Project Materials

Contributions to Iterative Algorithms for Nonlinear Equations in Banach Spaces

TABLE OF CONTENTS

Acknowledgements vi
Abstract viii
1 General Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Iterative algorithms for Hammerstein equations . . . . . . . . 1
1.3 Algorithms for common xed points . . . . . . . . . . . . . . 10
1.4 Algorithm for common solutions of three problems . . . . . . 27
1.5 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
I Approximation of Solution of Equations of Hammer-
stein Type 46
2 Strong Convergence Theorem for Approximation of Solutions of Equations of Hammerstein Type 47
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Approximation of Solutions of Generalized Equations of Ham-
merstein Type 55
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
II Iterative Algorithm for Common Fixed Points of a
xii
Family of Mappings 64
4 Strong Convergence Theorems for a Mann-Type Iterative
Scheme for a Family of Lipschitzian Mappings 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
III Algorithms for Common Solutions of Common Fixed
Point Problems for a Family of Nonlinear Maps; Variational Inequality Problems and Equilibrium Problems 72
5 An Iterative Method for Fixed Point Problems, Variational
Inclusions and Generalized Equilibrium Problems 73
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 Application to optimization problem . . . . . . . . . . . . . . 86
6 An Iterative Method for Non-expansive Semigroups, Variational Inclusions and Generalized Equilibrium Problems 88
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7 A New Iterative Scheme for a Countable Family of Relatively Non-expansive Mappings and an Equilibrium Problem
in Banach Spaces 104
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8 Strong Convergence Theorems for Nonlinear Mappings, Variational Inequality Problems and System of Generalized Mixed
Equilibrium Problems 114
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9 Conclusions and Future Work 132
9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9.2 Suggestions For Future Work . . . . . . . . . . . . . . . . . . 133

CHAPTER ONE

General Introduction

1.1 Introduction

The contributions of this thesis fall within the general area of nonlinear functional analysis, an area with vast amount of applicability in recent years, as such becoming the object of an increasing amount of study. We devote our
attention to three important topics within the area.

1. Approximation of solution of nonlinear equations of Hammerstein type.

2. Iterative algorithms for common xed points of a family of mappings

and,

3. Algorithms for common solutions of common xed point problems for a family of nonlinear maps; variational inequality problems; and equilibrium problems.

1.2 Iterative algorithms for Hammerstein equations

A nonlinear integral equation of Hammerstein type (see, e.g., Hammerstein [102]) is one of the form
u(x) +
Z

k(x; y)f(y; u(y))dy = h(x) (1.2.1)
where dy is a -nite measure on the measure space
; the real kernel k
is dened on

; f is a real-valued function dened on
R and is,
1
General Introduction 2
in general, nonlinear and h is a given function on
. If we now dene an
operator K by
Kv(x) =
Z

k(x; y)v(y)dy; x 2
;
and the so-called superposition or Nemytskii operator F by Fu(y) := f(y; u(y)) then, the integral equation (1.2.1) can be put in operator theoretic form as follows:
u + KFu = 0; (1.2.2)

where, without loss of generality, we have taken h 0. Interest in equation (1.2.2) stems mainly from the fact that several problems that arise in differential equations, for instance, elliptic boundary value problems whose linear parts possess Green’s functions can, as a rule, be transformed into the form (1.2.2). Among these, we mention the problem of the forced oscillations of nite amplitude of a pendulum (see, e.g., Pascali and Sburlan [152], Chapter IV).

Example 1.2.1 The amplitude of oscillation v(t) is a solution of the problem

d2v
dt2 + a2 sin v(t) = z(t); t 2 [0; 1]
v(0) = v(1) = 0;
(1.2.3)

where the driving force z(t) is periodical and odd. The constant a 6= 0 depends on the length of the pendulum and on gravity. Since the Green’s function for the problem
v
00(t) = 0; v(0) = v(1) = 0;
is the triangular function
k(t; x) =

t(1 􀀀 x); 0 t x;
x(1 􀀀 t); x t 1;
problem (1.2.3) is equivalent to the nonlinear integral equation
v(t) = 􀀀
Z 1
0
k(t; x)[z(x) 􀀀 a2 sin v(x)]dx: (1.2.4)
If Z 1
0
k(t; x)z(x)dx = g(t) and v(t) + g(t) = u(t);
then (1.2.4) can be written as the Hammerstein equation
u(t) +
Z 1
0
k(t; x)f(x; u(x))dx = 0;
where f(x; u(x)) = a2 sin[u(x) 􀀀 g(x)].

General Introduction 3

Equations of Hammerstein type play a crucial role in the theory of optimal control systems and in automation and network theory (see, e.g., Dolezale [90]). Several existence and uniqueness theorems have been proved for equations of the Hammerstein type (see, e.g., Brezis and Browder [20, 21, 22], Browder [29], Browder and De Figueiredo [33], Browder and Gupta [32], Chepanovich [47], De Figueiredo [91]).