i s section we apply the r e s u l t s of Section 3.1 t o t h e case i n which

f(t,x,y) = y

i n (3.1.1).

That is, we consider

the equivalent second-order s c a l a r equation

(3s2O1)

XI1

= @;(t,X,X').

Recall t h a t a ftmction a ( t ) E C(2)[0,t

of (3.2.1)

if

(3.2.2)

a"(t) 2 g(t,(Y('(t),(Y'(t)),

bbreover,

(Y

- is >

1

t

i s a lower solution

E

[O,tOl.

becomes a s t r i c t lower solution if, i n (3.2.2),

replaced by

>.

Similarly a f'unction

i s an upper solution of (3.2.1) (3.2.3)

0

V(t)

5

B(t) E C(2)[0,to]

if

g(t,B(t),B'(t)),

t

E

[O,tOl-

If i n (3.2.3) 5 i s replaced by <, then B is a s t r i c t upper solution. We now present a general existence r e s u l t , Notice t h a t i n Chapter l o t h e r techniques have been used t o prove similar types of theorems. TI-IEOm 3.2.1. (i)

Assume t h a t :

there e x i s t s an upper solution

160

@ ( t )and a lower

3.2. APPLICATION TO SECOND-ORDER EQUATIONS

solution a ( t ) E

of (3.2.1) with

-< B(t)

a(t)

be any open s e t containing the s e t

5x5

a(t)

S,

I

# gj

and such t h a t

with

I<

such t h a t

(x(to)Jx'(to)) E

n

with

2

y'

0

Let

5 t 5 to,

such t h a t ( x , x l ) E S1 fl S a ( 0 ) f o r some

B'(0)

be a closed connected subset of

S2

[ ( x , ~ ' ) : x'

x(t)

C(0)

' ( 0 ) f o r some

#9

Let

S2

e x i s t s a solution

Proof:

x' <~

S1 fl S (0) B

(y,y') E S1 fl Sg(0). C(to)

[(t,x,x'):

be a compact connected s e t i n

S1 fl S a ( 0 )

[O,tO].

(3) holds.

(ii) Let

Ix'

B(t)J

on

of (3.2.1)

arbitrary]

# 8.

such t h a t

(x(O),x'(O)) E S ~ ,

s2 with a ( t ) < x(t) < @ ( t ) On

We f i r s t prove t h e r e s u l t with

a(t)

Then there [Ojt0I. and

p(t)

assumed t o be s t r i c t lower and upper solutions, respectively.

(3) i s s a t i s f i e d the p o s s i b i l i t y of

Since

Theorem 3.1.1 i s eliminated. of

IE(Sl,O,t)

Let

Ct

C

which i n t e r s e c t s both

C(t)

Sa(t)

be a component and

P by

Define

P

I

B

< x1 -

(x,x') E S a ( t ) fl Ct such a l ( t ) and there e x i s t s ( y , y l ) E

ct n sB ( t ) t E [oJ51]* P

S (t).

[t;: there e x i s t s that

Notice

case ( i i ) i n

such t h a t

i s nonempty since

y'

0 E P.

-> B ' ( t )

Let

f o r a11

n = sup P.

It i s

s u f f i c i e n t t o show n = to. Exactly a s i n t h e proof of Theorem 3.1.1 it follows t h a t There e i t h e r e x i s t s

n

E

P.

We omit the d e t a i l s .

( x , x l ) E Cn fl Sa(n)

such t h a t

x = a ( n ) , x ' = a l ( n ) o r there e x i s t s

(y,y') E Cn fl S (n) B f o r i f not, then there e x i s t s y = p(n), y ' = @ ' ( n ) such t h a t nl E P. This i s impossible.

such t h a t

nl

>n

( y , y l ) E Cn fl SB (n) such The other case i s similar. I f

Consider then the case where that

y = p(n), y ' = p ' ( n ) .

161

3. TOPOLOGICAL METHODS

y ( t ) i s any solution of (3.2.1) w i t h y(n) = B(n), y ' ( n ) p'(n), then there e x i s t s an &l > 0 such that y ( t ) >

