General Iterative Methods for Equilibrium Problems and Infinitely Many Strict Pseudocontractions in Hilbert Spaces

Peichao Duan,Aihong Wang

doi:10.1155/2012/602513

Abstract

We propose an implicit iterative scheme and an explicit iterative scheme for finding a common element of the set of fixed point of infinitely many strict pseudocontractive mappings and the set of solutions of an equilibrium problem by the general iterative methods. In the setting of real Hilbert spaces, strong convergence theorems are proved. Our results improve and extend the corresponding results reported by many others.

Highlights

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H
We denote the set of fixed points of S by F S i.e., F S {x ∈ C : Sx x}
In this paper, motivated by the above facts, we introduce two iterative schemes and obtain strong convergence theorems for finding a common element of the set of fixed points of a infinite family of strict pseudocontractions and the set of solutions of the equilibrium problem 1.1

Summary

Introduction

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. We denote the set of fixed points of S by F S i.e., F S {x ∈ C : Sx x}. Mann 6 , Shimoji and Takahashi 7 considered iterative schemes for finding a fixed point of a nonexpansive mapping. Acedo and Xu 8 projected new iterative methods for finding a fixed point of strict pseudocontractions. Liu 2 considered a general iterative method for equilibrium problems and strict pseudocontractions. Wang 10 considered a general composite iterative method for infinite family strict pseudocontractions. In this paper, motivated by the above facts, we introduce two iterative schemes and obtain strong convergence theorems for finding a common element of the set of fixed points of a infinite family of strict pseudocontractions and the set of solutions of the equilibrium problem 1.1

Preliminaries

Main Result