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Contributions to Iterative Algorithms for Non-linear Equations in Banach Spaces

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TABLE OF CONTENTS

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 s of Hammerstein Type 46

2 Strong Convergence Theorem for Approximation of of s of Hammerstein Type 47

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.2 Main . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Approximation of of s of Hammerstein Type 55

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Main . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

II Iterative Algorithm for Common Fixed Points of a 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

III Algorithms for Common 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 Equilibrium Problems 73

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 Main . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3 Application to optimization problem . . . . . . . . . . . . . . 86

6 An Iterative Method for Non-expansive s, Variational Inclusions and Equilibrium Problems 88

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.2 Main . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8 Strong Convergence Theorems for Nonlinear Mappings, Variational Inequality Problems and System of Mixed Equilibrium Problems 114

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8.2 Main . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 , 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,

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.

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)].

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