Existence Theory for Nonlinear Integral and Integrodifferential Equations

225 Pages·1998·11.481 MB·English

by  Donal O’Regan, Maria Meehan (auth.)

Existence Theory for Nonlinear Integral and
Integrodifferential Equations
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 445
Existence Theory for
N online ar Integral
and Integrodifferential
Equations
by
Donal O'Regan
Department ofM athematics,
National Un iversity ofI reland,
Galway, Ireland
and
Maria Meehan
Department ofM athematics,
National University oflreland,
Galway,lreland
SPRINGER -SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-6095-0 ISBN 978-94-011-4992-1 (eBook)
DOI 10.1007/978-94-011-4992-1
Printed on acid-free paper
AII Rights Reserved
@1998 Springer Science+Business Media Dordrecht
Origina11y published by Kluwer Academic Publishers in 1998
Softcover reprint of the hardcover 1s t edition 1998
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
MM: To my parents Tony and Mary
DOR: In memory of my father, Cornelius
CONTENTS
Preface ................................................ ix
Chapter 1. Intro"duction and preliminaries .................... 1
Chapter 2. Existence theory for nonlinear Fredholm and Volterra
integrodifferential equations ................................... 14
Chapter 3. Solution sets of abstract Volterra equations ...... 37
Chapter 4. Existence theory for nonlinear Fredholm and Volterra
integral equations on compact intervals ........................ 46
Chapter 5. Existence theory for nonlinear Fredholm and Volterra
integral equations on half-open intervals ....................... 65
Chapter 6. Existence theory for nonlinear nonresonant operator
and integral equations ........................................ 109
Chapter 7. Existence theory for nonlinear resonant operator and
integral equations ............................................ 128
Chapter 8. Integral inclusions ............................. 144
Chapter 9. Approximation of solutions of operator equations on
the haH line .................................................. 157
Chapter 10. Operator equations in Banach spaces relative to the
weak topology ............................................... 176
Chapter 11. Stochastic integral equations .................. 191
Chapter 12. Periodic solutions for operator equations ...... 204
Index ................................................. 216
Preface
The theory of integral and integrodifferential equations has ad
vanced rapidly over the last twenty years. Of course the question of
existence is an age-old problem of major importance. This mono
graph is a collection of some of the most advanced results to date in
this field.
The book is organized as follows. It is divided into twelve chap
ters. Each chapter surveys a major area of research. Specifically,
some of the areas considered are Fredholm and Volterra integral and
integrodifferential equations, resonant and nonresonant problems, in
tegral inclusions, stochastic equations and periodic problems. We
note that the selected topics reflect the particular interests of the
authors.
Donal 0 'Regan
Maria Meehan
CHAPTER 1
INTRODUCTION AND PRELIMINARIES
1.1. Introduction
The aim of this book is firstly to provide a comprehensive existence the
ory for integral and integrodifferential equations, and secondly to present
some specialised topics in integral equations which we hope will inspire fur
ther research in the area. To this end, the first part of the book deals with
existence principles and results for nonlinear, Fredholm and Volterra inte
gral and integrodifferential equations on compact and half-open intervals,
while selected topics (which reflect the particular interests of the authors)
such as nonresonance and resonance problems, equations in Banach spaces,
inclusions, and stochastic equations are presented in the latter part.
Chapter 2 is devoted to the study of integrodifferential equations. In
Section 2.2 we present existence principles for integrodifferential equations
on the compact interval [0, T]. A nonlinear alternative of Leray-Schauder