=

p ( t ) , y ' ( t ) > p f ( t ) on (n,n+E1). This follows since p ( t ) i s a strict upper solution on [O,tO]. Moreover, f o r any t > n, any solution y ( t ) of (3.2.1) w i t h y(T) = p(T), y ' ( z ) < PI(:) satisfies y(t) < p(t) for < t < % + S and y ( t ) > p ( t ) on 6 < t C t forsome 6>0. Eythecontinuityof g, it follows t h a t there e x i s t s an E~ > 0 and p l > 0 such t h a t p"(t) < g(t,y,y') for a l l t E [ n , n + ~ ~and l a l l ( y , y f ) such t h a t d((y,y'), ( p ( t ) , B ' ( t ) ) ) < pl, where d denotes t h e Euclidean metric. By standard arguments there e x i s t an &3 > 0 and p2 > 0 such t h a t any solution y ( t ) of (3.2.1) w i t h

z-

y(n) = YoJ y ' ( n ) = YA with d((YoJY~),(p(n),B'(n)))< p2 s a t i s f i e s d(y(t>,y'(t)),(p(t),p'(t))) < p1 f o r a l l t E [ n , n + ~ ~ ] . Moreover, B"(t) < y " ( t ) f o r a l l t E [ n , n + m i n [ ~ ~ , ~ ~ l . Finally, there e x i s t s an &4 > 0 such that f o r any solution

~i

where (Y1,Yi) €Cn ~ l ( t )of (3*2*1)w i t h y l ( n ) = ~ 1 y9i ( n ) = and a((Y,,Yi>,(B(n),B'(n))) < P1 both y,(t) _> B(t) and y i ( t ) 5 p ' ( t ) are not possible f o r any t E [ n , n + &4 1. If we l e t E = min[& , E ,& 1, then it follows from the 1 2 3'&4 preceding observations t h a t Ct n S p ( t ) # # and y' 2 p ' ( t ) f o r some (y,yl) E Ct fl S ( t ) f o r a l l t E [O,n+&]. A similar B argument leads t o the conclusion t h a t Ct n S 2 ( t ) # $ and x ' < a ' ( t ) f o r some (x,xt> E ct n SCY(t) f o r t E [o,n+El. Then nl = min[n+&, n + F ] E P and nl > n, a contradiction.

-

Thus n = to and we conclude that IE(Sl,O,tO) contains a compact connected component intersecting both SCy( t o ) and Sp(to). Hence S2 fl IE(SIJO,tO) # $ and hence there e x i s t s a solution x ( t )

of (3.2.1) w i t h (x(O),x'(O)) E S1, ( x ( t o ) , and ( t , x ( t ) , x ' ( t ) ) E E. Using the same type of x f ( t o ) E) S2, proof' as i n Theorem 3.1.4, we may conclude that a ( t ) 5 x ( t ) 5 p ( t ) f o r a l l t E [O,tal.

162

3.2. APPLICATION TO SECOND-ORDER EQUATIONS

We now prove Theorem 3.2.1, B(t) 0

assuming t h a t

a(t)

a r e lower and upper solutions, respectively.


1, l e t

A (t) = a(t) - y

and

Y

and

For

+ y.

B ( t ) = B(t)

Y

Define

x >

g ( t , B ( t ) , x ' ) + x - B(t),

Q(t5 > x 5 B(t), x

g(t,a(t),x') + x-a(t),

< a(t).

Ix' I + 1 and l e t N be t h e number ( X J ' )fS1 associated with M i n ( ). Choose K = max(N, m a x Let

M = max

5

t Ero, t 0~ i

( l Q ' ( t ) l , I B ' ( t ) l > ) and define

I2K Ix' I 5 K

Ix' H(t,x,x') =

-

+

+ 2,

+ 1,

1< x '

(K+2 x ' ) G ( t , x , K + l ) ,

K

( K + 2 + x ' ) G ( t , x , - K - l),

-K- 2

<

X'

< K + 2, < - K - 1.

Eloreover A"(t) = ~ " ( t _>) g ( t , a ( t ) , ~ ' ( t ) > ) g ( t , a ( t ) , A;/(t)) - Y

Y

B"(t) = B"(t)

Y

Hence A

5

and

Y

g ( t , B(t), B ' ( t ) )

B

Y

X" =

(p,q)

(u,v)

E

union of

E

s n 1

S1,

S1

n

AY(t), A+(t)),

< g(t,

B(t), B+(t)) + y

= H(t,

By&), B;i(t))-

a r e s t r i c t lower and upper solutions of

(3.2.4) Let

= H(t,

H(t,x,x').

[ ( x , x ' ) : x = Q(o),

XI

2

B(0)l.

t h e l i n e segment from

(p,q)

[(y,yl): y = B(O),

t h e l i n e segment from

(u,v)

yi

to

and l e t

5 a'(0)1, Let

X

Y

be the

(p - ~ , q ) and

to

(u + y,v).

Consider now t h e BVP (3.2.4) with t h e boundary conditions (3.2.5)

(X(O),X'(O))

E

xY' 163

( x(t,>,x'

( t o >)

E

S2'

3 . TOPOLOGICAL METHODS

Let

E = [ ( t > X > X ' ) :O

Ix'I

<

Since

m1.

