General Viscosity Approximation Methods for Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces

Xue-Song Li,Jongkyu Kim,Nan-Jing Huang

doi:10.1155/2011/783502

Abstract

This paper is devoted to the strong convergence of two kinds of general viscosity iteration processes for approximating common fixed points of a nonexpansive semigroup in Hilbert spaces. The results presented in this paper improve and generalize some corresponding results in (X. Li et al., 2009, S. Li et al., 2009, and Marino and Xu, 2006).

Highlights

Let H be a real Hilbert space and A be a linear bounded operator on H
We recall that a mapping T : H → H is said to be contractive if there exists a constant α ∈ 0, 1 such that Tx − Ty ≤ α x − y for all x, y ∈ H
It is obvious that pseudocontractive mapping is more general than φ-strongly pseudocontractive mapping

Summary

Introduction

Let H be a real Hilbert space and A be a linear bounded operator on H. Throughout this paper, we always assume that A is strongly positive; that is, there exists a constant γ > 0 such that. We recall that a mapping T : H → H is said to be contractive if there exists a constant α ∈ 0, 1 such that Tx − Ty ≤ α x − y for all x, y ∈ H. Let f : H → H be a contractive mapping with coefficient α ∈ 0, 1 , T : H → H be a nonexpansive mapping, and A be a strongly positive and linear bounded operator with coefficient γ > 0. Marino and Xu 6 considered the general viscosity approximation process as follows: xt I − tA Txt tγ f xt , 1.8 where t ∈ 0, 1 such that t < A −1 and 0 < γ < γ /α. Li et al 5 claimed that the sequence {xn} generated by 1.11 converges strongly as tn → ∞ to x∗ ∈

F Γ which solves the following variational inequality:

Preliminaries

Main Results