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Báo cáo hóa học: " Research Article Approximating Common Fixed Points of Lipschitzian Semigroup in Smooth Banach Spaces"

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 Approximating Common Fixed Points of Lipschitzian Semigroup in Smooth Banach Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008 Article ID 363257 17 pages doi 10.1155 2008 363257 Research Article Approximating Common Fixed Points of Lipschitzian Semigroup in Smooth Banach Spaces Shahram Saeidi Department of Mathematics University of Kurdistan Sanandaj 416 Kurdistan 66196-64583 Iran Correspondence should be addressed to Shahram Saeidi sh.saeidi@uok.ac.ir Received 16 August 2008 Accepted 10 December 2008 Recommended by Mohamed Khamsi Let S be a left amenable semigroup let S T s s e S be a representation of S as Lipschitzian mappings from a nonempty compact convex subset C of a smooth Banach space E into C with a uniform Lipschitzian condition let _yn be a strongly left regular sequence of means defined on an S-stable subspace of l S let f be a contraction on C and let an fn and ỵn be sequences in 0 1 such that an pn Yn 1 for all n. Let xn 1 anf xn fnxn pnT yn xn for all n 1. Then under suitable hypotheses on the constants we show that xn converges strongly to some z in F S the set of common fixed points of S which is the unique solution of the variational inequality if - I z f y - z 0 for all y e F S . Copyright 2008 Shahram Saeidi. 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 Let E be a real Banach space and let C be a nonempty closed convex subset of E. A mapping T C C is said to be i Lipschitzian with Lipschitz constant l 0 if Tx - Ty l x - yỊỊ Nx y e C 1.1 ii nonexpansive if Tx - Tyll x - y Nx y e C 1.2 iii asymptotically nonexpansive if there exists a sequence kn of positive numbers satisfying the property limn k.n 1 and Tnx - TnyW kn x - y x y e C. 1.3 2 Fixed Point Theory and Applications Halpern 1 introduced the following iterative scheme for approximating a fixed point of a nonexpansive mapping T on C xn 1 anx 1 - an Txn n 1 2