5 t 5 t O JQ ( t ) - 1 < X <

H(t,x,x') = 0

[ x ' I _> K

for

+

B(t)

+

1,

2,

we may

apply t h e proof of the f i r s t h a l f of t h i s theorem, r e c a l l i n g that

A (t), B ( t )

a r e s t r i c t lower and upper solutions Y Y respectively, t o conclude t h e existence of a solution Xy(t)

of t h e BVP (3.2.4),

< B ( t ) for - Y

X (t)

Y

for

2

Ix'I

K

+

2

(3.2.5) with t h e property t h a t A ( t ) < Y t E [O,tO]. Also, since H(t,x,x') = O i X r ( t ) [5 K

it follows t h a t

Y

+

2

on

By Kamke's convergence theorem a subsequence of converges t o a solution

Xo(t)

of (3.2.4) on

[O,tO].

n]i=l

{X

[O,tO]

11 such

that

( xo(l)Jxi)(l))

(xo(o)~xi)(o)) ' 1 ' Also A

a(t)

l/n

For

5

Xo(t)

(t) = a(t)

a(t)

g(t,x,x').

5

B(t),

- (l/n)

-< x -< B(t),

'2'

since

5 X l / n (t) 5

B(t) + ( l / n ) = Blln(t).

Ix'I < K + 1 we have

It follows, from

(?),

that

H(t,x,x') =

IXi)(t)[ < N

-< K.

Hence

s a t i s f i e s t h e BVP and t h i s concludes t h e proof of meorem Xo 3.2.1. It i s not necessary t o assume

t

and lower surface at which eventually hook

= 0

S1

i n t e r s e c t s t h e upper

provided t h e r e e x i s t solutions

S1 onto t h e lower and upper surfaces.

More precisely, we have t h e following r e s u l t .

THEOREM 3.2.2. a, B

Assume (H ) holds r e l a t i v e t o

3

E.

Let

B(t)

be lower and upper solutions f o r (3.2.1) with a ( t ) on [O,tO]. L e t S1 be a compact connected s e t i n

C(0)

and l e t

that

S2 fl [ ( x , x ' ) : x '

S2 be a closed connected subset of arbitrary]

#

@.

If

s1 such that

[(xl(u)Jxi(u))] 164

C(tO) such

(x,(t),xi(t))

and ( x 2 ( t ) , x h ( t ) ) a r e solutions of (3.2.1) with (x2(0)Jx6(0))

-<

( xl(0),xi(O)), s ~ ( u )# @

3.2. APPLICATION TO SECOND-ORDER EQUATIONS

[(x,(V)Jx;(v))] n s 2 ( v ) # $ f o r some v E (OJt0IJ then t h e r e exists a s o l u t i o n x ( t ) of

f o r some

u

( o , ~ , I and

E

(3.2.1) with

< x(t) a(t) -

( x ( 0 ) J X ' ( 0 ) ) E S1j (X(t,),X' ( t o ) )

-< p ( t )

and

S2j

f o r a l l t E [OJtoI.

The proof i s similar t o t h a t o f Theorem 3.2.1

Proof: since

E

I (S , O , t )

E l Sp(t) for t

has a component i n t e r s e c t i n g

2 max[uJvl.

Sa(t)

and

I n Chapter 1, we have provided s u f f i c i e n t conditions f o r

(3) t o hold,

One such condition was Nagumo's condition.

We now o b t a i n a r e s u l t t o show how Nagumo's condition can be u t i l i z e d t o determine

S1 and S2.

lower and upper s o l u t i o n s of (3.2.1)

t

E

Let with

a(t)

and

a(t)

[ O J t O l . Define

A = max

I

Ia(0)

J

max

g(t,x,y)

J

Ia'(t)lJ

max lp'(t)l [OJt01

s a t i s f i e s Nagumo's condition lg(t>xJy)l

f o r a l l t E [,,to]'

-< B ( t )

IMO)- B(0) I

- B(to) I [OJt01

and assume

B ( t ) be

a(t)

5

(P(lYl)

-< x < B(t),

Letting

165

IyI

<

w

such t h a t

for

3. TOPOLOGICAL METHODS

we observe t h a t

THEOREM 3.2.3.

