WEAK AND STRONG CONVERGENCE OF ISHIKAWA'S ITERATION METHOD FOR LIPSCHITZ - HEMICONTRACTIVE MAPPINGS

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ABSTRACT

It is our purpose in this project to introduce a new family of mappings known as - hemicontractive mappings. Also, certain properties of the xed point set of the new class of mappings are demonstrated together with the relationship between the new class of mappings and other related families of mappings. Again, weak and strong convergence of Ishikawa's iteration to the xed point of Lipschitz -hemicontractive mappings are proved in this project.


In our strong convergence result, there is no compactness assumption imposed on the map and its domain, T and C respectively, neither is there any restriction imposed on the nonempty xed point set of the map. Our main strong convergence result extends a recent result of L. Maruster and S. Maruster [29] from the class of demicontractive mappings to our more general class of hemicontractive mappings.


TABLE OF CONTENTS

Abstract

CHAPTER ONE
1  General Introduction
1.1  Introduction
1.1.1    Guage function:
1.1.2    Duality map
1.1.3    Lipschitz continuous mapping
1.1.4    Contraction mapping
1.1.5    Nonexpansive mapping
1.1.6    Firmly-Nonexpansive Operators
1.1.7    Quasi-nonexpansive mappings
1.1.8    Nonspreading mappings
1.1.9    k  strictly pseudononspreading mappings:
1.1.10  Accretive Operators
1.1.11  Strictly Pseudocontractive Mappings
1.1.12  Strongly Pseudocontractive Maps
1.1.13  Pseudocontractive Maps
1.1.14  Hemicontractive Mappings
1.1.15  Strongly Hemicontractive Mappings
1.1.16  Demicontractive Mappings:
1.1.17  -Demicontractive Mappings
1.1.18  -Demicontractive Mappings
1.2   ITERATION PROCESSES
1.2.1    Picard Iteration
1.2.2    Krasnoselski Iteration
1.2.3    The Mann Iteration Process
1.2.4    The Ishikawa Iteration Process
1.2.5    Mann Iteration Process with Errors in the Sense of Liu
1.2.6    Ishikawa Iteration Process with Errors in the Sense of Liu
1.2.7    Mann and Ishikawa Iteration Process with Errors in the Sense of Xu
1.2.8    Halpern-Type Iteration Method
1.2.9    Hybrid Algorithm or the CQ Algorithm

CHAPTER TWO
2  Preliminaries
2.0.10  De nitions and Notations
2.1       Lemmas, Proposition and Theorem
2.2       ORGANIZATION OF THESIS

CHAPTER THREE
3  Main Results
3.1       -Hemicontractive mapping and Examples
3.2       Weak Convergence of Ishikawa's Iteration Method for Lipschitz -hemicontractive Mappings
3.3       Strong Convergence of Ishikawa's Iteration Method for -hemicontractive mappings
References

Chapter 1

General Introduction

1.1      Introduction

From the available records, it is obvious that the theory of xed points has become an e cient means for the study of nonlinear phenomena. Interestingly, xed point theory has gained wonderful applications in diverse elds of human endeavour. For instance, many physically signi cant problems are modelled by

du
+ Au(t) = 0; u(0) = u0:
(1:1)

dt


Where A is an accretive self map on a subset of suitable Banach space. At equilibrium state of such system, dudt = 0. Consequently, a solution of Au = 0 describes the equilibrium or stable state of the system. This result is very desirable in many applications, for example in; Economics, Biology, Chemistry, Engineering, Physics, to mention but a few. As a result, considerable research e orts have been devoted to methods of solving the equation Au = 0 when A is accretive. Since, generally, A is nonlinear, there is no prede ned formula for solution of the equation. Thus, a standard technique is to introduce a pseudocontractive map, T de ned by : T = I A where I is the identity map. It is then clear that Ker(A) = F ix(T ) . Consequent upon this, the interests of numerous researchers including me have been attracted to the study of xed point theory for pseudocontractive maps and their related families of maps .


Several   xed point theorems have been derived in the last century, each focusing on a speci c class of mappings and some iterative scheme such as Picard iterative scheme, Krasnoselski iterative scheme [23] , Mann iterative scheme [25], Ishikawa iterative scheme [20], and many other iterative algorithms.



As remarked in Martinez and Xu [28]: Ishikawa iteration scheme in the light of Ishikawa

[20]  is indeed more general than Mann. But, research has been concentrated on the latter due probably to the reasons that the formulation of Mann is simpler than that of Ishikawa and that a convergence theorem for Mann may possibly lead to a convergence theorem for Ishikawa provided that the control sequence, n satis es certain appropriate conditions.

In spite of these advantages of Mann over Ishikawa, we remark with sense of challenge, that all e orts to use the Mann iteration method to approximate a xed point of a Lipschitz

pseudocontractive mappings de ned even on compact convex subset of a Hilbert space proved abortive. Chidume and Mutangandura [11] produced an example of a Lipschitz pseudocontractive self map of a compact convex subset of a real Hilbert space with a unique xed point for which no Mann sequence converges. Provision of the above counter example made several researchers to resort to other iterative schemes especially Ishikawa's scheme for the approximation of xed points of Lipschitz pseudocontractive mappings.

The Ishikawa's scheme was introduced in 1974 and was rstly used by Ishikawa [20] to achieve strong convergence to the xed points of Lipschitz pseudocontractive self map-pings of compact convex subsets of real Hilbert space. The above result is indeed laudable but the compactness condition is really very strong.

Later in 2008, Zhou [64] established the hybrid Ishikawa algorithim for Lipschitz pseudo-contractive mappings which assured strong convergence to the xed points of Lipschitz pseudocontractive mappings without compactness assumption.

Similarly in 2009, Yao et al [61] introduced the hybrid Mann algorithim which guarantees strong convergence to the xed points of Lipschitz pseudocontractive mappings without compactness assumption.

In their own contributions in 2011, Tang et al [56] generalised the hybrid Ishikawa iterative.....

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