Nonlinear Analysis: Hybrid Systems 4 (2010) 743–754 Contents lists available at ScienceDirect Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs Hybrid pseudoviscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings L.-C. Ceng a , Q.H. Ansari b,c , J.-C. Yao d,∗ a Department of Mathematics, Shanghai Normal University, Shanghai 200234, China b Department of Mathematics & Statistics, King Fahd University of Petroleum & Minerals, P.O. Box 1169, Dhahran, Saudi Arabia c Department of Mathematics, Aligarh Muslim University, Aligarh, India d Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan article info Article history: Received 24 May 2008 Accepted 3 May 2010 Keywords: Hybrid pseudoviscosity approximation schemes Equilibrium problem Fixed point Nonexpansive mapping Strongly positive linear bounded operators abstract In this paper, we introduce hybrid pseudoviscosity approximation schemes with strongly positive bounded linear operators for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of infinitely many nonexpansive mappings in the setting of Hilbert spaces. We prove the strong convergence of the sequences generated by our scheme to a solution of an equilibrium problem which is also a common fixed point of infinitely many nonexpansive mappings. Our results can be treated as extension and improvement of the corresponding results appeared in the literature in the recent past. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction Let H be a real Hilbert space with inner product 〈., .〉 and norm ‖·‖, C be a nonempty closed convex subset of H and φ : C × C → R be a real-valued bifunction. The equilibrium problem (for short, EP) for φ : C × C → R is to find u ∈ C such that φ(u,v) ≥ 0, ∀v ∈ C . (1.1) The set of solutions of (1.1) is denoted by EP (φ). It is a unified model of several problems, namely, variational inequality problems, complementarity problems, saddle point problems, optimization problem, fixed point problem, etc.; see, for example, [1,2] and the references therein. In the past two decades, the theory of equilibrium problems has been extensively studied in the literature. In most of the papers appeared in ninety’s on the equilibrium problems, the existence of their solutions and applications have been studied; see, for example, [3,4,1,5,6,2] and the references therein. In the past few years, several people have started working on the solution methods to find the approximate solutions of the equilibrium problems and their generalizations; see, for example, [7–12] and the references therein. Very recently, motivated by the work of Combettes and Hirstoaga [8], Moudafi [13] and Tada and Takahashi [10], Takahashi and Takahashi [11] proposed an iterative scheme by the viscosity approximation method for computing a common element of the set of solutions of EP and the set of fixed points of a nonexpansive mapping in the setting of Hilbert spaces. They also studied the strong convergence ∗ Corresponding author. E-mail addresses: zenglc@hotmail.com (L.-C. Ceng), qhansari@kfupm.edu.sa, qhansari@sancharnet.in (Q.H. Ansari), yaojc@math.nsysu.edu.tw (J.-C. Yao). 1751-570X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2010.05.001