Assume

g(t,x,y)

s a t i s f i e s Nagumo's

a(t),B(t) E C(2)[0,to] with a ( t ) 5 B ( t ) . If S1 is any closed connected subset of C ( 0 ) such t h a t S1 fl S # 0, S1 n S4 # 0 and i f S2 is any closed 3 connected subset of C(to) such t h a t S2 n S # jd and 5 S2 n S6 # I ,then there e x i s t s a solution x ( t ) of (3.2.1) such t h a t (x(O),x' ( 0 ) ) E S1, ( x ( t o ) , x l ( t o ) ) f: S2 with condition with respect t o

a ( t >,< x ( t > 5 B ( t ) * Proof:

Choose

n

(o,xl,xi)

G

s1 n s3

and

(o,x2,x;)

E

Sq such t h a t both p i n t s belong t o t h e same component S1 of S1 n [(O,X,Y): 5 N l . Choose (O,B(O),x) ) E Sp(0) as follows. If x1 = p(O), l e t = xi; i f x1 < B(O), S1

c

choose

IY)

5 > N.

Let

%

be t h e l i n e segment joining 166

(0,xl,xi)

3.2. APPLICATION TO SECOND-ORDER EQUATIONS

to

(O,p(O),x'), 3-

where, i n t h e case when

p(O), L1 = I n a similar manner, choose ( O , ~ ( O ) , X & c )

[ ( O , X ~ , X ; )c I sl. by l e t t i n g

xk = xh

Scy(0)

x2

>a(O)

and take

( O , X ~ , X ~t )o observe

x1

x2

if

= a(0)

or

=

xc

<

-N

if

t o be t h e l i n e segment joining

L2

(O,a(O),xi).

Let

ST

= L1 U

zl

and

U L2

ST i s compact, connected, and i n t e r s e c t s both

[(O,a(O),x'): x' 5 a ' ( O ) ] and {(O,p(O),x'): x' 2 p'(0)). Pick (t , x , x ' ) E S2 n s5 and (tO>X6'Xi) E s2 n s6

-

0 5 5

such t h a t both points belong t o t h e same component S2 S2

n

[(tO,X,X'):

l i n e s given by and l e t

s;

= L

5

5 N(tO)]. Let [(t0,x5,y): y 2 x;l IX'I

u 8, u L6.

Consider the BVP (3.2.1) (3.2.6)

(O,X(O),X'(O)) E

L5

and

of and L6 be t h e half [(to,x6,Y): y

and

( t0,x(to),x'(t0))

,s;

E

5 xi1

*

S2'

By the construction of

ST and t h e assumptions on g(t,x,y), there e x i s t s a compact, connected subset C C C(to) of the funnel cross section F(tO,O,ST) = i n t e r s e c t s both

uses++ F(tO,O,s)

which

1

S p ( t o ) : y 2 p t ( t o ) ] and Hence, by Theorem 3.2.1 the al(t,)].

[(tO,x,y)

E

[(tO,x,y) E S2(t0): y 5 BVP (3.2.l), (3.2.6) has a solution

x ( t ) with

a(t)

-< x ( t ) -<

B b ) , I x ' ( t O ) l 5 N(tO) and ( t O , x ( t O ) , x ' ( t O ) E) S2* From Nagumo's condition Ix'(0)I 5 N and by t h e construction of Sr, (O,x(O),x'(O)) E S1. This concludes the proof of Theorem 3.2.3. EXA.MFTE 3.2.1.

A s an application of t h i s r e s u l t consider

the following BVP a r i s i n g i n the dynamics of c e r t a i n chemical reactions. (3.2.7) (3.2.8)

X" = -LIX'

x(0) = 0,

- L2k(x),

x ' ( 1 ) = -L1X(l).

167

3 . TOPOLOGICAL METHODS

The positive constants

and

L1

of t h e physical problem with on

[O,c]

such t h a t

k(x)

L2

k(x)

>0

for

depend on various parameters continuous and nonincreasing 0

<x
and

k ( c ) = 0.

Theorem 3.2.3 gives t h e existence of a solution x ( t ) of t h e BVP (3.2.7), (3.2.8) with 0 < x(t) < c since m(t) = 0 and @ ( t )= c are lower and upper solutions, respectively, and Nagumo's condition i s c l e a r l y s a t i s f i e d .

3.3 WAZEWSKI RETRACT METHOD "his section i s devoted t o an application of t h e Wazewski r e t r a c t method t o (3.2.1) with boundary conditions

- blxl(0),

(3.3.1)

L1(x) = rl,

L1(x)

= alx(0)

(3.3.2)

L2(x) = r2,

L2(x)

= a2x(to) +

b2x1(t0).

