Strong convergence theorems for system of equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense

Peichao Duan,Jing Zhao

doi:10.1186/1687-1812-2011-13

Abstract

Let be N uniformly continuous asymptotically λ i -strict pseudocontractions in the intermediate sense defined on a nonempty closed convex subset C of a real Hilbert space H. Consider the problem of finding a common element of the fixed point set of these mappings and the solution set of a system of equilibrium problems by using hybrid method. In this paper, we propose new iterative schemes for solving this problem and prove these schemes converge strongly. MSC: 47H05; 47H09; 47H10.

Highlights

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H.A nonlinear mapping S : C ® C is a self mapping of C
(1) S is uniformly Lipschitzian if there exists a constant L > 0 such that ||Snx − −Sny|| ≤ L||x − y|| for all integers n ≥ 1 and x, y ∈ C
(2) S is nonexpansive if ||Sx − Sy|| ≤ ||x − y|| for all x, y ∈ C

Summary

Introduction

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let C be a nonempty closed convex subset of a real Hilbert space H and T: C ® C a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence gn such that F(T) is nonempty and bounded. Hu and Cai [3] further considered the asymptotically strict pseudocontractive mappings in the intermediate sense concerning equilibrium problem They obtained the following result in a real Hilbert space. ([4]) Let C be a nonempty closed convex subset of a real Hilbert space H and T : C ® C a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense. ([4]) Let C be a nonempty subset of a Hilbert space H and T : C ® C an asymptotically - strict pseudocontractive mapping in the intermediate sense with sequence {gn}.

Take p

It is easy to get

It follows that

We also have