CHAPTER ONE
GENERAL INTRODUCTION
1.1 Background of the Study
The study of nonlinear functional analysis has been in existence since early twentieth century, with investigation into the existence properties of solutions of certain boundary value problems, arising in nonlinear partial differential equations. The study of the approximation of fixed point theory of nonex-pansive mappings, is applied to solutions of diverse problems such as solving variational inequality, equilibrium problems and finding solutions of certain evolution equations. Some authors have proposed several methods for ap-proximation of fixed points of generalizations of nonexpansive mappings. For example Qin and Su (2007) established strong convergence theorems for rela-tively nonexpansive mappings in Banach spaces. Again, a new hybrid iterative scheme called a shrinking projection method for nonexpansive mappings was also introduced by (Takahashi et al., 2008).
1.2 Statement of the Problem
Chidume and
Ofoedu (2007), constructed an iterative sequence for approxi-mation of common
fixed points of finite families of total asymptotically non-expansive mappings.
Then the question arises, can approximation of common fixed points of finite
families of quasi-Bregman total asymptotically nonexpan-sive mappings be
established? In this dissertation we answer this question by constructing an
iterative sequence for approximation of common fixed points, of quasi-Bregman
total asymptotically nonexpansive mappings.
Secondly, Ugwunnadi et al. (2014), proved a strong convergence theorem for quasi-Bregman strictly pseudocontractive mappings and equilibrium problems in reflexive Banach space. Again the question arises can strong convergence be established for quasi-Bregman total strictly pseudocontractive mappings and equilibrium problems in reflexive Banach space? We also answered this ques-tion by developing a new iterative scheme, by hybrid method and proved a strong convergence theorem for quasi-Bregman total asymptotically strictly pseudocontractive mappings and equilibrium problems in reflexive Banach spaces
1.3 Aim and Objectives
The aim of this research is to establish some fixed point theorems for Bregman nonexpansive mapping in Banach spaces.
The aim will be achieved through the following objectives:
- Construction of an iterative sequence for approximation of common fixed points of quasi-Bregman total asymptotically nonexpansive mappings.
- Development of a new hybrid iterative scheme and establishment of strong convergence theorem for quasi-Bregman total asymptotically strictly pseudocontractive mappings and equilibrium problems in reflexive Ba-nach spaces.
1.4 Scope and Limitations
This research covers asymptotically nonexpansive mappings, total asymptoti-cally nonexpansive mappings and total asymptotically strictly pseudocontractive mappings using Bregman distance. The mappings must be uniformly Li-Lipschitzian and uniformly asymptotically regular.
1.5 Research Methodology
The method used in this research, is by reviewing relevant and necessary ar-ticles, in literature; on fixed point theory, nonexpansive mappings, asymptot-ically nonexpansive mappings, total asymptotically nonexpansive mappings and total asymptotically strictly pseudocontractive mappings. These articles are reviewed extensively to cover researches done for Bregman distance in ar-bitrary Banach spaces.
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STUDY ON SOME FIXED POINT THEOREMS FOR BREGMAN NONEXPANSIVE TYPE MAPPING IN BANACH SPACES
