%0 Journal Article
%T A New Iterative Method for Finding Common Solutions of a System of Equilibrium Problems, Fixed-Point Problems, and Variational Inequalities
%A Jian-Wen Peng
%A Soon-Yi Wu
%A Jen-Chih Yao
%J Abstract and Applied Analysis
%D 2010
%I Hindawi Publishing Corporation
%R 10.1155/2010/428293
%X We introduce a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed point sets of an infinite family of nonexpansive mappings, and the solution set of a variational inequality for a relaxed cocoercive mapping in a Hilbert space. We prove strong convergence theorem. The results in this paper unify and generalize some well-known results in the literature. 1. Introduction Let be a real Hilbert space with inner product and induced norm . Let be a nonempty, closed, and convex subset of . Let be a countable family of bifunctions from to , where is the set of real numbers. Combettes and Hirstoaga [1] considered the following system of equilibrium problems: If is a singleton, problem (1.1) becomes the following equilibrium problem: The solutions set of (1.2) is denoted by . And clearly the solutions set of problem (1.1) can be written as . Problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; see for instance, [1¨C4]. Recall that a mapping of a closed and convex subset into itself is nonexpansive if We denote fixed-points set of by . A mapping is called contraction if there exists a constant such that A bounded linear operator on is strongly positive, if there is a constant such that for all . Combettes and Hirstoaga [1] introduced an iterative scheme for finding a common element of the solutions set of problem (1.1) in a Hilbert space and obtained a weak convergence theorem. Peng and Yao [2] introduced a new viscosity approximation scheme based on the extragradient method for finding a common element in the solutions set of the problem (1.1), fixed-points set of an infinite family of nonexpansive mappings and the solutions set of the variational inequality for a monotone and Lipschitz continuous mapping in a Hilbert space and obtained a strong convergence theorem. Colao et al. [3] introduced an implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed-points of infinite family of nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem. Saeidi [4] introduced some iterative algorithms for finding a common element of the solutions set of a system of equilibrium problems and of fixed-points set of a finite family and a left amenable semigroup of nonexpansive mappings in a Hilbert space and obtained some
%U http://www.hindawi.com/journals/aaa/2010/428293/