Random Three-Step Iteration Scheme and Common Random Fixed Point of Three Operators

Somyot Plubtieng,Poom Kumam,Rabian Wangkeeree

doi:10.1155/2007/82517

Abstract

We construct random iterative processes with errors for three asymptotically nonexpansive random operators and study necessary conditions for the convergence of these processes. The results presented in this paper extend and improve the recent ones announced by I. Beg and M. Abbas (2006), and many others.

Highlights

Probabilistic functional analysis has come out as one of the momentous mathematical disciplines in view of its requirements in dealing with probabilistic models in applied problems
Random fixed point theorems for random contraction mappings on separable complete metric spaces were first proven by Spacek [1]
The study of random fixed point theorems was initiated by Spacek [1] and Hans [3, 4]

Summary

Introduction

Probabilistic functional analysis has come out as one of the momentous mathematical disciplines in view of its requirements in dealing with probabilistic models in applied problems. In an attempt to construct iterations for finding fixed points of random operators defined on linear spaces, random Ishikawa scheme was introduced in [5]. Beg and Abbas [11] studied the different random iterative algorithms for weakly contractive and asymptotically nonexpansive random operators on arbitrary Banach spaces. Plubtieng et al [12] studied weak and strong convergence theorems established for a modified Noor iterative scheme with errors for three asymptotically nonexpansive mappings in Banach spaces. We study the convergence of three-step random iterative processes with errors for three asymptotically nonexpansive random operators in Banach spaces. Our results extend and improve the corresponding ones announced by Beg and Abbas [11], and many others

Preliminaries

Main results