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 Strong Convergence to Common Fixed Points for Countable Families of Asymptotically Nonexpansive Mappings and Semigroups | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 301868 15 pages doi 2010 301868 Research Article Strong Convergence to Common Fixed Points for Countable Families of Asymptotically Nonexpansive Mappings and Semigroups Kriengsak Wattanawitoon1 2 and Poom Kumam2 3 1 Department of Mathematics and Statistics Faculty of Science and Agricultural Technology Rajamangala University of Technology Lanna Tak Tak 63000 Thailand 2 Centre of Excellence in Mathematics CHE Si Ayuthaya Road Bangkok 10400 Thailand 3 Department of Mathematics Faculty of Science King Mongkut s University of Technology Thonburi KMUTT Bangmod Thrungkru Bangkok 10140 Thailand Correspondence should be addressed to Poom Kumam Received 15 April 2010 Accepted 11 October 2010 Academic Editor A. T. M. Lau Copyright 2010 K. Wattanawitoon 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. We prove strong convergence theorems for countable families of asymptotically nonexpansive mappings and semigroups in Hilbert spaces. Our results extend and improve the recent results of Nakajo and Takahashi 2003 and of Zegeye and Shahzad 2008 from the class of nonexpansive mappings to asymptotically nonexpansive mappings. 1. Introduction Throughout this paper Let H be a real Hilbert space with inner product and norm II II and we write xn x to indicate that the sequence xn converges strongly to x. Let C be a nonempty closed convex subset of H and let T C C be a mapping. Recall that T is nonexpansive if Tx - Ty x - yịị for all x y e C. We denote the set of fixed points of T by F T that is F T x e C x Tx . A mapping T is said to be asymptotically nonexpansive if there exists a sequence kn with kn 1 for all n limn .kkn 1 and Tnx - Tny kn x - y Vn 1 x y e C. Mann s iterative algorithm was
