Inequalities for Differential and Integral Equations (Mathematics in Science and Engineering, Volume 197)

Inequalities for Differential and Integral Equations I his is Volume 197 in M A T H E M A T I C S IN S C I E N C E A ...

Author: B. G. Pachpatte

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Inequalities for Differential and Integral Equations

I his is Volume 197 in M A T H E M A T I C S IN S C I E N C E A N D E N G I N E E R I N G Edited by William F. Ames, Georgia Institute of Technology A list of recent titles in this series appears at the end of this volume.

Inequalities for Differential and Integral Equations B.G. Pachpatte

Marathwada University, Aurangabad, India

ACADEMIC

PRESS

San Diego • Boston • New York • London • Sydney • Tokyo • Toronto

This book is printed on acid-free paper. Copyright © 1998 by ACADEMIC PRESS LIMITED All Rights Reserved. No part of this publication may be reproduced or transmitted in any form by photostat, microfilm, or any other means without written permission from the publishers A C A D E M I C PRESS INC 525B Street Suite 1900 San Diego, CA 92101, USA http://www.apnet.com ACADEMIC PRESS LIMITED 24- 28 Oval Road LONDON NW1 7DX, UK http://www.hbuk.co.uk/ap/ A catalogue record for this book is available from the British Library ISBN 0-12 - 543430- 8 Typeset by Laser Words, Madras, India Printed in Great Britain by MPG Books Ltd, Bodmin, Cornwall

98 9 9 0 0 0 1 0 2 0 3 E B 9 8 7 6 5 4 3 2 1

Contents

Preface

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

Introduction 1.

ix

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

Linear Integral Inequalities

1

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

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.2

The Inequalities of Gronwall and Bellman . . . . . . . . . . . . . .

9

1.3

S o m e Generalizations of the G r o n w a l I - B e l l m a n Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.4

Volterra-Type Integral Inequalities

17

1.5

The Inequalities of G a m i d o v and Rodrigues . . . . . . . . . . . .

25

1.6

S i m u l t a n e o u s Inequalities

30

1.7

Pachpatte's Inequalities

1.8

Integro-differential Inequalities . . . . . . . . . . . . . . . . . . . . . . .

44

1.9

Inequalities with Several Iterated Integrals

52

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

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

32

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

1.10

Inequalities Involving Product Integrals . . . . . . . . . . . . . . . .

62

1.11

Applications

75

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

1.11.1 Second-order Integro-differential Equations

2.

9

..........

75

1.11.2 Perturbation of Volterra Integral Equations . . . . . . . . . . .

78

1.11.3 Higher Order Integro-differential Equations . . . . . . . . . . .

81

1.11.4 Integral Equation Involving Product Integrals

84

1.12

Miscellaneous Inequalities

1.13

Notes

.........

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

86

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

96

N o n l i n e a r Integral Inequalities I . . . . . . . . . . . . . . . . . . . . . . . .

99

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

2.2

Inequalities Involving C o m p a r i s o n

2.3

The Inequalities of Bihari and L a n g e n h o p . . . . . . . . . . . . . .

107

2.4

Generalizations of G r o n w a l I - B e l l m a n - B i h a r i Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

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

. ....

100

CONTENTS

vi 2.5

Inequalities with Volterra-Type Kernels . . . . . . . . . . . . . . . .

126

2.6

Inequalities with Nonlinearities in the Integral . . . . . . . . . . .

135

2.7

Pachpatte's Inequalities I . . . . . . . . . . . . . . . . . . . . . . . . . . .

148

2.8

Pachpatte's Inequalities II . . . . . . . . . . . . . . . . . . . . . . . . . .

158

2.9

Integro-differential Inequalities . . . . . . . . . . . . . . . . . . . . . . .

171

2.10

Inequalities with Iterated Integrals . . . . . . . . . . . . . . . . . . . .

181

2.11

Applications

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

186

2.11.1 Second Order Nonlinear Differential Equations . . . . . . . .

187

2.11.2 Perturbed Integro-differential Equations . . . . . . . . . . . . .

192

2.11.3 Higher Order Integro-differential Equations . . . . . . . . . . .

198

2.11.4 Estimates of the Solutions of Certain Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203

2.12

Miscellaneous Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . .

206

2.13

Notes

218

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

3. Nonlinear Integral Inequalities II . . . . . . . . . . . . . . . . . . . . . . . .

221

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221

3.2

Dragomir's Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221

3,3

Pachpatte's Inequalities l . . . . . . . . . . . . . . . . . . . . . . . . . . .

227

3.4

The Inequalities of Ou-lang and Dafermos

233

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

3,5


236

3.6

Pachpatte's Inequalities III . . . . . . . . . . . . . . . . . . . . . . . . . .

243

3.7

Pachpatte's Inequalities IV . . . . . . . . . . . . . . . . . . . . . . . . . .

251

3.8

The Inequalities of Haraux and Engler

267

3.9

Pachpatte's Inequalities V . . . . . . . . . . . . . . . . . . . . . . . . . .

270

3,10

Inequalities Involving Iterated Integrals . . . . . . . . . . . . . . . .

276

3.11

Applications

291

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

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

3.11,1 Volterra Integral Equations

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

291

3,11.2 On Some Epidemic Models . . . . . . . . . . . . . . . . . . . . . .

294

3.11.3 Certain Integral and Differential Equations . . . . . . . . . . .

297

3.11.4 Certain Integro-differential and Differential Equations

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

301

3.12


303

3.13

Notes

320

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

CONTENTS

4.

vii

Multidimensional Linear Integral Inequalities

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

323

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

323

4.2

Wendroff's Inequality

323

4,3


327

4,4


334

4,5


343

4.6

Snow's Inequalities

354

4.7

Generalizations of Snow's Inequalities

4.8


374

4.9

Inequalities in Several Variables

396

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

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

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

4,10 Young's Inequality and its Generalizations 4.11

Applications

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

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

364

409 417

4.11,1 Hyperbolic Partial Integro-differential Equations

.......

417

4.11.2 Non-Self-Adjoint Hyperbolic Differential and Integro-differential Equations . . . . . . . . . . . . . . . . . . . . .

5.

424

4.11.3 Perturbations of Hyperbolic Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

428

4.11.4 Simultaneous Integral Equations in Two Variables . . . . . .

437

4.12


439

4.13

Notes

456

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

Multidimensional Nonlinear Integral Inequalities

..........

459

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

459

5.2

Generalizations of Wendroff's Inequality . . . . . . . . . . . . . . .

460

5.3

Wendroff-type Inequalities

468

5.4

Generalizations of Pachpatte's Inequalities . . . . . . . . . . . . .

479

5,5


485

5.6


495

5.7

Inequalities in M a n y Independent Variables

508

5.8


527

5.9


537

5,10

Pachpatte's Inequalities V . . . . . . . . . . . . . . . . . . . . . . . . . .

544

5.11

Applications

556

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

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

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

5.11,1 Hyperbolic Partial Differential Equations . . . . . . . . . . . . .

556

ooe

CONTENTS

VIII

5.11.2 Hyperbolic Partial Integro-differential Equations

.......

5.11.3 Higher Order Hyperbolic Partial Differential Equations . . .

559 561

5.11.4 Multivariate Hyperbolic Partial Integro-differential Equations

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

5.12

M i s c e l l a n e o u s Inequalities

5.13

Notes

References Index

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

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

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

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

562 565 588 591 609

Preface

Inequalities have played a dominant role in the development of all branches of mathematics, and they have a central place in the attention of many mathematicians. One reason for much of the successful mathematical development in the theory of ordinary and partial differential equations is the availability of some kinds of inequalities and variational principles involving functions and their derivatives. Differential and integral inequalities have become a major tool in the analysis of the differential and integral equations that occur in nature or are constructed by people. A good deal of information on this subject may be found in a number of monographs published during the last few years. Most of the inequalities developed so far in the literature, which provide explicit known bounds on the functions appearing in differential, integral and other equations, perform quite well in practice and hence have found widespread acceptance in a variety of applications. Because of this, it is not surprising that numerous studies of new types of inequalities have been made in order to achieve many new developments in various branches of mathematical science and engineering practice. In the past few years, various investigators have discovered many useful and new integral inequalities through research into various branches of differential and integral equations, where inequalities are often the basis of important lemmas for proving various theorems or for approximating various functions. Many of the inequalities have only recently appeared in the literature, and cannot yet be found in books. The aims of this monograph are to provide a systematic study of integral inequalities, which find numerous applications in the theory of various classes of differential and integral equations, to be a valuable reference for researchers into differential and integral equations and to serve as a graduate textbook. Taking into account the vast literature, the choice of material for a book devoted to integral inequalities is a difficult task. In this book an attempt has been made to present all the core material as well as those inequalities that have been recently discovered and have proven most useful in applications. In the author's view, this book is vital reading for mathematical ix

x

PREFACE

analysts, pure and applied mathematicians, physicists, engineers, computer scientists and graduate students in those disciplines. I wish to express my sincere thanks to Professor William F. Ames for his constant interest in this work and inviting me to write this monograph for his series Mathematics in Science and Engineering. It is a pleasure to record my gratitude to the members of the book production department of Academic Press for their helpful cooperation. I am also indebted to my family members for their encouragement, understanding and patience during the writing of the book.

B. G. Pachpatte

May 1997

Introduction

The importance of inequalities has long been recognized in the field of mathematics. The mathematical foundations of the theory of inequalities were established in part during the 18th and 19th century by mathematicians such as K.F. Gauss (1777-1855), A.L. Cauchy (1789-1857) and P. L. Chebyshev (1821-94). In the years thereafter the influence of inequalities has been immense, and the subject has attracted many distinguished mathematicians, including H. Poincar6 (1854-1912), A. M. Lyapunov (1857-1918), O. HiSlder (1859-1937) and J. Hadamard (1865-1963). In a subject like inequalities, which has applications in every part of mathematics, historical and bibliographical questions are particularly troublesome. Often, apart from in particular applications, inequalities are regarded as auxiliary in character and are usually obtained by ad hoc methods rather than as a consequence of some underlying theory of inequalities. It is not easy to speak of the origin of a familiar inequality used in various applications. It is quite likely to occur first as an auxiliary proposition without explicit statement in the study of a certain branch of mathematics and then be rediscovered many years later by a number of authors. In the case of many inequalities the proofs provided by the original authors have been reproduced or a variety of proofs have been given.

2

INTRODUCTION

Analysis has been the dominant branch of mathematics for the last three centuries, and inequalities are the heart of analysis. Although inequalities play a fundamental role in all branches of mathematics, the development of the subject as a branch of modem mathematics came in the 20th century through the pioneering work Inequalities by G. H. Hardy, J. E. Littlewood and G. P61ya, which appeared in 1934. This theoretical foundation, which has been developed by other mathematicians, has in turn led to the discovery of many new inequalities and interesting applications in various fields of mathematics. This century has seen considerable and fruitful research in the field of inequalities and their applications in various branches of mathematics. The subject of inequalities is endlessly fascinating and remains a remarkably active and challenging area of current research, despite its relatively long history. Although various introductory and survey monographs on inequalities have appeared in the past few decades, the subject is sufficiently broad and active that there appears always to be room, indeed possibly always to be a need, for yet another one. In fact, it is fair to attribute the main driving force of the rapid development of this field to the users rather than to the inventors of inequalities. Its theory is beautiful, its techniques powerful, and its impact upon science and technology most profound. Since 1934, when the key work of Hardy et al., Inequalities, was published, several papers devoted to inequalities have been published which deal with new inequalities that are effective in many fields of applications of mathematics. It appears that in the theory of inequalities, the three fundamental inequalities, namely, the A M - G M inequality, the H61der (in particular, Cauchy-Schwarz) inequality and the Minkowski inequality, have played dominant roles. A detailed discussion of these inequalities can be found in the book Inequalities by E. F. Beckenbach and R. Bellman, which appeared in 1961, and the book Analytic Inequalities by D. S. Mitrinovid, published in 1970, see also Agarwal and Pang (1995), and Mitrinovid et al. (1991, 1992). In the theory of inequalities, an enormous amount of effort has been devoted to the sharpening of classical approaches and to the discovery of new ones. In the course of discovering a new approach to inequalities, one is naturally inspired by the classical one. In return, the classical approach may be used to establish more inequalities. The modem theory of inequalities, as

INTRODUCTION

3

well as the continuing and growing interest in this field, have deep classical roots which have retained great importance for several centuries at the core of mathematics. Foremost among the mathematical challenges in modem science and technology is the field of nonlinear differential equations and the theory of such equations has challenged most of the world's greatest mathematicians, providing a tremendous stimulus for many of the modern developments of mathematics. The importance of differential equations for understanding many physical and mathematical problems was first recognized by Sir Isaac Newton in the 17th century, and he used them in his study of the motion of particles and planets. The development of the subject as a branch of modem mathematics came in the 19th and 20th centuries through the pioneering work of a number of mathematicians, notably, Birkhoff, Cauchy, Lyapunov, Picard, Poincar6, Riemann, Bendixion and many others. Nonlinear differential equations arise in essentially every branch of modem science, engineering and mathematics. However, in only a very few special cases is it possible to obtain useful solutions to nonlinear equations via analytical calculations. In practice, a nonlinear equation is usually approximated by a linear equation which can be solved explicitly, and the solution of the linear equation is taken as an approximation to the solution of the nonlinear equation. It should be noted that not all nonlinear problems can be approximated suitably by linear problems. In recent years, several approaches and methods have been developed for the study of nonlinear systems which cannot be approximated sufficiently well by tractable linear systems. The study of nonlinear differential equations is a fascinating question which is at the very heart of the understanding of many important problems of the natural sciences. It has already drawn a great deal of attention, not only among mathematicians themselves, but from various other disciplines as well. The mathematical description of dynamical processes encountered in physical, biological and applied sciences requires the investigation of ordinary, functional and partial differential equations. There exists a vast literature on the study of ordinary and partial differential equations. A good deal of information in this regard may be found in the monographs by Bellman (1953), Bellman and Cooke (1963), Coddington and Levinson (1955), Halanay (1966), Coppel (1965), Hale (1977), Hartman (1964), Hille

4

INTRODUCTION

(1969), Lakshmikantham and Leela (1969), Reid (1971), Sansone and Conti (1964), Szarski (1965), Walter (1970), Courant and Hilbert (1962), Garbedian (1964), Sobolev (1964), Mikhailov (1978) and Sneddon (1957) to mention but a few. Integral equations have been of considerable significance in the history of mathematics and have held a central place in the attention of mathematicians during the last few decades. The beginning of the theory of integral equations can be traced back to N. H. Abel who found an integral equation in 1812 starting from a problem in mechanics. In 1895, V. Volterra emphasized the significance of the theory of integral equations. In 1900, I. Fredholm made his famous contribution to integral equations, which led to a fascinating period in the development of mathematical analysis. During the past few years, integral equations have proved to be of immense use in applied fields, such as automatic control theory, network theory and the dynamics of nuclear reactors. The mathematical literature provides a good deal of information, and an excellent account of this subject may be found in the monographs by Burton (1983), Miller (1971), Corduneanu (1973, 1977, 1991), Gripenberg et al. (1990), Krasnoselskii (1964) and Tricomi (1957), to mention a few. Many problems, arising in a wide variety of application areas, give rise to mathematical models which form boundary value problems for ordinary or partial differential equations. The foremost desire of an investigator is to solve the problem explicitly. If little theory is available and no explicit solution is readily obtainable, generally the ensuing line of attack is to identify circumstances under which the complexity of the problem may be reduced. In the past few years the growth of this theory has taken beautiful and unexpected paths and will continue with great vigour in the next few decades. Integral inequalities that give explicit bounds on unknown functions provide a very useful and important device in the study of many qualitative as well as quantitative properties of solutions of nonlinear differential equations. One of the best known and widely used inequalities in the study of nonlinear differential equations can be stated as follows. If u is a continuous function defined on the interval J = [or, c~ + h] and t

O < u(t) < / [ b u ( s ) + a] ds, Ol

t ~ J,

INTRODUCTION

5

where a and b are nonnegative constants, then 0 < u(t) < a h e bh,

t ~ J.

This inequality was found by Gronwall (1919) while investigating the dependence of systems of differential equations with respect to a parameter. In fact the roots of such an inequality can be found in the work of Peano (1885-86), who proved explicitly the special case of the above inequality with a = 0. Its usefulness is felt from the very beginning when one encounters the most elementary theorem on the existence and uniqueness of solutions and their dependence on parameters and initial values. After the discovery of the integral inequality resulting from Gronwall (1919), a number of mathematicians have shown their considerable interest to generalize the original form of this inequality. During the period 1919-75 a large number of papers appeared in the literature which were partly inspired by the Gronwall inequality and its applications. An extensive survey of integral inequalities of the Gronwall type which are adequate in many applications in the theory of differential and integral equations may be found in Beesack (1975). The year 1973 marked a new beginning in the theory of such inequalities due to the discovery of the following remarkable inequality by Pachpatte (1973a). If u, f , g are nonnegative continuous functions on R+ -- [0, c~), uo > 0 is a constant and

t 6R+, o

then

u(t) < uo

IJ 1+

0

f ( s ) exp

[f(cr) + g(cr)] d~r

ds

]

,

t 6R+.

There are two interesting features related to this inequality. First, in the special case when g - 0, it reduces to the well-known generalization of Gronwall's inequality given by Bellman (1943); and second, it is applicable to the more general situations for which the earlier inequalities do not apply

6

INTRODUCTION

directly. In the years following the publication of the above inequality, Pachpatte made his fundamental contribution to the theory of such inequalities, providing ready and powerful tools in investigating various problems in the theory of differential and integral equations. Over the last twenty years the study of such inequalities has undergone formidable development, and various investigators have discovered many useful and new inequalities centred on the above inequality which will be an indispensable source for a long time to come. One of the most useful inequalities in the development of the theory of differential equations is given in the following theorem. If u, f are nonnegative continuous functions on R+ = [0, r uo > 0 is a constant and t

U 2 (t)

< u 2 + 2 / f (s)u(s)ds,

t 6R+,

0

then

t 1,1

u(t) < uo + / f (s) ds,

t ~ R+.

tg

0

77 It appears that this inequality was first considered by Ou-Iang (1957), while investigating the boundedness of solutions of certain second-order differential equations. In the literature there are many papers which make use of this inequality to obtain global existence, uniqueness, stability, and other properties of the solutions of various nonlinear differential equations. The importance of this inequality stems from the fact that it is applicable in certain situations in which other available inequalities do not apply directly. Although stimulating research work related to differential and integral inequalities arising in the theory of differential and integral equations has been undertaken in the literature, it seems that the importance of various generalizations and extensions of the above inequality have been overlooked by the investigators. Motivated and inspired by the great and enduring success enjoyed by this inequality in various applications, in a series of recent papers Pachpatte (1994a-c; 1995a,b,e; 1996a-e; 1997; in press a-j) has established a number of new inequalities related to the above inequality, which can be used as handy tools in the study of certain new classes of differential, integral and integro-differential equations. The growth of the theory and applications of

INTRODUCTION

7

these inequalities are still under development and promise to open up new vistas for research in the near future. In the study of global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity, Engler (1989) used the following slight variant of the inequality given by Haraux (1981). If a E L 1(0, T, R+), u0 > 0 is a constant, and the function u: [0, T] --+ [1, cx~) satisfies the inequality u(t) < uo

(j 1+

a(s)u(s) log u(s) ds

0

)

O v(x) exp

( / ) -k(t)

h(s) ds

,

ct _< x _< t _ < / ~ .

(1.3.14)

x

[2

Proof: Let I o

t

z(x) = u(t) q- k(t) / h(s)v(s) ds,

t~ < x
in both (1.4.1) and (1.4.2). The inequality given in Theorem 1.4.1 includes as a special case the inequality given in Theorem 1.2.2. Further we note that Willett (1965) considered the inequality in Theorem 1.4.1 and obtained the upper bound for u under the assumption that either k(t, s) or (O/Ot)k(t, s) is degenerate or directly separable in the following sense: n

k(t, s) < Z

hi(t)gi(s),

i=1

or a similar relation holds for (O/Ot)k(t, s). A The following theorem presents slight variants of the inequality given by Norbury and Stuart (1987) which are sometimes applicable more conveniently. Theorem 1.4.2 Let u and k(t, s) be as in Theorem 1.4.1 and k(t, s) be

nondecreasing in t for each s ~ J.

19

LINEAR INTEGRAL INEQUALITIES

(i) If t

u(t) < c + f k(t, s)u(s) ds,

t 6 J,

(1.4.3)

Ol

where c > 0 is a constant. Then u(t) < c exp ( tf k(t, s) ds) , t ~ J.

(1.4.4)

(ii) Let n(t) be a positive continuous and nondecreasing function for tEJ. If t

u(t) < n(t) + / k(t, s)u(s) ds,

t ~ J,

(1.4.5)

Ol

then u(t) < n (t) exp ( / k ( t , s) ds) , t ~ J.

(1.4.6) U]

Proof: (i) Fix any T, ot t. Unless otherwise stated, we will assume that f 6 L + (I) and k 6 L + (I • I), where I -- { t ' 0 < t < c~} and L+(I) -- {h" h ~ L2(I) and h(t) > 0 for t 6 I}. Define kl(t, s) - k(t, s) and for n > 2, t

kn (t, s) --

f k(t, ff)kn-1 (o', s)dcr. s

By the resolvent kernel we mean n

H (t, s) -- Z

ki(t, s).

i=1

This series converges in L2(I x I) norm. For completeness we state without proof the principal theorem for Volterra equations.


23

Theorem 1.4.4 Let f E L2(I), k ~ L2(I • I); then the equation (1.4.17) has a unique L2 solution given by t 1"8

u(t) -- f (t) + / H (t, s) f (s) ds,

a.e. on I.

0

D For the proof of this theorem the reader is referred to Tricomi (1957). We will need the following Lemmas proved in Beesack (1969). L e m m a 1.4.1 Let f E L+(I) and k E L+(I x I) in equation (1.4.17), then u(t) > 0 a.e. on I. A

Lemma 1.4.2 Let v E L + (I) and let t

v(t) < f (t) + / k(t, s)v(s) ds,

a.e. on I.

0

If u(t) is the unique L2 solution of(1.4.17), then v(t) < u(t) a.e. on I. A Thompson (1971) gave the following inequality. Theorem 1.4.5 Let f i E L~ (I) and let ki E L+ (1 • 1), and let ui be the unique L2 solution of t f~

ui(t) -- f i(t) "Jr-/ ki(t, s)ui(s) ds, 0

where i -- 1, 2 . . . . . n. If n

F(t) -- Z

f i(t) for

t EI

i=1

and n

k(t, s) - Z

ki(t, s) for

(t, s) E I • I,

i=1

then the unique L2 solution u of t

u(t) -- F(t) + / k(t, s)u(s) ds, 0

24


exists, and n

Z Ui(t) < u(t) a.e. on

I.

i=1

E] P r o o f : Since f i E

L+(I) and ki E L+(I x I), then F ~ L2(I) and k u(t) exists. The proof that ~-'~in___lui(t) __0 a.e. on I for i = 1, 2, . . . , n. Therefore

From Lemma 1.4.1, ui(t) n

/,

t

n

Un(t) "k-v(t) < ~ f i(t) -Jr ] ~ ki(t, s)[Un (s) + 1)(s)] ds, ,/

i=1

0

i=1

or t

Un(t) + v(t) < F(t) + f k(t, S)[Un(S) -]- V(S)] d s , a.e. on I. 0 Therefore, from Lemma 1.4.2 we have un(t)+ v(t)< u(t). But by our induction assumption n-1

~ ui(t) O,

> 0, from (1.7.41) we observe that

g-~ k , - ~ ] z(t)

(1.7.39)

1 z'(t))' ~-~ - ~ ]

I 1

z'(t) > 0 and z(t)

)1 dr

< p(t) + g(t) k,-~

z2(t)

z'(t)

42


i.e.

1 (z'(t)~'

'

< p(t).

z(t)

(1.7.42)

By taking t - a in (1.7.42) and integrating it with respect to a from 0 to t we have t

(z'(t)/f (t))' < g(t) / p(a)da. z(t)

(1.7.43)

-

0

z'(t)/f(t)>

Since observe that

0,

z'(t)> 0

(

and

)

z(t)>

0, as above from (1.7.43) we

/

z' (t)/ f (t) < Z(t) _ g(t)

t

p(a) da.

(1.7.44)

0

By taking t -

r in (1.7.14) and integrating from 0 to t we have

z'(t)z(t)- 0 is an arbitrary small constant, and subsequently pass to the limit as e --+ 0 to obtain (1.7.35).

I Theorem 1.7.6

tions definedon J. (i) If

Let u, v, p, f, g and h be nonnegativecontinuousfunc-

u(t) > v(s)_ p(t) [~s f (~r)v(~r)dcr+ ~ f (cr) (~ g(r)v(r) dr) d~r] (1.7.47)

43


for ot < s < t < fl, then

[ j

u(t) > v(s) 1 + p(t)

)]

-1

f (o.) exp

[ f ( r ) p ( t ) -k- g(r)] dr

do.

s

(1.7.48)

for a < s < t V(S)- p(t) [~ f (o")V(o")do" +

+j,.,. s [r

d,) d,]

(1.7.49)

forot < s < t < fl, then

{ j

u(t) > v(s) 1 + p(t)

f

(r) exp

p(t) f (o")do"

s

x

[ j 1+

g(o.) exp

[h(~) + g(~:)] d~

do

]

dr

-1

, (1.7.50)

17

forot < s < t < fl. V1

Proof: (i) For fixed t in the interval J we define for c~ < s < t

m(s)--u(t)+p(t) [~f(a)v(a)da-k- f f(a) (~g(r)v(r)dr)da] (1.7.51) From (1.7.51) we have

m'(s) - - p ( t ) f (s) (v(s) + f g(r)v(r)dr) which in view of

v(s) < m(s)

implies

m'(s)>-p(t)f(s)(m(s)-t-!ig(r)m(r)dr).

(~.7.5e)

44


If we put t F(S)

(1.7.53 )

-- m(s) + / g(r)m(r) dr, S

it follows from (1.7.52), (1.7.53) and the fact that m(s)< r(s) that the inequality

r'(s) + [f (s)p(t) + g(s)]r(s) > O, is satisfied, which implies the estimation for r(s) such that

r(s) < u(t)exp

(/

[f (r)p(t) + g(r)]dr

)

,

(1.7.54)

since r(t)= u(t). Using (1.7.54) in (1.7.52) we have

mI(s)>-p(t)u(t)f(s)exp(~[f(r)p(t)+g(r)]dr).

(1.7.55)

Now by taking s = cr in (1.7.55) and integrating it from s to t and substituting the bound on re(s) in v(s) < re(s) we get the desired inequality in (1.7.48). The proof of (ii) can be completed by following the same arguments as in the proof of inequality in (i) with suitable modifications. For details, see Pachpatte (1977a). II

1.8 Integro-differential Inequalities Integral inequalities involving functions and their derivatives have played a significant role in the developments of various branches of analysis. Pachpatte (1977b, 1978a, 1982a) has given some integral inequalities of the Gronwall-Bellman type involving functions and their derivatives which are useful in certain applications in the theory of differential and integrodifferential equations. This section gives some of the inequalities obtained in Pachpatte (1977b, 1978a, 1982a) and the inequalities that are slight variants of those given therein. Theorem 1.8.1 Let u, uI, a, b and c be nonnegative continuous functions

defined on R+.

45


(i) If b > 1 and t

u'(t) < a(t) + b(t) / c(s)(u(s) + u'(s)) ds, t ~R+,

(1.8.1)

0

then t

d(t) < a(t) + b(t) f c(cr)[A(cr) q- b(o-)B(cr)] dcr, t 6 R+,

(1.8.2)

0

where t

A(t) -- u(O) + a(t) + f a(s) ds,

'

t 6 R+,

(1.8.3)

0

B(t) -- / c(s)A(s) exp

b(r)[c(r) + 1] dr

ds,

t ~ R+. (1.8.4)

0

(ii) If

(

j c(s)[u(s) ) + u'(s)]

u'(t) < a(t) + b(t) u(t) +

ds

,

t ~ R+,

(1.8.5)

0

then ut(t) < a(t) + b(t) [u(O)exp (

[b(s) -b c(s) q- b(s)c(s)] ds)

t

+ / a(s)(1 +

c(s))

0

x exp

d )as], !-I

Proof: (i) Define a function

m(t) by

t

m(t) -- / c(s)(u(s) Jr-uI(s)) ds, 0

(1.8.7)


46

then (1.8.1) can be restated as

u' (t) < a(t) + b(t)m(t).

(1.8.8)

Differentiating (1.8.7) and using (1.8.8) we have

m'(t) < c(t)(u(t) + a(t) + b(t)m(t)).

(1.8.9)

By taking t - s in (1.8.8) and integrating it from 0 to t we have t

t

u(t) 1 and (1.8.3) we have

m'(t) < c(t) [A(t) + b(t) (m(t) + f b(s)m(s)ds) ] Define a function

(1.8.11)

r(t) by t i-o

r(t) -- m(t) + /b(s)m(s)ds.

(1.8.12)

J

0

Differentiating (1.8.12) and using m'(t) and the fact that m(t) < r(t) we get

< c(t)[A(t)+ b(t)r(t)]

from (1.8.11)

r' (t) - m' (t) + b(t)m(t) < b(t)[c(t) +

1]r(t) +

c(t)A(t).

(1.8.13)

The inequality (1.8.13) implies the estimate

r(t) < B(t), where

B(t)

(1.8.14)

is given by (1.8.4). Using (1.8.14) in (1.8.11) we have

m'(t) < c(t)[A (t) + b(t)B(t)], which implies the estimate for

m(t)

such that

t in(t)

< / c(cr)[A (or) + b(cr)B(cr)] d~r. ,.I

0

Using (1.8.15) in (1.8.8) we get the desired inequality in (1.8.2).

(1.8.15)


47

(ii) Define a function z(t) by t

z(t) = u(t) + f c(s)(u(s) + u' (s)) ds.

