Methods in Nonlinear Integral Equations

205 Pages·2002·7.666 MB·English

by  Radu Precup (auth.)

METHODS IN NONLINEAR INTEGRAL EQUATIONS
Methods in Nonlinear
Integral Equations
by
Radu Precup
Department ofA pplied Mathematics,
Babe§-Bolyai University,
Cluj, Romania
Springer-Science+Business Media, B.Y.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
Printed on acidjree paper
ISBN 978-90-481-6114-0 ISBN 978-94-015-9986-3 (eBook)
DOT 10.1007/978-94-015-9986-3
All Rights Reserved
© 2002 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2002.
Softcover reprint of the hardcover 1st edition 2002
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.
To Rodica and Barbu with love
Contents
Preface xi
Notation xiii
o
Overview 1
I FIXED POINT METHODS 11
1 Compactness in Metric Spaces 13
1.1 Hausdorff's Theorem. . . . . 13
1.2 The Ascoli-Arzela Theorem . 15
1.3 The Frechet-Kolmogorov Theorem 18
2 Completely Continuous Operators on Banach Spaces 25
2.1 Completely Continuous Operators 25
2.2 Brouwer's Fixed Point Theorem. 28
2.3 Schauder's Fixed Point Theorem . 32
3 Continuous Solutions of Integral Equations via Schauder's
Theorem 35
3.1 The Fredholm Integral Operator 35
3.2 The Volterra Integral Operator . 37
3.3 An Integral Operator with Delay 39
4 The Leray-Schauder Principle and Applications 43
4.1 The Leray-Schauder Principle ........... . 43
4.2 Existence Results for Fredholm Integral Equations 45
4.3 Existence Results for Volterra Integral Equations 50
Vll
viii Contents
4.4 The Cauchy Problem for an Integral Equation with Delay 53
4.5 Periodic Solutions of an Integral Equation with Delay . . 55
5 Existence Theory in £P Spaces 61
5.1 The Nemytskii Operator ... 61
5.2 The Fredholm Linear Integral Operator 64
5.3 The Hammerstein Integral Operator .. 68
5.4 Hammerstein Integral Equations . . . . 69
5.5 Volterra-Hammerstein Integral Equations 72
References: Part I 77
II VARIATIONAL METHODS 83
6 Positive Self-Adjoint Operators in Hilbert Spaces 85
6.1 Adjoint Operators .................. . 85
6.2 The Square Root of a Positive Self-Adjoint Operator 88
6.3 Splitting of Linear Operators in £P Spaces . . . . . . 90
7 The Fnkhet Derivative and Critical Points of Extremum 97
7.1 The Frechet Derivative. Examples ...... . 97
7.2 Minima of Lower Semicontinuous Functionals . 103
7.3 Application to Hammerstein Integral Equations 107
8 The Mountain Pass Theorem and Critical Points of
Saddle Type 111
8.1 The Ambrosetti-Rabinowitz Theorem . . . . . . . . . . 111
8.2 Flows and Generalized Pseudo-Gradients 117
8.3 Schechter's Bounded Mountain Pass Theorem 121
9 Nontrivial Solutions of Abstract Hammerstein Equations 129
9.1 Nontrivial Solvability of Abstract Hammerstein Equations 129
9.2 Nontrivial Solutions of Hammerstein Integral Equations 132
9.3 A Localization Result for Nontrivial Solutions. . . . . .. 137
References: Part II 145
ix
III ITERATIVE METHODS 149
10 The Discrete Continuation Principle 151
10.1 Perov's Theorem . . . . . . . . . . . . . . . . . . . . 151
10.2 The Continuation Principle for Contractive Maps on
Generalized Metric Spaces. . . . . . . . . . . . . . . 154
10.3 Hammerstein Integral Equations with Matrix Kernels 159
11 Monotone Iterative Methods 163
11.1 Ordered Banach Spaces . . . . . . . . . . . . . . . . 163
11.2 Fixed Point Theorems for Monotone Operators . . . 167
11.3 Monotone Iterative Technique for Fredholm Integral
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
11.4 Minimal and Maximal Solutions of a Delay Integral Equation 177
11.5 Methods of Upper and Lower Solutions for Equations of
Hammerstein Type . .. . . . . . . . . . . . . . . . . . . . . . 184
12 Quadratically Convergent Methods 195
12.1 Newton's Method ......... . 196
12.2 Generalized Quasilinearization for an Integral Equation with
Delay ............................... 201
References: Part III 211
Index 217
Preface
