Singular Differential and Integral Equations with Applications

412 Pages·2003·11.5 MB·English

by  Ravi P. Agarwal

SINGULAR DIFFERENTIAL AND INTEGRAL EQUATIONS WITH APPLICATIONS
Singular Differential and Integral
Equations with Applications
by
Ravi P. Agarwal
Department of Mathematical Sciences,
Florida Institute of Technology,
Melbourne, Florida, U.S.A.
and
Donal 0' Regan
Department of Mathematics,
National University of Ireland,
Galway, Ireland
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-6356-4 ISBN 978-94-017-3004-4 (eBook)
DOI 10.1007/978-94-017-3004-4
Printed on acid-free paper
Ali Rights Reserved
© 2003 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2003
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or
otherwise, without written permission from the Publisher, with the exception
of any material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work.
This book is dedicated to our daughters
Sheba Agarwal, Aoife Frances O'Regan
Contents
Preface 1x
Chapter 1 Differential Equations Singular
in the Independent Variable
1.1. Introduction 1
1.2. Preliminaries 2
1.3. Initial Value Problems 3
1.4. Boundary Value Problems 7
1.5. Bernstein-Nagumo Theory 13
1.6. Method of Upper and Lower Solutions 16
1. 7. Solutions in Weighted Spaces 25
1.8. Existence Results Without Growth Restrictions 29
1.9. Nonresonant Problems 38
1.10. Nonresonant Problems of Limit Circle Type 58
1.11. Nonresonant Problems of Dirichlet Type 66
1.12. Resonance Problems 89
1.13. Infinite Interval Problems I 107
1.14. Infinite Interval Problems II 133
Chapter 2 Differential Equations Singular
in the Dependent Variable
2.1. Introduction 144
2.2. First Order Initial Value Problems 145
2.3. Second Order Initial Value Problems 156
2.4. Positone Problems 163
2.5. Semipositone Problems 173
2.6. Singular Problems 178
2.7. An Alternate Theory for Singular Problems 181
Vlll
2.8. Singular Semipositone Type Problems 199
2.9. Multiplicity Results for Positone Problems 208
2.10. General Problems with Sign Changing Nonlinearities 210
2.11. Problems with Nonlinear Boundary Data 238
2.12. Problems with Mixed Boundary Data 247
2.13. Problems with a Nonlinear Left Hand Side 257
2.14. Infinite Interval Problems I 278
2.15. Infinite Interval Problems II 287
Chapter 3 Singular Integral Equations
3.1. Introduction 298
3.2. Nonsingular Integral Equations 299
3.3. Singular Integral Equations with a Special Class of Kernels 306
3.4. Singular Integral Equations with General Kernels 312
3.5. A New Class of Integral Equations 318
3.6. Singular and Nonsingular Volterra Integral Equations 325
Problems 337
References 379
Subject Index 401
Preface
In the last century many problems which arose in the science, engineer
ing and technology literature involved nonlinear complex phenomena. In
many situations these natural phenomena give rise to (i). ordinary differ
ential equations which are singular in the independent and/or dependent
variables together with initial and boundary conditions, and (ii). Volterra
and Fredholm type integral equations. As one might expect general exis
tence results were difficult to establish for the problems which arose. Indeed
until the early 1990's only very special examples were examined and these
examples were usually tackled using some special device, which was usually
only applicable to the particular problem under investigation. However in
the 1990's new results in inequality and fixed point theory were used to
present a very general existence theory for singular problems. This mono
graph presents an up to date account of the literature on singular problems.
One of our aims also is to present recent theory on singular differential
and integral equations to a new and wider audience. The book presents a
compact, thorough, and self-contained account for singular problems. An
important feature of this book is that we illustrate how easily the theory
can be applied to discuss many real world examples of current interest.
In Chapter 1 we study differential equations which are singular in the
independent variable. We begin with some standard notation in Section 1.2
and introduce LP-Caratheodory functions. Some fixed point theorems, the
Arzela-Ascoli theorem and Banach's theorem are also stated here. These
results are used throughout this monograph. In Section 1.3 we discuss
general existence theory for the initial value problems. First we state the
Picard-Lindelof, Peano's and a local existence theorem in the Caratheodory
setting, and then we prove some very general existence criteria. In Section
1.4 we provide general principles which can be used to establish existence
to second order boundary value problems in the Caratheodory setting. Sec
tion 1.5 uses the existence principles of Section 1.4 to present a Bernstein
Nagumo theory for the general Sturm-Liouville problems. In Section 1.6
we discuss the method of upper and lower solutions to singular second order
differential equations subject to Sturm-Liouville boundary data. In partic
ular we show that our results are easily applicable to a problem occuring
in circular membrane theory. Section 1. 7 discusses solutions to singular
X
boundary value problems in a weighted Banach space. In Section 1.8 we
establish existence criteria for boundary value problems when the nonlin