(1.8.16)

0

Differentiating (1.8.16) and using the facts that u'(t) O, pi(t) > O, i = 1, 2..... n,

f Pk-l(Sk-1)

"-'

"ilpk(sk)fk(Sk)

0

0

• dsk dsk_l . . . ds2 d s l ) t

t

Sl

= f pl(sl)f l(Sl)dSl + f pl(sa) f p2(s2)f 2(s2)ds2dsl + ' " 0

0 t

Sl

0 8n-2

+fpl(Sl)fp2(s2).../Pn--l(Sn--1) 0

0

0

Sn-1

X I Pn (sn)dsn dSn_l.., ds2 (iS1. o


54

In the following three theorems some basic inequalities given by Pachpatte (1988a, in press j) are presented. Theorem 1.9.1 Let u > O, h >_ O, rj(t) > O, j - l , 2 . . . . , n - l , g(t) > 0 be continuous functions defined on R+ and uo > 0 be a constant.

(al) If u(t) < uo + M[t, r, h(tn )U(tn )],

(1.9.1)

u(t) < uo exp(M[t, r, h(tn)]),

(1.9.2)

u(t) < uo + A[t, g, h(tn)U(tn)],

(1.9.3)

u(t) < uo exp(A[t, g, h(tn )]),

(1.9.4)

u(t) < uo + B[t, g, h(tn)u(tn)],

(1.9.5)

u(t) < uo exp(B[t, g, h(tn )]),

(1.9.6)

u(t) < uo + E[t, g, h(tn )U(tn )],

(1.9.7)

u(t) < uo exp(E[t, g, h(t, )]),

(1.9.8)

for t ~ R+, then

for t ~ R+. (a2) If

for t ~ R+, then

for t ~ R+. (a3) If

for t ~ R+, then

for t ~ R+. (a4) If

for t ~ R+, then

for t ~ R+. D Theorem

1.9.2

Letu>O, p>O, q>O, rj>O, j - - l , 2

.....

n

-

1,

g > 0 be continuous functions defined on R+ and uo > 0 be a constant.

(b~) If u(t) < uo + M[t, r, p(tn)(U(tn) + M[tn, r, q(Sn)U(Sn)])],

(1.9.9)

for t ~ R+, then u(t) < uo[1 + M[t, r, p(tn ) exp(M[tn, r, [p(s, ) + q(Sn )]])]],

(1.9.10)


55

for t ~ R+. (b2) If u(t) < uo+A[t,g, p(tn)(U(tn)+A[tn, g,q(Sn)U(Sn)])])],

(1.9.11)

for t E R+, then u(t) < uo[1 + A[t, g, p(tn)exp(A[tn, g, [p(Sn) + q(Sn)]])]],

(1.9.12)

f o r t ER+. (b3) If u(t) < uo + B[t, g, p(tn )(U(tn ) -4c-B[tn, g, q(Sn )U(Sn )])],

(1.9.13)

for t E R+, then u(t) < u0[1 + B[t, g, p(tn ) exp(B[tn, g, [p(Sn ) + q(Sn )]])]],

(1.9.14)

for t E R+. (b4) If u(t) < uo + E[t, g, P(Zn)(U(Zn) + E[zn, g, q(an)U(Crn)])],

(1.9.15)

for t ~ R+, then u(t) < uo[1 + E[t, g, p(rn)exp(E[zn, g, [P(an) + q(an)])]],

(1.9.16)

for t E R+.

Theorem 1.9.3 Let u, h, rj and g be as defined in Theorem 1.9.1. Let f and q be nonnegative continuous functions defined on R+.

(cl) If u(t) < f (t) + q(t)m[t, r, h(tn )U(tn )],

(1.9.17)

for t E R+, then u(t) < f (t) + q(t)M[t, r, f (tn)h(tn)]exp(M[t, r, q(tn)h(tn)]),

(1.9.18)

for t ~ R+. (c2) If u(t) 0 for i = 2, 3 . . . . . n, for t 6 R+, we observe that

Lnv(t)

v(t)

< [p(t) + q(t)] +

((d/dt)v(t))Ln-1 v(t)

vZ(t)


i.e.

d (Ln-lV(t))

--

dt

~'!

(1.9.28)

< [p(t) + q(t)].

v(t)

-

By setting t - - S n in (1.9.28) and integrating it from 0 to t and using the fact that Liv(O) = 0, for i -- 1, 2 . . . . . n - 1, we obtain t

(d/dt)Ln-zV(t) < rn-1 (t) f [p(Sn ) + q(s, )] ds,,.

v(t)

(1.9.29)

0

Again as above from (1.9.29) we observe that t

--d (tn-2V(t)) < rn-1 (t) f [p(Sn ) "k-q(Sn )] dsn. dt v(t) 0

(1.9.30)

By setting t - - S n - 1 in (1.9.30) and integrating it from 0 to t and using the fact that Liv(O) -- 0, i = 1, 2 . . . . . n - 2, we get Sn-1

t

f

Ln-2V(t) _ O, Pi > Ofor i -- 1, 2 . . . . . n, be continuous functions defined on R+ and uo > 0 is a constant.

n (/ s~

(dl) If

u(t)< U0 + Z

k=l

pl(S1)

p2(s2). 99 0

Sk-1p~(sk)f~(sk)u(sk)dsk

x /

o f o r t ~ R+, then

sk/2 Pk-l(Sk-1)

0

d s k - 1 . . , ds2 dsl

I,

(1.9.38)

LINEAR

INTEGRAL

(/

u(t) < uoexp

si

pl(Sl)

k=l

59

INEQUALITIES

p2(s2)...

0

is

Pk-l(Sk-1)

)]

0

sk

x /

pk(sk)fk(sk)dskdsk-1 ... ds2ds1

,

(1.9.39)

0

for t ~ R+. (d2) Let b(t) be a positive continuous and nondecreasing function defined for t ~ R+. If u(t) < b(t) + L ~

(JO Si

k=l

x

p2(s2)...

pl(Sl)

0

iS

Pk-l(Sk-1)

0

[,--kl-1Spk(sk)f k(sk)u(sk) dsk dsk-1. . . ds2 dsl I

(1.9.40)

0

for t ~ R+, then u(t) < b(t) exp

.1 x f

p2(s2)...

pl(sl) k=l

0

Pk-l(Sk-1) 0

)]

pk(Sk)fk(Sk) dsk dsk-1.., ds2 dsl

,

(1.9.41)

0

f o r t ~ R+. D Proof: (dl) It is sufficient to assume that u0 is positive, since the limiting argument can be used to treat the remaining case. Define a function z(t) by the fight-hand side of (1.9.38). Then differentiating z(t) and rewriting we have

z'(t)

pl(t)

- - Zl

(t),

('S Si ''i'

where Zl ( t ) - -

f 1 (t)u(t)

__ L

pe(se)

k=2

p3($3)

0

Pk-1 (Sk-1)

999

0

Sk-1p~(sk)f~(s~)u(sk)dskdsk-1.., ds3 ds2 I ,

• /

0

(1.9.42)

60 with

LINEAR INTEGRAL INEQUALITIES S1 =

t. From the definition of

Zl' ( t )

z2(t) "- ~

we observe that (1.9.43)

-- f 2(t)u(t) -- z2(t),

p2(t)

where

Zl(t)

n(oJ 'i p 3 ($3)

p4($4)

k=3

999

0

"i'

Pk-l(Sk-1)

0

l,iSk-1

x - - pk(sk)fk(Sk)U(Sk)dskdSk-1.., dsads3

I

,

0

with s2 -- t. Continuing in this way, we obtain ' (t) Zn-2

(1.9.44)

f n - 1 (t)u(t) -- Zn-1 (t),

-

Pn-l(t) where t

Zn-1 (t) -- f Pn (Sn)fn (Sn)U(Sn) dsn, 0 from the definition of Zn-1 (t) and using the fact that u(t) < z(t) it is easy to observe that ' (t) Zn-1

< Pn (t)fn (t).

z(t)

(1.9.45)

If H(t) is any cl-function defined on R + , H ( t ) > 0 for all t ~ R+ and H (0) = 0 and the function z(t) is defined as above, then it is easy to observe that (Medved, 1993) t

f

0

H'(s) ds > n ( t ) z(s) - z(t)'

(,)

for t 6 R+. Indeed, integrating by parts the left-hand side of (,) we obtain

t 0

t z(s)

z(t) +

0

z2(s)

ds>

for t 6 R+. From (,) and (1.9.45) it is easy to observe that

Zn-l(t) < z(t) --

t

Zn-1(Sn) dan Z(Sn)

i' 0

-

H(t) z(t)'

61


< f pn(Sn)fn(sn)dsn. Using (,), (1.9.44) and (1.9.46) and the fact that

Zn-2(t) < Z(t) --

(1.9.46)

u(t) < z(t) we observe

that

t

dsn_1 /0 Zn_2(Sn-1) Z(Sn-1)

-I- Pn-1(Sn-1)Zn-1(Sn-1) dsn-1 / 0 Pn-1(Sn-1)f n-1(Sn-1)U(Sn-1) Z(Sn-1) t

t

Pn-l(Sn-1)f n-l(Sn-1)dsn-1 -+-f Pn-l(Sn-1)Zn-l(Sn-1) Z(Sn_I) dsn-1 0 0 t

Pn-1(Sn-1)f n-1 (Sn-1) dsn-1 t Sn-1 +fPn_l(Sn-1) fpn(Sn)fn(sn)dsndsn-1. 0 0

(1.9.47)

Proceeding in this way we get Zl(t) < z(t)

~(fO

s~

P2 ($2)

k=2

P3 ( $ 3 ) . . .

o

sk/2 Pk- 1(Sk-1) o

Sk-1 x f pk(sk)fk(Sk)dsk dsk-1..,

ds3 ds2

I

(1.9.48)

,

0

From (1.9.48), (1.9.42) and using the fact that u(t) < z(t), we observe that

z'(t) pl(t)

n (~o fl (t)u(t) < z(t) Z k=2

s~ p2(s2)

Sk/2

p3(s3). 99 0

sk-1 • / pk(sk)fk(sk)dskdsk-1.., 0

Pk-1 (Sk-1)

0

) ds3ds2

,

62


i.e.

< Pl (t)fl (t) z~t~ -

+ Pl (t) 7C77

Sk-2

pz(s2) k-2

p3($3)

9 9 9

o

Sk-1

\

• iPk-l(Sk-1) iPlc(xk)flc(sk)dskdsk-1...dx3ds2 0

0

n(7

< P l ( t ) f l (t) q- P l (t) Z k=2

Sk-2 x iPk-l(Sk-1) 0

s/ p2(s2)

p3(s3) 9 9 9 0

Sk-1 j'pk(sk)fk(Sk)dskdSk-1...ds3ds2 0

I . (1.9.49)

By taking t = Sl in (1.9.49) and integrating it from 0 to t we get

(io

Z(t) < u0exp

ski

k=l

pl(Sl)

si

p2(s2)..

0

• / pk(&)fk(sj:)dskdsk-1 ...

9

is 0

ds2dsl

Pk-l(Sk-1)

)]

.

(1.9.50)

0

Using (1.9.50) in u(t) < z(t) we get the required inequality in (1.9.39). The proof of (d2) can be completed by using the idea of the proof of Theorem 1.7.4 and using the inequality given in (dl). II R e m a r k 1.9.1 We note that the inequalities given in Theorem 1.9.4 are inspired by the inequalities given by Medved (1993). For some other results analogous to the above, see the results given by Pachpatte (in press e, j). A

1.10 Inequalities Involving Product Integrals Helton (1977) gave two inequalities involving product integrals, which in turn are further generalizations of the inequality established by Pachpatte


63

(1973a, Theorem 1). In this section we shall give the inequalities established by Helton (1977). Following Helton (1977), for completeness, we began with some definitions and notations used in this section. Definitions and integrals are of the subdivision-refinement type, and functions are from R to R or R x R to R, where R denotes the set of real numbers. Further, interval functions are assumed to be defined only for elements {x, y} of R x R such that x < y. Lower-case letters are used to denote functions defined on R, and upper-case letters are used to denote functions defined on R x R. If h and G are functions defined on R and R x R, respectively, and {xi}ni=0 is a subdivision of some interval [a, b], then hi - h(xi) for i = 1, 2, . . . , n and Gi "-- G(xi-1, xi) for i - 1,2, . . . , n. The statement that fb G exits means there exists a number L such that, if E > 0, then there exists a subdivision D of [a, b] such that, if {Xi}in___.Ois a refinement of D, then

IL_s"i=1 Similarly, aI-Ib(1 + G) exists if there exists a number L such that, if e > 0, then there exists a subdivision D of [a, b] such that, if {xi }/no is a refinement of D, then n

L

-II(1 +

Gi)

< ~:.

i=1

Also, G has bounded variation on [a, b] if there exist a subdivision D of [a, b] and a number B such that, if {Xi}in___Ois a refinement of D, then 1Gi] < B . i=1

If G has bounded variation on [a, b], then f t G exists if and only if 1-IY(1 + G) exists for a < x _< y < b. X

For convenience in notation, we adopt the conventions that p

II i=p+l

p

1 and

Z

i=0.

i--p+l

These conventions simplify the representation of certain expressions that occur in this section.


64

Right and left integrals arise in this section. These are denoted by

fb G(u, v)h(v) and fb h(u)G(u, v), respectively. Suppose {xi}/=0n denotes a subdivision of some interval [a, b]. Then, the preceding fight and left integrals have approximating sums of the form n

n

~Gihi

and

~hi-lGi,

i=1

i=1

respectively. Through the section, several different functions are involved in fight or left integrals. For examples, integrals of the form

]

h(r)F(r,s) G(u,v)

and j

a

1" 1

G(u,v) v__b( +

+G)

a

arise. Here, the approximating sums are of the form

]

h(r)F(r, s) ai i= 1

Gi

and i= 1

f II ~

1

(1 + F + G) ,

xi

respectively. Representations involving fight and left integrals are necessary due to possible discontinuities of the functions involved. If fb G exists, then fb iG(u ' v ) - fv GI exists and is zero. This result is of use in switching between difference inequalities and integral inequalities. Additional background on product integration can be obtained from the references given in Helton (1977). In order to establish our main results, we need the following two lemmas given in Helton (1977).

Lemma 1.10.1 Suppose c is a positive constant, h is a bounded function

from R to R, each of F and G is a nonnegative function from R x R to R, each of fb F and fb G exists, each of b

h(u)G(u, v) and a

exists and

t

J

h(t) < c + f h(u)G(u, v) + a

h(r)F(r, s) G(u, v), a

65


for a < t < b. Then, if a < t < b and following inequality holds: h(t)
c

[ j/ 1+

aHU(1 + F + G) G(u, v)

]

.

a

Let d denote the positive number such that

d - h(t) - c

[

1+

/ 1

aHU(1 + F + G) G(u, v) a

We note that the existence of

f [aHU(1 71-F-JI-G)]

.

t

aHU(1-I-F q-G)and

G(/,/, v)

a

can be established from the existence of fb F and fb G. It follows from the existence of the integrals involved that there exists n is a refinement of D1 then a subdivision D1 of [a, t] such that, {Xi}i=0

f(aHU(1-.I-F+a))a(u,v)-~ a

(1-'[-Fjq-aj)

ai

< d(2c) -1

i=1 \ j = l

Let fl represent a nonnegative function of bounded variation from R x R to R such that, if {Xi}in=lis a subdivision of [a, t] and 1 < i < n, then 2(IFil + IGil) < ]~i.

72


There exists a subdivision D2 of [a ' t] and a number B such that, if is a refinement of D2, then l~I(1 +/~i)

0 and a(t) are continuous functions defined on I and r(t) is continuously differentiable on I. Many authors have studied the behaviour of solutions of special versions of equations (A) and (B) with different viewpoints. In the following theorems we present results on the boundedness and asymptotic behaviour of the solutions of equation (A) under some suitable conditions on the functions involved in (A) and on the solutions of (B). These results are slight variants of the main results given by Pachpatte (1975e).


76

Theorem 1.11.1 Suppose that the functions k and f in (A) satisfy the conditions

Ik(t, s, x)l ~ h(s)lxl,

(1.11.1)

If (t, x, z)l _< g(t)(Ixl + Izl),

(1.11.2)

where h and g are real-valued continuous functions defined on I and ft~ h(s) ds < oo, ft~ g(s) ds < c~. If Xl (t) and x2 (t) are bounded solutions of equation (B), then the corresponding solution x(t) of equation (A) is bounded on I. E] Proof: Let Xl (t) and x2(t) be solutions of equation (B) such that r(t)[Xl (t)Xtz(t) - x'1(t)xz(t)] --- 1,

(1.11.3)

and suppose that x(t) is any solution of (A). Then by using the variation of constants formula, the solution x(t) of (A) is given by t

x(t) - ClXl (t) -~- c z x 2 ( t ) - + - / [ X l ( s ) x z ( t ) - Xl (t)x2(s)] +.1

to

(s

+ to

where Cl and c2 are constants. From (1.11.4) and using (1.11.1) and (1.11.2) we obtain Ix(t)l < c + f_

,~ + _

h~,~>lx(,~>l,~

,~

L I

to

to

to

where c >_ 0 and M >_ 0 are constants and the upper bounds for ]ClXl( t ) + czxz(t)l and Ixl (s)x2(t)- Xl (t)xz(s)l. Now an application of Theorem 1.7.1 yields Ix(t)l ___c

[] 1+

to

Mg(s) exp

(])

[Mg(o-) + h(o-)l d~

ds

]

.

to

The above estimation implies the boundedness of the solution x(t) of (A) on I.

I

77


Theorem 1.11.2 Suppose that the functions k and f in (A) satisfy the conditions Ik(t, s, x)l ~ h(s)lxl,

(1.11.5)

If (t, x, z)l 0 is a constant, h and g are real-valued continuous functions defined on I and ft7 h(s) exp(-ots) < cx~, ft7 g(s) exp(-2ots) ds < cx~. If Xl (t) and xz(t) are the solutions of (B) such that Ixi(t)l Ofor i = 1, 2 are constants, then the corresponding solution x(t) of (A) approach zero as t --+ cx~. 0

Proof: Let xl(t) and x2(t) be solutions of (B) satisfying the condition (1.11.3) and suppose x(t) is any solution of (A). Then by using the variation of constants formula, any solution x(t) of (A) is given by the integral equation (1.11.4). From (1.11.4) and (1.11.5)-(1.11.7), we obtain /-I

t

Ix(t)[ < c exp(-ca) + exp(-ca) / mg(s) exp(-ots)lx(s)[ ds to

+ exp(-oet) f_ M

da

, /

to

to

)

ds (1.11.8)

From (1.11.8) we have t

Ix(t)l exp(ctt) < c + f Mg(s) exp(-2as)lx(s)l exp(c~s) ds to

+ f Mg(s) exp(-2cts) to

x exp(cttr) d a ) ds.

(j to

h(tr) exp(-ctcr)lx(cr)l

78


Now an application of Theorem 1.7.1 yields Ix(t)l exp(ca) < c

[j 1+

Mg(s)exp(-2oes)

to

xexp(f[Mg(~)exp(-2o~)+h(~)exp(-oe~)]d~)ds] Now, multiplying both sides of the above inequality by exp(-ca) and then taking the limit as t --~ ec, we get the desired result, and hence the proof of the theorem is complete.

l

1.11.2. Perturbation of Volterra Integral Equations In this section we shall study the stability, exponential asymptotic stability and the growth of the solutions of a system of Volterra integral equations of the form r

x(t) -- f (t) +

fib(t, s)x(s) + F(t,

s, x(s))] ds,

(P)

0

under the assumption that the unperturbed linear system t

y(t) - f (t) + f k(t, s)y(s) ds,

(L)

0

has certain stability properties. Here x, y, f and F are in R', the ndimensional Euclidean space with Euclidean norm I" I, and k is an n • n matrix. We assume that f is continuously differentiable on R+, and that k(t, s) and F(t, s, x) are continuous for t, s 6 R+, Ixl < c~, and F(t, s, x) is continuously differentiable in t. For the study of existence of solutions of the above equations, see Burton (1983), Cordunneau (1973, 1977, 1991) and Miller (1971). We begin with the stability definitions that we use later in this section (see Pachpatte (1976a)). The following definitions are stated for (P). Of course, we can apply them to (L) as well, with the same initial values x(0) = y ( 0 ) = f ( 0 ) = x0.

79


Definitions (1) The system (P) is called globally uniformly stable on R+ if there exists a constant M > 0 such that Ix(t)l _< M for Ix01 < ~ . (2) The system (P) is called exponentially asymptotically stable on R+ if there exist positive constants M and u such that Ix(t)l < Mix01 e x p ( - ~ t ) for Ix01 sufficiently small. (3) The system (P) is called uniformly slowly growing on R+ if and only if for every ot > 0 there exists a constant M > 0, possibly depending on ~, such that Ix(t)l _< Mix01 exp(ca), for Ix01 < c~. A It is known (Bownds and Cushing, 1975) if U(t, s) is an n x n matrix (called the resolvent kernel) which solves the matrix equation t

U(t, s) -- I + f k(t, r)U(r, s)dr, s

for 0 < s < t < cx~, where I is the n x n identity matrix, then the solution of (P) is given by

,(j

x(t) = y(t) -t-

U(t, s)

F(s, s, x(s)) +

0

~ F(s, z, x(r)) dr

)

ds,

0

(1.11.9) where t i " $

y(t) = g(t, O)xo + / U(t, s)f'(s)ds,

(1.11.10)

0

is a solution of the linear system (L). Our approach and arguments lead us to the following assumptions on the perturbation function F" (H1) (H2)

IF(t, t,x(t))l < p(t)lx(t)l, 0 -xrF(t, s,x(s)) < p(t)q(s)lx(s)l, Of

or

(H3)

O--F(t, s x(s)) < p(t) exp(-ot(t - s))q(s)lx(s)l, Ot

or

(H4)

0

-~ F(t, s, x(s)) < p(t) exp(ot(t - s))q(s)lx(s)l,

80


for 0 < s < t < c~, where ot is a positive constant and p and q are continuous functions defined on R+ such that OO

(H5 )

OO

/ p(s)ds < cxz,

/ q(s)ds < cx~.

0

0

We now state and prove the following theorems given by Pachpatte (1976a), which deals with the preservation of stability of (P). Theorem 1.11.3 Let the resolvent kernel U(t, s) satisfy the condition

IU(t,s)l 0 is a constant. If I

t

u(t) < c + f f (s)[p(s)u(s) + q(s)] ds, o

t ~ R+,

then for t 6 R+,

u(t)
0, b > 0, and y > 0 are constants and b < }//2. Then

u(t) < ( a / b ) ( y -

8)exp[-8(t-

cr)],

t ~ J0,

where 8 - (?,2 _ 2b?,)l/a.

1.12.8 Coppel (1965) Let u be a continuous function in J, and suppose t~

u(t) < a exp[- ?'(/~

I

t)] + /

b exp(-yIt-

sl)u(s)ds,

t ~ J,

ior 19l

where a >_ 0, b >_ 0, and y > 0 are constants and b < },/2. Then

u(t) 0 be a decreasing continuous function, for t > cr and cr sufficiently


90

large, in such a way that OO

(X)

1

Suppose that u is a nonnegative continuous function such that gu is bounded and t

oo

.(~ o-, where k > 0 is a constant. Then, for t ~ R+, u(t) < (M/(1 - fl))exp

( j ) 1/(1 - fl)

f(s)ds

,

t

where

CX~

(7

O"

1.12.10 Mahmudov and Musaev (1969) Let the function u be positive and continuous on [0, T] and satisfy the inequality t

T

~,~ ~_I,,, +/~l,S,U,S, + ~,,,,s,~s+ / ~ , s , u , s ,

+ ~2(t, s)] ds,

tl

0

0

where qbi(t) > O, ~i(t, s) >__0, i -- 1, 2, and f ( t ) > 0 are continuous functions and I"

Q(t) -

/1 + exp I_

Then

(i)J ~1 (s) ds

]J (i)J ] ~l(S) ds

0

u(t) < (1/(1 - Q ( t ) ) ) [1 + exp

~2(s) ds < 1.

0

~1 (s) Os

~1 (s) ds

0

T

• / $2(s)N(s)ds + N(t), 0


91

where T

N(t) = f (t) + f tTtl (t, s) + q/2(t, s)] ds 0

+ exp

(i )r 4~1(s) ds

4h ( r ) f (r) dr

0

+exp (~4h(s)ds)) f dpl(r)

1.12.11 Let u, p and q be nonnegative continuous functions defined on

J, k(t, s)

be

a continuous and nonnegative function on the rectangle [:]" c~ _< s < t _< fl

and nondecreasing in t for each s ~ J. If t f~

u(t) 0 or En < O. Then the sequence {rn } converges uniformly on [or, t31] to a function r which is the maximal solution of (2.2.1) if ~:n > O, or the minimal solution o f (2.2.1) if ~n < O. A

The basic comparison theorem used in the theory of differential equations can now be formulated. Theorem 2.2.2 Let F(t, x) be continuous in an open set D containing a point (or, xo), and suppose that the initial value problem r' = F(t, r),

r(ot) = xo,

(2.2.6)

has a maximal solution r = r(t) with domain ot < t < ~1. If x is any differentiable function on lot, 131] such that (t, x(t)) ~ D f o r t ~ [or, ill] and x'(t) < F(t, x(t)),

ot < t < ill,

x(ot) < xo,

(2.2.7)

then x(t) < r(t),

~ < t < ill.

(2.2.8)

102

NONLINEAR INTEGRAL INEQUALITIES I

Moreover, the result remains valid if 'maximal' is replaced by 'minimal' and < is replaced by > in both (2.2.7) and (2.2.8). D

The proof of this theorem, which depends on the above two lemmas, may be found in Szarski (1965, Theorem 9.5, p. 27), Lakshmikantham and Leela (1969, Theorem 1.4.1, p. 15), Walter (1970, Theorem 10, p. 68), Beesack (1975, Theorem 6.2, p. 37) and Bainov and Simeonov (1992, Theorem 6.3, p. 57). Since the proofs of Lemmas 2.2.1 and 2.2.2 and Theorem 2.2.2 are essentially contained in the above references, the explicit proofs will not be given here. Brauer (1963) obtained bounds for solutions of a system of ordinary differential equation of the form x' - f (t, x),

(P)

where x and f are n-dimensional vectors and 0 _< t < oo. The symbol I" I will be used to denote any convenient vector norm. It is assumed that f (t, x) is continuous for 0 _< t < oo, Ixl < oo, but no assumptions are required on f to ensure the uniqueness of solutions of (P), as the arguments do not require uniqueness. The following comparison theorem is well known and used considerably in the literature (Bainov and Simeonov, 1992; Beesack, 1975; Brauer, 1963; Brauer and Wong, 1969; Lakshmikantham and Leela, 1969; Szarski, 1965; Walter, 1970). Theorem 2.2.3 Suppose that there exists a continuous nonnegative function w(t, r) on 0 < t < oo, 0 < r < oo, such that If(t, x)l < w(t, Ixl),

0 < t < oo,

Ixl < oo.

(2.2.9)

(i) Let x(t) be a solution of (P) and r(t) be the maximum solution of the scalar equation r' - w(t, r),

(2.2.10)

with r(O) - Ix(0)l, then x(t) can be continued to the right as f a r as r(t) exists, and

Ix(t)l < r(t), f o r all such t.

(2.2.11)

N O N L I N E A R INTEGRAL INEQUALITIES I

103

(ii) Let x(t) be a solution o f (P) and let u(t) be the minimum solution o f u' - - w ( t , u), with u(0) --

Ix(0)l.

(2.2.12)

Then, f o r all t > 0 such that x(t) exists and u(t) > O, we

have

Ix(t)l >__u(t).

(2.2.13) [-1

Theorem 2.2.3 involves a comparison between solutions of the system (P) and the scalar equations (2.2.10) and (2.2.12). As noted in (Brauer, 1963, p. 36), the bounds given in (2.2.11) and (2.2.13) are sometimes difficult or impossible to calculate explicitly. However, the comparison method provides not only a powerful tool for obtaining bounds for solutions, but also a unified approach to many such problems. Brauer (1963) gave explicit bounds when the function w(t, r) has the form ;~(t)dp(r). In this case the condition (2.2.9) is replaced by I f ( t , x)l ~ )~(t)~b(Ixl).

(2.2.14)

Then the equations (2.2.10) and (2.2.12) lead us to consider the separable first order equations r' -- )~(t)dp(r),

(2.2.15)

u' -- -)~(t)c~(u),

(2.2.16)

and

which can be solved explicitly. We define J ( r ) -- f o ds/4)(s). If this integral diverges at zero, the lower limit of integration can be replaced by any fixed > 0, but we will use the lower limit zero for convenience. If f ~ ds/4)(s) R < c~, the function J maps the half line [0, ee) onto the interval [0, R), with J ( e c ) - R. The function J has a positive derivative 1/4~(r), and is therefore monotonic increasing. Thus the inverse function j - 1 exists and is monotonic increasing on [0, R). If r(t) is a solution of (2.2.15) with r(0) -- r0, then

~(s) ds 0

j ro

~(s) -- J ( r ) - J(ro).

