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 Weak Convergence Theorems for a Countable Family of Strict Pseudocontractions in Banach Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 632137 16 pages doi 2010 632137 Research Article Weak Convergence Theorems for a Countable Family of Strict Pseudocontractions in Banach Spaces Prasit Cholamjiak1 and Suthep Suantai1 2 1 Department of Mathematics Faculty of Science Chiang Mai University Chiang Mai 50200 Thailand 2 Centre of Excellence in Mathematics CHE Si Ayutthaya Road Bangkok 10400 Thailand Correspondence should be addressed to Suthep Suantai scmti005@ Received 2 June 2010 Accepted 16 September 2010 Academic Editor Massimo Furi Copyright 2010 P. Cholamjiak and S. Suantai. 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 investigate the convergence of Mann-type iterative scheme for a countable family of strict pseudocontractions in a uniformly convex Banach space with the Frechet differentiable norm. Our results improve and extend the results obtained by Marino-Xu Zhou Osilike-Udomene Zhang-Guo and the corresponding results. We also point out that the condition given by Chidume-Shahzad 2010 is not satisfied in a real Hilbert space. We show that their results still are true under a new condition. 1. Introduction Let E and E be a real Banach space and the dual space of E respectively. Let K be a nonempty subset of E. Let J denote the normalized duality mapping from E into 2E given by J x f e E x f x 2 f 2 for all x e E where denotes the duality pairing between E and E . If E is smooth or E is strictly convex then J is single-valued. Throughout this paper we denote the single valued duality mapping by j and denote the set of fixed points of a nonlinear mapping T K E by F T x e K Tx x . Definition . A mapping T with domain D T and range R T in E is called i pseudocontractive 1 if for all x y e D T there exists j x - y e
