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: STRONG CONVERGENCE TO COMMON FIXED POINTS OF NONEXPANSIVE MAPPINGS WITHOUT COMMUTATIVITY ASSUMPTION | STRONG CONVERGENCE TO COMMON FIXED POINTS OF NONEXPANSIVE MAPPINGS WITHOUT COMMUTATIVITY ASSUMPTION YONGHONG YAO RUDONG CHEN AND HAIYUN ZHOU Received 11 June 2006 Revised 27 July 2006 Accepted 2 August 2006 We introduce an iteration scheme for nonexpansive mappings in a Hilbert space and prove that the iteration converges strongly to common fixed points of the mappings without commutativity assumption. Copyright 2006 Yonghong Yao 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 Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. A mapping T of C into itself is said to be nonexpansive if Tx - TyUllx - yll for each x y e C. For a mapping T of C into itself we denote by F T the set of fixed points of T. We also denote by N and R the set of positive integers and nonnegative real numbers respectively. Baillon 1 proved the first nonlinear ergodic theorem. Let C be a nonempty bounded convex closed subset of a Hilbert space H and let T be a nonexpansive mapping of C into itself. Then for an arbitrary x e C 1 n 1 yn 0 Tix 0 0 converges weakly to a fixed point of T. Wittmann 9 studied the following iteration scheme which has first been considered by Halpern 3 x0 x e C Xn 1 an 1x 1 - ữn 1 Txn n 0 where a sequence an in 0 1 is chosen so that limn o an 0 y0 1 an o and y0 1 an 1 - an 00 see also Reich 7 . Wittmann proved that for any x e C the sequence Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006 Article ID 89470 Pages 1-8 DOI FPTA 2006 89470 2 Nonexpansive mappings without commutativity assumption xn defined by converges strongly to the unique element Px e F T where P is the metric projection of H onto F T . Recall that two mappings S and T of H into itself are called commutative if ST TS for all x y e H. .
