Approximation common fixed point of asymptotically quasi-nonexpansive-type mappings by the finite steps iterative sequences

Jing Quan,Xian Jun Long,Shih-Sen Chang

doi:10.1155/fpta/2006/70830

Abstract

The purpose of this paper is to study sufficient and necessary conditions for finite-step iterative sequences with mean errors for a finite family of asymptotically quasi-nonexpansive and type mappings in Banach spaces to converge to a common fixed point. The results presented in this paper improve and extend the recent ones announced by Ghost-Debnath, Liu, Xu and Noor, Chang, Shahzad et al., Shahzad and Udomene, Chidume et al., and all the others.

Highlights

Introduction and preliminariesThroughout this paper, we assume that E is a real Banach space, F(T), D(T), and N denote the set of fixed points of T, the domain of T, and the set of positive integers, respectively.Definition 1.1
It follows that if F(T) is nonempty, quasi-nonexpensive mappings, asymptotically nonexpensive mappings, asymptotically quasi-nonexpensive mappings, and asymptotically nonexpensive type-mappings are all special cases of asymptotically quasi-nonexpensive-type mappings
Chang et al [2] complement, improve, and perfect all the above results and obtained some necessary and sufficient conditions for the Ishikawa iterative sequence with mixed errors of asymptotically quasi-nonexpansive-type mappings in Banach spaces to converge to a fixed point in Banach spaces

Summary

Introduction and preliminaries

Throughout this paper, we assume that E is a real Banach space, F(T), D(T), and N denote the set of fixed points of T, the domain of T, and the set of positive integers, respectively. Liu [6,7,8] extended the above results and proved some sufficient and necessary conditions for the Ishikawa iterative sequence or the Ishikawa iterative sequences with errors for asymptotically quasi-nonexpansive mappings to converge to a fixed point. Chang et al [2] complement, improve, and perfect all the above results and obtained some necessary and sufficient conditions for the Ishikawa iterative sequence with mixed errors of asymptotically quasi-nonexpansive-type mappings in Banach spaces to converge to a fixed point in Banach spaces. The purpose of this paper is to study sufficient and necessary conditions for finitestep iterative sequences with mean errors for a finite family of asymptotically quasinonexpansive-type mappings in Banach spaces to converge to a common fixed point.

Main results

Approximation of common fixed point