104


Thus t

J(r) -- J(ro) 4- f ~(s) ds, 0

and

r(t) = j_l (j(ro) + ~ )~(s)ds)

(2.2.17)

This solution exists as long as J(ro)+ fo )~(s)ds is in the domain [0, R) of j - 1 . This requires J(ro) + fo )~(s)ds < R or fo ~.(s)ds < fro ds/4~(s). It follows that the solution r(t) of (2.2.15) with r(0) - r0 exists on the interval [0, T), where T is defined by f o Z ( s ) d s - f r o ds/dp(s). If fro ds/qb(s)c~, the solution exists on 0 < t < cxz. If f o )~(s)ds < fro ds/~(s)< o0, then J(ro)+ fo ~.(s)ds < R, which implies r(t) < c~, and the solution r(t) remains bounded on 0 < t < c~. Since r(t) is monotonic increasing, this implies that r(t) tends to a limit as t --+ c~. Equation (2.2.16) can be solved by the same approach. It is found that the solution u(t) of (2.2.16) with u ( 0 ) - r0 is given by u(t) -- j-1

J(ro) - f X(s) ds] 0

defined so long as J ( r o ) - fo )~(s)ds > 0. Thus u(t) exists on the interval [0, r], where fo ~(s) ds - fo ~ ds/dp(s). If f ~ )~(s) ds < fo ~ ds/qb(s), u(t) exists on 0 < t < c~z, and if f ~ )~(s)ds < fo ~ ds/q~(s), u(t) > 0 on 0 < t < c~. To avoid misinterpretation, recall that the convergence of the integral fo ~ ds/qb(s) has been assumed. A great many papers have been written on the various generalizations and extensions of the above given comparison theorems. In the following theorem we present a comparison theorem given by Opial (1960). Theorem 2.2.4 Let J -- [or, fl), and f" J x R n ~ R n be continuous and

satisfy f (t, x) < f (t, y)for t ~ J and x, y ~ map x" J --+ R n satisfies

Rn

with x
0 is an arbitrary small constant, and subsequently pass to the limit as E --+ 0 to obtain (2.3.2). The subinterval 0 < t < tl is obvious. II The inequality given in Theorem 2.3.1 is now known in the literature as Bihari's inequality. However, the above inequality seems to have appeared first in the work of LaSalle (1949). The case k - 0 of Bihari's inequality was dealt with in LaSalle (1949). Both LaSalle (1949) and Bihari (1956) applied their results to uniqueness questions for differential equations. R e m a r k 2.3.1 The choice of the point r0 6 R+ does not affect the final result (2.3.2). If ~0 6 R+ and F -

G(r) =

w(s) -io

,

r ~ R+,

109


then G(r) -- G(r) - G(?o), so ~ - l ( v ) -- G -1 (v + G(~0)). Consequently,

-~-1

( / ) ( / ) -G(k) -t-

f (s) ds

-- G -1

G(k) -t-

0

f (s) ds

,

0

so that the bound in (2.3.2) is independent of the choice of the point r0 ~ R+. A R e m a r k 2.3.2 In some cases we may have w(0) = 0 at the end-point of the interval, say (0, ~ ) with w continuous on [0, e~) and w > 0 on (0, ee). To be specific, suppose w is also nondecreasing in (0, e~) and f > 0 on R+. Then Theorem 2.3.1 is valid for functions u > 0 continuous on R+, even if u(~) = 0 for some T e R+. To see this, let E > 0 and u~(t)= u ( t ) + ~ > 0 for all t ~ R+. Then t

uE(t) < (k + E) + / f (s)w(uE(s)

i

E)

ds

qor

0 t

< (k + E) + / f (s)w(uE(s))ds, 0

and it follows that

uE(t) < G -1

[

G(k + e) +

/ ] f (s) ds

.

0

Letting E ~ 0, we obtain (2.3.2). A R e m a r k 2.3.3 Theorem 2.3.1 remains valid if < is replaced by > in both (2.3.1) and (2.3.2). Taking w(u) = u, G(u) - log u, we obtain Bellman's inequality, given in Theorem 1.2.2. A Langenhop (1960) proved the following inequality, which provides an explicit lower bound for an unknown function. T h e o r e m 2.3.2 Let u, f and w be as defined in Theorem 2.3.1 and t

u(t) > u(cr) - / f (s)w(u(s)) ds, ~r

0 < cr < t < c~,

(2.3.8)

110


then u(t) > G -1

(

G(u(cr)) -

/ ) f (s) ds

0 < cr < t _< t2,

(2.3.9)

(y

where G, G -1 are as defined in Theorem 2.3.1 and t2 ~ R+ is chosen so that t

G(u(cr)) - /

f (s) ds ~ Dom(G-1),

ff

for all t ~ R+ lying in 0 < cr < t < t2.

[2] Proof: We write (2.3.8) in the form t

U(O')

< u(t) + /

f (s)w(u(s)) ds,

O < cr < t < ~ ,

,.J

o"

and let t 6 R+ be fixed. Continuing with the function u(cr), ~r 6 R+ and proceeding as in the proof of Theorem 2.3.1, we get the desired inequality in (2.3.9). We omit the details. II

We note that the above result is also valid if > is replaced by < in both (2.3.8) and (2.3.9). Moreover, as pointed out by Beesack (1975, p. 14), the inequality (2.3.8) implies (2.3.9) even when w is not monotonic nondecreasing on R+. This follows from a general result given by Walter (1970, p. 44); see also Redheffer (1964). Constantin (1990a,b) obtained the following interesting and useful inequalities, similar to those of Bihari and Langenhop, which in fact are motivated by the inequality established by Morro (1982). Theorem 2.3.3 Let g, h and k be continuous functions on the interval [to, tl], to, tl E R, tl > to, g > 0, kh > 0 and k, g E cl([t0, tl], R), w: R+ --+ R+ a continuous, monotonic nondecreasing function so that there exists L > 0 with w(x) > Lx for each x ~ R+ and w ( O ) = O. If a continuous positive function u has the property that t

u(t) 0 and w is monotonic nondecreasing. We observe that because kh > 0 we have ky > 0 for t > to and so

g'(t) g'(t) < 0 =~ < O, w(g(t) + k(t)y(t)) g'(t) > 0 =~

g'(t) g'(t) g'(t) < < w(g(t) + k(t)y(t)) - L(g(t) + k(t)y(t)) - Lg(t)

Hence we have that g'(t)

g'(t) < max [ 0 , w(g(t) + k(t)y(t)) L Lg(t) }"

(2.3.13)

112

NONLINEAR INTEGRAL INEQUALITIES I It is now obvious that for t > to

k'(t) < O,

y(t) > O=~

y(t)k'(t) < O, w(g(t) d- k(t)y(t)) -

k'(t) > O,

y(t) > O=~

y(t)k' (t) y(t)k' (t) < w(g(t) + k(t)y(t)) - L(g(t) + k(t)y(t)) y(t)k'(t) < - Lk(t)y(t)

---

k'(t) Lk(t)'

k'(t) > O,

y(t) < O=~

y(t)k'(t) 0, r0 > 0,

(2.4.32)

and t t is defined so that the existence condition of the right-hand part of the inequality (2.4.31) should be assured. [~ Proof: From (2.4.30) we have t

u(t) < g(t) + k(t) f max{hi (s), h2(s)}[u(s) + H(u(s))] ds. 0

We consider w" R+--+ R+ defined by w ( x ) - x + H(x) and obtain that w(x) > x, x ~ R+, w is continuous monotonic nondecreasing and w(0) - 0. Applying Theorem 2.3.3 we deduce the inequality (2.4.31).

I The following generalization of Bihari's inequality is used by Talpalaru (1973). Theorem 2.4.5 Let u, f : I --->R+ and k(t, s) : I x I --->R+, to < s < t,

be integrable functions. Let w(r) be a positive, continuous and nondecreasing function defined for r > O. If u(t) < c +

j[

to

f (s)w(u(s)) +

J

]

k(s, cr)w(u(a))da ds,

to

for all t > to, where c is a nonnegative constant, then

(2.4.33)


U'' j[ j ds
0 is continuous for 0 < s < t < c~ and w ' R + --+ R+ is continuous. In this section we wish to give some results which provides upper bounds on the inequalities of the form (2.5.1). We first present the following inequality needed in the proof of our next theorem, which is given by Constantin (1992). Theorem 2.5.1 Let a, b be continuous and positive functions defined on R+ and w" R+ --+ R+ be a continuous monotonic nondecreasing function such that w(O) - 0 and w(x) > 0 for x > O. If u is a positive differentiable function on R+ that satisfies

u'(t) < a(t)w(u(t)) + b(t), then we have

l[o(u,o,+

t ~ R+,

]

+ f a(s) ds 0

(2.5.2)

,

(2.5.3)

for the values of t, so that the existence of the right-hand part of (2.5.3) should be assured, where

127

NONLINEAR INTEGRAL INEQUALITIES I ds w(s) '

G(r) --

(2.5.4)

r > 0 , r0 > 0.

ro

D Proof: From (2.5.2) we have t

t

u(t) 0 for u > 0, and satisfies

V-I~(u) ~

~(12-1U) for v > 1, u > 0.

(2.5.14)

The class J was introduced in Dhongade and Deo (1973), but with the condition that (2.5.14) holds for v > 0, u > 0. It was noted by Beesack (1975, p. 65) and Beesack (1977, p. 207) that 3" then becomes merely the trivial class of linear functions; q~(u)- 4~(1)u. The definition (2.5.14) was adopted to avoid this triviality. For a further discussion concerning the class ~ , see Beesack (1975, 1977, 1984a,b). Beesack (1984a) proved and made use of the inequalities given in the following theorem. Theorem 2.5.3 Let u and p be nonnegative continuous functions defined

on R+, g(t, s) > 0 is continuous for 0 < s < t 0 for u > 0 and such that (2.5.1) is satisfied. Fix ro > O, and define G by G(r) --

j~

w(s)'

r > O.

(2.5.15)

ro

(In case w(O) > O, or in case w(O+) is finite, one may take ro = 0). Let G -1 denote the inverse function of G. Then if ~ and-p are defined by (2.5.13), we have u(t) < G_l [G(~(t)) + ~ g(t,s)dsl

O_ O. We start with the following fundamental inequality given by Willett and Wong (1965).

136


Let u, a, b and k be nonnegative continuous functions defined on J, let p > 1 be constant, and suppose T h e o r e m 2.6.1

lip

u(t) < a(t) + b(t) ( / k(s)uP (s) ds)

t 6 J.

(2.6.1)

Then k(s)e(s)a p (s) ds u(t) < a(t) + b(t)

1 - [1 -

e(t)]l/p

t ~ J,

'

(2.6.2)

where e(t) - exp ( - f k(s)bP (s) ds) "

(2.6.3)

El P r o o f : Define a function

v(t)

on J by t

v(t) -- e(t) f k(s)uP(s) ds.

(2.6.4)

Ol

It follows from (2.6.1) and (2.6.4) that

u(t) O, z(x) is a nonincreasing function of x for x > 0. Thus we may replace f t bP(s)k(s)v(s ) ds in (2.6.6) by a larger quantity and still have a valid inequality. It is easy to see from the definition of v(t) given in (2.6.4) that

f b p (s)k(s)v(s) ds < Ol

(/)(/

b p (s)k(s)e(s) ds

k(s)u p (s) ds

)

t

- (1 - e(t)) / k(s)uP(s)ds.

(2.6.7)

0d

The conclusion

(2.6.2) follows

upon

substituting

(2.6.4)

and (2.6.7)

in (2.6.5). II

138


R e m a r k 2.6.1 A special form of this inequality with k(t)= 1 was proved first by Willett (1964). Moreover, the continuity hypothesis was replaced by the usual Lebesgue integrability in Willett and Wong (1965). A The following theorem, in which the nonlinear function up in Theorem 2.6.1 is replaced by a general function is given by Gollwitzer (1969). Theorem 2.6.2 Let u, f, g and h be nonnegative continuous functions

on the interval [a, b], G(u) be a continuous, strictly increasing, convex and submultiplicative function for u > 0, limu~oo G(u) = cxz, a(t), ~(t) be continuous functions on [a, b], a(t), ~(t) > O, o~(t) + ~(t) = 1, and u(t) V(S)- g(t)G -1 ( / h(cr)G(v(~) ) d~

(2.6.35)

+ f h(~) ( ~ k ( r ) G ( v ( r ) ) d r ) d ~ ) for a < s < t < b. Then /

u(t)

> c~(t)a -1

f

(o1-1(t)G(v(s)) ~ 1 + fl(t)G(g(t)f1-1 (t))

\

• f h(tr) exp

[

)}1 /

[h(r)~(t)G(g(t)~ -1 (t)) + k(r)] d'r do

,

s

(2.6.36)

fora < s < t < b. D


146

Proof: Rewrite (2.6.33) as / t

U(t) O, p > 0 and p ~ 1 are constants and suppose u(t) < c + f- tl

0

'

J(JO'

(s)u(s) ds -Jr-

(s)

g(ff)u p

da

ds,

t E

R+.

0

(2.7.1)

(i) If 0 < p < 1 and Eo(t) is defined by Eo(t) -- 1 + (1 - p)c p-1

fg(r)exp(j)

- ( 1 - p)

0

f(cr)dcr

dr,

0

then

u(t) < c

[j

f (s) exp

1+

)

f(cr) dcr

o

for t ~ R+. (ii) If 1 < p < cx~, then u(t) < c

EJ (Jo ) 1+

f(s)exp

0

f(cr)dcr

(Eo(s)) 1~(l-p) ds

J

(-Eo(s)) - 1 / ( p - 1 ) ds

,

]

,

(2.7.2)

(2.7.3)

for t ~ [0, or), where Eo(t) is defined by the right-hand side of the definition of Eo(t) for 1 0}. Moreover, if we assume that Eo(t) > 0 for all t ~ R+, then the inequality (2.7.3) remains valid for all t E R+. D

Pachpatte (1976b) established the following useful inequalities. Theorem 2.7.2 Let u, f , g and h be nonnegative continuous functions defined on R+ and c > 0 is a constant and suppose

150

NONLINEAR

u(t)

INTEGRAL INEQUALITIES

~_c+ f_i~s~s~s

_~s~u~s~

+

I

~,u~o~

ds,

r

0

0

0

(2.7.4)

for t ~ R+. Then +

u(t) < c exp ( fo [f (s) + cg(s)(E(s))-I exp

(2.7.5) for t ~ [0, 0~1 ), where

[f (cr) + h(cr)] do) dr,

E(t) -- 1 - c / g(r)exp

(0J

0

and O~1 ~" sup{t 6 R+ : E(t) > 0}. Moreover, if we assume that E(t) > 0 for all t ~ R+, then the inequality (2.7.5) remains valid for all t ~ R+. D

Theorem 2.7.3 Let u, f , g, h and c be as in Theorem 2.7.2 and p > O, p ~ 1 is a constant and suppose

~,~c+f__i~s~u~s,~s+~s~u~s~ t

( u (~s )+ h(cr)u(tr)

ft

0

0

dcr )

ds,

0

t 6 R+.

(2.7.6)

(i) If O < p < 1 and E1 (t) is defined by E1 (t) -- 1 -- p c p

j

g(r) exp

( j ) p

0

[f(a) + h(cr)] do" dr,

(2.7.7)

0

then

(j)[ xexp +

u(t) < cexp

f (s)ds

1 + (1

-

p)c p

J

g(s)(El(S)) -1/p

0

]

1/(l-p)

(2.7.8)

NONLINEAR

INTEGRAL

INEQUALITIES

1~1

I

for t ~ [0, Or2), where ~ 2 --" sup{t 6 R+" El(t) > 0}. (ii) If 1 < p < c~, then u(t) < c exp

(0))[ f (s) ds

1 - (p - 1)c p

g(s)(-E1 (s)) -1/p

0

)

( p f ( r ) + h(r)) dr

x exp

J

ds

(2.7.9)

,

for t ~ [0, or3), where El(t) is defined by the right-hand side of (2.7.7)for 1 0 and (2) E1 (t) > 0 and the expression between [...] in (2.7.9) is positive for all t ~ R+, then the inequalities (2.7.8) and (2.7.9) remains valid for all t ~ R+. D Another interesting and useful inequality established by Pachpatte (1978b) is embodied in the following theorem. Theorem 2.7.4 Let u, f , g, h, c and p be as in Theorem 2.7.3 and suppose u(t) __- p f (t)m(t) - p(g(t) + h(t)). The inequality (2.7.18) and the fact that for

v(t)

re(t) - v-P(t)

(2.7.18)

implies the estimate

such that

v(t) < c(E2(t))-l/P exp (fo f (r)dr ) .

(2.7.19)

Using (2.7.19) in (2.7.14) we get

z'(t) 0}. (ii) If 1 < p < c~, then

u(t) < c

[j 1+

f (s) exp

f (r) dr

1 - (p - 1)c p

0

x f g(v)(Yl (Z'))-1 exp

0

(OJ)

[ p f (a) + h(a)] da

-I/(p-1)I , dr (2.7.32)

m

for t ~ [0, f13), where F1 (t) is defined by the right-hand side of (2.7.30)for 1 0 and (2) -if1 (t) > 0 and the expression between {...} in (2.7.32) is positive for all t ~ R+, then the inequalities (2.7.31) and (2.7.32) remains valid for all t ~ R+. D


la'/

Theorem 2.7.8 Let u, f , g, h, c and p be as in Theorem 2.7.4 and suppose

t

u(t) < c + / f (s)u(s) ds +

0

j

f (s)

(r

0

( j U(r) +

h(ff)u p+I (or) dcr

0

g('c)u p ('c)

)) dr

ds,

(2.7.33)

for t e R+. (i) If O < p < 1 and Fz(t) is defined by

tfig(r) +

F2(t) -- 1 - pc p

(j)

h(z)] exp

p

0

f (or) dcr

(2.7.34)

dr,

0

then

u(t) _< c

[j 0 1+

f (~) exp

(!){ f (z) dr

1 + (1 - p)c p

(j)

0

• fg(s)(F2(s)) -l/pexp

p

0

}

1/(l-p)

f ( r ) dr

ds

d ~ l , (2.7.35)

0

for t e [0, f14), w h e r e / ~ 4 - sup{t ~ R+" F2(t) > 0}. (ii) If 1 < p < oo, then

[j

u(t) < c

1+

f (~) exp

0

(!){

• f g(s)(--ff2(S)) -lIp exp 0

f (r) dr

(j) p

1 - (p - 1)c p

f ( r ) dr

0

ds

1

d~ , (2.7.36)

m

for t e [0,/~5), where F2(t) is defined by the right-hand side of (2.7.34)for 1 0 and (2) F2(t) > 0 and the expression between {...} in (2.7.36) is positive for all t ~ R+, then the inequalities (2.7.35) and (2.7.36) remain valid for all t ~ R+. D As noted above, the inequality in Theorem 2.7.5 is dealt with in Pachpatte (1977a) and the inequalities in Theorems 2.7.6 and 2.7.7 are dealt with in Pachpatte (1977c), while the inequality in Theorem 2.7.8 is motivated by the inequality given in Theorem 2.7.4. In fact, the inequalities in Theorems 2.7.1 to 2.7.8 can be considered as further generalizations of the Gronwall-Bellman inequality, and in the special cases when f ( t ) = 0 these inequalities reduce to various new inequalities which can be used very effectively in certain applications.

2.8 Pachpatte's Inequalities II The nonlinear integral inequalities of Bihari (1956) and Langenhop (1960) are often used to study the various problems in the theory of differential and integral equations. Pachpatte (1974d) established some new Bihari-like integral inequalities which can be used as handy tools in the study of certain classes of general integral and integro-differential equations for which the inequalities of Bihari and Langenhop do not apply directly. Since then, in a series of papers (Pachpatte 1975a,b,d,h, 1976c), Pachpatte has obtained a number of new results related to those given in Pachpatte (1974d), which are very effective in various applications. This section deals with some basic inequalities given in the above-mentioned papers. Pachpatte (1974d) gave the inequalities in the following two theorems. Theorem 2.8.1 Let u and g be nonnegative continuous functions defined

on R+. Let H (u) be a continuous nondecreasing function defined on R+ and H (u) > 0 on (0, c~). If

u(t) < c + f g(s) (u(s) + fo

ds,

for t ~ R+, where c > 0 is a constant, then for 0 < t < tl,

(2.8.1)

NONLINEARINTEGRALINEQUALITIESI

159

t [ j J

u(t) < c + / g ( s ) G -1

G(c) +

0

g(a) da

(2.8.2)

ds,

0

where

jf

G(r) -

ds

(2.8.3)

r>O, ro > O,

s +H(s) ro

and G -1 is the inverse of G, and t l ~ R+ is chosen so that t

G(c) + f g(a) da E

Dom(G -1),

0

for all t ~ R+ lying in the interval 0 < t < tl. E] Theorem 2.8.2 Let u, f and g be nonnegative continuous functions defined on R+. Let H(u) be a continuous nondecreasing and subadditive function defined on R+ and H (u) > 0 on (0, ocz). If

u(t) < f (t) +

j ( j g(s)

u(s) +

0

g(a)H(u(a))da

)

(2.8.4)

ds,

0

for t ~ R+, then for 0 < t < t2,

u(t) < f (t) + A(t) + / g ( s ) G

-1

0

[

G(A(t)) +

j ] g(a) da

ds,

(2.8.5)

0

where

A(t) --

j ( j g(r)

f(r) +

0

g(a)H(f(a))da

)

dr,

(2.8.6)

0

G, G -1 are as defined in Theorem 2.8.1 and t2 E R+ is chosen so that t

G(A(t))

+ f g(a) da E Dom(G -1), 0


160

for all t ~ R+ lying in the interval 0 < t < t2.

E3 The next two theorems deal with the slight variants of the inequalities given by Pachpatte (1976c) and (1975h) respectively. Theorem 2.8.3 Let u, h and k be nonnegative continuous functions

defined on R+. Let H(u) be a continuous nondecreasing function defined on R+ and H(u) > 0 on (0, oo). If

u(t) < c +

j ( j h(s)H

u(s) +

0

k(cr)H(u(cr)) do

)

(2.8.7)

ds,

0

for t ~ R+, where c > 0 is a constant, then for 0 < t < t3,

u(t) < c +

j ( [ j h(s)H

F -1

F(c) +

0

[h(tr) + k(tr)] do

0

])

ds,

(2.8.8)

where r

F(r) --

H(s)'

(2.8.9)

r > 0, ro > 0 ,

ro

F -1 is the inverse of F and t3 E R+ is chosen so that t

fth(~)

+ k(~)] do ~ Dom(F -1),

F(c) +

0

for all t ~ R+ lying in the interval 0 < t < t3. I-1 Theorem 2.8.4 Let u, f , g, h and k be nonnegative continuous functions

defined on R+. Let H(u) be a continuous nondecreasing subadditive and submultiplicative function defined on R+ and H (u) > 0 on (0, c~). If

u(t) < f (t) + g(t)

j(j h(s)H

0

u(s) + g(s)

k(tr)H(u(cr))dcr 0

)

ds,

(2.8.10)


161

for t ~ R+, then for 0 < t < t4, u(t) O, ro > O,

ro

and ~-1 is the inverse of ~, and t2 ~ R+ is chosen so that t

O(b(t)) + / h(s)w(a(s)) ds e Dom(O -1), 0

for all t ~ R+ lying in the interval 0 < t 0 for each fixed t ~ R+. If u(t) < p(t) + f__ (s)u(s) ds + 0

)

dcr ds

tl

0 \

+ qb ( / h ( s ) w ( s , u(s)) ds) ,

(2.8.30)

for t ~ R+, then u(t) 0 is a constant, If

172

NONLINEAR INTEGRAL INEQUALITIES I t

(2.9.1)

u'(t) < a + f b(s)u' (s)(u(s) + u'(s)) ds, 0

for t ~ R+, and E(t) is defined by t

(2.9.2)

E(t) -- 1 - [a + u(0)] f e ~ b(a)da, 0

then u'(t) < a exp

(

[a + u(0)]

J

e s b(s)(E(s)) -1 ds

0

)

,

(2.9.3)

for t ~ [0, fl), where fl = sup{t 6 R+ : E(t) > 0}. Moreover, if we assume that E(t) > Ofor all t ~ R+, then the inequality (2.9.3) remains valid for all t6R+. D Proof: Define a function z(t) by the fight-hand side of (2.9.1). Then z(O) = a, u'(t) 0 be a constant. If u'(t) < a + c

( j u(t) +

b(s)w(u(s) + u'(s)) ds

)

,

(2.9.21)

0

for t ~ R+, then for 0 < t < t2,

u'(t) < exp(ct) [a + cu(O) + c ~o exp(-cs)b(s)

XW

G 1 [G([a -+- (1 -+-c)u(0)])

]) ]

S

+ (1 + c) f b(cr)do"

ds ,

(2.9.22)

0

where G, G -1 are as defined in Theorem 2.9.3, and t2 ~ R+ is chosen so that t

G([a + (1 + c)u(0)]) + (1 + c) f b(cr)do 6 Dom(G -1), 0

for all t ~ R+ lying in the interval 0 < t < t2. D

The proof of this theorem can be completed by following the proofs of Theorems 2.9.2 and 2.9.3. Here we omit the details. The following variants of Theorem 2.9.3 can also be useful in certain applications. Theorem 2.9.5 Let u, u' and b be nonnegative continuous functions defined on R+, u(O) - 0 and a > 0 be a constant. Let w(u) be a continuous nondecreasing submultiplicative function defined on R+ and w(u) > 0 on (0, cr If tI-i

u'(t) < a + / b(s)w(u(s) + u'(s))ds, J

0

(2.9.23)

177


for t ~ R+, then for 0 < t < t3, u(t) + u'(t) < (1

[ j f2(a) +

+ t ) ~ -1

0

where

]

b(s)w(1 + s)ds ,

(2.9.24)

r

f2(r) --

r > 0, ro > 0,

w(s) '

(2.9.25)

ro

and f2 -1 is the inverse of f2 and t3 ~ R+ is chosen so that t

f2(a) + / b(s)w(1 + s)ds ~ Dom(g2 -1), 0

for all t ~ R+ lying in the interval 0 < t < t3. [3 Proof: As noted in the proof of Theorem 2.9.3 we assume that a is positive. Integrating (2.9.23) from 0 to t we have t

u(t) 0 for each fixed t ~ R+. If t

u'(t) < a + / b(s)w(s, u(s) + u'(s)) ds,

t 6 R+,

(2.9.28)

0

then u(t) + u'(t) < p(t) + q(t)r(t),

t 6 R+,

(2.9.29)

where p(t) -- a(1 + t), q(t) - (1 + t) and r(t) is the maximal solution of r'(t) = b(t)w(t, p(t) + q(t)r(t)),

r(O) - O,

(2.9.30)

existing for all t ~ R+. [2 Proof: By following the proof of Theorem 2.9.5 we have t

u(t) + u' (t) < p(t) + q(t) / b(s)w(s, u(s) + u' (s)) ds.

(2.9.31)

0

Define a function z(t) by t

z(t) - / b(s)w(s, u(s) + u' (s)) ds, 0

then z(0) - 0, u(t) + u'(t) < p(t) + q(t)z(t) and

z'(t) = b(t)w(t, u(t) + u'(t)) O, vo > 0 ,

vo

f2 -1 is the inverse of f2, and t l ~ R+ is chosen so that

f2

r(s)w(p(s)) ds

r(s)w(q(s)) ds ~ Dom(ff2-1),

40

for all t ~ R+ lying in the interval 0 < t < tl.

Theorem 2.9.8 Let u, u ~ u (n), a, b and c be as in Theorem 2.9. 7. Let w(t, v) be as defined in Theorem 2.9.6. If . . . . .

u(n)(t) < a(t) -k- b(t) ( E \ i=0

u(i)(t) 4- f c(s)w(s, u(n)(s))ds ) ,

(2.9.39)

0

for t ~ R+, then u(n)(t) < a(t) 4- b(t)(p(t) 4- q(t)R(t)),

(2.9.40)

for t ~ R+, where p(t), q(t) are as defined in Theorem 2.9.7 and R(t) is the maximal solution of R t(t) -- r(t)w(t, a(t) + b(t)[p(t) 4- q(t)R(t)]),

R(O) -- O,

(2.9.41)

existing on R+, in which r(t) is defined by (2.9.38) in Theorem 2.9.7.

O We note that the usefulness of the inequalities in Theorems 2.9.7 and 2.9.8 becomes apparent if we consider that u(0), u'(0) . . . . . u~"-l)(0), a, b and c are known and u, u t, . . . , u ~n) are unknown functions. These inequalities give bounds in terms of the known functions which majorizes u~")(t) and consequently u(t) after n times integration. The same is true about the other inequalities given in this section. For a number of other inequalities of the above type, see Li (1963), Lin (1981, 1990), Lin and Wu (1989) and Pachpatte (1977c, 1978a,c, 1980a, 1982a).


181

2.10 Inequalities with Iterated Integrals For the study of various properties of the solutions of higher order differential and integro-differential equations certain integral inequalities with several iterated integrals are very important in a variety of situations. This section presents some inequalities investigated by Pachpatte (in press e,j) with several iterated integrals. In what follows, the definitions and notations used in Section 1.9 are used to simplify the details of presentation. The following results established by Pachpatte (in press e,j) can be considered as further generalizations of some of the inequalities given in Section 1.9. These results are formulated only for the iterated integrals as defined by M[t, r, h] in Section 1.9. The results corresponding to the iterated integrals defined by A[t, g, h], B[t, g, h] and E[t, g, h] in Section 1.9 can be formulated similarly. Pachpatte (in press e,j) established the following inequalities, which can be considered as generalizations of Bihari's inequality given in Theorem 2.3.1 and Pachpatte's inequality given in Theorem 2.8.1. Theorem 2.10.1 Let u > O, h > O, rj > O, j -- 1, 2 . . . . . n - 1, be continuous functions defined on R+ and c > 0 be a constant. Let w(u) be a continuously differentiable function defined on R+, w(u) > 0 on (0, c~) and w'(u) > 0 f o r u ~ R+. If u(t) < c -t- M[t, r, h(tn )W(U(tn ))],

(2.10.1)

f o r t ~ R+, then f o r 0 < t < tl, u(t) < ~-l[f2(c) + M[t, r, h(tn)]],

(2.10.2)

where v

S2(v) -

w(s)

'

v > 0, vo > 0,

(2.10.3)

vo

~"2- 1 is the inverse of f2 and-i ~ R+ is chosen so that

f2(c) + M[t, r, h(tn)] ~ Dom(g2 -1), f o r all t ~ R+ lying in the interval 0 < t < -i. D

182


Theorem 2.10.2 L e t u > O, f > 0, h > O, rj > O, j -- 1, 2 , . . . , n - 1, be continuous functions defined on R+, c > 0 is a constant and let w(u) be as given in Theorem 2.10.1. If u(t) < c + M[t, r, f (tn )(U(tn ) -Jr-M[tn, r, h(sn )W(U(Sn ))])],

(2.10.4)

for t ~ R+, then for 0 < t < t, u(t) < c + M[t, r, f (tn)G -1 [G(c) + M[tn, r, [ f (Sn) -~- h(sn )]]]],

(2.10.5)

where ds

G(v) -

s + w(s)

,

v > 0, v0 > 0,

(2.10.6)

vO

G -1 is the inverse of G and-i ~ R+ is chosen so that G(c) + m[t, r, [ f (Sn) + h(sn)]] 6 Dom(G -1), for all t ~ R+ lying in the interval 0 < t < -i. D

Proofs of Theorems 2.10.1 and 2.10.2: We will give the details for the proof of Theorem 2.10.2 only; the proof of Theorem 2.10.1 can be completed similarly. Assume that c > 0 and define a function z(t) by the fight-hand side of (2.10.4). From the hypotheses, it is easy to see that w(u) is monotonically increasing on (0, ~ ) . From the definition of z(t) and using the fact that u(t) < z(t) we observe that

Lnz(t) < f (t)(z(t) + M[t, r, h(sn)W(Z(Sn))]), where Li, i -

(2.10.7)

O, 1, 2 . . . . , n are the operators as defined in Section 1.9.

