tailieunhanh - Báo cáo hóa học: " Research Article Approximating Fixed Points of Nonexpansive Mappings in Hyperspaces"

Tham khảo tài liệu 'báo cáo hóa học: " research article approximating fixed points of nonexpansive mappings in hyperspaces"', luận văn - báo cáo phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007 Article ID 50596 9 pages doi 2007 50596 Research Article Approximating Fixed Points of Nonexpansive Mappings in Hyperspaces Zeqing Liu Chi Feng Shin Min Kang and Jeong Sheok Ume Received 29 March 2007 Revised 27 August 2007 Accepted 13 September 2007 Recommended by Wataru Takahashi Two convergence theorems for the Ishikawa and Mann iteration sequences involving nonexpansive mappings in hyperspaces are established. Copyright 2007 Zeqing Liu et al. 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 X be a nonempty compact subset of a Banach space E ll H and let C X and CC X denote the families of all nonempty compact and all nonempty compact convex subsets of X respectively. It is well known that C X H is compact where H is the Hausdorff metric induced by ll H- For A B G CC X and t G R -TO ro let A B a b a G A b G B and let tA ta a G A . In the sequel we assume that X is a nonempty compact convex subset of E. Hu and Huang 1 proved that CC X H is a compact subset of C X H . It is clear that tA 1 - t B G CC X for all A B G CC X and t G 0 1 . That is CC X has convexity structure. Let I be a nonempty subset of CC X . A mapping T I H I H is said to be nonexpansive if H TA TB H A B for all A B G I. Within the past 20 years or so a few researchers have applied the Mann iteration method and the Ishikawa iteration method to approximate fixed points of nonexpansive mappings in several classes of subsets of Banach spaces. For details we refer to 2-11 . Recently Hu and Huang 1 established the following result. Theorem . Let X be a nonempty compact convex subset of a Banach space E ll ll and let I be a nonempty compact convex subset of CC X . Suppose that T I H I H is nonexpansive. Then for any A0