Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 417089, 15 pages doi:10.1155/2008/417089 Research Article A New Hybrid Iterative Algorithm for Fixed-Point Problems, Variational Inequality Problems, and Mixed Equilibrium Problems Yonghong Yao, 1 Yeong-Cheng Liou, 2 and Jen-Chih Yao 3 1 Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China 2 Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan 3 Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan Correspondence should be addressed to Jen-Chih Yao, yaojc@math.nsysu.edu.tw Received 29 August 2007; Accepted 6 February 2008 Recommended by Tomonari Suzuki We introduce a new hybrid iterative algorithm for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the variational inequality of a monotone mapping, and the set of solutions of a mixed equilibrium problem. This study, proves a strong convergence theorem by the proposed hybrid iterative algorithm which solves fixed-point problems, variational inequality problems, and mixed equilibrium problems. Copyright q 2008Yonghong Yao 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. 1. Introduction Let C be a nonempty closed convex subset of a real Hilbert space H. Recall that a mapping f : C → C is called contractive if there exists a constant α ∈ 0, 1 such that ‖f x - f y‖≤ α‖x - y‖ for all x, y ∈ C. A mapping T : C → C is said to be nonexpansive if ‖Tx - Ty‖≤‖x - y‖ for all x, y ∈ C. Denote the set of fixed points of T by FT . Let ϕ : C → R be a real-valued function and Θ : C × C → R be an equilibrium bifunction, that is, Θu, u 0 for each u ∈ C. The mixed equilibrium problem for short, MEP is to find x ∗ ∈ C such that MEP: Θ ( x ∗ ,y ) ϕy - ϕ ( x ∗ ) ≥ 0 ∀y ∈ C. 1.1 In particular, if ϕ ≡ 0, this problem reduces to the equilibrium problem for short, EP, which is to find x ∗ ∈ C such that EP: Θ ( x ∗ ,y ) ≥ 0 ∀y ∈ C. 1.2