An Iterative Method for Generalized Equilibrium Problems, Fixed Point Problems and Variational Inequality Problems

Qing-You Liu,Nan-Jing Huang,Wei-You Zeng

doi:10.1155/2009/531308

Abstract

AbstractWe introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of generalized equilibrium problems, the set of common fixed points of infinitely many nonexpansive mappings, and the set of solutions of the variational inequality for "Equation missing"-inverse-strongly monotone mappings in Hilbert spaces. We give some strong-convergence theorems under mild assumptions on parameters. The results presented in this paper improve and generalize the main result of Yao et al. (2007).

Highlights

Let C be a nonempty closed convex subset of a real Hilbert space H and let Φ : C × C → R be a bifunction, where R is the set of real numbers
The set of solutions for the problem 1.1 is denoted by Ω, that is, Ω {u ∈ C : Φ u, v Ψu, v − u ≥ 0, ∀v ∈ C}
Inspired and motivated by the works mentioned above, in this paper, we introduce an iterative process for finding a common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of GEP 1.1, and the solution set of the variational inequality problem for an α-inverse-strongly monotone mapping in real Hilbert spaces

Summary

Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H and let Φ : C × C → R be a bifunction, where R is the set of real numbers. Takahashi 14 introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert spaces.

Results

Conclusion