WEAK AND STRONG CONVERGENCE OF ISHIKAWA’S ITERATION METHOD FOR LIPSCHITZ – HEMICONTRACTIVE MAPPINGS

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WEAK AND STRONG CONVERGENCE OF ISHIKAWA’S ITERATION METHOD FOR LIPSCHITZ – HEMICONTRACTIVE MAPPINGS (STATISTICS PROJECT TOPICS AND MATERIALS)

 

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.

 

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

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WEAK AND STRONG CONVERGENCE OF ISHIKAWA’S ITERATION METHOD FOR LIPSCHITZ – HEMICONTRACTIVE MAPPINGS (STATISTICS PROJECT TOPICS AND MATERIALS)

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