Nonlinear Analysis 72 (2010) 99–112 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications Xiaolong Qin a , Yeol Je Cho b , Shin Min Kang c,∗ a Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea b Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea c Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea article info Article history: Received 24 January 2009 Accepted 12 June 2009 MSC: 4705 47H09 47J25 47N10 Keywords: Non-expansive mapping Fixed point Equilibrium problem Variational inequality problem Inverse-strongly monotone mapping abstract In this paper, we introduce a general iterative scheme for finding a common element of the set of common solutions of generalized equilibrium problems, the set of common fixed points of a family of infinite non-expansive mappings. Strong convergence theorems are established in a real Hilbert space under suitable conditions. As some applications, we consider convex feasibility problems and equilibrium problems. The results presented improve and extend the corresponding results of many others. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction and preliminaries Throughout this paper, we always assume that H is a real Hilbert space with inner product 〈·, ·〉 and norm ‖·‖, respec- tively. Let C be a nonempty, closed and convex subset of H. Let A : C → H be a nonlinear mapping. Recall the following definitions. (a) A is said to be monotone if 〈Ax − Ay, x − y〉≥ 0, ∀x, y ∈ C . (b) A is said to be strongly monotone if there exists a constant α> 0 such that 〈Ax − Ay, x − y〉≥ α‖x − y‖ 2 , ∀x, y ∈ C . For such a case, T is said to be α-strongly monotone. (c) A is said to be inverse-strongly monotone if there exists a constant α> 0 such that 〈Ax − Ay, x − y〉≥ α‖Ax − Ay‖ 2 , ∀x, y ∈ C . For such a case, A is said to be α-inverse-strongly monotone. ∗ Corresponding author. E-mail addresses: qxlxajh@163.com, ljjhqxl@yahoo.com.cn (X. Qin), yjcho@gsnu.ac.kr (Y.J. Cho), smkang@gnu.ac.kr (S.M. Kang). 0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.06.042