Common Fixed Points of Multistep Noor Iterations with Errors for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Mappings

S Imnang,S Suantai

doi:10.1155/2009/728510

Abstract

We introduce a general iteration scheme for a finite family of generalized asymptotically quasi-nonexpansive mappings in Banach spaces. The new iterative scheme includes the multistep Noor iterations with errors, modified Mann and Ishikawa iterations, three-step iterative scheme of Xu and Noor, and Khan and Takahashi scheme as special cases. Our results generalize and improve the recent ones announced by Khan et al. (2008), H. Fukhar-ud-din and S. H. Khan (2007), J. U. Jeong and S. H. Kim (2006), and many others.

Highlights

Let C be a subset of real Banach space X
In 1974, Senter and Dotson 4 studied the convergence of the Mann iteration scheme defined by x1 ∈ C, xn 1 αnT xn 1 − αn xn, ∀n ≥ 1, 1.5 in a uniformly convex Banach space, where {αn} is a sequence satisfying 0 < a ≤ αn ≤ b < 1 for all n ≥ 1 and T is a nonexpansive or a quasi-nonexpansive mapping
Inspired and motivated by these facts, we introduce a new iteration process for a finite family of {Ti : i 1, 2, . . . , k} of generalized asymptotically quasi-nonexpansive mappings as follows

Summary

Introduction

Let C be a subset of real Banach space X. In 1974, Senter and Dotson 4 studied the convergence of the Mann iteration scheme defined by x1 ∈ C, xn 1 αnT xn 1 − αn xn , ∀n ≥ 1, 1.5 in a uniformly convex Banach space, where {αn} is a sequence satisfying 0 < a ≤ αn ≤ b < 1 for all n ≥ 1 and T is a nonexpansive or a quasi-nonexpansive mapping. They established a relation between condition I and demicompactness. Our results generalize and improve the corresponding ones announced by Khan et al 6 , Fukhar-ud-din and Khan 5 , and many others

Preliminaries

Convergence Theorems in Banach Spaces

Convergence Theorems in Uniformly Convex Banach Spaces