Define a function m(t) by

m(t) = z(t) + M[t, r, h(sn)W(Z(Sn))].

(2.10.8)

From (2.10.8), (2.10.7) and the fact that z(t) < m(t) we observe that

Lnm(t) - Lnz(t) + h(t)w(z(t)) < f ( t ) m ( t ) + h(t)w(m(t)) < [ f ( t ) + h(t)](m(t) + w(m(t))).

(2.10.9)

NONLINEAR INTEGRAL INEQUALITIES I From

(2.10.9)

and

using

the

facts

that

m(t)>0,

183

Ln_lm(t)>O,

(d/dt)(m(t) + w(m(t))) > 0 for t 6 R+, it is easy to observe that Lnm(t) (Ln_lm(t))(d/dt)(m(t) + w(m(t))) < [ f (t) + h(t)] + m(t) + w(m(t)) (m(t) + w(m(t))) 2 ' i.e.

d ( Ln_lm(t) ) < [ f ( t ) + h(t)]. dt m(t) + w(m(t)) -

--

(2.10.10)

By setting t --Sn in (2.10.10) and integrating it from 0 to t and using the fact that Li_lm(O) - - 0 for i - 2, 3, . . . , n, we obtain t

(d/dt)Ln_2m(t) < rn-l(t) / [ f (Sn) -q- h(sn)]dsn. m(t) + w(m(t)) J o

(2.10.11)

Again, as above, from (2.10.11) we observe that t

--d ( L n - 2 m ( t ) ) dt m(t) + w(m(t))

< rn-l(t) f [f (Sn) -+-h(sn)]dsn. o

(2.10.12)

By setting t = s~-i in (2.10.12) and integrating it from 0 to t and using the fact that Li_2m(O) --- 0 for i -- 3, 4 . . . . . n, we get t

Sn-1

Ln-2m(t) < f r n - l ( S n - 1 ) f[f(Sn)-+-h(sn)]dsndSn_l. m(t) + w(m(t)) o o Continuing in this way we obtain t

$2

Sn-2

(d/dt)m(t) 0 and define a function m(t) by

(2.10.21)

m(t) -- c + M[t, p, h(tn )W(U(tn ))].

Then (2.10.18) can be restated as (2.10.22)

u(t) _ O, Pi > Of o r i = 1, 2 , . . . , n, be continuous functions defined on R+ and c > 0 is a constant and let w(u) be as in Theorem 2.10.1. I f

U(t) < C-I- ~__ k=l

(i "i pl(S1)

P2($2) . . .

0

"i'

Pk-l(Sk-1)

0

i,jSk-1

x __ p k ( s k ) f k ( s k ) w ( u ( s k ) ) d s k d s k - 1 . . ,

ds2dsl

I

,

0

[

f o r t E R+ with so = t, then f o r 0 1,

(2.11.1)

1 t

u ( t ) = Cl + c

1)-

(t -

f (t - s)f (s, u(s),

t > 1.

(2.11.2)

1

By (ii) and (2.11.1), (2.11.2) we obtain that .,

t

t

lut(t)l l.

1

Define a function x(t) by the right-hand side of (2.11.4). Then by (2.11.3) and (2.11.4) we obtain that t

t

x(t) < a(t) + f hl (s)g(x(s))ds + f h2(s)x(s)ds, 1

t >_ 1,

(2.11.5)

1

where

a(t) =

1 + ICll + Ic21 + f ha(s) ds,

t>l.

1

Fix T > 1. By (2.11.5) we obtain that t

t

x(t) < a(T) + f hi (s)g(x(s))ds + f h2(s)x(s)ds, 1

1

l0,

1

then limx__,~ G ( x ) = cx~. Then for every solution u(t) of (A) we have that u(t) = at + b + o(t) as t --+ cx~ where a, b are real constants.

D The following Corollary of Theorem 2.11.1 given in Tong (1982) is interesting in itself. C o r o l l a r y 2.11.1 Let f ( t , u) be continuous on D = {(t, u ) : t > 1, u 6 R}. Suppose there exist ~, u ~ C(R+,R+), w nondecreasing on R+, w(x) > 0 for x > 0, such that t>l,

If(t,u)[ < ~ ( t ) w ( ~ ) , and

/

oo

/as

u6R,

oo

~ ( s ) d s < c~,

1

w(s)

~

(X).

1

Then if u(t) is a solution of (B) we have that u(t) = at + b + o(t) as t --+ c~ where a, b are constants. A

192


2.11.2. Perturbed Integro-differential Equations This section is devoted to the study of asymptotic behaviour of the solutions of nonlinear integro-differenfial systems of the form t

y ' - - A ( t ) y + f ( t , y ) + f k(t, s, y(s)) ds,

(P)

to

where y, f , k are real n-vectors, A(t) is a continuous n x n matrix, whose real-valued components are defined and continuous for t ~ Rto = [to, c~), and f (t, y), k(t, s, y) are continuous functions defined for t, s ~ Rto and y e R n, R n the n dimensional Euclidean space. The norm of a matrix or a vector is defined as the sum of the absolute value of each of the components and is denoted by I1" II. Many papers have been devoted to a discussion of the asymptotic relationships between the solutions of (P) when the integral term is absent and those of the linear system

x' = A(t)x,

t ~ Rto.

(L)

Excellent accounts of this subject may be found in a number of standard texts on ordinary differential equations (Bellman, 1953; Coddington and Levinson, 1955; Halanay, 1966; Hartman, 1964; Lakshmikantham and Leela, 1969; Sansone and Conti, 1964) and a large number of papers appeared in the literature (Bihari, 1957; Brauer, 1964; Brauer and Wong, 1969; Hallam, 1969, 1970; Morchalo, 1984; Pachpatte, 1973b, 1974b, 1975f; Strauss and Yorke, 1967; Yang, 1984). We shall denote by X(t) the fundamental matrix of (L) which satisfies the initial condition X(to) = I, where I is the identity matrix. Let A(t) be an n x n matrix and ct(t) a positive function. We say that the systems (L) and (P) are asymptotically equivalent if, corresponding to each solution x(t) of system (L), there exists a solution y(t) of (P) with the property that

IlA(t)[x(t)- y(t)]ll = o(ot(t))

(t --+ c~),

(2.11.7)

and conversely, to each solution y(t) of (P) there corresponds a solution x(t) of (L) such that (2.11.7) holds. We shall now give the following result established by Talpalaru (1973).

193


Theorem 2.11.3 Suppose that the following conditions are satisfied:

(i) there exists a nonsingular continuous matrix A (t) for t e Rto such that IlA(t)X(t)l] < t~(t),

(2.11.8)

where or(t) is a positive and continuous function for t e Rto, (ii) there exists a continuous nonnegative function a(t) such that ]lx-l(t)f(t'Y)]]- 0 and u

f2(u)-

w(s)

-~ c~,

for u ~ oo,

(2.11.10)

uo

(iii) there exists a continuous, nonnegative function h(t, s) defined for to < s < t < c~ such that ][x-l(t)k(t's'Y)l[- 0 are constants. Suppose that the functions f and k in (H) satisfy t 6 R+,

If(t, y, z)l _ 0 there exists a constant N, which may depend on e such that Iz(t)l ~ N exp(~t),

t 6 R+.

The following result demonstrates that all the solutions of (H) grow more slowly than any positive exponential. Theorem 2.11.8 Suppose that IqS(t)l ~ c exp(/~t),

~ i=1

(2.11.32)

t 6 R+,

[dpi(t)cri(s)l < M exp(/~(t - s)),

0<s 0 is a constant, such that

(I) there is a map A

JUT

for n E U x V , a, a + b E A, and each eigenvalue of A + I A has a negative real part, where 11 . 11 denotes a convenient vector norm of an element in A,

(II) there is a positive constant 6 such that for n,n

+ o E U x V ,a E A.

We observe that this class of evolution equations is larger than that considered by Morro (1982) because it contains all functions of the form w(llbll) = Sllbll and also others (for example w(llbl1) = llbll llb112). Let the hidden variables, as in Morro (1982), be a, a b E A corresponding to the paths n,n o.We have

+ +

+

Denoting

+

Y = f ( n w, a r = f ( n ,a

+ b ) - f ( n ,a + b),

+ b ) - f(n, a) - Ab,


205

we obtain that

A = y +r +

Ab.

We find that

and an integration yields

Considering (I) and (11) and denoting by -m the real part of the eigenvalue of A with the greatest real part, it follows that

t

Let us assume that v ( t ) = IIb(t)Il,

k ( t ) = exp(-mt),

+

g(t> = exp(-m(t - t~>>llb(to>ll6

h(t) = exp(mt),

1

to

exp(-m(t - s))Ilw(s)ll ds.

Now an application of Theorem 2.3.3 due to Constantin (1990a,b) yields

206


where G , G-' are defined as in Theorem 2.3.3. The bound obtained in (2.11.38) can be used in studying the independence property of the hidden variables of the present value of the physical variables (Morro (1982)).

2.12 Miscellaneous Inequalities

2.12.1 Rogers (1974) Let k(t, s, z ) be a continuous Rn-valued function defined on [0, 11 x [0, 11 x R", nondecreasing in z componentwise for fixed t , s, and is uniformly Lipschitz in the last variable, and let g (t, r ) be a continuous Rn-valued function on [0, 11 x R" nondecreasing in r componentwise for fixed t. Suppose z ( t ) is a continuous solution of the integral inequality

then there is a in [0, 13 such that

and p ( t ) is the maximal solution on [0, a) of the integral equation r(t)=

where

J 0

L(t, s, r(s))ds,

L(t, s, r ) = k(t, s, g(s, r ) ) .

2.12.2 Pachpatte (19753) Let w1 E C[Z x R+, R+], w2 E C[Z x Z x R+, R,] and assume that w2(t, s, r ) is nondecreasing in r for each t , s E I , where Z = [to, co),to 2 0. Suppose that r ( t ) is the solution of the integro-differential equation

207


existing to the right of to, where +(t) 2 0 is a continuous function defined on I . Let m(t) 2 0 be a continuous function on I such that m(t0) 5 ro and

+

m’(t) i w1 ( t , m ( t ) )

s

w2(t, s,

fn

where m’(t) = limh+o+ sup(l/h)[m(t

m(s)

+ +(s) eaS)ds,

+ h ) - m(t)].Then

m(t) 5 r ( t ) , t

E R+.

2.12.3 Willett and Wong (1965) Let u, f and g be nonnegative continuous functions defined on R+, c > 0, p > 0, p # 1 be constants and t

t

0

(i) If 0 < p < 1, then t

xexp

(

-(1 - p )

/ )] f(a)da

ds

0

’’(’-’),

t E R+.

(ii) If 1 < p < 00, then

for t E [0, a),where a is the suprimurn of t E R+ for which the expression between [. . .] in (2.12.1) is positive. Moreover, if we assume that the expression between [. . .] in (2.12.1) is positive for all t E R+, then the inequality (2.12.1) remains valid for all t E R + .

208


2.12.4 Pachpatte (1974a) Let u, f , g and h be nonnegative continuous functions defined on R+ and c > 0, 0 0 is a constant and

0

for t E R+. Then


209

for t E [0, a),where

and (11= sup{t E R+: E ( t ) > O}. Moreover, if we assume that E ( t ) > 0 for all t E R+, then the inequality (2.12.2) remains valid for all t E R+.

2.12.6 Pachpatte (1974d)

Let u, f,g and h be nonnegative continuous functions defined on R+. Let H (u)be a continuous nondecreasing subadditive function defined on R+ and H ( u ) > 0 on ( 0 , ~ ) If .

\

S

for t E R+, then for 0 5 t 5 t,

where t

/

P

r

(2.12.3)

210


G-' is the inverse of G, and t

E R+

is chosen so that

2.12.7 Pachpatte (1976c) Let u and f be nonnegative continuous functions defined on R+, n ( t ) 2 1 is a monotonic nondecreasing continuous function defined on R+, and H E $ , where is the class of functions defined in Section 2.5. If

7, for t E R+ then for 0 5 t I

where

G is defined by (2.12.3) and G-' is the inverse of G, and t so that

E

(2.12.4) R+ is chosen

f(o)do E Dom(G-I), 0

for all t E R+ lying in the interval 0 I t 5 2.

2.12.8 Pachpatte (1976~) Let u, f and h be nonnegative continuous functions defined on R+, H E g,where $ is the class of functions defined in Section 2.5, w(u) be a continuous nondecreasing subadditive and submultiplicativefunction defined

211


on R+ and w(u) > 0 on (0, 00). Let p(t) 3 1, @(t)2 0 be continuous nondecreasing functions defined on R+ and @(O) = 0. If

for t

E

R+, then for 0 5 t 5 7,

where b(t) is defined by (2.12.4) in which G and G-' are as defined therein and (2.12.5) ro

I+-'is the inverse of $ and 1 E R+ is chosen so that 1

f(s)ds

E

Dom(G-I),

0

h(s)w(b(s)p(s)) ds for all t

E

h(s)w(b(s)) ds E Dam($-'),

R+ lying in the interval 0 I tI 7.

2.12.9 Pachpatte (1976c) Let u, f and h be nonnegative continuous functions defined on R+, H E g, where eF is the class of functions defined in Section 2.5, w(t, u ) be a nonnegative continuous monotonic nondecreasing function in u 2 0 for each fixed

212


t E R+, the functions p ( t ) 2 1, $(t) 2 0 be continuous and nondecreasing on R+, 4(0) = 0. If

for t E R+, then for t E Z c R+,

+ 4(r(t))l,

u ( t ) i b(t"(t)

(2.12.6)

where b(t) is defined by (2.12.4) in which G and G-' are as defined therein and Z is the largest interval of R+ on which the right-hand side of (2.12.6) exists, and r(t) is the maximal solution of

+ 4(r(t))l),

r'(t>= h(t)w(t,b ( t ) [ p ( t )

r(O) = 0,

existing on R+.

2.12.10 Pachpatte (1976g) Let u, f , g , h, w, p and 4 be as in Theorem 2.8.6 and a 2 1 be a constant. If

for t E R+, then for 0 5 t 5 t,

where (2.12.7)

NONLINEAR INTEGRAL INEQUALITIES I in which

e(t) --

exp

213

(j)

h(s)g p (s) ds ,

-

o

1~,

l~r-1 are as in Theorem 2.8.6 and 7 6 R+ is chosen so that

(0))

f (s)w(Q(s)p(s)) ds +

J

f (s)w(Q(s)) ds E D o m ( ~ - l ) ,

o

for all t 6 R+ lying in the interval 0 < t < t.

2.12.11 Pachpatte (1976g) Let u, f , g, h, p, ~b and constant. If

w(t, u)

be as in Theorem 2.8.7 and a > 1 be a

1/a u(t) 0,


216

where

Q2(t) - exp ( o~

f(s)(E1 (S)) -1

exp

(2.12.11)

for t ~ R+.

2.12.16 Pachpatte (1976b) Let u, f , g, h, w, p and ~p be as in Theorem 2.8.6. If

t

( /

u(t) < A(t) + (1/A(t)) f f (s)u(s) u(s)+ 0

for t 6 R+, where

)

g(cr)u(~r)dcr

ds,

0

A(t) is defined

by (2.12.9), then for 0 < t < ~,

])]

t

+ / h(s)w(Q2(s))ds

,

0

where Q2(t) is defined by (2.12.11), ap, ~f-1 are as defined in Theorem 2.8.6 and ~ 6 R+ is chosen so that

)/

h(s)w(Q2(s)p(s))ds + h(s)w(Q2(s))ds ~ Dom(ap -1),

(0J

0

for all t E R+ lying in the interval 0 < t < t.

2.12.17 Pachpatte (1976b) Let u, f , g, h, p, 4) and

w(t, u) be

as in Theorem 2.8.7. If

u(t) < B(t) + (1/B(t)) f f (s)u(s) (u(s) + fo g(a)u(cr)dcr)


217

for t ~ R+, where B(t) is defined by (2.12.10), then

u(t) < Q2(t)[p(t) 4-q~(r(t))],

t E R+,

where Q2(t) is defined by (2.12.11), and r(t) is the maximal solution of

r'(t) -- h(t)w(t, Q2(t)[p(t) + q~(r(t))])

r(O) = O,

existing on R+.

2.12.18 Pachpatte (1977b) Let u, f and g be nonnegative continuous functions defined on R+, and n (t) be a positive monotonic nondecreasing continuous function defined on R+. If t

u(t) O, 0

for t E R+, then

u(t) < Q3(t)n(t),

t E R+,

where Q3(t) = f (t)(E(t)) -1/2 for t 6 R+.

2.12.19 Pachpatte (1978c) Let u, u' and b be nonnegative continuous functions defined on R+ and t

u'(t) < a + / b(s)u(s)(u(s) + u'(s)) ds, o

t ER+,

218


where a is a positive constant. Then t

u'(t) 0}. Moreover, if we assume that E4 (t) > 0 for all t ~ R+, then the inequality (2.12.13) remains valid for all t ~ R+.

2.13 Notes The material included in Section 2.2 contains some basic comparison theorems used for studying the behaviour of solutions of a system of


219

differential equations. Theorems 2.2.1 and 2.2.2 in their present forms are taken from Beesack (1975); see also Bainov and Simeonov (1992). Theorem 2.2.3 is taken from Brauer (1963). Theorem 2.2.4 is due to Opial (1960) and Theorem 2.2.5 is taken from Lasota et al. (1971). The literature concerning the comparison theorems and their applications is particularly rich, and the reader is referred to the standard references (Lakshmikantham and Leela, 1969; Szarski, 1965; Walter, 1970). Section 2.3 contains the basic nonlinear integral inequalities which provides explicit bounds on the unknown functions, which are very effective in the development of the theory of systems of ordinary differential equations. Theorem 2.3.1 is due to B ihari (1956). The special version of Bihari's inequality was also obtained independently by LaSalle (1949). Theorem 2.3.2 is due to Langenhop (1960). Theorems 2.3.3 and 2.3.4 are taken from Constantin (1990b). The results given in Sections 2.4 and 2.5 are the further generalizations of the Gronwall-Bellman and Bihari inequalities mainly developed in order to apply them in certain applications. Theorems 2.4.1-2.4.3 are taken from Pachpatte (1975a). Theorem 2.4.4 is due to Constantin (1990b) and Theorem 2.4.5 can be found in Talpalaru (1973). Theorems 2.5.1 and 2.5.2 are due to Constantin (1992). Theorem 2.5.3 is due to Beesack (1984a) and Theorems 2.5.4 and 2.5.5 are taken from Beesack (1975, 1977) which basically involves the comparison principle. Section 2.6 contains some useful inequalities with nonlinearities in integrals. Theorem 2.6.1 is due to Willett and Wong (1965) which is more convenient in certain applications. Theorems 2.6.2 and 2.6.3 are due to Gollwitzer (1969). Theorems 2.6.4 and 2.6.5 are taken from Pachpatte (1984) which are further generalizations of the Greene's inequality (1977). Theorems 2.6.6 and 2.6.7 are due to Pachpatte (1975c,d,h). Section 2.7 deals with some specific nonlinear integral inequalities established by Pachpatte (1973a, 1976b, 1977a,c,d, 1978b), which can be used as ready and powerful tools in certain applications. Section 2.8 is devoted to the Bihari-like integral inequalities developed by Pachpatte (1974d, 1975a,b,h, 1976c), which can be used as ready tools in the study of more general versions of certain integral and integro-differential equations. The results in Section 2.9 deal with the integro-differential inequalities developed by Pachpatte (1977b, 1981a) which are very effective in

220


the study of certain differential and integro-differential equations. Section 2.10 is devoted to the inequalities involving several iterated integrals and investigated by Pachpatte in (in press e,j) which can be used as powerful tools in the study af certain higher order differential and integro-differential equations. Section 2.11 is devoted to the applications of certain inequalities in studying the qualitative properties of solutions of certain differential and integro-differential equations. Section 2.12 contains some miscellaneous inequalities which can also be used in some applications.

Chapter Three Nonlinear Integral Inequalities II 3.1 Introduction The fundamental role played by the integral inequalities in the development of the theory of differential and integral equations is well known. In the literature there are many papers on integral inequalities and their applications in the theory of differential and integral equations. Although stimulating research work related to integral inequalities used in the theory of differential and integral equations has been undertaken in the past few years, it appears that there are certain classes of differential and integral equations for which the earlier results on such inequalities do not apply directly. Motivated by this fact, various investigators have discovered some useful integral inequalities in order to achieve a diversity of desired goals. This chapter deals with some basic nonlinear integral inequalities which can be used as handy tools in the study of certain new classes of differential and integral equations. Some immediate applications and miscellaneous inequalities which can be used in some new applications are also presented.

3.2 Dragomir's Inequalities The integral inequalities which provide explicit bounds on unknown functions have proved to be very useful in the study of qualitative properties of the solutions of differential and integral equations. Dragomir (1987a,b) 221

NONLINEAR INTEGRAL INEQUALITIES II

222

obtained some generalizations of the well-known Gronwall-Bellman inequality which can be used as convenient tools in applications. In this section we shall give the inequalities established by Dragomir (1987a,b) and their subsequent generalizations obtained by Dragomir (1992). The main result given by Dragomir (1987a,b) is embodied in the following theorem. Theorem 3.2.1 Let u, a, b: I = [or, ~] --+ R+ be continuous functions. Let L : I • R+ --+ R+ be a continuous function such that 0 < L(t, x ) - L(t, y) < M(t, y ) ( x - y),

(L)

for t ~ I and x > y > O, where M" I • R+ --+ R+ be a continuous function. If t

(3.2.1)

u(t) < a(t) + b(t) / L(s, u(s)) ds, Ol

for t ~ I, then t

u(t) < a(t) + b(t) / L(s, a(s))exp

M(cr, a(cr))b(cr)dcr

ds,

(3.2.2)

Ol

for t E I. D Proof: Define a function z(t) by t

z(t) = f L(s, u(s)) ds,

t 6 I.

(3.2.3)

Ol

From (3.2.3), the fact that u ( t ) < a ( t ) + b(t)z(t) and the condition (L) it follows that z'(t) - L(t, u(t))

< L(t, a(t) + b(t)z(t)) -- L(t, a(t) + b(t)z(t)) - L(t, a(t)) + L(t, a(t)) < M(t, a(t))b(t)z(t) + L(t, a(t)).

(3.2.4)

223

NONLINEAR INTEGRAL INEQUALITIES II The inequality (3.2.4) implies the estimate

z(t) < f L(s, a(s))exp

M(tr, a(tr))b(cr)dcr ds.

(3.2.5)

Ol

Using (3.2.5) in

u(t)< a(t)+ b(t)z(t), we get the desired inequality in

(3.2.2). II The following two corollaries are evident by the above theorem. C o r o l l a r y 3.2.1 Let u, a and b be as defined in Theorem 3.2.1. Let G" I • R+ --+ R+ be a continuous function such that

0 < G(t, x ) - G(t, y) < N ( t ) ( x - y),

(G)

for t 6 I and x > y > 0, where N" I --+ R+ is a continuous function. If t

(3.2.6)

u(t) < a(t) + b(t) f G(s, u(s)) ds, Ol

for t 6 I, then

u(t) < a(t) + b(t) f G(s, a(s)) exp

N(a)b(tr) da

ds,

(3.2.7)

Ol

for t 6 I. A C o r o l l a r y 3.2.2 Let u, a, b,

H'R+ --~ R+ be

c ' I -+ R+ be continuous functions. Let

a continuous function such that

0 < H ( x ) - H (y) < Q ( x - y),

(H)

for t 6 I and x > y > 0, where Q is a nonnegative constant. If t

u(t) < a(t) + b(t) f c(s)H(u(s))ds, Ol

(3.2.8)

224


for t 6 I, Then

t

u(t) < a(t) + b(t) f_ c(s)H(a(s))exp

( Q / c(o")b(o")do") ds,

Ol

(3.2.9)

S

for t ~ I. A R e m a r k 3.2.1 By taking H ( u ) = u in Corollary 3.2.2, we get the inequality which in turn is a further generalization of the well known Gronwall-Bellman inequality. A Dragomir (1992) has given the following generalization of his earlier result in Theorem 3.2.1. Theorem 3.2.2 Let u, a, b: I ~ R+ be continuous functions. Let L: I x R+ ~ R+ be a continuous function and O:R+ ~ R+ be a continuous and strictly increasing function with ~(0) = 0 and

0 < L ( t , x ) - L(t, y) < M(t, y ) O - l ( x - y),

(Lo)

for t ~ I and x > y > O, where M" I • R+ ~ R+ is a continuous function and 7t-1 is the inverse of 7t. If u(t) < a(t) + Tt (b(t) / L(s, u(s)) ds) ,

(3.2.10)

for t ~ I, then

(/

u(t) < a(t) + 7t b(t)

L(s, a(s))exp

))

M(o", a(o"))b(o")do"

Ol

ds

,

(3.2.11)

for t ~ I. D Proof: Define a function z(t) by

t

z(t) -- / L(s, u(s)) ds. Ol

(3.2.12)

225


From (3.2.12), the fact that u(t) < a(t) + 7z(b(t)z(t)) and the condition (L~) it follows that z'(t) - L(t, u(t))

< L(t, a(t) + ~(b(t)z(t))) -- L(t, a(t) + q/(b(t)z(t))) - L(t, a(t)) + L(t, a(t)) < M ( t , a(t))ap -1 (~(b(t)z(t))) + L(t, a(t))

(3.2.13)

-- M(t, a(t))b(t)z(t) + L(t, a(t)). The inequality (3.2.13) implies the estimate

t

z(t) < f L(s, a(s)) exp

M(tr, a(tr))b(tr) dtr

ds.

(3.2.14)

Using (3.2.14) in u(t) < a ( t ) + 7z(b(t)z(t)) we get the desired inequality in (3.2.11). II The following two corollaries follow obviously from the above theorem. C o r o l l a r y 3.2.3 Let u, a, b and ap be as in Theorem 3.2.2. Let G" I x R+ ~ R+ be a continuous function such that (G~)

0 < G(t, x ) - G(t, y) < N(t)ap -1 ( x - y), for t 6 I and x > y > 0 and N" I ~ R+ is a continuous function. If

( j

u(t) < a(t) + 7z b(t)

G(s, u(s)) ds

)

(3.2.15)

,

Ol

for t ~ I, then

u(t) < a(t) + ~/

(j b(t)

G(s, a(s)) exp

N(~)b(cr) &r

ds

,

Ol

(3.2.16) for t ~ I. A

226


C o r o l l a r y 3.2.4 Let u, a, b and 7t be as defined in Theorem 3.2.2 and c" I --+ R+ is a continuous function. Let H" R+ --+ R+ be a continuous function such that

0 < H ( x ) - H(y) < O~r-1 ( x - y),

(H~)

for x > y > 0, where Q _> 0 is a constant. If

u(t) < a(t) + ~

(j b(t)

Ol

for t ~ I, then

u(t) 0 for u > O. Let w(u) be a continuous, subadditive, submultiplicative and nondecreasing function defined on R+ and w(u) > 0 for u > O. If u(t) < a(t) + b(t)G ( ~ L(s, w(u(s))) ds) ~

,

(3.3.9)

v

for t ~ R+, thenfor 0 < t < t, u(t) < a(t) + b(t)G (f2-1 [f2 ( fo L(s, w(a(s))) ds)

+ ~0 M(s, w(a(s)))w(b(s))ds] ) ,

(3.3.10)

230


where f2(r) --

w(G(s))'

r > 0, r0 > 0,

(3.3.11)

r0

~ - 1 is the inverse of ~2 and-i ~ R+ is chosen so that

f2

L(s, w(a(s))) ds

+

M(s, w(a(s)))w(b(s)) ds E Dom(f2 -1), 0

for all t ~ R+ lying in the interval 0 < t < t. V1 Proof: Define a function z(t) by t

z(t) = f L(s, w(u(s))) ds.

(3.3.12)

0

From (3.3.12) and using the fact that u(t) < a(t) + b(t)G(z(t)) and the conditions on the functions w and L we observe that t

z(t) < f L(s, w(a(s) + b(s)G(z(s)))) ds 0 t

< f L(s, w(a(s)) + w(b(s))w(G(z(s))))ds 0 t

-- f [L(s, w(a(s)) + w(b(s))w(G(z(s)))) - L(s, w(a(s))) 0

+ L(s, w(a(s)))] ds t

t

0 is a constant, then t

u(t) 0 and define a function z(t) by the fight-hand side of (3.4.1). Then z ( O ) - c 2, u(t) < ~/z(t) and

z'(t) - 2 f (t)u(t) < 2f(t)v/~-).

(3.4.3)

(z(t))-l/2z' (t) < 2 f (t).

(3.4.4)

From (3.4.3) we have


234

By setting t -- s in (3.4.4) and integrating it from 0 to t we get t

V/~

< c -+-f f (s) ds.

(3.4.5)

0

The desired inequality in (3.4.2) follows by using (3.4.5) in u(t) < 4~i-). If c > 0, we carry out the above procedure with c + E instead of c, where > 0 is an arbitrary small constant, and subsequently pass to the limit as E ~ 0 to obtain (3.4.2). II R e m a r k 3.4.1 It appears that this inequality was first given by OuIang (1957) while studying the boundedness of solutions of certain secondorder differential equations. In the past few years this inequality has been applied with considerable success to study the global existence, uniqueness, stability and other properties of the solutions of various nonlinear differential equations (Barbu, 1985; Brezis, 1973; Corduneanu, 1991; Haraux, 1981; Ikehata and Okazawa, 1990; Olekhnik, 1972; Tsutsumi and Fukuda, 1980). An interesting feature of this inequality lies in its fruitful utilization to the situations for which the other available inequalities do not apply directly. A Dafermos (1979) used the following inequality while attempting to establish a different connection between stability and the second law of thermodynamics. Theorem 3.4.2 Assume that the nonnegative functions u(t) ~ L~[0, s],

g(t)

~

L I[o, s] satisfy the inequality o"

U2(Cr) ~ M2u2(O) +

f [2~u2(t)+ 2Ng(t)u(t)] dt,

(3.4.6)

0

for cr ~ [0, s], where or, M and N are nonnegative constants. Then s

u(s) < Me~Su(O) + Ne ~s / g(t) dt.

