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 Contractions, Common Fixed Points, and Invariant Approximations | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 390634 10 pages doi 2009 390634 Research Article Weak Contractions Common Fixed Points and Invariant Approximations Nawab Hussain1 and Yeol Je Cho2 1 Department of Mathematics King Abdul Aziz University . Box 80203 Jeddah 21589 Saudi Arabia 2 Department of Mathematics Education and the RINS Gyeongsang National University Chinju 660-701 South Korea Correspondence should be addressed to YeolJe Cho yjcho@ Received 19 January 2009 Accepted 23 February 2009 Recommended by Charles E. Chidume The existence of common fixed points is established for the mappings where T is f 0 L -weak contraction on a nonempty subset of a Banach space. As application some results on the invariant best approximation are proved. Our results unify and substantially improve several recent results given by some authors. Copyright 2009 N. Hussain and Y. J. Cho. 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 and Preliminaries Let M be a subset of a normed space X II II . The set PM u 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 lly - u y e M . We denote N and cl M resp. wcl M by the set of positive integers and the closure resp. weak closure of a set M in X respectively. Let f T M M be mappings. The set of fixed points of T is denoted by F T . A point x e M is a coincidence point resp. common fixed point of f and T if fx Tx resp. x fx Tx . The set of coincidence points of f and T is denoted by C f T . 2 Journal of Inequalities and Applications The pair f T is said to be 1 commuting 1 if Tfx fTx for all x e M 2 compatible 2 3 if limn .xJITfXn - fTxn 0whenever xn is a sequence such that limn Txn limn fxn t for some t in M 3 weakly compatible if they
