Project Materials

GENERAL

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

Do You Have New or Fresh ic? Send Us Your Topic

Contributions to Iterative s for Non-linear Equations in s

OF CONTENTS

Acknowledgements vi

Abstract viii

1 General Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Iterative algorithms for Hammerstein equations . . . . . . . . 1

1.3 s for common xed points . . . . . . . . . . . . . . 10

1.4 for common solutions of three problems . . . . . . 27

1.5 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

I Approximation of Solution of Equations of Hammerstein Type 46

2 Strong Convergence Theorem for Approximation of Solutions of Equations of Hammerstein Type 47

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

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

3 Approximation of Solutions of Generalized Equations of Hammerstein Type 55

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

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

II Iterative for Common Fixed Points of a Family of 64

4 Strong Convergence Theorems for a Mann-Type Iterative Scheme for a Family of Lipschitzian 65

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Main . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

III s for Common Solutions of Common Fixed Point Problems for a Family of 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7 A New Iterative Scheme for a Countable Family of Relatively Non-expansive and an Equilibrium Problem in
s 104

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.2 Main . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

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

8 Strong Convergence Theorems for , Variational Inequality Problems and System of Generalized 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 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. s 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.

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 ’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 ’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)].

Not What You Were Looking For? Send Us Your Topic

INSTRUCTIONS AFTER PAYMENT

After making payment, kindly send the following:
  • 1.Your Full name
  • 2. Your Active Email Address
  • 3. Your Phone Number
  • 4. Amount Paid
  • 5. ect ic
  • 6. Location you made payment from

» Send the above details to our email; contact@premiumresearchers.com or to our support phone number; (+234) 0813 2546 417 . As soon as details are sent and payment is confirmed, your project will be delivered to you within minutes.

Leave a Reply

Your email address will not be published.

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Advertisements