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 and Strong Convergence of an Implicit Iteration Process for an Asymptotically Quasi-I-Nonexpansive Mapping in Banach Space | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 719631 13 pages doi 2010 719631 Research Article Weak and Strong Convergence of an Implicit Iteration Process for an Asymptotically Quasi-I-Nonexpansive Mapping in Banach Space Farrukh Mukhamedov and Mansoor Saburov Department of Computational Theoretical Sciences Faculty of Science International Islamic University Malaysia . Box 141 25710 Kuantan Malaysia Correspondence should be addressed to Farrukh Mukhamedov far75m@ Received 31 August 2009 Accepted 6 December 2009 Academic Editor Mohamed A. Khamsi Copyright 2010 F. Mukhamedov and M. Saburov. 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 the weak and strong convergence of the implicit iterative process to a common fixed point of an asymptotically quasi-I-nonexpansive mapping T and an asymptotically quasi-nonexpansive mapping I defined on a nonempty closed convex subset of a Banach space. 1. Introduction Let K be a nonempty subset of a real normed linear space X and let T K K be a mapping. Denote by F T the set of fixed points of T that is F T x e K Tx x . Throughout this paper we always assume that F T 0. Now let us recall some known definitions. Definition . A mapping T K K is said to be i nonexpansive if IITx - Ty x - y for all x y e K ii asymptotically nonexpansive if there exists a sequence Ãn c 1 to with limnto i 1 such that Tnx - TnyW ln x - y for all x y e K and n e N iii quasi-nonexpansive if Tx - p x - p for all x e K p e F T iv asymptotically quasi-nonexpansive if there exists a sequence pn c 1 to with limn .sfin 1 such that Tnx - pll pn x - pll for all x e K p e F T and n e N. 2 Fixed Point Theory and Applications Note that from the above definitions it follows that a nonexpansive mapping must be asymptotically .
