Strong Convergence Theorems of Viscosity Iterative Methods for a Countable Family of Strict Pseudo-contractions in Banach Spaces

Rabian Wangkeeree,Uthai Kamraksa

doi:10.1155/2010/579725

Abstract

For a countable family of strictly pseudo-contractions, a strong convergence of viscosity iteration is shown in order to find a common fixed point of in either a p-uniformly convex Banach space which admits a weakly continuous duality mapping or a p-uniformly convex Banach space with uniformly Gâteaux differentiable norm. As applications, at the end of the paper we apply our results to the problem of finding a zero of accretive operators. The main result extends various results existing in the current literature.

Highlights

Let E be a real Banach space and C a nonempty closed convex subset of E
Hu and Wang 11 obtained that the Mann iterative sequence converges weakly to a fixed point of λ-strict pseudo-contractions with respect to p in p-uniformly convex Banach spaces
Hu 15 obtained strong convergence theorems on a mixed iteration scheme by the viscosity approximation methods for λ-strict pseudo-contractions in puniformly convex Banach spaces with uniformly Gateaux differentiable norm

Summary

Introduction

Let E be a real Banach space and C a nonempty closed convex subset of E. Song and Zheng 4 studied the strong convergence theorem of Halpern iteration 1.5 for a countable family of nonexpansive mappings {Tn : C → C} satisfying some conditions in either a reflexive and strictly convex Banach space with a uniformly Gateaux differentiable norm or a reflexive Banach space E with a weakly continuous duality mapping. Hu and Wang 11 obtained that the Mann iterative sequence converges weakly to a fixed point of λ-strict pseudo-contractions with respect to p in p-uniformly convex Banach spaces. Hu 15 obtained strong convergence theorems on a mixed iteration scheme by the viscosity approximation methods for λ-strict pseudo-contractions in puniformly convex Banach spaces with uniformly Gateaux differentiable norm. Let E be a real p-uniformly convex Banach space with uniformly Gateaux differentiable norm, and C a nonempty closed convex subset of E which has the fixed point property for nonexpansive mappings. The results presented in this paper improve and extend the corresponding results announced by Hu and Wang 11 , Hu 15 , Aoyama et al 3 Song and Zheng 4 , and many others

Preliminaries

Main Results

10 This implies that

Some Applications for Accretive Operators