tailieunhanh - Báo cáo hóa học: " Research Article Strong Convergence of a Modified Halpern’s Iteration for Nonexpansive Mappings"

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 a Modified Halpern’s Iteration for Nonexpansive Mappings | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008 Article ID 649162 9 pages doi 2008 649162 Research Article Strong Convergence of a Modified Halpern s Iteration for Nonexpansive Mappings Liang-Gen Hu Department of Mathematics Ningbo University Ningbo 315211 China Correspondence should be addressed to Liang-Gen Hu hulianggen@ Received 12 September 2008 Accepted 9 December 2008 Recommended by Jerzy Jezierski The purpose of this paper is to consider that a modified Halpern s iterative sequence xn converges strongly to a fixed point of nonexpansive mappings in Banach spaces which have a uniformly Gateaux differentiable norm. Our result is an extension of the corresponding results. Copyright 2008 Liang-Gen Hu. 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 C a nonempty closed convex subset of E. We denote by J the normalized duality map from E to 2E E is the dual spaces of E defined by J x f e E x f llxll2 Ilf II2 Vx e E . A mapping T C C is said to be nonexpansive if Tx - Ty x - yịị for all x y e C. We denote by Fix T x e C Tx x the set of fixed points of T. In the last ten years many papers have been written on the approximation of fixed point for nonlinear mappings by using some iterative processes see . 1-18 . An explicit iterative process was initially introduced in 1967 by Halpern 3 in the framework of Hilbert spaces that is Halpern s iteration. For any u x0 e C the sequence xn is defined by xn 1 anu 1 - an Txn Vn 0 where an c 0 1 . He proved that the sequence xn converges weakly to a fixed point of T where an n a a e 0 1 . In 1977 Lions 8 further proved that the sequence xn converges 2 Fixed Point Theory and Applications strongly to a fixed point of T in a Hilbert space where an satisfies the .