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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 Halpern’s Iteration in CAT(0) Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 471781 13 pages doi 10.1155 2010 471781 Research Article Halpern s Iteration in CAT 0 Spaces Satit Saejung1 2 1 Department of Mathematics Faculty of Science Khon Kaen University Khon Kaen 40002 Thailand 2 Centre of Excellence in Mathematics CHE Sriayudthaya Road Bangkok 10400 Thailand Correspondence should be addressed to Satit Saejung saejung@kku.ac.th Received 26 September 2009 Accepted 24 November 2009 Academic Editor Mohamed A. Khamsi Copyright 2010 Satit Saejung. 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. Motivated by Halpern s result we prove strong convergence theorem of an iterative sequence in CAT 0 spaces. We apply our result to find a common fixed point of a family of nonexpansive mappings. A convergence theorem for nonself mappings is also discussed. 1. Introduction Let fX d be a metric space and x y e X with l d x y . A geodesic path from x to y is an isometry c 0 l X such that c 0 x and c T y. The image of a geodesic path is called a geodesic segment. A metric space X is a uniquely geodesic space if every two points of X are joined by only one geodesic segment. A geodesic triangle A x1 x2 x3 in a geodesic space X consists of three points x1 x2 x3 of X and three geodesic segments joining each pair of vertices. A comparison triangle of a geodesic triangle A x1 x2 x3 is the triangle à x1 x2 x3 A x1 x2 x3 in the Euclidean space R2 such that dfxi xj dR2 xi xf for all i j 1 2 3. A geodesic space X is a CAT 0 space if for each geodesic triangle A A x1 x2 x3 in X and its comparison triangle A A x1 x2 x3 in R2 the CAT 0 inequality d x y dR2 x y 1.1 is satisfied by all x y e A and x y e A. The meaning of the CAT 0 inequality is that a geodesic triangle in X is at least thin as its comparison triangle in the
