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 of Modified Halpern Iterations in CAT(0) Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 869458 11 pages doi 2011 869458 Research Article Strong Convergence of Modified Halpern Iterations in CAT 0 Spaces A. Cuntavepanit1 and B. Panyanak1 2 1 Department of Mathematics Faculty of Science Chiang Mai University Chiang Mai 50200 Thailand 2 Materials Science Research Center Faculty of Science Chiang Mai University Chiang Mai 50200 Thailand Correspondence should be addressed to B. Panyanak banchap@ Received 28 November 2010 Accepted 10 January 2011 Academic Editor Qamrul Hasan Ansari Copyright 2011 A. Cuntavepanit and B. Panyanak. 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. Strong convergence theorems are established for the modified Halpern iterations of nonexpansive mappings in CAT 0 spaces. Our results extend and improve the recent ones announced by Kim and Xu 2005 Hu 2008 Song and Chen 2008 Saejung 2010 and many others. 1. Introduction Let C be a nonempty subset of a metric space X d . A mapping T C C is said to be nonexpansive if d Tx Ty d x y x y e C. A point x e C is called a fixed point of T if x Tx. We will denote by F T the set of fixed points of T. In 1967 Halpern 1 introduced an explicit iterative scheme for a nonexpansive mapping T on a subset C of a Hilbert space by taking any points u x1 e C and defined the iterative sequence xn by xn 1 anu 1 - an Txn for n 1 where an e 0 1 . He pointed out that the control conditions C1 limnan 0 and C2 X1 an TO are necessary for the convergence of xn to a fixed point of T. Subsequently many mathematicians worked on the Halpern iterations both in Hilbert and Banach spaces 2 Fixed Point Theory and Applications see . 2-11 and the references therein . Among other things Wittmann 7 proved strong convergence of the Halpern .
