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 Theorems of Modiﬁed Ishikawa Iterations for Countable | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 483497 25 pages doi 2009 483497 Research Article Strong Convergence Theorems of Modified Ishikawa Iterations for Countable Hemi-Relatively Nonexpansive Mappings in a Banach Space Narin Petrot 1 2 Kriengsak Wattanawitoon 3 4 and Poom Kumam2 3 1 Department of Mathematics Faculty of Science Naresuan University Phitsanulok 65000 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 Bangkok 10140 Thailand 4 Department of Mathematics and Statistics Faculty of Science and Agricultural Technology Rajamangala University of Technology Lanna Tak Tak 63000 Thailand Correspondence should be addressed to Poom Kumam Received 17 March 2009 Accepted 12 September 2009 Recommended by Lech Gorniewicz We prove some strong convergence theorems for fixed points of modified Ishikawa and Halpern iterative processes for a countable family of hemi-relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space by using the hybrid projection methods. Moreover we also apply our results to a class of relatively nonexpansive mappings and hence we immediately obtain the results announced by Qin and Su s result 2007 Nilsrakoo and Saejung s result 2008 Su et al. s result 2008 and some known corresponding results in the literatures. Copyright 2009 Narin Petrot 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 C be a nonempty closed convex subset of a real Banach space E. A mapping T C C is said to be nonexpansive if Tx-Ty x-y for all x y e C. We denote by F T the set of fixed points of T that is F T x e C x Tx . A mapping T is .