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 Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008 Article ID 167535 14 pages doi 2008 167535 Research Article Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces Jong Soo Jung Department of Mathematics Dong-A University Busan 604-714 South Korea Correspondence should be addressed to Jong Soo Jung jungjs@ Received 13 January 2008 Revised 5 April 2008 Accepted 3 May 2008 Recommended by Mohammed Khamsi Let E be a reflexive Banach space with a uniformly Gateaux differentiable norm. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Let C be a nonempty closed convex subset of E f C C a contractive mapping or a weakly contractive mapping and T C C nonexpansive mapping with the fixed point set F T 0. Let xn be generated by a new composite iterative scheme yn inf xn 1 -kn Txn xn 1 1 - fn yn fnTyn n 0 . It is proved that xn converges strongly to a point in F T which is a solution of certain variational inequality provided that the sequence Ãn c 0 1 satisfies lim . 0 and XX1in .pn c 0 a for some 0 a 1 and the sequence xn is asymptotically regular. Copyright 2008 Jong Soo Jung. 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. Recall that a mapping f C C is a contraction on C if there exists a constant k G 0 1 such that f x -f y k x - y x y G C. We use Sc to denote the collection of mappings f verifying the above inequality. That is SC f C C f is a contraction with constant k . Note that each f G SC has a unique fixed point in C. Now let T C C be a nonexpansive mapping recall that a mapping T C C is nonexpansive if Tx - Ty x - yịị x y G C and F T denote the set of fixed points
