Weak and Strong Convergence Theorems of an Implicit Iteration Process for a Countable Family of Nonexpansive Mappings

Kittikorn Nakprasit,Weerayuth Nilsrakoo,Satit Saejung

doi:10.1155/2008/732193

Abstract

AbstractUsing the implicit iteration and the hybrid method in mathematical programming, we prove weak and strong convergence theorems for finding common fixed points of a countable family of nonexpansive mappings in a real Hilbert space. Our results include many convergence theorems by Xu and Ori (2001) and Zhang and Su (2007) as special cases. We also apply our method to find a common element to the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem. Finally, we propose an iteration to obtain convergence theorems for a continuous monotone mapping.

Highlights

Let H be a real Hilbert space with inner product ·, · and norm ·, and let C be a nonempty subset of H
We denote by F T the set of all fixed points of T
If C is bounded closed convex and T is a nonexpansive mapping of C into itself, F T is nonempty see 1

Summary

Introduction

Let H be a real Hilbert space with inner product ·, · and norm · , and let C be a nonempty subset of H. Xu and Ori 2 introduced the following implicit iteration process to approximate a common fixed point of a finite family of nonexpansive mappings {Ti}Ni 1: an initial point x0 ∈ C, Fixed Point Theory and Applications x1 α1x0 1 − α1 T1x1, x2 α2x1 1 − α2 T2x2, xN αN xN−1 1 − αN TN xN , xN 1 αN 1xN 1 − αN 1 T1xN 1. N}.They proved that this process converges weakly to a common fixed point of {Ti}Ni 1. To obtain a strong convergence theorem, Zhang and Su 3 modify iteration processes 1.3 by the implicit hybrid method for a finite family of nonexpansive mappings {Ti}Ni 1: an initial point x0 ∈ C, x0 ∈ C is arbitrary, yn αnxn 1 − αn Tnzn, zn βnyn 1 − βn Tnyn, 1.4. We establish weak and strong convergence theorems for finding common fixed points of a countable family of nonexpansive mappings in a real Hilbert space. We propose an iteration to obtain convergence theorems for a continuous monotone mapping

Preliminaries

Weak convergence theorems

Strong convergence theorems

Equilibrium problems

Convergence theorem for monotone mappings