Without loss of generality, we may assume b ,b > 0. Let 1 2 m(t) and B(t) be lower and upper solutions of (3.2.1) such that (3.3.3)

~ ~ ( 50 ri, )

(3.3.4)

Li(B)

We s h a l l assume t h a t

g

2

ri,

i = 1,2, i = 1,2.

s a t i s f i e s Negumo's condition, t h a t is,

As an application of Theorem

1.4.1, we obtain t h e following

lemmas.

168

3.3. WAZEWSKI RETRACT METHOD

U M M A 3.3.1.

Assume t h a t

Then, f o r every

> 0,

c

t h e r e e x i s t s a constant

t h a t each s o l u t i o n x ( t )

t

E

I

satisfies Nagmots condition.

g

satisfying

a(t)

M(c)

-< x ( t ) 5

such

p(t)

for

[O,tO] and which s a t i s f i e s

C

Jx'(sol

5

c

for

some

s0 E I

a l s o has t h e property

Assume t h a t t h e r e e x i s t s a s t r i c t lower

LEMMA 3.3.2.

solution a ( t )

and s t r i c t upper s o l u t i o n

t o ( 3 . 2 . 1 ) J s a t i s e i n g (3.3.3), all t

E

(3.2.1)

(3.3.4)

p(t)

with

with respect

< p(t) for

a(t)

Then, t h e r e e x i s t s a s o l u t i o n x o ( t )

[O,tO].

defined on

[O,h+&] C [O,tO] f o r some h _> 0 ,

of &

> 0,

a(t) < xo(t) 5 p(t) for a l l x o ( t ) > p ( t ) f o r t E (h,h+E].

s a t i s f y i n g (3.3.1) and such t h a t

t E [O,h], xo(h) = @(h) and S i m i l a r l y t h e r e e x i s t s a s o l u t i o n z o ( t ) s a t i s f y i n g (3.3.1), e x i s t i n g on an i n t e r v a l [O,k+ 61 C [O,tO] f o r some k _> 0, 6

> 0,

such t h a t

zo(h) = a ( h ) Proof:

and

a(t)

xo(0) = @(O),

zo(t)].

t

for

for a l l t

We show t h e existence of

y i e l d s t h e existence of letting

-< Zo(t) -< @ ( t )

zo(t) < a ( t )

E

xo(t)

E [ O J ~ ] ,

[k,k+61.

[ a similar proof

#

F i r s t assume bl

we o b t a i n from (3.3.4)

that

0;

then,

xA(0)

>

Hence t h e r e s u l t holds f o r h = 0.

p'(0).

Let

bl

#

0

and consider a s o l u t i o n s ( t )

s a t i s f y i n g , f o r each i n t e g e r

n,

xn(o) = r a 11'

169

x;(o> = n.

of (3.2.1)

3. TOPOLOGICAL METHODS

> M(B),

For N

there e x i s t s

s

0

E

[O,tO) satisfying %(so) =

and a ( t ) c % ( t ) < B(t), t E (O,so). Assume not; then e i t h e r cy(t) < % ( t ) < @ ( t ) f o r a l l t E (O,tO) o r there B(so)

T < to such t h a t x,(Tn) = a(Tn), a ( t ) < x n ( t ) < n for t E (O,%). I n e i t h e r case, there e x i s t s a

exists a

B(t), tNE [O,tO] such t h a t lXrj(tN)I 5 B. [In the second case, we make use of the f a c t < a r ( t N ) , since cy(t) i s a s t r i c t lower solution. ]

s(5)

Lemma 3.3.1,

since xP;(O)

= N,

ls(t)l-<

f o r t E [O,tNl and we have a contradiction f o r N > M(B). M(B)

Thus f o r some h E [O,t,I,a(t) 5 B(t) for a l l t E [O,hl and %(h) = p(h). Since p ( t ) i s a s t r i c t upper solution, we have

> B'(h). Thus, there e x i s t s an E > 0 such t h a t % ( t ) > p(t), t E (h,h+&]. The proof i s complete. Xp;(h)

U M M A

3.3.3.

by a parameter

simultaneously.

x(t,p)

Let

Let

p.

x(t,p)

open exterior of

U Int

where

t

M

=

C

be continuous i n

t

and

Let C be a simple closed curve i n the

plane and denote by I n t C C

be a family of functions indexed

and x ( t , p )

(t,x)

and E k t C the open i n t e r i o r and

Suppose t h a t f o r each

C.

p

p, (O,x(O,p))

is defined on an i n t e r v a l

E

[O,TM]

( T ,x(Tp,p)) E Ext C. Define y(p) = t where CI CI sup[t > 0 : ( t , x ( t , p ) ) E I n t CI. Then y is bwer semi-

continuous.

