A generalization and improvement of Chidume theorems for total asymptotically nonexpansive mappings in Banach spaces

Shih-Sen Chang,Chi Kin Chan,Jong Kyu Kim,H W Joseph Lee

doi:10.1186/1029-242x-2012-37

Abstract

Abstract The purpose of this article is to establish some new approximation theorems of common fixed points for a countable family of total asymptotically quasi-nonexpansive mappings in Banach spaces which generalize and improve the corresponding theorems of Chidume et al. and others. 2000 AMS Subject Classification: 47J05; 47H09; 49J25.

Highlights

Throughout this article, we assume that E is a real Banach space, C is a nonempty closed convex subset of E
We use F(T) to denote the set of fixed points of a mapping T, and use R and R+ to denote the set of all real numbers and the set of all nonnegative real numbers, respectively
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as a generalization of the class of nonexpansive mappings

Summary

Introduction

Throughout this article, we assume that E is a real Banach space, C is a nonempty closed convex subset of E. Suppose that there exist Mi, M∗i > 0 such that ζi(λi) ≤ M∗i λi whenever li ≥ Mi, i = 1, 2, ..., m and that one of T1, T2, ..., Tm is compact, {xn} converges strongly to some p ∈ F It is our purpose in this article to construct a new iterative sequence much simpler than (1.5) for approximation of common fixed points of a countable family of total asymptotically nonexpansive mappings and give necessary and sufficient conditions for the convergence of the scheme to common fixed points of the mappings in arbitrary real Banach spaces. The results presented in the article generalize and improve the corresponding results of Chidume et al [13,14,15] and unify, extend and generalize the corresponding result of [3,4,5,6,7,9,10,11,12,19]

Preliminaries

Main results