(3.4.7)

0

V1

235


The proof of this theorem is straightforward in view of the proof of Theorem 3.4.1 given above. The details are omitted here. Dafermos (1979) also used the following more general version of the above inequality in the proof of one of his main results. Theorem 3.4.3 Assume that the nonnegative functions u(t) ~ L~[0, s] and g(t) ~ LI[0, s] satisfy the inequality

f[(2• t7

U2(t7) < M2u2(0) -'l--

+

4fla)u2(t) + 2Ng(t)u(t)] dt,

(3.4.8)

0

for a ~ [0, s], where fl, y, M and N are nonnegative constants. Then s

u(s) < M exp(ots + fls 2)u(0) --[-N exp(ots +

fls 2) f

g(t) dt,

(3.4.9)

0

where ot -- y + fl/y. 89

Proof: We define a nonnegative function z(a) by

ft(2y (7

Z2(O") -- M2u2(0) -[-

+ 4fla)u2(t) + 2Ng(t)u(t)] dt,

(3.4.10)

0

for o- ~ [0, s], and we note that or

2z(a)z' (a) -- (2?, + 4flo-)u 2(o-) + 2Ng(a)u(a) + 4fl f u 2(t)dt 0

< (2ol + 4flo-)z2 (a) + 2Ng(a)z(a).

(3.4.11)

Hence

z'(a) < (a + 2fla)z(a) + Ng(a).

(3.4.12)

Integrating the differential inequality (3.4.12) under the initial condition z(O) = Mu(O) we arrive at (3.4.9). The proof is complete. II R e m a r k 3.4.2 We note that Dafermos (1979) gave very interesting results which establish different connections between stability and the second law of thermodynamics. The proofs of the main results given in Dafermos


236

(1979) are based on the applications of the inequalities given in Theorems 3.4.2 and 3.4.3. A

3.5 Pachpatte's Inequalities II In the past few years the inequality given in Theorem 3.4.1 has been used considerably in the study of qualitative properties of the solutions of certain abstract differential, integral and partial differential equations. Inspired by the important role played by the inequalities given in Section 3.4, Pachpatte (1994c, 1995a, in press h) investigated some useful generalizations and variants of the inequalities given in Theorems 3.4.1-3.4.3, which can offer greater versatility in certain applications. In this section we shall present some fundamental inequalities established by Pachpatte (1994c, 1995a, in press h). Pachpatte (1995a) obtained the following generalization of the inequality given in Theorem 3.4.1. Theorem 3.5.1 Let u, f and g be nonnegative continuous functions defined on R+ and c be a nonnegative constant. If t

f i b

u2(t) < c 2 + 2 / [ f ( s ) u 2 ( s ) + g(s)u(s)]ds, t

(3.5.1)

/

0

for t ~ R+, then

(3.5.2) for t ~ R+, where t

p(t) -- c + f g(s) ds,

(3.5.3)

0

f o r t 6 R+.

D and define a function z(t) by the fight-hand side of (3.5.1). Then z(0) - c 2, u(t) < ~ and P r o o f : Let c > 0

z'(t) = 2[f (t)u 2(t) + g(t)u(t)]

NONLINEAR INTEGRAL INEQUALITIES II < 2~[f(t)V~

+ g(t)].

237 (3.5.4)

From (3.5.4) we have

z'(t) < 2 [ f ( t ) V / ~ + g(t)].

4705-

(3.5.5)

By taking t - s in (3.5.5) and integrating from 0 to t we have t

V~

< p(t) -t- f f (s)v/~(s) ds,

(3.5.6)

t.I

0

where p(t) is defined by (3.5.3). Since p(t) is positive and monotonic nondecreasing in t, by applying Theorem 1.3.1, we have

V~z(t) < p(t) exp ( fo f (S) ds ) .

(3.5.7)

Using (3.5.7) in u(t) < ~(-i), we get the desired inequality in (3.5.2). The proof of the case when c > 0 can be completed as mentioned in the proof of Theorem 3.4.1. I

Pachpatte (1995a) also established the following inequality. Theorem 3.5.2 Let u, f , g and c be as in Theorem 3.5.1. Let w(u) be a

continuous, nondecreasing function defined on R+ and w(u) > 0 on (0, ~). if t

u2(t) < c 2 + 2/[f(s)u(s)w(u(s)) + g(s)u(s)] ds,

(3.5.8)

t,I

0

for t ~ R+, then for 0 < t < t, u(t)

< ~2 - 1

[ / ] f2(p(t)) +

f (s) ds ,

(3.5.9)

0

where p(t) is defined by (3.5.3) and ~2(r)- f

ds w(s) '

ro

r > O, ro > O,

(3.5.10)


238

~"~-1 is the inverse of f2 and-i 9 R+ is chosen so that t

f2(p(t)) 4- f f (s) ds 9 Dom(g2 -1), 0

for all t 9 R+ lying in the interval 0 < t 0 and define a function z(t) by the fight side of (3.5.8). Then z(0) - c 2, u(t) O,ro > O,

(3.6.11)

NONLINEAR INTEGRAL INEQUALITIES II ~'2 - 1

245

is the inverse of g2 and t l E R+ is chosen so that ~2(p(t)) + ~ f (s) (~o g(a) da) ds e Dom(f2_l )

for all t e R+ lying in the interval 0 0 and define a function z(t) by the fight-hand side of (3.6.1). Then z(0) = c 2, u(t) 0 and define a function z(t) by the fight-hand side of (3.7.1). Then z ( 0 ) = c 2, u(t) < ~/~t) and

z' ( t ) < 2 x / ~ w

( t, x / ~

).

(3.7.16)

254 Differentiating ~

NONLINEAR INTEGRAL INEQUALITIES II and using (3.7.16) we have

d z'(t) dt ( x / ~ ) - 2.v/~-~ < w(t, v / ~ ) .

(3.7.17)

Now a suitable application of Theorem 2.2.2 to (3.7.17) and (3.7.3) yields

X~

< r(t),

t e R+,

(3.7.18)

where r(t) is the maximal solution of (3.7.3). Now by using (3.7.18) in u(t) < ~ we get the desired inequality in (3.7.2). The proof of the case when c > 0 can be completed as mentioned in the proof of Theorem 3.4.1. (a2) By assuming that c > 0, defining a function z(t) by the fight-hand side of (3.7.4) and following the same steps as in the proof of (al) we have d dt(V~)

_< f ( t ) x / ~ + w(t, X / ~ ) .

(3.7.19)

From (3.7.19) it is easy to observe that t

t

(3.7.20) 0

0

Define a function m(t) by t

m(t) --

c +

f w(s

ds,

(3.7.21)

0

then (3.7.20) can be restated as t

X / / ~ < m(t) + / f ( s ) v / ~ ) ds.

(3.7.22)

0

Since m(t) is positive and monotonic nondecreasing for t e R+, by applying Theorem 1.3.1 yields

X~

< A(t)m(t),

t ~ R+,

(3.7.23)

where A(t) is defined by (3.7.6). From (3.7.21) and using (3.7.23) we observe that m'(t) < w(t,A(t)m(t)), (3.7.24)

255


for t 6 R+. Now a suitable application of Theorem 2.2.2 to (3.7.24) and (3.7.7) yields m(t) < r(t), t ~ R+, (3.7.25) where r(t) is the maximal solution of (3.7.7). From (3.7.23) and (3.7.25) we have

v/~

< A(t)r(t),

t ~ R+.

(3.7.26)

Now using the fact that u(t) 0 can be completed by following the limiting arguments. II Pachpatte (1995a) has proved the following inequality. Theorem 3.7.2 Let u, f and h be nonnegative continuous functions defined on R+, and Cl and c2 be nonnegative constants. If

(j

u 2(t) 0 and define a function z(t) by

Z2 ( t )

--

(j)( c2 + 2

f(s)u(s) ds

0

c2 + 2

J ) h(s)u(s) ds

0

.

(3.7.30)

256


Differentiating (3.7.30) and using the fact that

u(t) < z(t)

we observe that

z'(t) < [c2h(t) + c~f (t)] + 2 [h(t) (fo f (Cr)z(cr)dcr)

(3.7.31)

By taking t = s in (3.7.31) and integrating it from 0 to t we have

(3.7.32) where p(t) is defined by (3.7.29). Since p(t) is positive and monotone nondecreasing for t 6 R+, from (3.7.32) we observe that

< 1+ 2 p(t) -

j[ (? z.,. h(s)

f (or)p(cr) dcr + f (s)

)]

h(cr) z(cr) dcr p(cr)

ds.

o

(3.7.33) Define a function v(t) by the fight-hand side of (3.7.33). Then using the facts that z(t)/p(t) < v(t) and v(t) is monotonic nondecreasing for t ~ R+, we observe that

(3.7.34)

The inequality (3.7.34) implies the estimate

v,<exp(2Ehs(yo Now by using (3.7.35) in (3.7.33) and the fact that desired inequality in (3.7.28).

) u(t) < z(t)

(3.7.35) we get the


257

If C1, C2 are nonnegative, we carry out the above procedure with C1 -Iand c2 + E instead of Cl and c2, where E > 0 is an arbitrary small constant and subsequently pass to the limit as e --~ 0 to obtain (3.7.28).

II Pachpatte (1995d) proved the following useful inequality. Theorem 3.7.3 Let u, f and g be nonnegative continuous functions defined on R+ and Cl and c2 be positive constants. If

u(t)
O, for all t e R+, where t

t

do, 0

0

Q(t)-exp(/[clg(cr)+c2f(~r)]d~),

(3.7.38)

for t ~ R+, then u(t) < (1/H(t)) ClC2Q(t),

(3.7.39)

for t E R+. [2] P r o o f : Define a function z(t) by

z(t) -'-

(j)( Cl +

f (s)u(s) ds

0

c2 +

J ) g(s)u(s) ds

.

(3.7.40)

0

Differentiating (3.7.40) and using the facts that u(t) < z(t) and z(t) is monotonic nondecreasing for t e R+, we observe that

z'(t) < [clg(t) + c 2 f (t)]z(t) + R(t)z2(t).

(3.7.41)

The inequality (3.7.41) implies the estimate for z(t) such that

z(t) O f or u > O. If t

u2(t)

< c 2 --[- 2E[t,

a, b, u] + 2 f h(s)u(s)w(u(s))ds,

(3.7.47)

o for t e R+, then for 0 < t < -i,

u(t) O, ro > O,

w(s) '

(3.7.49)

ro

f2 -1 is the inverse of f2 and t ~ R+ is chosen so that t

f2(c) + / h(s)w(Q(s)) ds ~ Dom(f2 -1), o

for all t ~ R+ lying in the interval 0 < t < t. (b3) Let w(t,r) be a nonnegative continuous function defined for t ~ R+, 0 < r < oc, and monotonic nondecreasing with respect to r for any fixed t 6 R+. If t

u2(t)

< c 2 --[- 2E[t,

a, b, u] + 2 / u(s)w(s, u(s)) ds,

(3.7.50)

0

for t ~ R+, then u(t) < Q(t)r(t),

t ~ R+,

(3.7.51)

where Q(t) is defined by (3.7.46) and r(t) is a maximal solution of / ( t ) - w(t, Q(t)r(t)),

r(O) - c,

(3.7.52)

for t ~ R+. 77

260


Proof: It is sufficient to assume that c is positive, since a standard limiting argument can be used to treat the remaining case. (bl) Let c > 0 and define a function z(t) by the fight-hand side of (3.7.44). Differentiating z(t) and using the fact that u(t) < ~ we observe that

z
0 and defining a function z(t) by the right-hand side of (3.7.50) and the function re(t) by t

m(t) --

c +

f w(s, 0

(3.7.71)


263

and following the same steps as in the proof of (bl) we have < Q(t)m(t),

(3.7.72)

t ~ R+,

where Q(t) is defined by (3.7.46). From (3.7.71) and (3.7.72) we observe that (3.7.73)

m'(t) < w(t, O(t)m(t)).

Now a suitable application of the comparison Theorem 2.2.2 to (3.7.73) and (3.7.52) yields m(t) < r(t),

(3.7.74)

t ~ R+,

where r(t) is the maximal solution of (3.7.52). Using (3.7.74) in (3.7.72) and then the fact u(t) < 4~z-~, we get the desired inequality in (3.7.51). II The following theorem deals with the integro-differential inequalities established by Pachpatte (1996a). Theorem 3.7.5 Let u, u I and a be nonnegative continuous functions defined on I = [to, ~ ) , to > 0 with u(to) = k, a constant and c a nonnegative constant. (Cl) I f t

(u' (t)) 2 < C2 _+_2 f a(s)u'(s)(u(s) + u' (s)) ds,

(3.7.75)

to

f o r t ~ I, then

u(t) < k +

j[

c + (k + c)

to

j (J) a(r)exp

to

[1 + a(cr)] dcr

to

dr

]

ds,

(3.7.76) f o r t E I. (c2) Let L" I • R+ • R+ --+ R+ be a continuous function which satisfies the condition 0 < L(t, u, v) - L(t, ~, ~) < M(t, ~, ~)[(u - ~) + (v - ~)],

(3.7.77)

264


for t 6 I and u >-~ > O, v > ~ > O, where M" I x R+ • R+ --+ R+ is a continuous function. If t

(d (t)) 2
1, uo > 0 be constants, where I - [0, T]. If

t u(t) < uo + a /

u(s) log(a + u(s)) ds,

(3.8.1)

0 f o r t ~ I, then

hi(t) ~ (a + b/0) exp(at),

(3.8.2)

f o r t ~ I. D

Proof: Define a function z(t) by the fight-hand side of (3.8.1). Then z ( 0 ) - u0, u(t) 1 be a constant, where I - [0, T], R1 -- [1, c~). I f u(t) < c

[j 1+

a(s)u(s) log u(s) ds

0

] ,

(3.8.9)

f o r t ~ I, then

Ig(t) S r ~ s ~ > 0 0

Sn- 1

i rn-l(Sn-1) S rn(Sn) 0 0

• dsn dsn-1 ... ds3 ds2 t

$3

Sn-2

Sn- 1

+r2(t)fr3(s3)fr4(s4)...frn-l(Sn-1)/rn(Sn)

0

0

0

0

• dsn dsn-1 ... ds4 ds3 9149176

t -~- In_ 1(t) f r n (s n ) dsn 0 Jr- rn (t). Our first result contains the inequalities established by Pachpatte (1993, 1996). T h e o r e m 3.10.1 Let u > O, a > O, b > O, h > O, rj > O, j - 1, 2 , . . . ,

n - 1 be continuous functions defined on R+. Let f " R 2 --+ R+ be a continuous function which satisfies the condition 0 < f (t, x ) - f (t, y) < k(t, y ) ( x - y),

(3.10.1)

for t ~ R+ and x > y > O, where k" R 2 --+ R+ is a continuous function. (al) If u(t) O, l i m u ~ g(u) = cx~, g-1 denote the inverse function of


278

g, a(t), fl(t) be continuous and positive functions on R+ and a(t) + fl(t) - 1.

If u(t) < a(t) + b(t)g-l(M[t, r, f (tn, g(U(tn)))]),

(3.10.4)

for t ~ R+, then u(t) < a(t) + b(t)g-l(M[t, r, f (tn, a(tn)g(a(tn)ot-l(tn)))] X exp(M [t, r, k(tn, a(tn)g(a(tn)a -1 (tn)))fl(tn)g(b(tn)/3 -1 (tn))]), (3.10.5)

f o r t E R+. (a3 ) Let w(u) be a continuous, nondecreasing, subadditive and submultiplicative function defined on R+ and w(u) > 0 for u > O. If u(t) < a(t) + b(t)M[t, r, f (tn, w(U(tn)))],

(3.10.6)

for t E R+, then for 0 < t < tl, u(t) < a(t) + b(t)G -1 (G(M[t, r, f (tn, w(a(tn)))]) + M[t, r, k(tn, w(a(tn)))w(b(t,))]),

(3.10.7)

where 12

G(v) --

w(s)

v > 0, v0 > 0,

(3.10.8)

vO

G -1 is the inverse of G and t l E R+ is chosen so that G(M[t, r, f (tn, w(a(tn)))]) -k- M[t, r, k(tn, w(a(tn)))w(b(tn))] E Dom(G -1), for all t e R+ lying in the interval 0 < t < tl. Proof: (al) Define a function z(t) by

z(t) -- M[t, r, f (tn, U(tn))].

(3.10.9)

From (3.10.9) and using the fact that u(t) < a(t) + b(t)z(t), and the condition (3.10.1), we observe that

Z(t) < M[t, r, f (tn, a(tn) + b(tn)Z(tn))]


279

-- M[t, r, f (tn, a(tn))] + M[t, r, f (tn, a(tn ) + b(tn)Z(tn))] -- M[t, r, f (tn, a(tn))] < M[t, r, f (tn, a(tn))] -k- M[t, r, k(tn, a(tn ))b(tn )Z(tn )] (3.10.10)

< c(t) + M[t, r, k(tn, a(tn))b(tn)Z(tn)],

where c(t) = E + M[t, r, f (tn, a(tn))], in which E > 0 is an arbitrary small constant. Since c(t) is positive and monotonic nondecreasing in t 6 R+, from (3.10.10) we observe that

c(t)

< 1+ M

-

t, r, k(tn, a(tn))b(tn)c(tn

"

(3.10.11)

t, r, k(tn, a(tn))b(tn)c(tn )

9

(3.10.12)

Define a function m(t) by

re(t) -- 1 + M

From (3.10.12) it is easy to observe that

z(t) Lnm(t) -- k(t, a ( t ) ) b ( t ) ~

(3.10.13)

c(t)'

where Li, i - 0, 1, 2 . . . . . n are the operators as defined in Section 1.9. Using the fact that z(t)/c(t) < m(t) in (3.10.13) we have

Ln m(t) < k(t, a(t))b(t)m(t).

(3.10.14)

From (3.10.14) and using the same steps as in the proof of Theorem 1.9.1 part (bl) below the inequality (1.9.27) up to (1.9.32) we have

m(t) < exp(M[t, r, k(tn, a(tn))b(tn)]).

(3.10.15)

Using (3.10.15) in (3.10.11) we get

z(t) < c(t) exp(M[t, r, k(tn, a(tn ))b(tn )]).

(3.10.16)

Now, by letting 6--+ 0 in (3.10.16) we get

z(t) < M[t, r, f (tn, a(tn ))] exp(M[t, r, k(tn, a(tn ))b(tn )]). Now by using (3.10.17) in u ( t ) < a ( t ) + b ( t ) z ( t ) inequality in (3.10.3).

(3.10.17)

we get the desired

280

NONLINEAR INTEGRAL INEQUALITIES II (a2) Rewrite (3.10.4) as

u(t) < a(t)a(t)a-l (t) + ~(t)(b(t)~-l (t))g-l (M[t, r, f (tn, g(U(tn )))]). (3.10.18) Since g is convex, submultiplicative and monotonic, from (3.10.18) we have

g(u(t)) < t~(t)g(a(t)~ -1 (t)) + ~(t)g(b(t)~ -1 (t)) (3.10.19)

x M[t, r, f(tn, g(U(tn)))].

The estimate in (3.10.5) follows by first applying the inequality given in part (al) to (3.10.19) and then applying g-1 to both sides of the resulting inequality. (a3) Define a function z(t) by (3.10.20)

Z(t) -- M[t, r, f (tn, w(U(tn )))].

From (3.10.20) and using the fact that u(t) < a(t) + b(t)z(t) and the conditions on the functions w and f we observe that

z(t) < M[t, r, f (tn, w(a(tn ) -k- b(tn )Z(tn )))] < M[t, r, f (tn, w(a(tn )) + w(b(tn ))W(Z(tn)))] = M[t, r, f (tn, w(a(tn)))] -+-w(b(tn ))W(Z(tn ) ) ) ]

--

+

M[t, r, f (tn, w(a(tn))

M[t, r, f (tn, w(a(tn ) ) ) ]

< M[t, r, f (tn, w(a(tn)))] + M[t, r, k(tn, w(a(tn)))w(b(t.))w(z(tn))].

(3.10.21)

For an arbitrary T 6 R+, it follows from (3.10.21) that

z(t) < M[T, r, f (tn, w(a(tn)))] + M[t, r, k(tn, w(a(tn)))w(b(tn))W(Z(tn))],

0 < t < T. (3.10.22)

It is sufficient to assume that M[T, r, f ( t , w(a(tn)))] is positive, since the standard limiting argument can be used to treat the case when M[T, r, f ( t , w(a(tn)))] is nonnegative. Define a function m(t) by


281

m(t) = M[T, r, f (tn, w(a(tn)))] + M[t, r, k(tn, w(a(tn)))w(b(tn))W(Z(tn))],

0 < t < T. (3.10.23)

From (3.10.23) and using the fact that z(t) monotonic character of w, we observe that t

t2

< m(t) for 0 < t < T and the

tn-2

mt(t) y > O, where k" R 2 --+ R+ is a continuous function. If u 2 (t) < c 2 -b 2M[t, r, f (tn)U(tn)L(tn, U(tn )) -[- g(tn )U(tn )],

(3.10.45)

for t ~ R+, then u(t) < p(t) + q(t) exp(M[t, r, f (tn)k(tn, p(tn))]),

(3.10.46)

for t ~ R+, where p(t) is defined by (3.10.41) and q(t) - M[t, r, f (tn)L(tn, p(tn))],

(3.10.47)

for t ~ R+. D

Proof: It is sufficient to assume that c is positive, since the limiting argument can be used to treat the remaining case.

N O N L I N E A R I N T E G R A L INEQUALITIES II

286

(cl) Let c > 0 and define a function z(t) by the fight-hand side of (3.10.39). Then z(0) - c 2, u(t) < ~ and

LnZ(t) 0, (d/dt)z(t) > O, L~_lz(t) > 0 for t 6 R+, we observe that

Lnz(t) < 2 [ f ( t ) ~ ~ / ~ --

+ g(t)] + ~ ((d/dt)z(t))Ln-lZ(t)

(z(t))3/2

'

i.e.

d (Ln-lZ(t))

< 2[f(t)~/~

--

dt

,g~

-

+ g(t)].

(3 10.49)

By setting t - - t n in (3.10.49) and integrating it from 0 to t and using the fact that Ln-lZ(O) -- 0 for n -- 2, 3 . . . . . we obtain t

,./,(D

< am_ 1(o f tf (in ,/Z(tn + (tn dtn .

-

(3 . 1 0 . 5 0 )

o Again, as above from (3.10.50) we observe that t

d (Ln-2Z(t)) < 2rn- (t) f [f(tn)V/Z(tn) -k- g(tn)]dtn -dtk, ~ 1

(3.10.51)

0

By setting t - tn-1 in (3.10.51) and integrating from 0 to t and using the fact that Ln-zz(O) = 0, for n = 3, 4 . . . . . we get t

,/~

O, g > O, ri > O, for i = 1, 2 . . . . , n - 1, be continuous functions defined on R+ and let p > 1 be a constant. If u p (t) < c + M[t, r, g(tn)U(tn)],

(3.10.65)

for t ~ R+, where c > 0 is a constant, then u(t) 0 and define a function z(t) by the fight-hand side of (3.10.65). Then z(O) -- c, u(t) < ~

and

Lnz(t) = g(t)u(t) < g(t)(/Z~).

(3.10.67)

From (3.10.67) and using the facts that z(t)> 0, (d/dt)[Pv~-(t)] > 0 and

Ln-lZ(t) > 0 for t 6 R+, we observe that Znz(t)

[ ( d / d 0 ( P z~~)]Zn-1 z(t) [ Zp~ / ~ ] 2 '

< g(t) +

-

I nlZ t'l

ioeo -

-

dt

< g(t).

Pz4~

(3.10.68)

-

By setting t -- t~ in (3.10.68), integrating from 0 to t and using the fact that

Ln-lZ(O) - - 0 for n -- 2, 3 , . . . ,

we obtain t

Ln-lZ(t) < f g(tn)dtn ~ , 0

which implies t

(d/dt)Ln-2Z(t) < rn-l(t) f g(tn)dtn. "z./705

(3.10.69)

0

Again, as above from (3.10.69), we observe that

2z,, -

/ < rn-l(t)

-

at[

~

t

-

g(tn)dtn.

(3.10.70)

0

By setting t -- tn-1 in (3.10.70) and integrating it from 0 to t and using the fact that Ln-2z(O) = 0 for n --- 3, 4 . . . . .

Ln2Zt,/

we get

t

1 is a constant. Okrasinski (1980) studied the problem of existence and uniqueness of the solutions of the variant of equation (E) written in the form up = k , u + L , p > 1, (3.11.27) where k, L are known smooth functions depending on physical parameters and the convolution on the fight-hand side is well defined. For an interesting discussion concerning the occurrence of equation (3.11.27) in the theory of water percolation phenomena and its physical meaning, see Okrasinski (1980) and some of the references cited therein. Pachpatte (1994a) studied the boundedness and asymptotic behaviour of the solutions of equation (E)

298

N O N L I N E A R INTEGRAL INEQUALITIES II

by using the inequality investigated by himself in Theorem 3.10.4. Here the results given in Pachpatte (1994a) are presented. In what follows it is assumed that every solution u(t) of (E) under discussion exists on R+. We assume that the functions f , k and g in (E) satisfy the conditions If(t)l _< Cl,

Ik(t, s) I
0, c2 > 0, c3 > 0, ca > 0 are constants and ~ is as defined above.

301

N O N L I N E A R INTEGRAL INEQUALITIES II

3.11.4 Certain Integro-differential and Differential Equations In this section we present the applications of the inequalities established by Pachpatte (1995a, 1996a) in Theorems 3.6.1 and 3.7.5 to obtain bounds on the solutions of certain integro-differential and differential equations given in Pachpatte (1995a, 1996a). As a first application we obtain a bound on the solution of the integrodifferential equation

x' (t) - F

( j t, x(t),

k(t, s, x(s)) ds

o

)

- h(t),

x(O) = xo,

(A)

where h" R+ ~ R, k" R2+ x R ~ R, F" R+ x R 2 --~ R are continuous functions. Here we assume that the solution x(t) of (A) exists on R+. Multiplying both sides of equation (A) by x(t), substituting t = s and then integrating from 0 to t we have

x2t, +2jo

(sxs,

as (3.11.46)

We assume that

Ik(t, s, x(s))l < f (t)g(s)lx(s)[,

(3.11.47)

IF(t, x(t), v)l ~ f (t)lx(t)l + Ivl,

(3.11.48)

where f and g are real-valued nonnegative continuous functions defined on R+. From (3.11.46)-(3.11.48) we observe that Ix(t)l 2 _< 12012+ 2

j[

f(s)lx(s)l

0

( j Ix(s)l +

g(r)lx(r)l dr

0

)

-I

+ Ih(s)llx(s)l I ds.

(3.11.49)

_1

Now an application of the inequality given in Theorem 3.6.1 part (al) yields Ix(t)l < pl(t)

[ / (0r 1+

f(s)exp

0

[f(r) +g(r)]dr

ds

]

,

(3.11.50)

302


where t

Pl (t) -Ix01 + f Ih(s)l ds, 0

for t ~ R+. The inequality (3.11.50) gives the bound on the solution x(t) of equation (A) in terms of the known functions. As a second application we obtain bounds on the solutions of differential equations of the form x" = f (t, x, x'),

(P)

with the given initial conditions

x(to) - k,

x'(to) - c,

(P0)

where f : I x R x R --+ R is a continuous function and k, c are real constants, I = [to, ec), to > 0. Here we assume that the solution x(t) of (P)-(P0) exists on R+. Multiplying both sides of (P) by x'(t), substituting t = s and then integrating it from to to t and using (P0) we have t

(X' (t)) 2

--

c 2 -~

2 f x' (s)f (s, x(s), x' (s)) ds.

(3.11.51)

to

We assume that the function f in (P) satisfies the condition

If (t, x, x')l ~ a(t)(Ixl -t- Ix'l),

(3.11.52)

If (t,x,x')l ~ a(t)L(t, Ixl, Ix'l),

(3.11.53)

or

where a(t) is a real-valued nonnegative continuous function defined on I and L" I x R+ x R+ ~ R+ is a continuous function satisfying the condition

0 < L(t, u, v) - L(t,-a, ~) < M(t, ~, ~)[(u - ~) -t- (v - ~)],

(L)

for t 6 1 and u > ~ > 0 , v>~>0, where M ' I x R + x R + - + R + continuous function. From (3.11.51) and (3.11.52) we observe that

is a

t

Ix'(t)l 2 ~ Icl 2 + 2 f a(s)lx'(s)l(Ix(s)l + Ix'(s)l)ds. to

(3.11.54)


303

Now an application of Theorem 3.7.5 part (cl) yields

Ix(t)l _< Ikl +

j[

Icl + (Ikl + Icl)

to

J

a(r)exp

to

(/)

[1 + a(~)]d~

dr

to

]

ds,

(3.11.55) for t ~ I. Furthermore, from (3.11.51) and (3.11.53) we observe that t

Ix' (t)l 2 ~ Icl 2 + 2 f a(s)lx' (s)lt(s, Ix(s)l, Ix'(s)l) as.

(3.11.56)

to

Now an application of the inequality given in Theorem 3.7.5 part (c2) yields

' (/

Ix(t)l _< Ikl + Icl f exp to

[1 + a(r)M(z, Ikl, Icl)] dr

)

ds

to

+ f f a ( r ) L ( r , lkl,,cl) to to

xexp(~[l+a(~)M(~,lkl, lcl)]d~)drds,

(3.11.57)

for t ~ I. The inequalities (3.11.55) and (3.11.57) gives us the bounds on the solutions x(t) of equations (P)-(P0) in terms of the known functions.