Nonlinear integral equations represent the major source of nonlinear op
erators and, by this, equally, the motor and the testing ground of a vast
domain of nonlinear analysis, namely the theory of nonlinear operators.
This book is intended to present some of the most useful tools of nonlin
ear analysis in studying systems of integral equations (integral equations in
Rn).
The book is divided into three parts: fixed point methods, variational
methods, and iterative methods. Each part contains a number of chap
ters of theory and applications. The methods we deal with are: Schauder's
fixed point theorem; the Leray-Schauder principle; direct variational meth
ods; the mountain pass theorem; the discrete continuation principle; mono
tone iterative techniques; methods of upper and lower solutions; Newton's
method; and the generalized quasilinearization method.
We note that the selected topics reflect the particular interests of the
author and that many contributions to the subject, as well as some other
important treatment methods of the nonlinear integral equations have not
been included for the sake of brevity.
The lists of references, one for each part, include only referenced titles;
they are just intended to guide the reader through the enormous existing
literature and not to provide a complete bibliography on the subject.
The presentation is essentially self-contained and leads the reader from
basic concepts and results to current ideas and methods of nonlinear anal
ysis.
We hope the book will be of interest to graduate students, and theoret
ical and applied mathematicians in nonlinear functional analysis, integral
equations, ordinary and partial differential equations and related fields.
Radu Precup
Notation
R+ set of all nonnegative real numbers
Rn set of all n-tuples x = (Xl, X2, ... , xn)
R+
set of all x E Rn with Xi ~ 0 for all i
x n x; ) 1/2
Ixl Euclidian norm of ERn, Ixl = ( i~
n
(x, y) Euclidian inner product in Rn, (x,y) = 2: XiYi,
i=l
also denoted by X· Y
x5,y natural order relation in R n : Xi 5, Yi for all i
x<y strict order relation in R n : Xi < Yi for all i
x5,a for x E Rn and a E R : Xi 5, a for all i
Lebesgue measure in Rn
J-L
X,Y,Z metric spaces, real Banach or Hilbert spaces
X* dual space of X
1.lx
1·1 norm in X, also denoted by
(., .) inner product in a Hilbert space; also for u E X
and u* E X*, (u*,u) is the value of u* at u, u* (u)
Br (u;X) open ball {v EX: lu - vi < r} ( Br (u) for short)
Br (u; X) closed ball {v E X; lu - vi 5, r} (Br (u) for short)
U, intU closure of U, interior of U
au boundary of U : au = U\ int U
convA convex hull of A
convA closed convex hull of A
c
k (O;Rn) set of k-times continuously differentiable functions
u: 0 --t Rn (0 C RN open)
c
k (0; Rn) space of all functions u E Ck (0; Rn), u = (U1, ... , un)
such that DaUi admits a continuous extension to 0 for
N
all i and a = (a1, ... , aN) with lal = 2: aj 5, k. Here
j=l
Da = alai / axr1 ... axc;.r
xiv
lul ma~ lu (x)1 (0 C RN bounded open, u E C (0; Rn))
oo
xEn
Ck(O)
stands for Ck (0; R)
Ck [a,b] stands for Ck ([a, b])
V(O;Rn) space of all measurable functions u : 0 --+ R n with
In lu (x)IP dx < 00 (0 C RN open, 1:::; p < 00)
1·l norm in V (0; Rn), lul = Un lu (x)IP dx)l/P
p p
(., ·)2 inner product in L2 (0; Rn) ,
In
(u, v)2 = (u (x) , v (x)) dx
Loo (0; Rn) space of all measurable functions u : 0 --+ Rn for which
there is a constant c with lu (x)1 :::; c for a.e. x E 0
1·100 norm in Loo (0; Rn),
lul = inf {c: lu (x)1 :::; c a.e. on O}
oo
V(O) stands for V (0; R) (1:::; p :::; 00)
V (a, b; Rn) stands for V (0; Rn) with 0 = (a, b)
V (a, b) stands for V (a, b; R)
V(O;X) space of all stongly measurable functions u : 0 --+ X
with lulx E V (0)
Basic inequality:
lin in
u (x) dxl :::; lu (x)1 dx, u E L1 (0; Rn) ,
where 0 C RN is open, 1.1 is the Euclidian norm in Rn, and
in (in in
u (x) dx = udx) dx, ... , (x) dX) .
Un
Here u = (U1, U2, ... , un).