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**ABSTRACT**

The work of Osilike and Isiogugu, Nonlinear Analysis, 74 (2011), 1814-1822 on weak and strong

convergence theorems for a new class of k-strictly pseudononspreading mappings in real Hilbert

spaces is reviewed. We studied in detail this new class of mappings which is more general than

the class of nonspreading mappings studied by Kurokawa and Takahashi, Nonlinear Analysis 73

(2010) 1562-1568. Many incisive examples establishing the relationship of the class of k-strictly

pseudononspreading mappings and several other important classes of operators are presented. In-

teresting properties of k-strictly pseudononspreading mappings and weak and strong convergence

theorems for approximation of its xed points which appeared in the cited work of Osilike and

Isiogugu were studied and presented.

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**TABLE OF CONTENTS**

Acknowledgment i

Certication ii

Approval iii

Abstract v

Dedication vi

1 Introduction 1

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

1.2 Background of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Literature Review 4

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Preliminaries 11

3.1 Denition of some terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Basic facts in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.2 Demiclosedness Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Main Result 20

4.1 k-strictly Pseudononspreading Mapping . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Properties of k-strictly Pseudononspreading Mappings . . . . . . . . . . . . . . . . 22

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4.2.1 Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2.2 Proposition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3 Weak and strong convergence theorem for nonspreading-type mappings in a Hilbert

space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Summary 35

**CHAPTER ONE**

INTRODUCTION

1.1 General Introduction

The content of this thesis falls within the general area of functional analysis, in particular, nonlinear

operator theory; an area which has attracted the attention of prominent mathematicians due to its

diverse applications in numerous elds of science. In this thesis, we concentrate on an important

topic in this area { Weak and strong convergence theorems of nonspreading type mappings in a

Hilbert space.

In this chapter, the background of our research work will be given; this will reveal how relevant our

work is. Then in chapter two, we shall review the research work carried out in the area of research

described in this thesis. Some basic denitions and fundamental tools we used in our work will be

given in chapter three as preliminaries, while our main results will be presented in chapter four.

In chapter ve, conclusions will be given.

1.2 Background of study

Fixed point theory has been an important area of mathematics due to its applications in several

areas of research such as in Optimization, Economics, and Evolution Equations, to mention but a

few. For example, consider the problem of nding the equilibrium points of the system described

by the following equation

du

dt

+ Au(t) = 0 (1.1)

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where A : D(A) H ! H is a nonlinear map and H a real Hilbert space. At equilibrium,

du

dt = 0. Thus, the original problem is reduced to the problem of nding solutions of the equation:

Au = 0 (1.2)

i.e, nding the zeros of A. Several problems arising in Reservoir Engineering, Economics, Physics

to mention but a few, can be modelled in the form of equation (1.2). Since generally A is nonlinear,

there is no closed form solution of this equation. To solve equation (1.2), where A is a multi-valued

monotone map, Felix Browder dened an operator T : H ! H by T := I A, where I is the

identity map on H. He called such T a pseudocontraction. It is easy to see that xed points of

T correspond to zeros of A which in turn correspond to equilibrium points of dynamical system

described by equation(1.1). As a result of this, the study of xed point theory of pseudocontractive

maps and their types has attracted the interest of numerous scientists and researchers.

Kohsaka and Takahashi introduced an important class of mappings called the class of nonspread-

ing mappings. They obtained a xed point theorem for a single valued nonspreading mapping

in Banach space. Furthermore, they obtained a common xed point theorem for a commutative

family of nonspreading mappings in Banach space.

Let C be a nonempty closed and convex subset of a real smooth, strictly convex and re exive

real Banach space, a map T : C ! C is nonspreading if 8 x; y 2 C

(Tx; Ty) + (Ty; Tx) (Tx; y) + (Ty; x): (1.3)

where (x; y) = kxk2 2hx; j(y)i + kyk2; 8 x; y 2 E and J : E ! 2E

dened by

Jx = fj(x) 2 E : hx; j(x)i = kxkkj(x)kg; 8x 2 E, where h:; :i is the duality pairing between

x 2 E and j(x) 2 E, so that hy; j(x)i = (j(x))(y).

We observe that if E is a real Hilbert space, then j is the identity and

(x; y) = kxk2 2hx; yi + kyk2 = kx yk2.

Thus, if C is a nonempty closed and convex subset of a real Hilbert space, then

T : C ! C is nonspreading if

2kTx Tyk2 kTx yk2 + kTy xk2; 8 x; y 2 C: (1.4)

It is shown in Lemma 3.22 that inequality 1.4 is equivalent to

kTx Tyk2 kx yk2 + 2hx Tx; y Tyi 8x; y 2 C.

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S. Lemoto and W. Takahashi obtained some fundamental properties for nonspreading mapping

in a Hilbert space. Furthermore, they studied the approximation of common xed points of non-

expansive mappings and nonspreading mappings in a Hilbert space.

Y. Kurokawa and W. Takahashi obtained a weak convergence theorem of Bailon’s type for non-

spreading mapping in Hilbert space, using an idea of mean convergence they proved a strong

convergence theorem for nonspreading mapping in a Hilbert space.

Osilike and Isiogugu [Osilike and Isiogugu, 2011] introduced a new class of nonspreading type

mappings which is more general than the class studied by Kurokawa and Takahashi. Following the

terminology of Browder and Petryshn they called a mapping

T : C ! C k strictly pseudononspreading if there exists k 2 [0; 1) such that

kTx Tyk2 kx yk2 + kkx Tx (y Ty)k2 + 2hx Tx; y Tyi 8x; y 2 C (1.5)

Our main focus in this thesis, is to review the work done by Osilike and Isiogugu in their paper

titled “Weak and strong convergence theorem for nonspreading type mapping in Hilbert space”

which appeared in Nonlinear Analysis, Vol 74 (2011), 1814-1822.

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