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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 Asymptotically Pseudocontractions, Banach Operator Pairs and Best Simultaneous Approximations | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 812813 11 pages doi 10.1155 2011 812813 Research Article Asymptotically Pseudocontractions Banach Operator Pairs and Best Simultaneous Approximations N. Hussain Department of Mathematics King Abdulaziz University P.O. Box 80203 Jeddah 21589 Saudi Arabia Correspondence should be addressed to N. Hussain nhusain@kau.edu.sa Received 3 December 2010 Accepted 12 January 2011 Academic Editor Mohamed Amine Khamsi Copyright 2011 N. Hussain. 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. The existence of common fixed points is established for the mappings where T is asymptotically -pseudo-contraction on a nonempty subset of a Banach space. As applications the invariant best simultaneous approximation and strong convergence results are proved. Presented results are generalizations of very recent fixed point and approximation theorems of Khan and Akbar 2009 Chen and Li 2007 Pathak and Hussain 2008 and several others. 1. Introduction and Preliminaries We first review needed definitions. Let M be a subset of a normed space X II II . The set PM Ù x e M x - u dist u M is called the set of best approximants to u e X out of M where dist u M inf y - u y e M . Suppose that A and G are bounded subsets of X. Then we write rG A inf supIIa - gII geG aeA centG A g0 e G sup ữ - gcll tg A 1.1 The number rG A is called the Chebyshev radius of A w.r.t. G and an element y0 e centG A is called a best simultaneous approximation of A w.r.t. G. If A u then rG A dist u G and centG A is the set of all best approximations PG u of u from G. We also refer the reader to Milman 1 and Vijayraju 2 for further details. We denote by N and cl M wcl M 2 Fixed Point Theory and Applications the set of positive integers and the closure weak closure of a set M in X
