Nonlinear Integral Equations in Abstract Spaces

350 Pages·1996·8.85 MB·English

by  Dajun Guo

Nonlinear Integral Equations in Abstract Spaces
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 373
Nonlinear
Integral Equations
in Abstract Spaces
by
Dajun Guo
Shandong University, P.R. China
V. Lakshmikantham
Florida Institute ofTechnology,
Department ofA pplied Mathematics,
Melbourne, U.S.A.
and
Xinzhi Liu
University ofWaterloo, Ontario, Canada
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Guo, Dajun.
Nonl inear integral equations in abstract spaces 1 by DaJun Guo,
V. Lakshmikantham, Xinzhi L1u.
p. cm. -- <Mathematics and its appl1cations; 373l
Includes bibliographical references.
ISBN 978-1-4612-8547-2 ISBN 978-1-4613-1281-9 (eBook)
DOI 10.1007/978-1-4613-1281-9
1. Nonlinear integral equations. 2. Banach spaces.
I. Lakshmikantham, V. II. L1u, Xinzhi. III. Title. IV. Series,
Mathematics and its applications <Kluwer Academic Publishersl ; v.
373.
QA431 . G83 1996
515' .45--dc20 96-26714
Printed on acid-free paper
Ali Rights Reserved
© 1996 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1996
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.
CONTENTS
PREFACE
VII
1 PRELIMINARIES 1
1.0 Introduction 1
1.1 Mean Value Theorems 2
1.2 Measure of Non compactness and Some Fixed Point Theo-
rems 9
1.3 Comparison Theorem for Volterra Integral Equations 26
1.4 Linear Integral Equations in Banach Spaces 33
1.5 Relations between Differential Equations and Integral Equa-
tions in Banach Spaces 43
1.6 Notes and Comments 51
2 NONLINEAR INTEGRAL EQUATIONS IN
BANACH SPACES 53
2.0 Introduction 53
2.1 Equations of Fredholm Type 54
2.2 Equations of Volterra Type 91
2.3 Equations of Hammerstein Type 141
2.4 An Equation Modeling Infectious Disease 161
2.5 Notes and Comments 171
3 NONLINEAR INTEGRO-DIFFERENTIAL
EQUATIONS IN BANACH SPACES 173
3.0 Introduction 173
3.1 First Order Integro-Differential Equations 174
3.2 Second Order Integra-Differential Equations 196
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NONLINEAR INTEGRAL EQUATIONS
Vl
3.3 Integra-Differential Equations on Unbounded Domains 233
3.4 Notes and Comments 239
4 NONLINEAR IMPULSIVE INTEGRAL
EQUATIONS IN BANACH SPACES 241
4.0 Introduction 241
4.1 Impulsive Equations of Volterra Type 242
4.2 Impulsive Equations of Fredholm Type 265
4.3 Impulsive Integra-Differential Equations 275
4.4 Notes and Comments 331
REFERENCES 333
PREFACE
Many problems arising in the physical sciences, engineering, biology and ap
plied mathematics lead to mathematical models described by nonlinear integral
equations in abstract spaces. The theory of nonlinear integral equations in ab
stract spaces is a fast growing field with important applications to a number of
areas of analysis as well as other branches of science. This book is devoted to a
comprehensive treatment of nonlinear integral equations in abstract spaces. It
is the first book that is dedicated to a systematic development of this subject,
and it includes the developments during recent years.
Chapter 1 introduces some basic results in analysis, which will be used in later
chapters. Chapter 2, which is a main portion of this book, deals with nonlin
ear integral equations in Banach spaces, including equations of Fredholm type,
equations of Volterra type and equations of Hammerstein type. Some applica
tions to nonlinear differential equations in Banach spaces are given. We also
discuss an integral equation modelling infectious disease as a typical applica
tion. In Chapter 3, we investigate the first order and second order nonlinear
integro-differential equations in Banach spaces including equations of Volterra
type and equations of mixed type. Chapter 4 is devoted to nonlinear impulsive
integral equations in Banach spaces and their applications to nonlinear impul
sive differential equations in Banach spaces. For the sake of illustrating the
abstract results, several examples on both finite and infinite systems of scalar
integral equations, differential equations, integra-differential equations and im
pulsive equations are given. The notes at the end of each chapter indicate the
sources of the ideas developed. References are included at the end for further
study. This book may be used as a text for graduate students as well as a
reference book for pure and applied mathematicians, scientists in physics and
engineering, researchers in the areas of integral equations, differential equations
and dynamical systems, and graduate students in these disciplines.
We wish to thank Shoun Martin, Diane Crouse and Helen Warren for their
excellent typing of the original manuscript. We would also like to thank Ann
Puncher for formatting the final version of this book.
Vll
NONLINEAR INTEGRAL EQUATIONS
Vlll
Finally, the third author wishes to gratefully acknowledge the support of the
National Sciences and Engineering Research Council of Canada.
Dajun Guo
V. Lakshmikantham
Xinzhi Liu
1
PRELIMINARIES
1.0 INTRODUCTION
This chapter introduces some basic results of nonlinear analysis, which will be
used in the remaining chapters.
Section 1.1 deals with the mean value theorems which are useful in vanous
areas. As an application, a necessary and sufficient condition is given for a
map to be nondecreasing.
In section 1.2, we introduce the concept of measure of noncompactness and
give some fixed points theorems. The concept of fixed point index for strict set
contraction is also discussed.
Section 1.3 considers the comparison theorem for Volterra integral equations.
We also give a comparison theorem for ordinary differential equations.
Section 1.4 deals with linear integral equations in Banach spaces. We give some
existence and uniqueness theorems of solutions in spaces C and L2 for both
equations of Fredholm type and Volterra type.
Finally, in Section 1.5, we discuss the relations between differential equations
and integral equations in Banach spaces. We prove that the initial value prob
lem is equivalent to a nonlinear Volterra equation and the boundary value
problem is equivalent to a nonlinear Fredholm equation.
1