Viscosity-Projection Method for a Family of General Equilibrium Problems and Asymptotically Strict Pseudocontractions in the Intermediate Sense

Dao-Jun Wen

doi:10.1155/2013/608241

Abstract

In this paper, a Meir-Keeler contraction is introduced to propose a viscosity-projection approximation method for finding a common element of the set of solutions of a family of general equilibrium problems and the set of fixed points of asymptotically strict pseudocontractions in the intermediate sense. Strong convergence of the viscosity iterative sequences is obtained under some suitable conditions. Results presented in this paper extend and unify the previously known results announced by many other authors.

Highlights

Let H be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖, respectively
Let C be a nonempty closed and convex subset of a real Hilbert space H and T : C → C be a uniformly continuous asymptotically λ-strictly pseudocontractive mapping in the intermediate sense with a sequence {kn} such that Fix(T) is nonempty and bounded
In this paper, inspired and motivated by research going on in this area, we introduce a new viscosity-projection method for a family of general equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense, which is defined in the following way: x1 ∈ C, C1 = C, un

Summary

Introduction

Let H be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖, respectively. Let C be a nonempty closed and convex subset of a real Hilbert space H and T : C → C be a uniformly continuous asymptotically λ-strictly pseudocontractive mapping in the intermediate sense with a sequence {kn} such that Fix(T) is nonempty and bounded. In this paper, inspired and motivated by research going on in this area, we introduce a new viscosity-projection method for a family of general equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense, which is defined in the following way: x1 ∈ C, C1 = C, un. Our purpose is to extend the viscosity-projection method with a Meir-Keeler contraction to the case of a family of general equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense, and to obtain a strong convergence theorem by using the proposed schemes under some appropriate conditions. Results presented in this paper extend and unify the corresponding ones of [10,11,12,13, 16]

Preliminaries

Main Results

F F A2 A1