If, i n addition, each M c t i o n

x(t,p)

has the

t

t + h ), P' CI l.l then y i s continuous. (Lemma 3.3.3 is another version of the Wazewski method.) additional property t h a t

THEOREM 3.3.1.

B(t)

(t,x(t,p))

E

Ext C

for

E

(t

Assume t h a t there e x i s t s an upper solution

-

and a lower solution a ( % ) of (3.2.1) with cy(t) < B(t)

Let f f o r a l l t E [O,tOl, s a t i s w i n g (3.3.3) and (3.3.4). satisfy Nagumo's condition with respect t o cy and B. Then there is a solution x ( t ) of t h e B" (3.2.l), (3.3.1), (3.3.2) 170

3.3. WAZEWSKI RETRACT METHOD

Proof:

We may assume t h a t (3.3.5)

since we may redefine

g

o u t s i d e of

holds f o r a l l x D

E

R

so t h a t t h e Nagwno’s

condition w i l l hold. that g

We f i r s t prove Theorem 3.3.1 under t h e stronger hypotheses w ( t ) and B(t) a r e s t r i c t lower and upper s o l u t i o n s and

i s l o c a l l y Lipschitzian.

We w i l l then show how these

hypotheses may be weakened. Consider the s e t of s o l u t i o n s of (3.2.1) and (3.3.1) such that a ( 0 ) < x(0) < p(0). These s o l u t i o n s form a one-parameter family of functions a r e connected.

{x(t,p)],

where t h e s e t of values of

Define a map cp:

P

--f

( t p , x ( t p , PI),

t i s defined i n Lemma 3.3.3 i n which C P Jordan curve forming t h e boundary of t h e region x ( t , p) i s continuous i n both t and p since Lipschitz and thus applying Lemma 3.3.3 we have continuous. Thus cp i s continuous and {rp(p)} Combining Lemma 3.3.2 w i t h t h e connectedness of have the existence of v1 and p2, such t h a t

is the

where

Since

CY

and

p

0

are strict solutions

171

D.

g

Now i s locally

y: p -+ t is CL i s connected. {cp(p)}, we

3. TOPOLOGICAL METHODS

The solution x ( t , p o )

i s a solution of BVP (3.2.l),

(3.3.2). We no longer assume From (3.3.6),

g

(3.3.1),

s a t i s f i e s a Lipschitz condition.

there e x i s t s an

>0

E~

such t h a t i f

<

0

E

<

then

There e x i s t s a sequence of l o c a l l y Lipschitzian functions

E gn ( t , x, Y 1Inzl M

2 [O,tO]X R

.

Ign(t,x,y) Pick N

which converge t o

g(t,x,y)

Assume t h a t

- g(t,x,y) I 5

1/n

so large such t h a t

for 1/N

5

n

> N, a

(t,x,y)

[0,t0I X R

2

and

E~

1/< ~ inf[Iatt - g ( t , a , a l ~ ,

Thus, f o r

uniformly on

IB"

-g(t,~,~f)Il,

p w i l l be s t r i c t lower and s t r i c t

and

upper solutions for (3.3.7)

XI'

= gn(t,x,x'),

where Ign(t,x,y)

I 5 h ( IY I )

+

(l/n)*

By o u r previous analysis, t h e r e e x i s t s a function s ( t ) s a t i s f y i n g (3.3.7), (3.3.1), and (3.3.2). From t h e mean value theorem, there e x i s t s a such t h a t

Ix'(T)I

5

K/to

Choose M = M(K)

= K1.

so large t h a t

SMiqq K1

Then, t h e r e e x i s t s

N1,

sds

'K*

such t h a t f o r 172

n

> N1

T

E

[O,tO]

E ~ ,

3.3. WAZEWSKI RETRACT METHOD

s

s ds + (l/nl

h(s) K1

As i n the proof of Theorem 1.4.1,

> K.

we have f o r

t

Ix;(t)l

IM(K)

for

all

Ixn(t)I

IK

for

all t

n

> N1j

E [OJtO].

Since E

[OJtO]J

it follows t h a t Ixi(t)

I I max[lg(tJxJy)I 1x1

Therefore t h e sequence

5 K,

[O,to],

+ 1:

I MI. for n > max(N,N1)

IYI

{xn(t)?,

subsequence which converges i n t h e

C(l)

function i s then a solution of (3.2.1)

norm.

has a

The l i m i t

and s a t i s f i e s (3.3.1)

and (3.3.2). We now remove t h e r e s t r i c t i o n t h a t a ( t ) and B(t) are s t r i c t lower and upper solutions. Let p: [O,tO] -+ (O,tO] satisfy

p"(t)

i = 1,2,

and

-< 0

for a l l t

E

[O,tOl Li(p) 2 0

a,(t) = a ( t )

Define 173

for

i = 1,2,

for

i = 1,2.

for

- EP(t).