3.12 Miscellaneous Inequalities

3.12.1 Dragomir (1987a) Let u, a, b'I--+ R+ be continuous functions, where I = [or,/3). Let D" I • R+--+ R+ be a continuous function such that D is differentiable on (~, ~) • (0, c~), (O/Ox)D(t,x) is nonnegative on (or,/3) x (0, c~) and there exists a continuous function P'I x R+ --+ R+ such that (O/Ox)D(t,u) < P(t, v) for any t ~ (c~,/~) and u > v > 0. If /.i

U(t)

t

< a(t) -t- b(t) /D(s, u(s)) ds, oJ

Ol


304

for all t ~ I, then t

u(t) < a(t) 4- b(t) f D(s, a(s)) exp

P(a, a(cr))b(cr) dcr ds,

Ol

for all t ~ I.

3.12.2 Pachpatte (in press k) Let u, a, b, c, k(t, s), (O/Ot)k(t, s), L and M be as in Theorem 3.3.4. Let g(u) be a nonnegative continuous strictly increasing convex and submultiplicative function for u > 0, limu~oo g(u) = oo, g-1 denote the inverse function of g, a(t), fl(t) be continuous and positive functions on R+ and c~(t) + fl(t) - 1. If

u(t) 0 be a constant. Let L" R 2 ~ R+ be a continuous function which satisfies the condition

0 < L(t, x ) - L(t, y) < k(t, y)(x- y), for t ~ R+ and x > y > 0, where k" R2+ ~ R+ is a continuous function.

(L)

305


(i) If t

u2(t) ~ C2 -~- 2 /

u(s)[f (s)u(s) + g(s)L(s, u(s))]

ds,

0

for t ~ R+, then

+ f g(s)L(s, c)exp

[f (a) + g(a)k(a,

c)] da

ds,

0

for t ~ R+. (ii) If t

U2 (t) < C2 q-

2 f u(s)[f (s)u(s) + g(S)U p (S)] ds, 0

for t ~ R+, where 0 1 be constants. If oo

up(t)

< cp + p

ftf(s)u~(s) + g(s)u(s)]ds, t

306


for t e R+ and f o f ( s ) d s < cx~,

u(t) < exp

f o g(s)ds < ~ , then

(7)[ f (s) ds

x exp

(

-(p-

c p-1 + (p - 1)

7

g(s)

t

1)

j

f(a)d~r

ds

s

for t E R+. (ii) Let C1 >__ 0, C2 _> 0 be constants. If

(j)(

u2 (t) O,

)]

ds,

310


for all t 6 R+, then

u(t) < exp

f (s) ds

Jg(s)L(s) )]

c 1-~ + (1 - or)

0

x exp

(

- ( 1 - or)

j

1/(1-c~)

f(cr) dtr

ds

0

for t ~ R+, where

L(t)=[1/(B(t))lm]cexp(fo[f(cr)+h(cr)]dcr) , for t ~ R+. (ii) If

( + j h(tr)u(cr)dtr)]ds, L(t) t ( j ) 1)/ g(s)L(s)

u2(t) 1 a constant and B(t),

D(t) - -

c 1-~ -

exp

(og -

0

defined in part (i), and

(or - 1)

f (tr) do-

ds > 0,

0

for t ~ R+, then

u(t) < [1/(D(t))l/(~-l)]exp(fo f (S)ds) , for t 6 R+.

3.12.10 Pachpatte (1996a) Let u, u', a and b be nonnegative continuous functions defined on I = [to, c~), to _> O, with (i) If

(u'(t)) 2

_ 0 a constant.

t ( j 2 f a(s)u'(s) u(s)+ u'(s)+ to

to

) b(r)u'(r) dr

ds,

311

NONLINEAR INTEGRAL INEQUALITIES II for t e I, then

u(t) uo>O, c>O, where

a =

minteR+if(t)

< 0

and uo, c are constants; (H5)

u ~ C([a, ~), R1), u(t) >

uo > 1, c > 1, where a, uo, c are as in (H4),

R1 --- [1, c~); (H6)

w 6 C([uo, c~), R+), nondecreasing,

w(u) >

0 on (uo, c~) and

w(uo) = 0. Consider the following delay integral inequalities t

u2(t) < c 2 + 2

+

h(s)u(cr(s))]ds,

(A)

0

+ h(s)u(tr(s))] ds,

(B)

u2(t) < c2 + 2 ~ [f (s)u2(tr(s)) (~o + h(s)u(cr(s))] ds, for t 6 [0, cr

(L)

with the condition

u(t)=~(t), t6[O,a],

~(cr(t)) 0

and o r ( t ) < 0 .

(Q) (i) The inequality (A), with (Q) and assumptions (H1), (H2), (Ha) and (H6), implies

313


u(t) < G -1

E(/ G

c +

h(s) ds o

)/ ] +

f (s) ds

,

0 < t

1, uo > 0 are constants, where I - [0, T]. If t

u(t) < uo + f f (s)u(s)[log(a + u(s)) + g(s)] ds,

0 for t 6 I, then

for t 6 I, where t

q(t) -- log(a + uo) + f f (s)g(s) ds,

0 forte/.

3.12.14 Pachpatte (in press d) Let u, b, c" I ~ R+ be continuous functions and a > 1, u0 > 0 are constants, where I - [0, T]. If

H(t)

< uo + [__ t.J

(

log(a + u(s)) +

0

/

c(r) log(a + u(r))

0

for t ~ I, then

u(t)_ 1, u0 >_ 0 be constants, where I - [0, T]. Let f (u) be the same function as in Theorem 3.9.1. If

u(t) < uo + / b(s)u(s) ( / c(r)f (log(a + u(r)))dr) ds,

0 for t 6 I, then for 0 _< t _< ~,

u,t) [exp

a]

where F, F -1 are as defined in Theorem 3.9.1 and 7 6 I is chosen so that

F(log uo) + f b(s)

c(r) d r

ds 6 D o m ( F -1),

0

for all t ~ I lying in the interval 0 < t < 7.

3.12.17 Pachpatte (1995c) Let u, a, b, pi, i - 1, 2 . . . . . n, be nonnegative

continuous functions defined

on R+. Let F[t, Pl, P2 . . . . , Pn-1, Pn] be as defined in Section 3.10.

316

NONLINEAR INTEGRAL INEQUALITIES II (i) If t

u(t) < a(t) + b(t) f F[s, Pl, P2..... Pn-1, pnU] ds, 0

for t ~ R+, then t

u(t) < a(t) + b(t) / F[s, Pl, P2, . '.,

Pn-1,

pna]

0

x e x p ( ~ F [ Z , pl, P2.... ,Pn_l, pnb]d'c) ds, for t ~ R+. (ii) Let g, g - l , ~,/~ be as in (a2) in Theorem 3.10.1. If

u(t) < a(t) + b(t)g -1 ( ~ f[s, Pl, P2.... , Pn-1, prig(u)] ds) , for t ~ R+, then

u(t) < a(t) + b(t)g -1 ( / F [ s ,

Pl, P2 . . . . ,

Pn-1, pnotg(aot-1)]

x e x p ( ~ F [ Z , pl, P2..... Pn_l, Pn~g(bfl-1)]d-r)ds) , for t 6 R+. (iii) Let w, G, G -1 be as in (a3) in Theorem 3.10.1. If t

u(t) < a(t) + b(t) / F[s, Pl, P2, . .. , Pn-1, pnW(U)] ds, 0

for t 6 R+, then for 0 < t < t l,

[ J

u(t) < a(t) + b(t)G -1 G(c(t)) +

]

F[s, Pl, P2.... , Pn-1, pnw(b)] ds ,

0


317

where t

c ( t ) - f F[s, Pl, P2 . . . . . Pn-1, pnw(a)] ds, o and t l ~ R+ is c h o s e n so that t

G(c(t)) + f F[s, Pl, P2 . . . . , pnw(b)l ds ~ D o m ( G - 1 ) , o for all t E R+ lying in the interval 0 < t < t l.

3.12.18 Pachpatte (in press e) Let u >_ O, f i >__O, Pi >_ 0 for i -- 1, 2 . . . . .

n be c o n t i n u o u s functions

defined on R+ and c >_ 0 is a constant. (i) If U2(t) < C2 _+_

(1

2 ~__. k--1

pl(S1)

Si

Pk-l(Sk-1)

p2(s2) 9 9 9

0

0

;,kiS-1

• - - pk(sk)fk(Sk)U(sk)dskdsk-1 ... d s 2 d s l

I

,

o for t 6 R+ with so -- t, then

n

bt(t) < C -~- Z k--1

pl(S1)

is p 2 ( s 2 ) . 9 9is 0

f,ikS-1

• - - pk(sk)fk(Sk)

Pk-l(Sk-1)

0

dsk

dsk-1.., ds2dsl

) ,

o for t 6 R+. (ii) Let w(u) be a c o n t i n u o u s n o n d e c r e a s i n g function defined for u > 0 and w(u) > 0 for u > 0 and w'(u) > 0 for u > 0. If u2(t) < c 2 q- 2 ~.._.. k=l

;,k-1 iS

pl(S1)

Pk-l(Sk-1)

p2(s2) 9 9 9 0

0

• - - pk(sk)fk(sk)U(Sk)W(U(Sk))dsk d s k - 1 . . , d s 2 d s l o

I

,

318


[

for t 6 R+, with so -- t, then for 0 _< t _< 7,

u(t) < G -1 G(c) +

si

pl (Sl)

p2(s2)...

k=l

is Pk-l (Sk-1)

)]

0

0

ski

x I Pk(Sk)fk(sk) dsk dsk-1..,

ds2 dsl

,

0

where

G(r)- f

ro

ds w(s) '

r>

0, ro > 0,

G -1 is the inverse of G and t 6 R+ is chosen so that

n (~o

G(c) + Z

Pl (sl)

k=l

is

p2(s2) 9 9 9

0

sk/2

Pk-1 (sk-1)

0

Sk-1

• I pk(sk)fk(Sk)dskdSk-1 ...

ds2dsl

I

E Dom(G-1),

o for all t ~ R+ lying in the interval 0 < t _< ~.

3.12.19 Pachpatte (in press j) Let

u > O, p >_O, q >_O, ri

> 0 for i -- 1, 2 . . . . , n - 1 be continuous func-

tions defined on R+, c _> 0 be a constant and n > 2 be a natural number. Let

M[t, r, h]

be as defined in Section 1.9.

(i) If

U2(t) < c2 q--2M[t, r, p(tn)U(tn)(U(tn) + M[tn, r, q(tn)U(tn)])], for t ~ R+, then

u(t) < for t 6 R+.

c[1 +

M[t, r, p(tn)exp(M[tn, r, (p(tn) + q(tn))])]],


(ii) Let

w(u) be a

319

continuous nondecreasing function defined on R+ and

w(u) > 0 for u > 0 and w'(u) > 0 for u > 0. If u2(t)

< c 2 -[-

2M[t, r, p(tn)U(tn)(U(tn) -t- M[tn, r, q(Sn)W(U(Sn))])],

for t 6 R+, then for 0 < t _< t l,

u(t) < c + M[t, r, p ( t n ) G - l [ G ( c ) -1- M[tn, r, (p(Sn) k- q(Sn))]]], where v

G(v) --

f vo

as

s + w(s)

v>0,

v0 > 0 ,

G -1 is the inverse of G and t l E R+ is chosen so that

G(c) + M[t, r, (p(Sn) -[- q(Sn))] E D o m ( G -1), for all t 6 R+ lying in the interval 0 < t < tl.

3.12.20 Pachpatte (1992) Let u" R+ ~

R1 -- [1, oo), b" R+ --+ R+, ri" R+ ~

(0, c~), i -- 1, 2 . . . . . n - 1, be continuous functions and u0 > 1 is a constant. Let w(u) be a continuously differentiable function defined on R+ and w(u) > 0 for u > 0 and w'(u) > 0 for u > 0. Let M[t, r, h] be as defined in Section 1.9. (i) If

u(t) _0 for x, y ~ R+, then u(x, y) < E(x, y) exp ( / fo c(s, t) ds dt ) ,

(4.2.2)

for x, y ~ R+, where E(x, y) -- [a(x) + b(0)][a(0) + b(y)]/[a(O) + b(0)],

(4.2.3)

for x, y ~ R+. D Proof: Define a function z(x, y) by the fight-hand side of (4.2.1). Then z(O, y) -- a(O) + b(y), z(x, O) - a(x) + b(O), u(x, y) 0, Zx(X, y) > O, Zy(X, y) >__O, we observe that

Zxy (X, y) < c(x, y)+ z(x, y)

Zx (X,

y)Zy (X, y)

z2(x, y)

i.e.

0 ((O/ax)z(x,Y)) 0 can be weakened to k > 0. If k = 0, then the result holds with k = E > 0, and the conclusion follows by letting E --+ 0. A An interesting and useful generalization of Wendroff' s inequality is given in the following theorem.

Let u(x, y), n (x, y) and c(x, y) be nonnegative continuous functions defined for x, y ~ R+, and let n (x, y) be nondecreasing in each variable x, y ~ R+. If T h e o r e m 4.2.2

x

y

u(x,y) 0 for x, y ~ R+. From (4.2.9) we

observe that

x y

u(x,y)< l + / / c ( s , t ) u ( s , t ) d s d t . n(x, y) n(s, t) 0

(4.2.11)

0

Now an application of a special version of Theorem 4.2.1 yields the required inequality in (4.2.10). If n (x, y) - 0, then from (4.2.9) we observe that x y

u(x,y) 0 are continuous functions for x, y R+, having derivatives such that a'(x) > O, b'(y) > 0 for x, y E R+, then x

y

Uxy(X,y) O, b' (y) > Ofor x, y ~ R+, and M > 0 is a constant, then

Uxy(X, y)

< E(x, y)exp ( f /o[ M + (l + M)c(s,t)]dsdt)

(4.3.34)

for x, y ~ R+, where E(x, y) is defined by (4.2.3) in Theorem 4.2.1. D Proof: Define a function z(x, y) by the right-hand side of (4.3.33). Then z(O, y) -- a(O) + b(y), z(x, O) - a(x) -t- b(O), and

Zxy(X, y) = M[uxy(X, y)+ c(x, y)(u(x, y)+ Uxy(X, Y))I.

(4.3.35)

Using the facts that Uxy(X, y) < z(x, y) from (4.3.33) and Mu(x, y) < z(x, y) from the definition of z(x, y), in (4.3.35) we see that the inequality

Zxy(X, y) < [M + (1 + M)c(x, y)]z(x, y),

334

MULTIDIMENSIONAL LINEAR INTEGRAL INEQUALITIES

is satisfied, which implies the estimate for

z(x, y)

such that

z(x, y) < E(x, y)exp "( . i j o[ M\ + (l o+ M)c(s,t)]dsdt) Now by using (4.3.36) in in (4.3.34).

Uxy(X,y) < z(x, y),

(4.3.36)

we get the required inequality

II R e m a r k 4.3.2 We note that in the special case when a(x) + b(y) - k, for x, y 6 R+, in (4.3.33), where k > 0 is a constant, the bound obtained in (4.3.34) reduces to

Uxy(X,y) < k exp ( fo fo [M + (l + M)c(s, t)] ds dt ) , for x, y 6 R+. The requirement k > 0 can be weakened to k > 0 as noted in Remark 4.2.1. A

4.4 Pachpatte's Inequalities II During the past few years some new inequalities of the Wendroff type have been developed which provide a natural and effective means for the further development of the theory of partial integro-differential and integral equations. This section presents some inequalities of the Wendroff type given by Pachpatte (1980b) and analogous inequalities which can be used in the study of qualitative properties of the solutions of certain integro-differential and integral equations. Pachpatte (1980b) established the following inequality. Theorem 4.4.1 Let u(x, y), f (x, y) and g(x, y) be nonnegative contin-

uous functions definedfor x, y ~ R+. If u(x,

( )

y) < a(x) + b(y) + _ _ f (s, t) u(s, t)

st

0 0

+fJ'g(a,O)u(a,o)dado 0 0

dsdt,

(4.4.1)


33b

for x, y ~ R+, where a(x) > O, b(y) > 0 are continuous functions for x, y R+, having derivatives such that a'(x) > O, b'(y) > 0 for x, y ~ R+, then x y

u(x,y)0, 8x

f (x, y)

~yZ(X, y) > 0

and

z(x, y) > O,

-

from (4.4.34) we observe that

0 (Zxy(X,y)) 0 8 ( 0 (Zxy_(x,y))) Ox f (x, y) -~yZ(X,y) Oy -~x \ f (x, y) < p(x, y)+ z2(x, y) Z(X, y) i.e.

a (Zxy(X,y)) 0 -~x f(x, y) Oy z(x, y)

< p(x, y). -

Now, keeping x fixed in (4.4.35), set y from 0 to y to obtain the estimate

(4.4.35)

0 and integrate with respect to r/

0 (Zxy(X,y)) y Ox f (x, y) < / p(x, O)do. z(x, y)

(4.4.36)

0

Since

Zxy(X, y)

> O,

f (x, y) -

Zx(X, y) >_0

and

z(x, y)

> O,


343

again as above from (4.4.36) we observe that

Zxy(x, Y)) 3 3x

Y

f (x, y) z(x, y)

< f p(x, rl) do. -

(4.4.37)

0

Keeping y fixed in (4.4.37), set x = a and integrate with respect to a from 0 to x to obtain the estimate x

Zxy(X,y)

< f (x,y)f f p(a,o)dado.

z(x, y) -

Since Zx(X,y) observe that

y

(4.4.38)

0 0

> O, Zy(X, y) > 0

z(x, y)

and

> 0,

from

/o / (jj)

3 3xZ(X'Y) < f(x, y) Oy z(x, y) -

p(a, ~7)da do

\0

9

(4.4.38)

we

(4.4.39)

0

Now, keeping x fixed in (4.4.39), set y = t and integrate with respect to t from 0 to y to obtain the estimate 0

-~xZ(X'Y) < j f (X, t) ( j r z(x, y) 0

\0

p(a, ri) da do) dt.

(4.4.40)

0

Keeping y fixed in (4.4.40), set x = s and integrate with respect to s from 0 to x to obtain the estimate

z(x, y) < exp ( f jo f (s, t) \o ( j j o p(a, o) da do) ds dt)

(4.4.41)

Now by using (4.4.41) in (4.4.31) we get the required inequality in (4.4.26). The proof of the case when u0 is nonnegative can be carded out as above with u0 + ~ instead of u0, where E > 0 is an arbitrary small constant and subsequently passing to the limit as ~ --+ 0 to obtain (4.4.26).

II

4.5 Pachpatte's Inequalities III In the qualitative analysis of some classes of partial differential and integro-differential equations, the bounds provided by the inequalities in

344


earlier sections are inadequate and it is necessary to seek some new Wendroff-like inequalities in order to achieve a diversity of desired goals. In this section we shall give some Wendroff-like integral inequalities established by Pachpatte (1988b, c) which can be used as ready tools in the study of certain fourth-order partial differential and integro-differential equations. First, we specify some notations and definitions which will be used in this section. The ordinary first and second derivatives of a function r(x) defined for x > 0 are denoted by r'(x) and r"(x) respectively. The partial derivatives of a function m(x, y) with respect to the variables x and y are denoted by mx(X, y) and my(X, y) or (O/Ox)m(x, y) and (O/Oy)m(x, y) respectively. The second-order partial derivatives of the function re(x, y) with respect to the variables x and y are denoted by mxx(X, y) and myy(X, y) or (02/OxE)m(x, y) and (02/Oy2)m(x, y) respectively. The other partial derivatives of re(x, y) can be denoted in the usual way. For x, y ~ R+, and some function h(x, y) defined for x, y ~ R+, we set

A[x,y,h(sl, tl)]-ffffh(sl, tl)dtldtdslds. x

0

s

0

y

0

t

0

Let m > 1, n > 1 be integers. We denote the differential operators D]' and D~' by

D~Z(X, y)

Onz(x, y) --

-

,

-

OX n

0 m2 Z ( x ,

y)

Omz(x, y) ~ Oy ,m

--

where z(x, y) is some function defined for x, y 6 R+. For x, y 6 R+, we set x Sn-1

S1 Y tin-1

B[x,y,h(s,t)]--ff ...f// 0

0

0

0

0

tl

...fh(s,t) 0

x dt dtl ... dtm-1 ds dsl ... dsn-1, where so = x and to = y. It is easy to observe that A[x, y, h] = A[y, x, h] and B[x, y, h] = B[y, x, h]. A two independent variable Wendroff-like integral inequality established by Pachpatte (1988c) is given in the following theorem. T h e o r e m 4.5.1 Let a(x), b(y), p(x) and q(y) be positive and twice

continuously differentiable functions defined for x, y ~ R+, a'(x), b'(y),


345

p'(x) and q'(y) be nonnegative for x, y ~ R+ and define c(x, y) -- a(x) + b(y) + yp(x) + xq(y) for x, y ~ R+. Let u(x, y) and h(x, y) be nonnegative continuous functions defined for x, y ~ R+ and u(x, y) < c(x, y) + A[x, y, h(sl, tl)U(S1, tl)],

(4.5.1)

for x, y ~ R+. (i) If a"(x), p"(x) are nonnegative for x ~ R+, then u(x, y) < c(O, y)exp

IX( )jj cx(O, y) c(O, y)

+

a" (Sl) + yp" (Sl) C(SII"~

dslds

0 0

+ A[x, y,

h(sl,tl)]] ,

(4.5.2)

for x, y ~ R+. (ii) If b"(y), q"(y) are nonnegative for y ~ R+, then

[ (Cy(X,O))ffb"(tl)-l-xq"(tl)_t_ dtl dt c(x, 0) C(tl, 0)

u(x, y) O, Zy(X, y) > O, z(x, y) > 0 for x, y 6 R+, we observe that y

19 ( Zxx(x, y) ) < p" (x)

oy \

yS

-c(x,

+ f h(x, tl) dtl.

(4.5.14)

0

By keeping x fixed in (4.5.14), set y - t and then integrating with respect to t from 0 to y and using (4.5.6) and (4.5.8) we have y

Zxx(X, y) < z(x, y) -

c(x, O)

t

+

h(x, tl)dtl dt.

(4.5.15)

0 0

As above, from (4.5.15) and the fact that Zx(X, y) > O, z(x, y) > 0 for x, y R+, we observe that

Zx(X,y) ) < a" (x) + yp" (x) Ox z(x, y) c(x, O)

y

t

tl d ld 0 0

(4.5.16)


347

Now, keeping y fixed in (4.5.16), we set x -- s l, and then integrating with respect to s mfrom 0 to x and using (4.5.5) and (4.5.7) we have

Zx(X, y) < cx(O' y) + f a" (sl) + yp" (sl) dsl z(x, y) - c(O, y) c(sl~-~ 0

x y t

+f/fh(sl,

tl)dtldtdsl.

(4.5.17)

0 0 0

By keeping y fixed in (4.5.17), set x - s and then integrating with respect to s from 0 to x and using (4.5.5) we obtain

z(x, y) < c(O, y)exp

[x(Cx(0' y)) s f f d 1 ( S+l ) + y p " ( S l ) d s l d

c(O, y)

+ A[x, y, h(sl, tl)l

]

r

O)

0 0

.

(4.5.18)

Using (4.5.18) in u(x, y) < z(x, y) we obtain the inequality (4.5.2). The proof of the inequality (4.5.3) follows by rewriting the definition of

z(x, y) in (4.5.4) in the form z(x, y) -- c(x, y) 4- A[y, x,

h(s1, t l ) / / ( s 1 , t l ) ] ,

and closely looking at the proof of the inequality (4.5.2) given above with suitable modifications. We omit the details.

II The following theorem deals with a more general Wendroff-like inequality established by Pachpatte (1988c). Theorem 4.5.2 Let a(x), b(y), p(x), q(y), a' (x), b' (y), p' (x), q' (y), c(x, y),

u(x, y) and h(x, y) be as in Theorem 4.5.1. Let k(x, y) be a nonnegative continuous function defined for x, y E R+ and u(x, y) < c(x, y) + A[x, y, h(s1, tl)(U(Sl, tl) + a[sa, tl, k(~rl, O1)u(crl, r/l)])], for x, y ~ R+.

(4.5.19)

348


(i) If a"(x) and p" (x) are nonnegative for t ~ R+, then x

u(x, y) < c(O, y) + XCx(O, y)+

s

f f ta"<s )+ yp"(Sl)]dsl ds 0 0

+ A[x, y, h(sl, tl)F(sl, tl)],

(4.5.20)

for x, y ~ R+, where

F(x, y)--c(O, y)+ exp

[x(Cx(O,y)) + / f \ c(O, y)

a"(s3)+yp"(s3) c(s3, O)

ds3 ds2

0 0 d- A[x, y, h(crl,

~1) -~"k(trl, 01)]] ,

(4.5.21)

for x, y ~ R+. (ii) If b" (y) and q"(y) are nonnegative for y ~ R+, then y

u(x, y) _ 0 there. Let Po(xo, Yo) and P(x, y) be two points in D such that ( x - x o ) ( y - yo) > 0 and let R be the rectangular region whose opposite corners are the points Po and P. Let v(s, t; x, y) be the solution of the characteristic initial value problem L[v] = vst - b(s, t)v - O,

v(s, y) - v(x, t) - 1,

and let D + be a connected subdomain of D which contains P and on which v > 0 (Figure 4.1). I f R C D + and u(x, y) satisfies x

y

(4.6.1)

u(x,y) O, s

t

ffB(cr, O)V(cr,O)dcrdO

,lTVll -

x

y

s

t

x

y

1

< ~ max IlV(cr, 0)11,

(4.6.15)

if we restrict s and t to be close enough to (x, y). Then IIV~+~ - V ~ I I - I I T ( W k - Vk-1)ll 1

< ~ max IIV~ - Vk-1 II < "'" < 2 -k max IIV~ - V011. Since V n + l - VO 71-~-'~=0(Vk+l- Vk), Vn+l i s the nth partial sum of a matrix series dominated in norm by a convergent geometric series, namely max IIV1 - V0 II ~ = 0 2-~. Therefore the matrix sequence {Vn } converges uniformly on the domain where (4.6.15) holds. Since each V, is continuous, the limit function V (s, t; x, y) is also. To see that V is a solution to (4.6.14), note that I + T V -I + T(lim Vn) = I + lim TVn -- lim(I + T V n ) - - lim Vn+l -- W. To see that V is unique, suppose W also a solution. Then V - W -- T ( V - W), so 1

IIv - w II - IIT(V - W)II _< ~ max IIV - W II, which is possible only if IIV -

W ll - - 0 , i.e. V -

W.

Now suppose B(s, t) > O. Then, V > O, T V > O, since (s - x)(t - y) > O. Since V0-= I > 0, it follows by induction that Vn > 0 for all n. But then the limit function satisfies V(s, t;x, y) > 0 also. II

Proof of Theorem 4.6.2: Let x

y

z(x,y)-/'fB(s,t)u(s,t)dsdt.

xo Yo

(4.6.16)


362 Then

Zxy

-- B(x, y)u(x, y) and since B > 0 and (4.6.10) holds, Zxy

--

B u < B ( a + z),

or

L[z] -- Zxy

--

(4.6.17)

Bz < Ba.

This is a hyperbolic vector partial differential inequality for z. The initial conditions for z are z(xo, y ) -

(4.6.18)

z(x, Yo) - O.

The operator L is self-adjoint. We note that for any z, v ~ C 2, v TL[z] - z rL[v] -- v rzxy -- V T B z - ZTvxy + Z r B v .

The terms here are all scalars, and since B is symmetric, the second and fourth terms on the fight cancel. The fight-hand side is -- --(zrvy)x

T -Jr- Z x Vy -I- ( v r z x ) y

T

-- VyZx

(4.6.19)

-- - - ( z r v y ) x -t- (vrzx)y.

For P0 and P as required in the hypothesis we label the directed sides and comers of the rectangle R as shown in Figure 4.4.

AC2

l

P ( x , y)

mo(xo, yo) C4 Figure 4.4

363


Using s and t as the independent variables in identity (4.6.19), integrating over R and using Green's theorem, we get

f f (vrL[z] - zrL[v])dsdt - f f (-(zrvt)s + (VrZs)t)dsdt R

R

[ vrzs ds + zrvt dt ,.I

c

=-

/

vTZsdS-- /

C2-[-C4

zTvtdt.

C1 -1-C3

(4.6.20) This holds for any functions in C 2. For any z 6 C 2 which also satisfies the initial conditions (4.6.18), z on C3 and

0

z - Zs - 0 on Ca. Thus (4.6.20) reduces to

g(vrLtzl-zrLtvldsdt=-fVrzsds-fzrvtdt. R

C2

(4.6.21)

C1

Now suppose the vectors vi(s, t;x, V(s, t; x, y) of Lemma 4.6.1. Then L [ v

y) are the columns of the matrix i] - - 0 and vi(s, y) -- vi(x, t) - - e i, the ith column of the identity matrix. Thus vti __ 0 o n C 1, s o ( 4 . 6 . 2 1 ) r e d u c e s to

viTL[z] ds dt - R

--

f

i

e Zs d s

-- zi(P),

C2

where the subscript refers to the component of the vector. Using the matrix V this becomes P P

z(P) - - / / g

r (s, t; x, y)L[z] ds dt.

R

For z defined by (4.6.16), since (4.6.17) holds and since V > 0 by Lemma 4.6.1, we get

z(P) < / / V T B a ds dt.

(4.6.22)

R

This gives an upper bound for the integral term in (4.6.10) so that (4.6.11) results. The proof is complete. II

364


The matrix V is a generalization of a scalar Riemann function, and when u, a and B are scalars it reduces to the Riemann function relative to the point P(x, y) for the operator L. Note that, by the method of proof, if equality were to hold in (4.6.10) there would have been equality in (4.6.17) and (4.6.22) regardless of the nonnegativity of V, so the fight-hand side of (4.6.11) is the solution to the Volterra integral equation corresponding to (4.6.10). Since the right-hand side of (4.6.11) is a solution of the inequality (4.6.10) and is an upper bound for all such solutions, it is the maximal solution of (4.6.10).

