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Báo cáo hóa học: "Research Article A Hybrid Extragradient Viscosity Approximation Method for Solving Equilibrium Problems and Fixed Point Problems of Infinitely Many "

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 Hybrid Extragradient Viscosity Approximation Method for Solving Equilibrium Problems and Fixed Point Problems of Infinitely Many | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 374815 32 pages doi 10.1155 2009 374815 Research Article A Hybrid Extragradient Viscosity Approximation Method for Solving Equilibrium Problems and Fixed Point Problems of Infinitely Many Nonexpansive Mappings Chaichana Jaiboon and Poom Kumam Department of Mathematics Faculty of Science King Mongkut s University of Technology Thonburi Bangkok 10140 Thailand Correspondence should be addressed to Poom Kumam poom.kum@kmutt.ac.th Received 25 December 2008 Accepted 4 May 2009 Recommended by Wataru Takahashi We introduce a new hybrid extragradient viscosity approximation method for finding the common element of the set of equilibrium problems the set of solutions of fixed points of an infinitely many nonexpansive mappings and the set of solutions of the variational inequality problems for -inverse-strongly monotone mapping in Hilbert spaces. Then we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality which is the optimality condition for a minimization problem. Results obtained in this paper improve the previously known results in this area. Copyright 2009 C. Jaiboon and P. Kumam. 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. Recall that a mapping T of H into itself is called nonexpansive see 1 if Tx - Ty x - y for all x y e H. We denote by F T x e C Tx x the set of fixed points of T. Recall also that a self-mapping f H H is a contraction if there exists a constant a e 0 1 such that f x - f y a x - y for all x y e H. In addition let B C H be a nonlinear mapping. Let PC be the projection of H onto C. The classical variational inequality which is denoted by VI C B is to