3. TOPOLOGICAL METHODS

and l e t

where

6 ( & ) i s t h e modulus o f c o n t i n u i t y of

g(t,x,y)

domain D' = [ ( t , x J y ) : t E [ 0 , t 0 I ,

1x1 5 K, IyI i s defined as before and S > max (M(K), "Pt

5

K

E [0

[O,tOl I p ' ( t ) l ) .

Since

and

Ig(t,c?Jc?')- g(t,ct&Jc?;)l <

y& ( t , a E Ja E' ) E

to1 l a w I >

<

we have

= -2

g ( t , a , a ' ) = B(t,CY&,CYL) E

where

laE(t)-a(t)l + la;(t)-al(t)l

SU%€

Moreover, since

SIJ

on t h e

E

- 6(E).

Hence

a; > a"

aE i s a s t r i c t l o w e r s o l u t i o n o f

Thus

Similarly

BE i s a s t r i c t upper s o l u t i o n of (3.3.8).

Furthermore,

as

-> g ( t , a , a ' ) > gE(t,aE,a;).

E

+O.

g (t,xJy) E

Therefore

converges t o

g (t,x,y) E

g(t,x,y)

uniformu

satisfies Nagumo's condition

and as before t h e r e e x i s t s a s o l u t i o n x E ( t J x , y ) of (3.3.8), (3.3.1),

and (3.3.2).

Moreover, 174

Ex,}

converges i n t h e

C(l)

E

3.4. GENERALIZED DIFFERENTIAL EQUATIONS

The l i m i t i n g f'unction i s then a s o l u t i o n of the BVP (3.2.1), (3.3.1), and (3.3.2). This concludes the proof of Theorem 3.3.1.

norm as

E

+ 0.

REMARK 3.3.1.

The significance of t h i s r e s u l t i s that we can show existence of solutions of t h e Neumann problem, that i s , when a1 = a2 0 i n (3.3.1) and (3.3.2). Observe t h a t except f o r t h i s very important case Theorem 3.3.1 i s a consequence of Lemma 3.3.2 and Theorem 3.2.2. The following example i l l u s t r a t e s Theorem 3.3.1. EXAMPLE 3.3.1.

Assume r1

=

and. satisfies Nagumo's condition.

r2 = 0 Let

and

(3.3 . l o ) then, from (3.3.9)

c > 1/E

t € [O,

11

e x i s t and satisfy

gx

Then the boundary value problem (3.2.1), a solution. To see t h i s l e t Ig(t,O,O)

g: [0,1]xR2 - t R

(3.3.1),

(3.3.2) has

I;

we have

g(t,O,O) -g(t,-C,O) _> CE. Using (3.3.10 ) , we then obtain

g(t,-C,O) < 0. Thus a ( t ) = -C i s a b w e r solution and similarly ~ ( t =) c i s an upper s o l u t i o n f o r (3.2.1), (3.3.1), and (3.3.2). By Theorem 3.3.1, there exists a solution of t h e BVP (3.2.l), (3 03 el), (3 e3.2). 3.4

GENERALIZED DIFFERENTIAL EQUATIONS

I n t h i s section we develop a Wazewski theorem f o r con-

tingent d i f f e r e n t i a l equations and. u t i l i z e it t o prove 175

3. TOPOLOGICAL METHODS

existence theorems f o r BVP's. The following preliminaries w i l l prove usef'ul i n t h e discussion.

Let

(cc(Rn)) be t h e c o l l e c t i o n of a l l

c(Rn)

nonempty compact (compact and convex) subsets of

x

and A,B E c(Rn),

E Rn

and

let

For

Rn.

q(x,B) = i n f [ l l x - b ( ( : b c B]

q(A,B) = sup[q(x,B): x E A].

Then

d(A,B)

I

max[q(B,A),q(A,B)] i s t h e Hausdorff metric on c(Rn) and n c(R ) i s a complete metric space. Let V be a subset of R x R n and denote t h e points of V by

p = (t,x).

continuous (USC) at a

6

>0

such t h a t

If we replace

continuous

> 0,

p E V

i f , f o r each

E

IlQ-pII

<

q(F(Q),F(p))

implies

6

q(F(Q),F(p)) by

d(F(Q),F(p)),

X be an open subset of

Let

be USC.

