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Research Article Strong Convergence Theorems for a Generalized Equilibrium Problem with a Relaxed Monotone Mapping and a Countable Family of Nonexpansive Mappings in a Hilbert Space | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 230304 22 pages doi 10.1155 2010 230304 Research Article Strong Convergence Theorems for a Generalized Equilibrium Problem with a Relaxed Monotone Mapping and a Countable Family of Nonexpansive Mappings in a Hilbert Space Shenghua Wang 1 Giuseppe Marino 2 and Fuhai Wang1 1 School of Applied Mathematics and Physics North China Electric Power University Baoding 071003 China 2 Dipartimento di Matematica Universita della Calabria 87036 Arcavacata di Rende Italy Correspondence should be addressed to Shenghua Wang sheng-huawang@hotmail.com Received 15 March 2010 Accepted 20 June 2010 Academic Editor Naujing Jing Huang Copyright 2010 Shenghua Wang et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We introduce a new iterative method for finding a common element of the set of solutions of a generalized equilibrium problem with a relaxed monotone mapping and the set of common fixed points of a countable family of nonexpansive mappings in a Hilbert space and then prove that the sequence converges strongly to a common element of the two sets. Using this result we prove several new strong convergence theorems in fixed point problems variational inequalities and equilibrium problems. 1. Introduction Throughout this paper let R denote the set of all real numbers let N denote the set of all positive integer numbers let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let S C C be a mapping. We call S nonexpansive if Sx- Sy x- y x y e C. 1.1 The set of fixed points of S is denoted by Fix S . We know that the set Fix S is closed and convex. Let C X C R be a bifunction. The equilibrium problem for is to find z e C such that b z ý 0 Vy e C. 1.2 2 Fixed Point Theory and Applications The set of all .