type will be used to establish an existence principle for the operator equa
tion
y'(t)=Vy(t), a.e. tE[O,T]
(1.1.1) {
=
yeO) Yo.
Two special cases of (1.1.1) will then be considered. Firstly, we examine
(1.1.1) when
(1.1.2) Vy(t) = f(t,y(t),Ky(t)) , a.e. t E [O,T],
where K is either the Fredholm integral operator
loT
=
Ky(t) K.(t,s, y(s)) ds,
or the Volterra integral operator
lot S,
K yet) = K.(t, y(s)) ds.
Secondly, we consider (1.1.1) when
(1.1.3) Vy(t) = h(t) + Ky(t), a.e. t E [O,T],
1
D. O’Regan et al., Existence Theory for Nonlinear Integral and Integrodifferential Equations
© Kluwer Academic Publishers 1998
2 Introduction and preliminaries
where on this occasion, J( is the Fredholm integral operator defined by
loT
Ky(t) = k(t,s)g(s,y(s))ds,
or the Volterra integral operator defined by
lot
](y(t) = k(t,s)g(,~,y(s»)ds.
Existence principles are presented for V in both cases (1.1.2) and (1.1.3).
Section 2.3 deals with the operator equation
y'(t) = Vy(t), a.e. t E [0, T)
( 1.1.4) {
yeO) = Yo,
where this time we are working on the half-open interval [0, T), 0 ~ T ~ 00.
Two existence principles are presented for (1.1.4). The first relies on the
Schauder-Tychonoff Theorem, while the second is based on an idea of Lee
and O'Regan [5]. In the latter existence principle, V must be a Volterra
operator, which basically means that if V is defined as in (1.1.2) or (1.1.3),
the operator J( in each case must be a Volterra integral operator.
The integrodifferential equation
y'(t)=h(t)+I~k(t,,~)g(y(s)ds, a.e.tE[O,T]
( 1.1.5) {
=
yeO) Yo
will be discussed in Section 2.4. Here hELl [0, T] and the kernel k is of
negative type. In [3], existence of a solution of (1.1.5) is guaranteed if, for
all y E R, G(y) defined by G(y) = It g(s)ds, satisfies limlyl-+oo (i(y) = 00,
and if there exists a positive, nondecreasing function u, such that Ig(y)1 ~
u(G(y)). A similar result is also discussed in [7]. In Section 2.4 we present a
variation of this result, but we also obtain a new existence result for (1.1.5).
Our result includes the case when limy-+_oo (i(y) = -00. For example, for
o
~ m ~ n, we can consider functions of the form
A result of Pachpatte [8], where 9 satisfies a growth condition and k is not
necessarily of negative type, concludes Chapter 2.
Chapter 1
Chapter 3 discusses the topological structure of the solution set of the
abstract Volterra equation
x'(t) = V x(t) a.e. t E [0, Tj
{
x(o) = Xo.
By placing mild conditions on the operator V we will show in Section
3.2 that the set of solutions is a R6 set (Le. recall a set is R6 if it is
the intersection of a decreasing sequence of nonempty, compact absolute
retracts). We illustrate out theory by discussing in detail the solution set
of differential and integral equations.
Existence theory for nonlinear Fredholm and Volterra integral equations
on compact intervals is the topic of Chapter 4. Many papers on this topic
have appeared in the literature, and we include some of the more advanced
results here; firstly for completeness, and secondly, since many of our results
on the half-open interval (to be discussed in chapter 5) rely on knowledge
of solutions on the compact interval.
In Section 4.2, existence principles are presented for the Fredholm inte
gral equation
iT
+ yes»~
(1.1.6) yet) = h(t) k(t, s)g(s, ds,
and the Volterra integral equation
it
+
(1.1.7) yet) = h(t) k(t,s)g(s,y(s»ds,
when both are defined on [0, Tj. We first consider the case when h E
£P[O, T], 1 ~ p < 00, and put conditions on k and g to yield a solution
y E LP[O,Tj of (1.1.6) and (1.1.7). We then consider h E C[O,Tj and estab
lish conditions under which (1.1.6) and (1.1.7) have solutions y E C[O, T].
In Section 4.3, we present two results which establish the existence of a
solution y E C[O, TJ of the Volterra integral equation
it
+
(1.1.8) yet) = h(t) k(t, s )g(s, y(s» ds, t E [0, T].
The first result is based on an idea of Brezis and Browder [1], for Fred
holm integral equations. For this result we assume that the nonlinearity g,
satisfies a monotonicity condition, that is,
there exists R> 0, with y get, y) 2': 0, for Iyl 2': R.