4.7 Generalizations of Snow's Inequalities Ghoshal and Masood (1974a,b) obtained some generalizations of Snow's inequalities given in Snow (1971, 1972) which can be used as tools in the study of various properties of the solutions of some self-adjoint partial differential equations of the hyperbolic type. This section presents the results given by Ghoshal and Masood (1974a,b). A generalization of Theorem 4.6.1 established by Ghoshal and Masood (1974a) is contained in the following. Theorem 4.7.1 Let u(x, y), g(x, y), h(x, y), p(x, y) and q(x, y) be continuous functions with h(x, y) > 0 on a domain D and Po(xo, yo) and P(x, y) be two points such that (x - xo)(y - Yo) > 0 on D. Let R denote the rectangular region whose opposite corners are Po and P. Let v(s, t; x, y) be the solution of the characteristic initial value problem M[v] -- 0,

(4.7.1)

where M is the adjoint operator of the operawr L defined by L[w] -- Wst + aWs + bwt + cw,

(4.7.2)

where a = - h ( x , y)q(x, y), b - - h ( x , y)p(x, y), c - - h ( x , y). The function v(s, t; x, y) is called the well-known Riemann function for the partial differential operator L and satisfies all the properties of a Riemann function for an operator with continuous coefficients. Let D + be a connected subdomain of D which contains P and on which v > 0 (Figure 4.5). If R C D + and u(x, y) satisfies


365

x

u(x, y) < g(x, y)+ p(x, y ) I h(s, y)u(s, y)ds xo Y

x y

(4.7.3)

+q(x,y)/h(x,t)u(x,t)dt+ffh(s,t)u(s,t)dsdt, yo

xo yo

then u(x, y) also satisfies Y

x

u(x, y) 0 (Figure 4.5). If R C D + and u(x, y) satisfies x

y

u(x, y) < g(x, y) + p(x, y) f h(s, y)u(s, y)ds + q(x, y) f h(x, t)u(x, t)dt xo x

Yo

y

(4.7.21)

+r(x,y)ffh(s,t)u(s,t)dsdt, xo yo where the inequality holds componentwise, then u(x, y) satisfies x

y

u(x, y) < g(x, y) + p(x, y) f h(s, y)u(s, y)as -t- q(x, y) f h(x, t)u(x, t)at xo x

Yo

y

(4.7.22)

+r(x,Y)ffVr(s,t)h(s,t)g(s,t)dsdt. xo yo

Here the relation holds for every column of V (s, t; x, y). Further if q(x, y) -- O, then x

u(x, y) 0 (Figure 4.1). If R C D + and u(x, y) satisfies u(x, y) < a(x, y) + b(x, y) [ x ~ / c(s, t)u(s, t)ds dt

+ -il,,-x-,,-y p(s, t) x0 Yo

(

i

k X o YO

q(~,f O)u(~, ) ,)d~d0

,

(4.8.12)


379

then u(x, y) also satisfies

s

+ .(s ,) f f

} ] a(~, r/)[c(~, r/) + q(~, r/)]v(~, r/; s, t) d~ dr/ ds dt .

xo Yo

(4.8.13) 89 Proof: Define a function 4~(x, y) such that x

y

4~(x,y)-ffc(s,t)u(s,t)dsdt xo Yo

xo Yo

Xo Yo

then (P(xo, y) -- (p(x, Yo) = 0, and

u(x, y) 0 and let R be the rectangular region whose opposite corners are the points Po and P. Let v(s, t; x, y) be the solution of the characteristic initial value problem L[v] - Vst - [1 + b(s, t)]v - O,

v(s, y) - v(x, t) - 1,

and let D + be a connected subdomain of D which contains P and on which v > 0 (Figure 4.1). I f R C D + and u(x, y) satisfies x P

y

P

Uxy(X, y) < a(x, y) + / / b ( s , ~J

t)[u(s, t) + Ust(S, t)] ds dt,

(4.8.30)

lkl

Xo Yo


U(X,

y~ 0 (Figure 4.1). If R C D + and u(x, y) satisfies x

Uxy(X, y) < a(x, y) +

xy

y

f / b ( s ,~u(s ,, + ,s,(s ,,1 as dt xo yo

xo Yo

[r Lxo Yo

+ u~,7(~,0)] d~drll ds dt,

(4.8.39)

then u(x, y) also satisfies u(x,y) 0 and let R be the rectangular region whose opposite corners are the points Po and P. Let V(s, t; x, y) be the solution of the characteristic initial value problem

M[V] = 0,

(4.8.41)

where M is the adjoint operator of the operator L defined by L[~]

-

-

![tst -11-al~s q-a2~t + a3~,

(4.8.42)

in which al -- -bcq, a2 - - b c p , a3 - - [ g + bc(r + h)]. Let W(s, t; x, y) be the solution of the characteristic initial value problem N[W] -- O,

(4.8.43)

where N is the adjoint operator of the operator T defined by T[q~] - qbst d- bldps + b2~t d- b3dp,

(4.8.44)

in which bl - -bcq, b2 - - b c p , b3 - - b c ( r - h). Let D + be a connected subdomain of D which contains P and on which V > 0 and W > 0 (Figure 4.5). If R C D + and u(x, y) satisfies u(x, y) < a(x, y ) + b(x, y)

[j p(x, y)

c(s, y)u(s, y)ds xo

MULTIDIMENSIONAL LINEAR INTEGRAL INEQUALITIES y

x

391

y

+q(x,y)fc(x,t)u(x,t)dt+r(x,y)fJ'c(s,t)u(s,t)dsdt Yo

xo Yo

+ h(x, y ) f f ~ ( s , , ~

c(e, o)u(~, o)d~do

xo Yo

dsdt ,

\ xo Yo

(4.8.45)


[j

u(x, y) < a(x, y) + b(x, y) p(x, y)

c(s, y)u(s, y)ds xo

Y

+ q(x, y) f c(x, t)u(x, t)dt + r(x, y)Q(x, y) Yo

(4.8.46) xo Yo

where

xy

Q(x, y) -

ff w

[ t; x, y)c(s, t) a(s, t) + b(s, t)h(s, t)

xo Yo

ds dt. \xo

(4.8.47)

Yo

Further, if q(x, y ) - O, then u(x,y) 0. Hence by multiplying (4.10.6) throughout by v and using (4.10.4) and (4.10.10), we obtain (4.10.3).

II R e m a r k 4.10.1 We observe that the problem (4.10.1) defines precisely the so-called Riemann function for the operator L. The existence and regularity property of v can be deduced from Courant and Hilbert (1962) (see also Copson, 1958; Garbedian, 1964). Indeed, (4.10.1) is equivalent to the integral equation /,b

x

b(o)v(o; x) do.

v(~;x)-- 1 + / ,.I

A The following two theorems given by Bondge and Pachpatte (1980a) extend to the case of n independent variables the quite general results established by Pachpatte (1979a). Theorem 4.10.2 Suppose dp(x), a(x), b(x), c(x) and cr(x) are nonnegative continuous functions defined on f2 C e n. Let v(~; x) be a solution of the

characteristic initial value problem

412

MULTIDIMENSIONALLINEAR INTEGRAL INEQUALITIES ( -1- ) nV ~ , . . . ~ n (~; x) - [b(~) 4- c(~)]v(~; x) - 0 in ~, (4.10.11) v(~; x ) - 1 o n

~i -

xi,

1 0 for all ~ E D +. If D C D + and

. [ /

rX

r

xo

xo

]

xo

(4.10.12) then

dp(x) < a(x) +

J [ b(p)

i

a(p) 4- tr(p) 4-

xo

{a(~)c(~) + b(~)

xo

x [a(~) 4- tr(se)]}v(~; p)dse] dp.

(4.10.13)

D Proof: Define a function u(x) such that

X

X[i

u(x) -- f b(p)dp(p) dp + f b(p) x0

u(x) -- 0 on

x0

Xi

--

XO,

]

c(se)q~(s e) d~ dp,

or(p) +

(4.10.14)

xo

1 < i < n,

then we have

D1.. Dnu(x)-b(x)[dp(x)-t-cr(x)-t-Tc(~)dp(~)d~ 1,

(4.10.15)

xo

Using the fact that ~b(x) < a(x) + u(x) in (4.10.15) we have

D 1 . . . Dnu(X) 0. Now multiplying (4.10.19) throughout by v and using (4.10.14) and (4.10.22), we obtain X

+ b(~)[a(~) + cr(~)]}v(~; x)d~.

O, w > 0 for all ~ ~ D +. If D C D + and

X ( i k(~)4)(~) ) d~

qb(x) 0 is arbitrary, then the bound obtained in (4.11.7) reduces to

lu(x, Y)I ~ ~

{/j

[

w(s, t; x, y) [c(s, t) + p(s, t)]

1+

xo Yo s

t

+ p(r ,) xo Yo

+ q(~, 0)]v(~, rl; s, t) a~ dr/] as at } .

(4.11.8)

In this case we note that (4.11.8) implies not only the boundedness but the stability of the solution u(x, y) of (4.11.1)-(4.11.2) if the bound obtained on the fight-hand side in (4.11.8) is small enough. As a second application, we discuss the uniqueness of the solution of the nonlinear hyperbolic partial integro-differential equation (4.11.1) with the given boundary conditions (4.11.2). We assume that the functions f , k

420


and h in (4.11.1) satisfy

If(x, y, u ) -

(4.11.9)

f (x, y, u)l-< c(x, y)lu - ul,

(4.11.10)

Ik(x, y, s, t, u) - k(x, y, s, t,-a)l < q(s, t)lu - ul,

Ih(x, y, u, r) - h(x, y, ~, V)l _< p(x, y)[]u - KI + Ir - VI], (4.11.11) where c(x, y), p(x, y) and q(x, y) are nonnegative continuous real-valued functions defined on a domain D C R+ x R+. The problem (4.11.1)-(4.11.2) is equivalent to the Volterra integral equation (4.11.6). Now if u(x, y) and ~(x, y) are two solutions of the problem (4.11.1)-(4.11.2), then we have x

y

u - n - f f {f (s, t, u) - f (s, t, ~)} ds dt

xoYo ffx y ( (s,t,u,ff xo Y0

- h

)

xo Yo

( JJ s, t,-~,

k(s,

)}

t, ~, rl, ~)d~ dr1

xo Yo

(4.11.12)

ds dt.

Using (4.11.9)-(4.11.11) in (4.11.12) we have x

y

lu--al 0 is arbitrary, then from (4.11.39) we have

Iz(x, y) - u(x, y)l _< ~0

{

1 + MoQ2(x, y) at-

JJ

g(s, t)e2(s, t)ds dt

xo Yo

xo lf~,~,sYg(~, + M0 - - - -

t)Qz (~,

] } t)d~ dt ds ,

(4.11.40)

x0 Y0

If in (4.11.40) the expression in braces is bounded and e0 --+ 0, then we obtain Iz(x, y) - u(x, Y)I --+ 0, which gives the equivalence between the solutions of (4.11.25)-(4.11.26) and (4.11.27)-(4.11.28). Note that Theorem 4.8.8 can be used to study the stability boundedness and continuous dependence of the solutions of (4.11.25)-(4.11.26) and (4.11.27)-(4.11.28) by following arguments similar to those given in Section 4.11.1 (see also Pachpatte, 1979a, 1980c) with suitable modifications. Further, note that the integral inequality established in Theorem 4.8.8 can be used to study the similar problem for nonlinear non-self-adjoint partial differential and integro-differential equations of the form

Uxy(X, y) = (bo(x, y)u(x, Y))x + bo(x, y)F(x, y, u(x, y)),

(4.11.41)

and

Zxy(X, y)

= (bo(x, y)z(x, Y))x + bo(x, y)F(x, y, z(x, y)) + H

(. jj , y,

k(x, y, s, t, z(s, t)) ds dt

)

, (4.11.42)

xo Yo

with the given boundary conditions and some suitable conditions on the functions involved in (4.11.41) and (4.11.42).

428

MULTIDIMENSIONALLINEAR INTEGRAL INEQUALITIES

4.11.3 Perturbations of Hyperbolic Partial Differential Equations This section deals with the variation of constants formulae established by Pachpatte (1982b) (taking into consideration the correction suggested by Beesack (1987) that the integral term involved in the function f in Pachpatte (1982b) must be absent) which gives the relationship between the solutions of a Bianchi-type partial differential equation and its perturbed partial integro-differential equation in three independent variables. Applications of these formulae investigated in Pachpatte (1982b) by using the inequality given in Theorem 4.9.2 are also given. In fact, the results given in Pachpatte (1982b) are motivated by the important role played by the variation of constants formulae in the theory of ordinary differential and integral equations; see Bemfeld and Lord (1978) and the references cited therein. Consider the nonlinear hyperbolic parial differential equation L[u] = f (x, y, z, u),

(D) u(xo, y, z) = u(x, Yo, z) = u(x, y, zo) = uo,

and its perturbed hyperbolic partial integro-differential equation

L[v] = f (x, y, z, v) 4- g

(X jjj , y, z, v,

k(x, y, z, s, t, r, v) ds dt dr

)

xo Yo zo

v(xo, y, z) = v(x, Yo, z) = v(x, y, zo ) = uo,

(E) where L[m] = mxyz 4- almxy 4- a2myz 4- a3mzx -I- a4mx 4- a5my + a6mz 4- a7m,

(x, y, z) ~ A = [xo, a] x [Yo, b] • [zo, c], u, v ~ C[A, R], ai E C[A, R], i -1, 2 . . . . . 7, k 6 C[A 2 x R, R], f 6 C[A x R, R], g 6 C[A x R x R, R], u0 is a constant, in which R denotes the set of real numbers. By a solution of equation (D) we mean a function u which satisfies (D) and is continuous in A along with the derivatives, Ux, Uy, Uz, Uxy, Uyz, Uzx and Uxyz. Similarly, we can define a solution v of equation (E). We use u = u(x, y, z, xo, Yo, zo, uo)


429

and v = v(x, y, z, xo, Yo, zo, uo) to denote the solutions of (D) and (E) respectively. A useful nonlinear variation of constants formula established by Pachpatte (1982b) is embodied in the following theorem. Theorem 4.11.1 Suppose that equation (D) admits a unique solution u = u(x, y, z, xo, Yo, zo, uo). Suppose also that 0 cp(x, y, z, xo, Yo, zo, uo) -- :---u(x, y, z, xo, Yo, zo, uo), Ouo exists and is continuous f o r x > xo, y > Yo, z > zo and that t~ -1 (X, y, z, xo, Yo, zo, uo ) exists f o r all x > xo, y > Yo, z > zo. If w = w(x, y, z) is a solution of

Wxyz -- ~ - I (x, y, z, xo, yo, zo, w) [A(x, y, z, xo, yo, zo, w)

+g

(, JiJ , y,z,v,

k(x, y , z , s , t, r, v ) d s d t d r ) ] , (4.11.43)

xo yo zo

w(xo, y, z) -- w(x, Yo, z) - w(x, y, zo) -- uo, where A(x, y, z, xo, Yo, zo, w) = -[al[UywWx + UxwWy + UwwWxWy "1-"UwWxy]

-'l-"a2[UzwWy --1-UywWz --1--UwwWyWz --1--UwWyz] -+- a3[UxwWz -k- UzwWx -+- UwwWzWx -+- UwWzx] -1- a4UwWx + a5UwWy -t--a6UwWz

-'[- {UyzwWx -'[- UzxwWy -l- UxywWx -~- UzwwWxWy _~- UxwwWyWz _~_UywwWzWx qt_ UzwWxy + UxwWyz -l'- UywWzx -'1- (WyzW x + WzxWy + WxyWz)Uww + UwwwWxWyWz}],

(4.11.44)

then any solution v - v(x, y, z, xo, Yo, zo, uo ) o f (E) satisfies the relation

430


v(x, y, z, xo, Yo, zo, uo ) , y, z, xo, Yo, zo, uo +

~-U

/JJ

~b-1 (S, t, r, xo, Yo, zo, w(s, t, r))

xo Yo zo

x [A(s, t, r, xo, Yo, zo, w(s, t, r)) + g(s, t, r, v(s, t, r, xo, Yo, zo, uo),

s

fff

] )

r

k(s, t, r, ~, O, P, v(~, rI, p, xo, Yo, zo, uo))d~ do dp

ds dt dr

,

xo Yo zo

(4.11.45) as f a r as w(x, y, z) exists f o r x >_ xo, y >_ Yo, z _> zo. D P r o o f : Let u(x, y, z, xo, Yo, zo, uo) be any solution of (D) existing for x >_ xo, y >_ Yo, z >_ zo. The method of variation of parameters requires determining a function w(x, y, z) so that v(x, y, z, xo, Yo, zo, uo) -- u(x, y, z, xo, Yo, zo, w(x, y, z)), (4.11.46) w(xo, y, z) -- w(x, Yo, z) -- w(x, y, zo) -- uo, is a solution of (E). Differentiating (4.11.46) partially with respect to z, y, x respectively we have Vz -- Uz + UwWz,

(4.11.47)

1;y - - Uy + UwWy ,

(4.11.48)

Vx -- Ux + UwWx,

(4.11.49)

Differentiating (4.11.47), (4.11.48), (4.11.49) partially with respect to x, z, y respectively we have Vzx - - Uzx ~ UxwW z ~ UzwWx -F- UwwWzWx 9 UwWzx,

(4.11.50)

Vy z - - Uyz ~ UzwWy -Jr- UywW z -Jr- UwwWyWz "t- UwWy z,

(4.11.51)

Vxy - - Uxy -~- UywWx -t- UxwWy -Jr- UwwWxWy Jr- UwWxy.

(4.11.52)


431

Differentiating (4.11.52) partially with respect to z we have Vxyz -- llxyz + UyzwWx + tlzxwWy + UxywWz + tlzwwWxWy -t- UxwwWyWz -P UywwWzWx ~ UzwWxy -t- UxwWyz -t- UywWzx -t- (WyzWx + WzxWy -t- WxyWz)llww

(4.11.53)

+ UwwwWxWyWz + UwWxyz.

Multiplying both sides of (4.11.46), (4.11.47), (4.11.48), (4.11.49), (4.11.50), (4.11.51), (4.11.52) by a7, a6, a5, a4, a3, a2, al respectively and adding in (4.11.53) we have (4.11.54)

L[v] -- L[u] - A(x, y, z, xo, Yo, zo, w) + UwWxyz,

where A(x, y, z, xo, Yo, zo, w) is as defined in (4.11.44). From (D), (E) and (4.11.54) we have f (x, y, z, v) + g

(X jjj , y, z, v,

k(x, y, z, s, t, r, v) ds dt dr

)

xo Yo zo

-- f (x, y, z, u) - A (x, y, z, xo, Yo, zo, w) -4- UwWxyz,

which because of (4.11.46) and the fact that 4b-1(X, y, Z, X0, Y0, Z0, U0) exists reduces to !"

Wxyz -- t~ -1 (X, y, z, xo, Yo, zo, w) IA(x, y, z, xo, Yo, zo,

w)

/,

+ g (x, y, z, v(x, y, z, xo, Yo, zo, uo ), \

ill

y, z, s, t, r,

v s,

r, xo, yo, zo,

.

xo Yo zo

(4.11.55) The solutions of (4.11.55) then determine w(x, y,z). Consequently, if w(x, y, z) is a solution of (4.11.55), then v(x, y, z, xo, Yo, zo, u0) given by (4.11.46) is a solution of (E). From (4.11.55), w(x, y, z) must satisfy the

432


integral equation x

Y

z

w(x,y,z)-uo+J'J'f~-l(s,t,r,

xo, yo, zo, w(s,t,r))

xo Yo zo

x [A(s, t, r, xo, Yo, zo, w(s, t, r)) /

+ g (s, t, r, v(s, t, r, xo, Yo, zo, uo), "k

fff

k(s, t, r, ~, rl, p, v(~, O, P, xo, Yo, zo, uo))

xo Yo zo

x d ~ d r / d p ) ] dsdtdr.

(4.11.56)

Substituting (4.11.56) in (4.11.46) we obtain (4.11.45). Note that v(x, y, z, xo, Yo, zo, uo) exists for those values of x > xo, y > Yo, z > zo for which the solution w(x, y, z) of (4.11.43) exists.

II Another interesting and useful representation formula established by Pachpatte (1982b) is given in the following theorem. Theorem 4.11.2 Under the assumption of Theorem 4.11.1 the following

relation is also valid. v(x, y, z, xo, Yo, zo, uo ) -- u(x, y, z, xo, Yo, zo, uo ) x y z

+/ffB(x,y,z,

xo, yo, zo, w(s,t,r))dsdtdr

xo Yo zo x

Y

+fff4

z

(x,y,z, xo, yo, zo, w(s,t,r))

xo Yo zo

x ~b-1(s, t, r, xo, Yo, zo, w(s, t, r))


433

x [A(s, t, r, xo, Yo, zo, w(s, t, r))

+ g (s, t, r, v(s, t, r, xo, Yo, zo, uo), s

t

r

f f f k(s, t, r, ~, O, p, v(~, O, p, xo, yo, zo, uo)) xo Yo zo

X d~ do d p ) ] ds dt dr, (4.11.57)

where A(x, y, z, xo, Yo, zo, w) is as given in (4.11.44) and B(x, y, z, xo, Yo, zo, w ) -

UwwwWsWtWr

+ (WtrWs -]- WsrWt -~- WstWr)Uww.

(4.11.58)

D Proof: For xo x0 > 0, y > Y0 > 0, z > z0 > 0 and In01 < c ~ . Theorem 4.11.3 Let the solution u(x, y, z, xo, yo, zo, uo) of (D) be globally uniformly stable. Assume that the hypotheses of Theorem 4.11.1 hold and the functions involved in (E) satisfy Ik(x, y, z, s, t, r, u(s, t, r, xo, Yo, zo, w(s, t, r)))l < q(s, t, r)lw(s, t, r)l,

(4.11.61)

Idp-1 (x, y, z, xo, yo, zo, w(x, y, z))[A(x, y, z, xo, Yo, zo, w(x, y, z)) + g(x, y, z, u(x, y, z, xo, yo, zo, w(x, y, z)), u)]l (4.11.62)

p(x, y, z)[Iw(x, y, z)l + I~1],

where p(x, y, z) and q(x, y, z) are real-valued nonnegative continuous functions defined on A and abc

abc

f f J"

fff

xo Yo zo

xo Yo zo

p(s, t, r)ds dt d r < c~,

q(s, t, r)ds dt d r < cx~. (4.11.63)

Then any solution v(x, y, z, xo, Yo, zo, uo ) of (E) is bounded on A.

D Proof: By Theorem 4.11.1 any solution v(x, y, z, xo, Yo, zo, uo) of (E) satisfies (4.11.46), where w(x, y, z) given by (4.11.56) is a solution of (4.11.43). Using (4.11.56), (4.11.61) and (4.11.62) we have


xYz

Iw x,y,z)l 0 is a constant. The proof is complete. II

Theorem 4.11.4 Assume that the hypotheses of Theorem 4.11.2 hold and the functions involved in (4.11.57) satisfy

abc

fff xo Yo zo

IB(x, y, z, xo, yo, zo, w(s, t, r))l ds dt d r < M*,

(4.11.66)

14~(x, y, z, xo, Yo, zo, w(s, t, r))4~-1 (s, t, r, xo, Yo, zo, w(s, t, r))l ~ N, (4.11.67)

436


Ik(x, y, z, s, t, r, v(s, t, r, xo, yo, zo, u0))l

(4.11.68)

< q(s, t, r)lv(s, t, r, xo, Yo, zo, u0)l,

[A(x, y, z, xo, Yo, zo, w(x, y, z)) + g(x, y, z, v(x, y, z, xo, Yo, zo, uo), v)l

(4.11.69)

< p(x, y, z)[lv(x, y, z, xo, Yo, zo, u0)l + I~11,

where M* and N are positive constants and p(x, y, z) and q(x, y, z) are realvalued nonnegative continuous functions defined on A and abc

abc

ff.fNp(s,t,r)dsdtdr O, g' (t) >_ 0 for x 6 [0, 1], t 6 [0, T], r (x) exists and nonnegative for x ~ [0, 1], then

u(x, t) O, g'(t) _> O, for x ~ [0, 1], t 6 [0, T], then u(x, t) < c(x, O)exp -

f ' ( t l ) + xg'(tl) dtl + c ( O , t~)

JJJ

p(sl, tl)dsl dsdtl

ooo

]

,

for x 6 [0, 1], t 6 [0, T].

4.12.5 Pachpatte (unpublished manuscript) Let ~b(x), f ( t ) , g(t) be positive and continuously differentiable functions for x ~ [0, 1], t ~ [0, T] and define c(x, t) = ~p(x) + xg(t) + f (t), for x [0, 1], t ~ [0, T]. Let u(x, t), p(x, t) and q(x, t) be nonnegative continuous functions defined for x ~ [0, 1], t ~ [0, T] and

x s t (

U(X,t) 0, g' (t) >_ 0 for x 6 [0, 1], t E [0, T], r (x) exists and nonnegative for x 6 [0, 1], then

u(x,t) 0, f ' ( t ) > 0, g'(t) > 0 for x E [0, 1], t 6 [0, T], then

t

u(x, t) < c(x, O) +

f tf' (tl) + xg' (tl)] dtl 0

t

x

s

+ff/p(sl,

tl)Q2(tl, sl)dsldsdtl,

000 for x ~ [0, 1], t 6 [0, T], where

Q2(x, t) - c(x, 0)exp t

000 for x E [0, 1], t ~ [0, T].

x

S2

/ f ' ( c r ) + xg' (or) c(0, ~r)

d~r

ds.d,.J "]

444


4.12.6 Pachpatte (unpublished manuscript) Let r fi(t), gi(t) for i = 1, 2 be positive and continuously differentiable functions for x e [0, 1], t e [0, T] and define ci(x, t) = 4)i(x) + xgi(t) +

fi(t), i = 1, 2, for x ~ [0, 1], t ~ [0, T] and c(x, t) = Cl (x, t) + c2(x, t), q~(x) = ~bl(X) + q~2(x), f(t) = f l ( t ) + f2(t), g(t) = gl(t) -k- g2(t), for x e [ 0 , 1 ] , t ~ [0, T]. Let u(x,t),v(x,t),hi(x,t),i = 1 , 2 , 3 , 4 , be nonnegative continuous functions defined for x ~ [0, 1], t e [0, T] and define

h(x, t) -- max{[hl (x, t) -+-h3(x, t)], [h2(x, t) q- h4(x, t)]}, for x ~ [0, 1], t ~ [0, T] and x

s

t

u(x,t) O, g'(t) > O, for x ~ [0, 1], t ~ [0, T], ~p"(x)

~(x, t) = exp(/z(x +

exists and nonnegative for x ~ [0, 1], then

[

u(x, t) < c(O, t)exp /z(x + t) + x -

(cx'~162 *''sl' c(O,

x

s

t

]

+fffh(sa, ta)dtadsads, 0 0 0

t)

+

~

C(S1, O)

0

0

dsl

ds


IX (cx(O't)) ~tt(S1) c--~i t) + i f C(Sl, o) ds1 00

v(x, t) O, f ' ( t ) > O, g'(t) > O, for x 6 [0, 1], t 6 [0, T], then

u(x, t) < c(x, O)exp -

txs

[/

tx(x + t) +

0

f ' ( a ) + xg'(a) do c(O, a)

]

- ] - f f f h ( s l , tl)dsldsdtl ooo Vt u(x, t)

< c(x, O) exp I f f ' ( a ) + xg' (a) [Jo c(O, a)

,

txs

]

dffq-fffh(Sl, 000

tl)dsldsdtl

,

for x ~ [0, 1], t ~ [0, T].

4.12.7 Let u(x, y), Ux(X, y), Uy(X, y), Uxy(X , y) and c(x, y) be nonnegative continuous functions defined for x, y ~ R+, and u(x, O) = u(O, y) -- O. If

Uxy(X, y)

< a(x) + b(y) + M

[ ]] u(x, y) +

c(s, t)Ust(S, t)ds dt

00

]

,

for x, y 6 R+, where a(x) > O, b(y) > O, a'(x) > O, b'(y) > 0 are continuous functions defined for x, y 6 R+, and M > 0 is a constant, then

Uxy(X, y)

0, b(y) > 0, d (x) >_ 0, b'(y) >_ 0 be continuous functions defined for

x,y~R+. (i) If

st

)

0 0

for x, y 6 R+ then

Uxy(X,y)< a(x) + b(y) + f

xy

[

f ~(s,,~ ~a(O~+b(t)][a(s) + b(O)]

-

a(O) 4- b(O) 0 0

for x, y ~ R+. (ii) Let p(x, y) > 1 and M > 0 be a constant. If

Uxy(X,y) 0 there, and nondecreasing in both variables x and y. Let Po(xo, yo) and P(x, y) be two points in D such that (x - xo)(y - Yo) > 0 and let R C D be the rectangular region whose opposite comers are the points P0 and P. Let v(s, t; x, y) be the solution of the characteristic initial value problem

L[v] = Vst - b(s, t)[c(s, t) + p(s, t)]v - O, v(s, y) = v(x, t) = 1, and let D + be a connected subdomain of D which contains P and on which v > 0 (see Figure 4.1). If R C D + and

u(x, y)

satisfies

u(x, y) < a(x, y) + b(x, y) f f c(s, t) (u(s, t) + b(s, t) xo yo

xo yo

)


449


[

u(x, y) < a(x, y) 1 + b(x, y)

xf f

JJ{

c(s, t) 4- b(s, t)c(s, t)

xo Yo

,~ + p(~,,>]v(~,,; ~,,> da

do

ds dt

.

xo Yo

4.12.12 Bainov and Simeonov (1992, p. 119) Let x, c~ belong to a domain D C R n, and ot 0,

q(x) >_0 be continuous functions in D. Let v(s; x) be the solution of the characteristic initial value problem

( - 1)nVs(S; x) - q(s)b(s)v(s; x) -- 0 in D, v(s, x) - 1 on

Si :

Xi,

i - 1, 2 , . . . , n,

and let D + be a connected subdomain of D containing x, on which v > 0 for all s 6 D +. If [ct, x] C D + and x

u(x) 0 for u > O. If x

u(x, y) < a(x) + b(y) +

y

ff 0

(5.2.1)

t)g(u(s, t)) ds dt,

0

for x, y ~ R+, where a(x) > O, b(y) > O, a'(x) > O, b'(y) > 0 are continuous functions defined for x, y ~ R+, then for 0 < x < Xl, 0 < y < Yl, u(x, y)

0,

r0>0,

(5.2.14)

G -1 is the inverse function of G and x2, y2 are chosen so that x

y

G(A(x, y)) + f f c(s, t)g(b(s, t)) ds dt ~ Dom(G -1), 0

0

for all x, y lying in the subintervals 0 O, b'(y) > O, g'(u) > 0 for x, y, u R+, the functions a(x), b(y), g(u) are monotonically increasing on (0, c~).

