Study on Some Fixed Point Theorems for Bregman Nonexpansive Type Mapping in Banach Spaces Chapter One Aim and 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.

CHAPTER TWO

LITERATURE REVIEW

 Asymptotically Nonexpansive Mappings

Gobel and Kirk (1972) intoroduced the class of asymptotically nonexpansive mappings as a generalization of the class of nonexpansive mappings. If C is a nonempty closed convex bounded subset of a uniformly convex Banach space E and T is a self mapping of C which is asymptotically nonexpansive in the intermediate sense, then T has a fixed point, (Kirk, 1974). However the class of mappings which are asymptotically nonexpansive in the intermediate sense contains the class of asymptotically nonexpansive mappings.

A modified Mann iteration to approximate fixed points of asymptotically non-expansive mappings in uniformly convex Banach spaces was introduced by (Schu, 1991) Osilike and Aniagbosor (2000) and Shahzad and Udomene (2006) obtained weak and strong convergence theorem for finding a fixed point of asymptotically nonexpansive mappings.

A more general class of mappings called total asymptotically nonexpansive mappings was introduced by Albert et al. (2006) and studied method of ap-proximation of fixed points of mappings belonging to this class. Several au-thors are constructing iterative sequences for finding the fixed point of total asymptotically nonexpansive mappings.(such as Chidume and Ofoedu (2007) and Yolacan and Kizitunc (2012) )

Chidume and Ofoedu (2007) constructed the system (2.1.1) for the approxi tion of common fixed points of finite families of total asymptotically nonexpan-sive mappings, and gave necessary and sufficient conditions for the convergence of the scheme to common fixed points of the mappings in arbitrary real Banach spaces. A sufficient condition for convergence of the iteration process to a common fixed point of mappings under the same setting was also established in real uniformly convex Banach spaces.

Bregman Nonexpansive Mappings

Iterative methods for approximating fixed points of nonexpansive, quasi non-expansive mappings and their generalizations have been studied by various authors such as Browder (1967) and Halpern (1967), in Hilbert spaces. But extending this theory to Banach spaces encountered some difficulties because the useful examples of nonexpansive operators in Hilbert spaces are no longer nonexpansive in Banach spaces (e.g., the resolvent RA = (I rA) 1; for r > 0, of a monotone mapping A : C ! 2H and the metric projection PC onto a nonempty, closed, and convex subset C of H).

To overcome these difficulties, Bregman (1967) discovered an effective tech-nique using the so called Bregman distance function Df (:;

Order Now