Finding Common Solutions of a Variational Inequality, a General System of Variational Inequalities, and a Fixed-Point Problem via a Hybrid Extragradient Method

Lu-Chuan Ceng,Sy-Ming Guu,Jen-Chih Yao

doi:10.1155/2011/626159

Abstract

AbstractWe propose a hybrid extragradient method for finding a common element of the solution set of a variational inequality problem, the solution set of a general system of variational inequalities, and the fixed-point set of a strictly pseudocontractive mapping in a real Hilbert space. Our hybrid method is based on the well-known extragradient method, viscosity approximation method, and Mann-type iteration method. By constrasting with other methods, our hybrid approach drops the requirement of boundedness for the domain in which various mappings are defined. Furthermore, under mild conditions imposed on the parameters we show that our algorithm generates iterates which converge strongly to a common element of these three problems.

Highlights

Let H be a real Hilbert space with inner product ·, · and norm ·
Based on the relaxed extragradient method and viscosity approximation method, Yao et al 30 proposed and analyzed an iterative algorithm for finding a common element of the solution set of the general system 1.3 of variational inequalities and the fixed-point set of a strictly pseudocontractive mapping in a real Hilbert space H
Motivated by the above work, in this paper, we introduce an iterative algorithm for finding a common element of the solution set of the variational inequality 1.1, the solution set of the general system 1.3 and the fixed-point set of the strictly pseudocontractive mapping S : C → C via a hybrid extragradient method based on the well-known extragradient method, viscosity approximation method, and Mann-type iteration method, that is, zn PC xn − λnAxn, yn αnQxn 1 − αn PC PC zn − μ2B2zn − μ1B1PC zn − μ2B2zn, 1.8 xn 1 βnxn γnyn δnSyn, ∀n ≥ 0, where {λn} ⊂ 0, ∞, {αn}, {βn}, {γn}, {δn} ⊂ 0, 1 such that βn γn δn 1 for all n ≥ 0