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 On Strong Convergence by the Hybrid Method for Equilibrium and Fixed Point Problems for an Inifnite Family of Asymptotically Nonexpansive Mappings | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 798319 20 pages doi 2009 798319 Research Article On Strong Convergence by the Hybrid Method for Equilibrium and Fixed Point Problems for an Inifnite Family of Asymptotically Nonexpansive Mappings Gang Cai and Chang song Hu Department of Mathematics Hubei Normal University Huangshi 435002 China Correspondence should be addressed to Gang Cai caigang-aaaa@ and Chang song Hu huchang1004@ Received 17 April 2009 Accepted 9 July 2009 Recommended by Tomonari Suzuki We introduce two modifications of the Mann iteration by using the hybrid methods for equilibrium and fixed point problems for an infinite family of asymptotically nonexpansive mappings in a Hilbert space. Then we prove that such two sequences converge strongly to a common element of the set of solutions of an equilibrium problem and the set of common fixed points of an infinite family of asymptotically nonexpansive mappings. Our results improve and extend the results announced by many others. Copyright 2009 G. Cai and C. S. Hu. 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 Hilbert space H. A mapping T C C is said to be nonexpansive if for all x y e C wehave Tx-Ty x-y . It is said to be asymptotically nonexpansive 1 if there exists a sequence kn with kn 1 and limn. kn 1 such that Tnx - Tny kn x - y for all integers n 1 and for all x y e C. The set of fixed points of T is denoted by F T . Let ộ C X C R be a bifunction where R is the set of real number. The equilibrium problem for the function ộ is to find a point x e C such that ộ x y 0 Vy e C. The set of solutions of is denoted by EP ộ . In 2005 Combettes and Hirstoaga 2 introduced an iterative scheme of finding the .
