International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 391839, 9 pages
Research Article

A New Iterative Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

Received 16 December 2008; Accepted 9 April 2009

Academic Editor: Jie Xiao

Copyright © 2009 Suwicha Imnang and Suthep Suantai. 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.


Let X be a real uniformly convex Banach space and C a closed convex nonempty subset of X. Let {Ti}i=1r be a finite family of nonexpansive self-mappings of C. For a given x1C, let {xn} and {xn(i)}, i=1,2,,r, be sequences defined xn(0)=xn,xn(1)=an1(1)T1xn(0)+(1-an1(1))xn(0),xn(2)=an2(2)T2xn(1)+an1(2)T1xn+(1-an2(2)-an1(2))xn,,xn+1=xn(r)=anr(r)Trxn(r-1)+an(r-1)(r)Tr-1xn(r-2)++an1(r)T1xn+(1-an(r)(r)-an(r-1)(r)--an1(r))xn,n1, where ani(j)[0,1] for all j{1,2,,r}, n and i=1,2,,j. In this paper, weak and strong convergence theorems of the sequence {xn} to a common fixed point of a finite family of nonexpansive mappings Ti(i=1,2,,r) are established under some certain control conditions.