Fixed Point Theory and Applications
Volume 2007 (2007), Article ID 59262, 11 pages
Research Article

Strong Convergence of Cesàro Mean Iterations for Nonexpansive Nonself-Mappings in Banach Spaces

Rabian Wangkeeree

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Received 9 March 2007; Accepted 12 September 2007

Academic Editor: Wataru Takahashi

Copyright © 2007 Rabian Wangkeeree. 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 E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from E to E*, C a nonempty closed convex subset of E which is also a sunny nonexpansive retract of E, and T:CE a non-expansive nonself-mapping with F(T). In this paper, we study the strong convergence of two sequences generated by xn+1=αnx+(1αn)(1/n+1)j=0n(PT)jxn and yn+1=(1/n+1)j=0nP(αny+(1αn)(TP)jyn) for all n0, where x,x0,y,y0C, {αn} is a real sequence in an interval [0,1], and P is a sunny non-expansive retraction of E onto C. We prove that {xn} and {yn} converge strongly to Qx and Qy, respectively, as n, where Q is a sunny non-expansive retraction of C onto F(T). The results presented in this paper generalize, extend, and improve the corresponding results of Matsushita and Kuroiwa and many others.