Two New Iterative Methods for a Countable Family of Nonexpansive Mappings in Hilbert Spaces

Shuang Wang,Changsong Hu

doi:10.1155/2010/852030

Abstract

AbstractWe consider two new iterative methods for a countable family of nonexpansive mappings in Hilbert spaces. We proved that the proposed algorithms strongly converge to a common fixed point of a countable family of nonexpansive mappings which solves the corresponding variational inequality. Our results improve and extend the corresponding ones announced by many others.

Highlights

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H
Recall that a mapping T : C → C is said to be nonexpansive if T x−T y ≤ x−y, for all x, y ∈ C
A mapping F : H → H is called k-Lipschitzian if there exists a positive constant k such that

Summary

Introduction

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. He proved that {xn} generated by 1.8 converges strongly to the unique solution of variational inequality Fx, x − x ≥ 0, x ∈ F T . Let H be a Hilbert space, C a closed convex subset of H, and T : C → C a nonexpansive mapping with F T / ∅, if {xn} is a sequence in C weakly converging to x and if { I − T xn} converges strongly to y, I − T x y.

Results

Conclusion