.

i s upper semi-

F: V + c(Rn)

A mapping

RxRn

there e x i s t s then

and l e t

<

E.

is

F

F: X +cc(Rn)

A s o l u t i o n of t h e generalized d i f f e r e n t i a l equation

(3 4.1)

x ' E F(t,x)

9

i s a function

cp: I + R

n

which is absolutely continuous on

each compact s u b i n t e r v a l of everywhere on For

I

and

cp' ( t ) E F(t,cp(t))

almost

I.

E C X

and

po = (to,xo),

an accumulation point of

at p0 i s D+(E,pO) = [y E Rn: t h e r e e x i s t s {(tn,xn)3 C E, (tn,xn) + (to,xo), tn > to, ( x n - x O ) / ( t n - t O ) + y as n + m I . The negative E,

E

t h e p o s i t i v e contingent of

contingent and contingent of

similar manner. then w r i t e

If

+ D cp(to)

E

E

at

p

0

a r e defined i n a

i s t h e graph of a function

instead of

+ D (E,pO).

cp: I +Rn,

A solution of t h e

contingent equation

(3.4.2)

Dx C F ( t , x )

i s a continuous function

F(t,cp(t))

f o r a l l t E I.

cp: I + R n

such t h a t

A function

176

cp(t)

Dcp(t) C i s a solution

3.4. GENERALIZED DIFFERENTIAL EQUATIONS

of (3.4.1) if and only i f it i s a s o l u t i o n of (3.4.2). The b a s i c theory f o r generalized equations hold; namely, t h e Peano's e x i s t e n c e theorem, e x t e n d a b i l i t y of s o l u t i o n s , and t h e Kamke convergence theorem are true f o r generalized equations, and we shall not expound on t h i s f'urther. Denote t h e maximal i n t e r v a l of existence of a s o l u t i o n D Before s t a t i n g a Wazewski theorem f o r cp' generalized equations, we w i l l need t h e following d e f i n i t i o n .

cp(t)

of (3.4.1) by

DEFDIITION 3.4.1.

A s e t A c X i s ( p o s i t i v e l y ) weakly i n v a r i a n t

with r e s p e c t t o (3.4.1) if f o r each po = ( t O , x o ) E A , t h e r e i s a solution V(t) of ( 3 . 4 . l ) w i t h cp(to) REMARK 3.4.1.

(t,cp(t)) E A

A closed set

v a r i a n t if and only if

LEMMA 3.4.1.

=Xo,

on

D cp

I [tO,m).

i s p o s i t i v e l y weakly in-

E

+ D (E,pO) fl F(pO) f 6

f o r all po

E

E.

are r e l a t i v e l y closed, X with X = E U E2. 1 i s p o s i t i v e l y weakly i n v a r i a n t .

Assume

and E2 1 p o s i t i v e l y weakly i n v a r i a n t subsets of Then H = E fl E2 1 Proof:

Let

two s o l u t i o n s and an Let

a> 0

L(t)

E

po = ( t O , x O ) E H.

qi(t)

By hypotheses, t h e r e e x i s t

of (3.4.1) with (t,cpi(t))

such t h a t

be t h e segment j o i n i n g

E

(pi(to) = xo, Ei

to 5 t < to + a.

for

(t,cpl(t))

i = 1,2,

to

(t,cp,(t)

)

( E n~ E~). Then x ( t ) = a(t)cpl(t) + (1-ct(t))cp2(t), 0 5 a ( t ) ,< 1. Choose a sequence {t,); + t n + t o , and a ( t n ) +ao. Then and l e t

(t,x(t)) E L(t)

n

We now can choose a subsequence

{t,]

t h e l i m i t t h e l e f t s i d e belongs t o s i d e equals

aovl + ( 1 - a o ) v 2

where

177

of

{t,}

Df(H,pO) vi

E

such t h a t i n

and t h e r i g h t

D+cpi(to).

3 . TOPOLOGICAL METHODS

By convexity, D+(H,pO) H

cy

n F(pO) # fl

v + (1-wo)v2 i s i n F(pO) and thus 0 1 and as a r e s u l t , by our previous remark,

i s p o s i t i v e l y weakly i n v a r i a n t .

This concludes t h e proof of

Lemma 3.4.1. The following i d e a s w i l l be needed f o r t h e Wazewski theorem. For

vc

po = ( t O J x O ) E

t h e zone of emission r e l a t i v e t o

X,

is

Ev(po) = [(T,y): y = cp(T), cp(t) with cp(to) = xo and ( t , c p ( t ) ) E V and

21,

v

i s a s o l u t i o n of (3.4.1)

t between to

for a l l

and t h e r i g h t zone of emission r e l a t i v e t o

'

is

V

E;(Po) = [(.JY>: ( T ~ Y )E Ev(Po), >_