Positive Solutions of Differential, Difference and Integral Equations

425 Pages·1999·12.96 MB·English

by  Ravi P. Agarwal

Positive Solutions of Differential,
Difference and Integral Equations
Positive Solutions
of Differential,
Difference and Integral
Equations
by
Ravi P. Agarwal
Department ofM athematics.
National University of Singapore.
Singapore
Donal O'Regan
Department ofM athematics.
National University ofI reland.
Galway. Ireland
and
Patricia J. Y. Wong
Division ofM athematics.
Nanyang Technological University.
Singapore
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5153-0 ISBN 978-94-015-9171-3 (eBook)
DOI 10.1007/978-94-015-9171-3
Printed on acid-free paper
AU Rights Reserved
© 1999 Springer Science+-Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1999
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
Dedicated to our mothers
Godavari Agarwal
Eileen O'Regan
Phuah Liew Hong
Contents
Preface ix
Ordinary Differential Equations
1. First Order Initial Value Problems 1
2. Second Order Initial Value Problems 11
3. Positone Boundary Value Problems 19
4. Semi-positone Boundary Value Problems 29
5. Semi-infinite Interval Problems 40
6. Mixed Boundary Value Problems 47
7. Singular Boundary Value Problems 63
8. General Singular and Nonsingular Boundary Value Problems 86
9. Quasilinear Boundary Value Problems 106
10. Delay Boundary Value Problems 110
11. Coupled System of Boundary Value Problems 119
12. Higher Order Sturm-Liouville Boundary Value Problems 131
13. (n,p) Boundary Value Problems 190
14. Focal Boundary Value Problems 210
15. General Focal Boundary Value Problems 222
16. Conjugate Boundary Value Problems 236
Difference Equations
J 7. Discrete Second Order Boundary Value Problems 261
18. Discrete Higher Order Sturm-Liouville Boundary Value Problems 279
19. Discrete (n,p) Boundary Value Problems 315
20. Discrete Focal Boundary Value Problems 325
21. Discrete Conjugate Boundary Value Problems 353
Integral and Integrodifferential Equations
22. Volterra Integral Equations 370
23. Hammerstein Integral Equations 381
24. First Order Integrodifferential Equations 386
References 395
Authors Index 412
Subject Index 417
Preface
In analyzing nonlinear phenomena many mathematical models give rise
to problems for which only nonnegative solutions make sense. Therefore,
since the publication of the monograph Positive Solutions of Operator Equa
tions in the year 1964 by the academician M.A. Krasnosel'skiit, hundreds
of research articles on the theory of positive solutions of nonlinear problems
have appeared. In this vast field of research, particular attention has been
focused on the Sturm-Liouville boundary value problem
y" + >..f(x, y, yl) = 0, a < x < b, >.. > 0
{ oy(a) + (3yl(a) =
Cl
=
'"Yy(b) - tSy'(b) C2.
For example the special case
y" + >..q(x)g(y) = 0, 0 < x < 1
{
y(O) = 0 = y(l)
and its n-dimensional analog
du + >..q(x)g(u)
{
ulan
=
0
whose positive solutions have been studied and constructed arise in a vari
ety of interesting applications such as gas diffusion through porous media,
thermal self-ignition of a chemically active mixture of gases in a vessel.
With q(x) = 1 the cases g(y) = 0(1 + (3 - y)e-'Y/Y b large), g(y) = e-1/lyl
=
and g(y) (3(r - y )e-k/(1+y), respectively, arise in models in catalytic
theory, chemically reacting systems and adiabatic tubular reactors. The
special case q(x) = q(constant» 0, g(y) = eUY occurs in applications in
volving the diffusidn of heat generated by positive temperature-dependent
=
sources. For instance, when 1 it arises in the analysis of Joule losses
(J'
in electrically conducting solids as well as in frictional heating. This model
was proposed by Gelfand in the year 1959.
The last twenty years have witnessed hundreds of research articles guar
anteeing the existence of the positive solutions to
=
y"+>..q(X)g(x,y) 0, O<x<l
{
y(O) = Cl 2: 0, y(l) = 0,
The theory of the above boundary value problem with q(x) > 0, x E (0,1)
rely upon the form and/or properties ofthe function g(x, y). The cases when
x
9 is superlinear, sublinear, 9 2': O(Positone), 9 :::; O(Semi-Positone), and 9
is singular at y = 0 have been discussed extensively. In this classification
two significant examples are: the Emden-Fowler problem
y" + )..q(x)y-1' = 0, 0 < x < 1, 'Y > 0
{
y(O) = 0 = y(l)
which appears in various branches of fluid dynamics; and the Thomas
Fermi problem
{ y" - x-1/2y3/2 = 0, 0 < x < 1
y(O) = 1, y(l) = 0
which was derived in the year 1927 for determining the electrical potential
in an atom.
The boundary value problem
{ y " + -N--1Y, +)..g ( x, y ) = 0, 0 < x < 1, N > 2
y'(O) =x 0, 'Yy(l) - 8y'(1) = C2
finds applications in combustion theory, and models various physiological
processes. For example, when N = 3, 'Y > 0, 8 = -1, g(x,y) = -ay/(y +
k), a > 0, k > 0 it arises in the study of steady-state oxygen diffusion
in a cell with Michaelis-Menten kinetics. The solution y is a measure of
the oxygen tension and x = 1 corresponds to the boundary of the cell
membrane. If g(x, y) = ke-{3y, k > 0, (3 > 0 heat conduction in the human
brain is modeled. Here 9 is the heat production rate per unit volume, y
