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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 A New Strong Convergence Theorem for Equilibrium Problems and Fixed Point Problems in Banach Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 572156 14 pages doi 10.1155 2011 572156 Research Article A New Strong Convergence Theorem for Equilibrium Problems and Fixed Point Problems in Banach Spaces Weerayuth Nilsrakoo Department of Mathematics Statistics and Computer Faculty of Science Ubon Ratchathani University Ubon Ratchathani 34190 Thailand Correspondence should be addressed to Weerayuth Nilsrakoo nilsrakoo@hotmail.com Received 5 June 2010 Revised 28 December 2010 Accepted 20 January 2011 Academic Editor Fabio Zanolin Copyright 2011 Weerayuth Nilsrakoo. 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 sequence for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a Banach space. Then we study the strong convergence of the sequences. With an appropriate setting we obtain the corresponding results due to Takahashi-Takahashi and Takahashi-Zembayashi. Some of our results are established with weaker assumptions. 1. Introduction Throughout this paper we denote by N anW the sets of positive integers and real numbers respectively. Let E be a Banach space E the dual space of E and C a closed convex subsets of E. Let F C X C R be a bifunction. The equilibrium problem is to find x e C such that F x y 0 Vy e C. 1.1 The set of solutions of 1.1 is denoted by EP F . The equilibrium problems include fixed point problems optimization problems variational inequality problems and Nash equilibrium problems as special cases. Let E be a smooth Banach space and J the normalized duality mapping from E to E . Alber 1 considered the following functional f E X E 0 to defined by Ax y x 2 - 2 x Jy y 2 x y e E . 1.2 2 Fixed Point Theory and Applications Using this