earity has no growth restriction. The ideas rely on the notion of upper
and lower surfaces for the problem, i.e., surfaces having a particular form
and on which the nonlinearity has a given sign. In Section 1.9 we dis
cuss nonresonant problems for second order differential equations together
with Sturm-Liouville, Neumann and periodic boundary date. Nonresonant
problems of limit circle type and Dirichlet type are considered in Sections
1.10 and 1.11 respectively. In Section 1.12 we examine systematically res
onant problems involving second order differential equations together with
Dirichlet, mixed, Neumann and periodic boundary data. In Sections 1.13
and 1.14 we establish general existence theory for boundary value prob
lems over infinite intervals. In particular our theory includes a discussion
of problems arising (i). in the study of plasma physics, (ii). in determining
the electrical potential in an isolated neutral atom, (iii). in modelling phe
nomena which arises in the theory of shallow membrane caps, (iv). in the
theory of colloids, and (v ). in the flow and heat transfer over a stretching
sheet.
In Chapter 2 we provide existence theory for differential equations which
are singular in the dependent variable. In Section 2.2 we present general
existence principles for first order singular initial value problems where our
nonlinearity may change sign. Second order singular initial value problems
are studied in Section 2.3. In Section 2.4 we provide existence criteria for
second order singular positone boundary value problems. In Section 2.5 we
prove a very general existence result for second order semipositone prob
lems. Using the method of upper and lower solutions in Section 2.6 we
present as existence result for second order singular problems where the
nonlinearity may change sign. A very general existence theory for second
order singular positone problems and problems with sign changing nonlin
earities is developed in Section 2. 7. In Sections 2.8 and 2.9 we use fixed
point theory in cones to establish, respectively, an existence criterion for
singular semipositone type problems, and multiplicity results for positone
problems. In Section 2.10 we discuss in detail second order singular bound
ary value problems where our nonlinearity involves y' and may change sign.
An existence theory for second order singular problems involving nonlinear
boundary data is presented in Section 2.11. In particular we show that our
theory is easily applicable to a problem occuring in the membrane response
of a spherical cap. In Section 2.12 we study second order singular boundary
value problems with sign changing nonlinearities involving mixed bound
ary data. Section 2.13 provides existence theory for second order singular
boundary value problems involving differential equations with a nonlinear
left hand side. In Sections 2.14 and 2.15 we provide general existence theory
Xl
for singular boundary value problems over infinite intervals. In particular
our theory includes a discussion of problems arising (i). in the unsteady
flow of a gas through a semi-infinite porous medium, and (ii). in the theory
of draining flows.
In Chapter 3 we discuss some recent results for Fredholm and Volterra
integral equations, which deal with the existence of positive (and possibly
multiple) solutions of certain classes of these equations. In Section 3.2
we provide some existence results for the nonsingular Fredholm integral
equations. The existence of a positive solution of the singular Fredholm
integral equations is considered in Section 3.3. A certain class of kernels is
examined and an application to a certain boundary value problem is then
discussed. The results of Section 3.3 place rather restrictive conditions on
the kernels, but allow us to consider a rather large class of functions which
appear in the integral equation. In Section 3.4 we consider the opposite
scenario, the conditions on the kernels are quite general, but the class of
functions which appear in the integral equation is restricted. In Section
3.5 we examine a new class of singular integral equation. The conditions
placed on the kernels are motivated by a problem in Homann flow. Here
for singular Volterra integral equations the existence of positive, continuous
solutions is examined. We continue with the singular Volterra integral
equation in Section 3.6. The existence theory presented in this section
relies on the ability to obtain solutions of certain integral inequalities.
Next, we list several problems which are directly related to the theory
we presented in earlier chapters. Most of these problems are taken from
the current literature and all the tools needed to discuss them are in this
book. Indeed we encouraged the reader to solve these problems without
consulting the literature. We assure the reader that their effort will indeed
be rewarding.
We hope this book is timely and will stimulate further research in this
area.
It is impossible to acknowledge individually colleagues and friends to
whom we are indebted for assistance, inspiration and criticism during the
preparation of this book. We must, however, express our appreciation and
thanks to Professors V. Lakshmikantham and Maria Meehan for their con
tinuous support. We also thank Sadhna for assistance in typing and proofs
reading.
Ravi P Agarwal
Donal O'Regan