is the absolute temperature and x is the radial distance from the center.
Evidently in these models only positive solutions are significant.
The positive solutions of boundary value problems on semi-infinite in
tervals such as
y"+)..q(x)g(x,y) = 0, O<x<oo
{ ay(a) + (3y'(a) = Cl
Y is bounded on [0,00); or lim y(x) exists; or lim y'(x) = 0
x-too x-too
are also of substantial interest.
In chapters 1 and 2 of this monograph we offer existence criteria for the
positive solutions of first and second order singular initial value problems.
Then in chapters 3 - 16 we present existence theory of the positive solutions
for all the above classes of boundary value problems involving second and
higher order ordinary differential equations.
Until about 1970, Difference Equations were considered as discretiza
tions of differential equations. Further, it was tacitly taken for granted that
xi
the theories of difference and differential equations are parallel. However,
striking diversities and wide applications reported in the last two decades
have made difference equations one of the major areas of research. In chap
ter 17 - 21 we shall offer discrete analogs of the several results presented in
earlier chapters.
In the last three chapters we develop existence theory of positive solu
tions of integral and integrodifferential equations.
We would like to take this opportunity to thank P. Eloe, J. Henderson,
V. Lakshmikantham and A. Peterson for their continuous help and assis
tance. Special thanks go to Sadhna for her constant encouragement and
support.
Ravi P Agarwal
Donal 0 'Regan
Patricia J.y' Wong
CHAPTER 1
First Order Initial Value Problems
1.1. Introduction.
In this chapter we shall offer two very general existence criteria for the
initial value problem
y' = q(t)f(t,y), 0 < t < T < 00
(1.1)
=
yeO) 0
where the nonlinearity will be allowed to change sign, and can have singu
= =
larities at y 0 and/or t O.
The literature on the existence of solutions of smooth initial value prob
lems of the form (1.1) is extensive, e.g. Agarwal and Gupta [6], Piccinini,
Stampacchia and Vidossich [256]. In recent years, much attention has been
paid to the study of singular initial value problems (singularities only at
t = 0) of the type (1.1) with f: [0, T] x R - R continuous and
q E L1 [0, T]. In fact, several interesting existence results have been estab
lished in Bobisud and O'Regan [70], Frigon and O'Regan [142], Lee and
O'Regan [193], Nkashama [226], O'Regan [247]. However, very little is
known when the nonlinearity f has a singularity at y = 0, i.e. when
f: [0, T] x (0,00) - R is continuous. A general theory was developed in
Bobisud and O'Regan [70] when f(t,y) = Iyl-ag(t,y) with > 0 and
Q
g: [0, T] x R - R continuous. Their technique involves transforming the
= =
problem with singularities at y 0 and t 0 to a problem which is only
o.
singular at t =
Here, we shall follow Agarwal and O'Regan [30] to examine the general
singular problem (1.1), i.e. when f: [O,T] x (0,00) - R continuous, and
the nonlinearity f is allowed to change sign. Our theory as a special case
includes the results proved in [70].
n
By a solution to (1.1) we mean a function y E C[O, T] C1 (0, T] which
satisfies the differential equation on (0, T) and the stated initial condition.
1.2. Preliminary Results.
To prove our main results we shall need the following:
Definition 1.1. Let -00 < a < b < 00. A collection of real valued
functions A = {Ii: [a, b] - R} is said to be
R. P. Agarwal et al., Positive Solutions of Differential, Difference and Integral Equations
© Springer Science+Business Media Dordrecht 1999
2 Chapter 1
1. uniformly bounded (bounded) if there exists a constant M > 0 with
Ifi(t)1 ~ M for all t E [a, b] and all fi E A, and
2. equicontinuous if for every 10 > 0 there exists 8 = 8(10) > 0 such that
ifto,tlE[a,b] then Ito-tIl <8 implies Ifi(tO)-li(tdl<€ for all
fi E A.
Theorem 1.1 (Arzela-Ascoli Theorem) [91,237]. A subset A of
C([a, b], JR) is relatively compact if and only if it is bounded and equicon
tinuous.
Theorem 1.2 (Existence Principle) [247]. With respect to the initial
value problem
yl = q(t)g(t,y), 0 < t < T
(1.2)
=
y(O) 0
suppose that g: [0, T] x JR JR is continuous, and
--t
(i) q E C(O, T] with q> 0 on (0, T] and JoT q(s)ds < 00.
(a) Assume there is a constant M, independent of >., with
Ilyll sup ly(t)1 =I M
tE[O,T]
for any solution y to
yl = >.q(t)g(t,y), 0< t < T
(1.3}.>.
=
y(o) 0
for each >. E (0,1). Then (1.2) has a solution.
(b) Assume there exists a constant M such that Ig( t, y) I < M for
t E [0, T] and y E JR. Then (1.2) has a solution.
1.3. Existence Results.
Througout this chapter we will assume f(t,y) = f(y) for t E [0, fl.
Our first main result is embodied in the following:
Theorem 1.3. Suppose in addition to condition (i) the following hold:
(ii) f: [O,T] x (0,00) --t JR is continuous,
°
(iii) If(t,y)1 ~ g(y) + h(y) on [O,T] x (0°,0 0) with 9 > continuous
and nonincreasing on (0,00), hi9 ? non decreasing on (0,00),
(iv) let n E {2, 3, ... } and associated with each n, we have a constan°t
=
Pn such that {Pn} is a nonincreasing sequence with limn Pn
-+DO
and such that for Tin ~ t ~ T we have f(t,Pn)? 0,