Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article An Extragradient Method for Mixed Equilibrium Problems and Fixed Point Problems | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 632819 15 pages doi 2009 632819 Research Article An Extragradient Method for Mixed Equilibrium Problems and Fixed Point Problems Yonghong Yao 1 Yeong-Cheng Liou 2 and Yuh-Jenn Wu3 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 Chung Yuan Christian University Chung Li 320 Taiwan Correspondence should be addressed to Yeong-Cheng Liou simplex_liou@ Received 2 November 2008 Revised 8 April 2009 Accepted 23 May 2009 Recommended by Nan-Jing Huang The purpose of this paper is to investigate the problem of approximating a common element of the set of fixed points of a demicontractive mapping and the set of solutions of a mixed equilibrium problem. First we propose an extragradient method for solving the mixed equilibrium problems and the fixed point problems. Subsequently we prove the strong convergence of the proposed algorithm under some mild assumptions. Copyright 2009 Yonghong 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 H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let p C R be a real-valued function and 0 C X C R be an equilibrium bifunction that is 0 u Ù 0 for each u e C. We consider the following mixed equilibrium problem MEP which is to find X e C such that 0 x y y y - y x 0 Vy e C. MEP In particular if p 0 this problem reduces to the equilibrium problem EP which is to find x e C such that 0 x y 0 Vy e C. EP Denote the set of solutions of MEP by Q and the set of solutions of EP by r. The mixed equilibrium problems include fixed point problems optimization problems variational 2 Fixed .