466

MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES

Define a function m(x, y) by x

m(x, y) -- a(x) + b(y) +

y

f f p(s, t)g(u(s, t)) ds dt,

(5.2.28)

0 0

then m(x, O) - a(x) + b(O), m(O, y) - a(O) + b(y) and (5.2.25) can be restated as x

y

u(x,y)

f2(u(x, y)) - a(s, t ) f f x

forO < x ~ s ~ s1,

O ~

d. ~ Dom(f2 -1 ),

y

y < t < tl. D

47'8

MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES

P r o o f : Since g'(u)> 0 on R+, the function g(u) is monotonically increasing on (0, c~). Rewrite (5.3.35) as

u(x, y ) < u ( s , t ) + a ( s , t ) f f b ( t r , x

rl)g(u(tr, 0)) dcr d0.

(5.3.37)

y

For fixed s and t in R+ we define for 0 < x < s, 0 < y < t, s

z(x, y) -- u(s, t) + a(s, t)

t

ff x

b(tr, o)g(u(tr, 0))dcr do,

(5.3.38)

y

then z(x, t) - z(s, y) - u(s, t), u(x, y) < z(x, y) and

Zxy(X, y ) = a(s, t)b(x, y)g(u(x, y)) < a(s, t)b(x, y)g(z(x, y)), i.e.

y) < a(s, t)b(x, y). g(z(x, y)) Zxy(X,

(5.3.39)

From (5.3.39) we observe that

Zxy(X, y) g' (Z(X, y))Zy (X, Y)Zx(X, y) < a(s, t)b(x, y) + g(z(x, y)) [g(z(x, y))]2 9

(5.3.40)

Here we note that Zx(X, y) and Zy(X, y) are nonpositive, which implies that Zx(X, y)zy(X, y) is nonnegative and hence (5.3.40) is true. Now (5.3.40) is equivalent to

0 (Zx(X'Y)) Oy g(z(x, y))

3) independent variables. The precise formulations of these results are very close to those of the results given in Sections 5.2-5.4 with suitable modifications. It is left to the reader to fill in the details where needed.

5.5 Pachpatte's Inequalities I In view of wider applications, Wendroff' s inequality given in Beckenbach and Bellman (1961) has been generalized and extended in various directions. The present section is devoted to the Wendroff-like inequalities investigated by Pachpatte (1988c, 1993, 1996d) in order to apply them in the study of certain higher order partial differential equations. In what follows we shall use the notations and definitions as given in Section 4.5 without further mention. Pachpatte (1988c) established the Wendroff-like inequalities in the following two theorems. Theorem 5.5.1 Let u(x, y) and h(x, y) be nonnegative continuous functions defined for x, y ~ R+. Let a(x), b(y), p(x), q(y) be positive and twice continuously differentiable functions defined for x, y ~ R+, a' (x), b' (y), p' (x), q' (y) are nonnegative for x, y ~ R+ and define c(x, y ) - a(x) + b(y) + yp(x) + xq(y), for x, y ~ R+. Let g be a continuously differentiable function defined on R+ and g(u) > 0 on (0, cx~), g'(u) > 0 on R+ and

u(x, y) < c(x, y ) + a [ x , y,h(sl, tl)g(u(sl, tl))],

(5.5.1)

holds for x, y ~ R+. (i) If a"(x), p"(x) are nonnegative for x >_ O, then for 0 < x < Xl, O 0, ro > 0,

s + g(s)

(5.5.8)

ro

H -1 is the inverse function of H and x3, Y3 are chosen so that x

H (c(O, y)) + x (

$2

Cx(O,y) ) . f+f a " ( s 3 ) + y p " ( s 3 ) ds3 ds2 c(O, y) -Jr-g(c(O, y)) c(s3, O) + g(c(s3, 0)) 0

0

+ A[x, y, h(s3, t3)] E D o m ( H -1), for all x, y lying in the subintervals 0 < x < x3, 0 < y < Y3 of R+. (ii) If b"(y), q"(y) are nonnegative for y ~ R+ then for 0 < x < x4, O< y < y4, y

u(x, y) < c(x, O) + yCy(X, O) -~-

t

f f tb"(tl)+ xq" (tl)] dtl dt 0

0

+ A[y, x, h(sl, tl)Q2(Sl, tl)],

(5.5.9)

488


in which

a2(x, y) - n -1

+ff

( c(x, O)cy,xo, ) + g(c(x, 0))

(c(x, 0)) + y

y t2

0 0

b"(t3)+xq"(t3) C(0, t3) + g(c(O, t3))

dt3 dt2

(5.5.10)

+ A[y, x, h(s3, t3)]] ,

where H, H -1 are as defined in (i) and x4, y4 are chosen so that

(Cy(X,O) )+ffff'(t3)+xq"(t3) H(c(x, 0)) + y c(x, O) + g(c(x, g(c(O, y t2

0))

c(0, t3) +

t3))

dt3 dt2

0 0

+ A[y, x, h(s3, t3)] E Dom(H-1), for all x, y lying in the subintervals 0 < x < x4, 0 < y < y4 of R+. D P r o o f of T h e o r e m 5.5.1" Since g'(u) >_ O, the function g(u) is monotonically increasing on (0, c~). Define a function z(x, y) by

z(x, y) = c(x, y) + A[x, y, h(sl, t l ) g ( u ( s 1 , t l ) ) ] .

(5.5.11)

From (5.5.11) it is easy to observe that z(0, y) -- c(0, y) -- a(0) + b(y) + yp(O),

(5.5.12)

z(x, O) -- c(x, O) = a(x) + b(O) + xq(O),

(5.5.13)

zx(O, y) -- cx(O, y) -- a'(O) + yp'(O) + q(y),

(5.5.14)

Zxx(X, O) -- Cxx(X, O) = a"(x),

(5.5.15)

Zxxy(X, O) -- Cxxy(X, O) "- p"(x),

(5.5.16)

and Zxxyy(X, y) -- h(x, y)g(u(x, y)).

(5.5.17)


489

Using u(x, y) < z(x, y) in (5.5.17) we have

Zxxyy(X, y) < h(x, y)g(z(x, y)). From (5.5.18)and using the facts that z(x, y) > 0, g'(u(x, y)) > 0 for x, y ~ R+, we observe that

(5.5.18)

Zy(X, y) > O, zxxy(X, y) > O,

Zxxyy(X, y) < h(x, y)4- Zxxy(X, y)g'(z(x, y))Zy(X, y) g(z(x, y)) [g(z(x, y))l 2 i.e.

O___(Zx~y(x,y) ) ay \ g(z(x, y))

O, Zx(X, y) > 0 for x, y ~ R+, we observe that

i9 ( z x ( x , y ) -~x \ g ( - ~ , y))

y

< a" (x) + yp" (x) g(c(x, 0))

t

(5.5.23) 0

0

490


Now keeping y fixed in (5.5.23), setting x - sl and then integrating with respect to Sl from 0 to x and using (5.5.12) and (5.5.14) we have

Zx(X,y) < cx(O, y) + / a"(Sl) + yp"(Sl) g(z(x, y)) - g(c(O, y)) g(C~-liO-)) dsl 0 x

y

t

+fffh(sl,

tl)dtldtdsl.

(5.5.24)

000 From (5.5.3) and (5.5.24) we have x

f2(z(x, y)) = Zx(X, y) < cx(O, y) + f a"(Sl)q-- yp"(sl) -~x g(z(x, y)) - g(c(O, y)) gi-C-~ll-0-)) dsl x y t

+J'ffh(sl,

tl)dtldtds1.

(5.5.25)

0 0 0

Now keeping y fixed in (5.5.25), setting x - - s and then integrating with respect to s from 0 to x and using (5.5.12) we obtain

a (z(x, y)) 0 on R+, the function g(u) is monotonically increasing on (0, ec). We assume that c > 0 and that the standard limiting argument can be used to treat the remaining case. Define a function m(x, y) by

m(x, y) -- c + A[x, y, p(sl, tl)g(u(sl, tl))].

(5.5.30)

Then (5.5.27) can be restated as X

S

u(x, y) < m(x, y) + f / f (sl, y)u(sl, y) dsl ds. 0 0

(5.5.3~)

492

MULTIDIMENSIONALNONLINEARINTEGRALINEQUALITIES

Clearly m(x, y) is positive and nondecreasing in both the variables x, y 6 R+. From (5.5.31) we observe that

u(x, y) < 1 + ~f f m(x, y) --

y________~)dsl U(S1, y) ds. f (sl, Y)m(sl,

(5.5.32)

0 0

Define a function z(x, y) by the fight-hand side of (5.5.32); then

z(O, y) and

--

1,

u(x, y) m(x, y)

< z(x, y)

u(x, y) Zxx(X, y ) - f (x, y ) ~ < f (x, y)z(x, y) m(x, y) -

i.e.

Zxx(X, y) < f(x, y). z(x, y)

(5.5.33)

Now by following the proof of Theorem 5.5.1 we obtain

z(x, y) < exp

f (sl, y)dsl ds

- Q(x, y).

(5.5.34)

Using (5.5.34) in (5.5.32) we have

u(x, y) < Q(x, y)m(x, y).

(5.5.35)

From (5.5.30) and (5.5.35) we have mxxyy(X,

y ) - p(x, y)g(u(x, y)) < p(x, y)g(Q(x, y)m(x, y)) < p(x, y)g(Q(x, y))g(m(x, y)).

(5.5.36)

Now by following the proof of Theorem 5.5.1 we obtain

m(x, y)

(i) If

V2

(H)

>_ 0 where k" R3+ --+ R+ is a continuous function.

u(x, y) < a(x, y) + b(x, y)B[x, y, h(s, t, u(s, t))],

(5.5.38)

for x, y ~ R+ then u(x, y) < a(x, y) + b(x, y)p(x, y)exp(B[x, y, k(s, t, a(s, t))b(s, t)]), (5.5.39) for x, y ~ R+, where p(x, y) -- B[x, y, h(s, t, a(s, t))],

(5.5.40)

for x, y E R+. (ii) Let F(u) be a continuous, strictly increasing, convex, submultiplicative function for u > O, l i m u ~ F(u) -- o0, F -1 denote the inverse function of F, and ~(x, y), fl(x, y) be continuous and positive functions for x, y ~ R+ and or(x, y) + fl(x, y) -- 1. If u(x, y) _ O, g(u) > 0 for u > 0 and g'(u) > 0 for u > 0 and g(u) is subadditive and submultiplicative for u > O. If u(x, y) < a(x, y) + b(x, y)B[x, y, h(s, t, g(u(s, t)))],

(5.5.43)

494


for x, y ~ R+, then for 0 < x < x0, 0 < y < Yo, u(x, y) < a(x, y ) + b(x, y)f2-1[f2(q(x, y)) d- B[x, y, k(s, t, g(a(s, t)))g(b(s, t))]],

(5.5.44)

where q(x, y) = B[x, y, h(s, t, g(a(s, t)))],

(5.5.45)

g2, f2 -1 are as defined in Theorem 5.5.1 and xo, Yo are chosen so that f2(q(x, y)) + B[x, y, k(s, t, g(a(s, t)))g(b(s, t))] 6 Dom(f2 -1), for all x, y lying in the subintervals 0 < x < xo, 0 < y < Yo of R+. 0 Proof: (i) Define a function z(x, y) by

z(x, y) - B[x, y, h(s, t, u(s, t))].

(5.5.46)

From (5.5.46) and using u(x, y) < a(x, y) + b(x, y)z(x, y) and the condition (H), we observe that

z(x, y) < B[x, y, h(s, t, a(s, t) + b(s, t)z(s, t))] -- p(x, y) + B[x, y, {h(s, t, a(s, t) + b(s, t)u(s, t)) - h ( s , t, a ( s ,

t))}]

< p(x, y) + B[x, y, k(s, t, a(s, t))b(s, t)z(s, t)],

(5.5.47)

where p(x, y) is defined by (5.5.40). Clearly p(x, y) is nonnegative for x, y R+. It is sufficient to assume that p(x, y) is positive, since the standard limiting argument can be used to treat the remaining case. Now since p(x, y) is positive and monotonic nondecreasing in x, y 6 R+, from (5.5.47) we observe that

z(x, y) < l + B [x, y, k(s, t, a(s, t))b(s, t) z(s, t) ] p(x, y) p(s, t) "

(5.5.48)

Define a function v(x, y) by the fight-hand side of (5.5.48), then it is easy to observe that

D~D~v(x, y) -- k(x, y, a(x, y))b(x, y) z(x, y_____~) p(x, y) 0 and W > 0 (Figure 4.5). IfR C D + and u(x, y) satisfies

[j

u(x, y) < a(x, y) + b(x, y)H -1 p(x, y)

c(s, y)H(u(s, y))ds xo

Y

+ q(x, y) / c(x, t)H(u(x, t))dt Yo x

y

x

y

+r(x,y)f./'c(x,t)H(u(s,t))dsdt+h(x,y)ffg(s,t) xo Yo

xo Yo

then u(x, y) also satisfies u(x, y) < H -1 [n(a(x, y)) + H(b(x, y)) [p(x, y ) f c(s, y)H(u(s, y))ds xo Y

+ q(x, y) / c(x, t)H(u(x, t))dt + r(x, y)Ql(X, y) Yo

xy -t-h(x, y)//g(s,t)Ql(s,t)dsdt

1] ,

(5.6.33)

xo Yo

where Q1(x, y) is defined by the right-hand side of (5.6.26) by replacing a(x, y) by H(a(x, y)) and b(x, y) by H(b(x, y)). Further, if q(x, y) - O, then u(x,y)_ 1.

(5.7.15)

For, if this were false for some n and some x ~ Go, then because

1 v(x ~ < a(x ~ < a(x ~ + - - u. (x~ n

it would follow from the continuity of v and

Un

that for some point z ~ Go

with z ~ x ~ we would have v(x) < Un (x) for x ~ G(x ~ z ) -

Un(z). But then v(Z) < a(z) +

f [ J

G(x~

k(z, t, v(t)) dt

{z} but v(z) -

514


'

/

< a(z) + - + n

k(z,

t, U n ( t ) )

dt

- - Un ( z ) .

G(x~

It follows that (5.7.15) must hold. Taking limits as n ~ oe, (5.7.14) now follows from Theorem 5.7.3.

II R e m a r k 5.7.3 If hypothesis (5.7.10) is deleted, then (5.7.14) will hold on any subset G(x ~ yO) C Go such that equations (5.7.11) and (5.7.12) have continuous solutions on the common domain G(x ~ yO), if such y 0 6 Go exists. This was the case considered in Headley (1974), and for N = 2 also by Rasmussen (1976). A In the further discussion, some useful integral inequalities in n independent variables given by Pachpatte (1981 c) are presented, which are motivated by a well-known integral inequality due to Wa2eski (1969). We use the same notation as given in Section 4.9 without further mention. Pachpatte (1981c) gave the following general version of Wa2eski's inequality given in Wa2eski (1969). Theorem 5.7.5 Let u(x) and a(x) be nonnegative continuous functions defined on ft. Let k(x, y, z) and W (x, z) be nonnegative continuous functions defined on f22 • R and g2 x R, respectively, and nondecreasing in the last variables, and k(x, y, z) be uniformly Lipschitz in the last variable. If

u(x) < a(x) + W

(j) x,

k(x, y, u(y))dy

,

(5.7.16)

xo

then u(x) 0 for u > 0 and satisfying (1/v)H (u) < H (u/v), for v > 1, u > O. If Theorem

j(j (/)

u(x) < f (x) +

g(y)

u(y) +

xo

+ W

x,

g(s)H(u(s))ds

)

dy

x0

k(x, y, u(y))dy

,

(5.7.47)

xo

for x ~ ~2, then for x ~ ~'22 C ~'2, u(x) < Ez(x)[f (x) + W(x, r(x))],

(5.7.48)

where x

[ / ] y

E2(x) -- 1 + f g(y)F -1

g(s)ds

F(1) +

x0

dy,

(5.7.49)

x0

in which o"

F(cr)

-

/

ds , s + H (s)

cr > 0, cro > 0,

(5.7.50)

o-0

F-1 is the inverse function of F and x

F(1) + / g ( s ) d s

Dom(F-1),

t] x0

for all x ~ f2e, and r(x) is a solution of the equation x

F(X)

/ k(x, y, E e ( y ) [ f (y) -F W(y, r(y))])dy,

(5.7.51)

tJ Xo

existing on f2. D The details of the proof of this theorem follow by an argument similar to that in the proof of Theorem 5.7.7, together with the proof of Theorem 5.4.1, and the details are omitted here.


523

Pachpatte (1981c) has established the following generalization of the integral inequality given by Young (1973). Theorem 5.7.9 Let u(x), a(x), b(x), c(x) and g(x) be nonnegative continuous functions defined on f2. Let f (x), k(x, y, z) and W(x, z) be as in Theorem 5.7.6. Let v(y; x) and e(y; x) be the solutions of the characteristic initial value problems (-- l)nvyl...y n (y; x) - [a(y)b(y) + a(y)g(y) + c(y)lv(y; x) - 0 in f2,

(5.7.52) v(y; x ) -

1 on Y i - xi,

1 0 for all y ~ D +. If D C D + and

X (j)

u(x) 0. Now multiplying (5.7.64) throughout by v and using (5.7.62) and (5.7.67), we obtain

g(s)]v(s; x)ds.

7t(x) < f [ b ( s ) + x0

Now substituting this bound on 7t(x) in (5.7.61) we obtain

Lz -- D1... Dnz(X)- [a(x)b(x)- c(x)]z(x)

_0 for u >_O. If x

y

a~

a~

u2(x, y) 0, r0 > 0,

ro ~'~-1

is the inverse of f2 and

Yl are chosen so that

Xl,

x

y

f2(c)+ffp(s,t)dtdseDom(S2-1), 0 0

for all x, y lying in the subintervals 0 < x < Xl, 0 0 for u > O. If uZ(x, y) < c 2 + 2B[x, y, f (s, t)u(s, t)g(u(s, t)) + h(s, t)u(s, t)], for x, y ~ R+, then for 0 < x
O, for x, y 6 R+, we observe that (see Theorem 4.5.4) From

(5.8.39)

and

m--1 n

( D z _ DlZ(X, Y) D2 \ ~/z(x, y)

) 0 are reals and r > 1) and (5.9.16), (5.9.12) we observe that FP(x, y) < 2P-l[exp(--plz(x + y))uP(x, Y) + vP(x, y)]

< 2P-1 [Cl -t--C2 -k- B[x, y, [hi (s, t) -at- h3(s, t)]u(s, t)] + B[x, y, [hz(s, t) + h4(s, t)]v(s, t)]].

(5.9.18)

Now using the fact that exp(-p/z(x + y)) < exp(-/z(x + y)) and (5.9.15) in (5.9.18) we observe that FP(x, y) < 2P-I(cl -+- c2) -+-B[x, y, 2p-lh(s, t)F(s, t)].

(5.9.19)

The bounds in (5.9.13) and (5.9.14) follow from an application of Theorem 5.9.1 to (5.9.19) and splitting. II

The inequalities in the following two theorems involving functions of n independent variables given by Pachpatte (1994a) can be used in the study of certain new classes of partial differential and integral equations. Theorem 5.9.3 Let F(x) > O, k(x) > 0 and bi(x) > 0 for i = 1, 2 . . . . . n - 1, be continuous functions defined for x ~ R~_ and let p > 1 be a constant. If FP(x) < c + M[x, b, k(y)F(y)], (5.9.20)

542


f o r x ~ R~_, where c > 0 is a constant, then F(x) < [r

_~_ ( ( p _

1)/p)M[x, b, k(y)]] 1~(p-l)

(5.9.21)

for x ~ R~_. [B Theorem 5.9.4 Let u(x) > O, v(x) > O, bi(x) > O, i - 1, 2 . . . . . n - 1, and hj(x) > O, i = 1, 2, 3, 4, be continuous functions defined for x ~ R~_ and let p > 1 be a constant. If c l, c2 and I~ are nonnegative constants such that

uP(x) < Cl + M[x, b, hi (y)u(y)] + M[x, b, h2(y)~(y)],

(5.9.22)

Vp (X) w > O, where k" R~_ x R+ --+ R+ is a continuous function. If U2(x) 0 for (x, y) 6 E, we observe that

D2D1v(x, y) D1v(x, y)D2v(x, y) < p(x, y)log v(x, y)+ v(x, y) Iv(x, y)]2 '


545

i.e.

y) ) < p(x, y)log v(x, y). v(x, y) -

D1 v(x,

D2

(5.10.5)

By keeping x fixed in (5.10.5), setting y = t, integrating from 0 to y and using the fact that DlV(X, 0) -- 0, we get y

D1 v(x, y) < / p(x, t) log v(x, t) dt, v(x, y) -

(5.10.6)

0

Keeping y fixed in (5.10.6) and setting x using the fact that v(0, y) - c, we obtain x

log v(x, y) < log c 4-

s, integrating from 0 to x and

y

ff

p(s, t)log v(s, t)dt ds.

(5.10.7)

0 0

Now a suitable application of the inequality given in Theorem 4.2.1 (see Remark 4.2.1) yields

logv(x, y) < (logc)exp ( / f o = log c Q~x'Y).

(5.10.8)


v(x, y)

< c Q(x'y).

(5.10.9)

Now using (5.10.9) in u(x, y ) < v(x, y) we get the required inequality in (5.10.2).

II Theorem 5.10.2 Let u" E -~ R1, p" E ~ R+ be continuous functions and c >_ 1 be a constant, where E and R1 are as defined in Theorem 5.10.1. Let g(u) be a continuously differentiable function defined on R+ and g(u) > 0 on (0, c~) and g'(u) > 0 on R+. If x

y

u(x, y) < c + f / p ( s , 0 0

t)u(s, t)g(log u(s, t)) dt ds,

(5.10.10)

546

MULTIDIMENSIONAL

NONLINEAR

for (x, y) ~ E, then for (x, y) u(x, y) < exp

INTEGRAL

~ E1 C E,

( [ JJ ~-1

INEQUALITIES

S2(logc) +

p(s, t) dt ds

0 0

])

,

(5.10.11)

where f2(r)- f

ds

g(s) '

(5.10.12)

r > O, ro > 0 ,

ro

~'2 - 1

is the inverse of f2 and (x, y) x

~ E1 C

E is chosen so that

y

~2(logc)+ffp(s,t)dtds~Dom(S2-1), 0

for (x, y)

0

6 E1 C E.

D Proof: Since g'(u)>_ 0 on R+, the function g(u) is monotonically increasing on (0, c~). Define a function v(x, y) by the fight-hand side of (5.10.10). By following the arguments as in the proof of Theorem 5.10.1 up to the inequality (5.10.7) we obtain x

log v(x, y) < log c +

y

f f p(s, 0

(5.10.13)

t)g(log v(s, t))dt ds.

0

Now a suitable application of Theorem 5.2.1 yields

log v(x, y) < ~_l lf2(log c) + ~ ~o p(s, t) dt ds]

(5.10.14)


v(x, y) < exp

( [ jj ~-I

~(log c) +

0 0

p(s, t) dt ds

l)

.

(5.10.15)

The desired inequality in (5.10.11) now follows by using (5.10.15) in u(x, y) < v(x, y). The subdomain E1 of (x, y) in E is obvious. II


547

Another interesting and useful inequality is embodied in the following theorem. Theorem 5.10.3 Let u" E --+ R1, p, q" E --+ R+ be continuous functions and c > 1 be a constant, where E and R1 are as defined in Theorem 5.10.1. Let g(u), g'(u) be as defined in Theorem 5.10.2 and furthermore we assume that g(u) is submultiplicative on R+. If

u(x, y) R1, p" E -~ (0, c~), q" E --> R+ be continuous functions and c >_ 1 be a constant, where E and R1 are as defined in

MULTIDIMENSIONAL NONLINEARINTEGRALINEQUALITIES 551 Theorem 5.10.1. If x y

u(x,y) 0 on R+, the function g(u) is monotonically increasing on (0, c~). Define a function z(x, y) by the fight-hand side of (5.10.35). Then

u(x, y) < z(x, y) and D2Dlz(x, y) 0,

Dlz(x, y) >_0,

0

From

(5.10.37) and using the D2z(x, y) >__0, we observe that

D2(Dlz(x'-Y))\z(x,y)

facts

that

z(x, y)

- w > O, where M'R3+ --+ R+ is continuous.

(i) If x y

u(x,y) 0 for u > 0 and g'(u) > 0 for u >_ O. (i) If ax(X, y), ay(X, y), axy(X, y) exist and are nonnegative continuous functions defined for x, y 6 R+ and

axy(X, y) < q(x, y)g(a(x, y)), for x, y 6 R+, where q(x, y) is a nonnegative continuous function defined for x, y 6 R+, and suppose further that

x y

b(s, t)g(u(s, t)) dt ds,

u(x,y) 0 for x ~ 11, and f(t), h(t) be continuously differentiable functions such that f ' ( t ) ___0, h'(t) > 0 for t ~ IT, then for 0 < x < Xl,

[

O_ O, h'(t) >_ 0 for x ~ I1, t E IT, then for 0 _< x __0, h'(t) >_ 0 for t ~ I t , then for 0 < x 0, ai(xi) >_0 for 1 x ~ Let H" R+ ~ R+ be continuously differentiable function with H(u) > 0 for u > 0 and H'(u) > 0 for u > 0.

f2 is defined as in Section 4.9. Let

(i) If x

U(X)< ~-~ai(xi)-+i----1

f p(s)n(u(s))ds,

x0

for x ~ f2, then for x ~ < x < x*,

iO(aixi,+alxo,)i=

u(x) < G -1 x1

al(Sl)

J

+ .o H(y~in=3ai(xi)+a2(xO)+al(Sl))

ds1

X ]

+ f p(s) ds , xo

where

G(r)- f

ds H(s)'

r > 0, r0 > 0,

ro

and G -1 is the inverse of G, and x* be chosen so that

) xj

G

ai(xi) + a l ( x 0) i=2

+

,

al(s1) x0

n(~-~in=3ai(xi) + a2(x 0) -{- al (s1))

x

+ f p(s) ds ~ D o m ( G -1), xo

for all x lying in the parallelepiped x ~ < x < x* in f2.

dsl


579

(ii) If

U(X)

naii + _r xp(s)( ju(s)+

i= 1

xO

n

/

x

ai(xi) +

i=1

p(y)H(u(y))dy ds, xo

for x 6 f2 then for x ~ _< x _< x**

U(X) < E

)

0, ro > 0,

ro

and W -1 is the inverse function of W, and x** is chosen so that

Xl

I

al(s1)

-Jr" _ (~-~n=3ai(xi) + a2(xO) + al(s1)) _.l_H~.,~--~i=3ai(xi) _l_ a2(x2) 0 .+.

dsl

x

+ / p(s) ds ~ D o m ( W -1), xo

for all x ~ f2 lying in the parallelepiped x ~ < x < x** in S2.

5.12.11 Singare and Pachpatte (1981) Let ~(x), a(x), on f2, and let

b(x) and c(x) be nonnegative continuous functions u(x) be a positive continuous function defined on f2.

defined

580


(i) Let H(r) be a positive, continuous, strictly increasing, convex and submultiplicative function for r > 0, l i m r ~ H ( r ) - c~. Let ct(x),/~(x) be positive continuous functions defined on ~2 with or(x)+/~(x) = 1. If

u(s) > dp(x) - a(s)H -1 I ~ h(tr)H (q~(tr)) dcr

+ / S b(~r)

c(~)H(q~(~)) d~

)1

dcr ,

x

is satisfied for x < s; x, s 6 f2, then

u(s) > ct(s)H -1 [~

{ 1 + f(s)H(a(s)f1-1 (s)) f b(tr) x

l(s))b(~) q- c(~)] d~

x exp

for x < s; x, s 6 ~2. (ii) Let G(r) be

)}'1 do

,

a positive, continuous, strictly increasing, subadditive and submultiplicative function for r > 0 with limr~o~ G(r)= cxz, and let G -1 denote the inverse function of G. If i"e

U(S) > dp(X)-a(s)G -1 I ~fb(cr)G(~(cr)) dcr

Vx

for x < s; x, s 6 fl, then

u(s) > G -1

[6(r {

x exp

for x 1 for x 6 f2 be a continuous and nondecreasing function in x. Let H(u) and H'(u) be as in Theorem 5.7.8. If

j ( j

u(x) < a(x) +

b(y)H

u(y) -t-

xo

c(s)H(u(s))ds

)

dy,

x0

for x ~ f2, then for x ~ f21 C f2,

u(x) < a(x)Q(x), where

Y

])

(1) + I[b(s) + c(s)] ds

Cds

H(s)

(**)

x0

x0

G(r)-

dy,

,

r > O, ro > O,

ro

G -1 is the inverse function of G and X

G(1)+f

[b(s) + c(s)] ds 6 Dom(G -1),

xo

for all x ~ f21 C f2.

5.12.13 Pachpatte (unpublished manuscript) Let u(x), b(x), c(x) and q(x) be nonnegative continuous functions defined on f2 and k > 1 is a constant. Let H(u), H'(u) be as in Theorem 5.7.8. Let g(u) be a continuously differentiable function defined for u > O, g(u)> 0 for u > 0 and g'(u) > 0 for u > 0 and g(u) is submultiplicative. If

X

u(x) __ V2 >__ 0, where k" R~_ x R+ --+ R+ is a continuous function. Let the notation M[x, p, q] be as defined in Section 5.9. (i) If

u(x) < a(x) + b(x)m[x, p, q(y, u(y))], for all x 6 R~_, then

u(x) < a(x) + b(x)M[x, p, q(y, a(y))] exp(M[x, p, k(y, a(y))b(y)]), for all x ~ R~_.

F(u) be a continuous, strictly increasing, convex, submultiplicaF(u) = c~, F -1 denotes the inverse function of F and or(x), fl(x) be continuous and positive functions defined for x ~ R~_ and c t ( x ) + fl(x) = 1. If (ii) Let

tive function for u > O, l i m u _ ~

u(x)

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