Fixed Point Theory and Applications
Volume 2007 (2007), Article ID 59262, 11 pages
doi:10.1155/2007/59262
Research Article
Strong Convergence of Cesàro Mean Iterations for Nonexpansive Nonself-Mappings in Banach Spaces
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.
Abstract
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:C→E a non-expansive nonself-mapping with F(T)≠∅. In this paper,
we study the strong convergence of two sequences generated by
xn+1=αnx+(1−αn)(1/n+1)∑j=0n(PT)jxn and
yn+1=(1/n+1)∑j=0nP(αny+(1−αn)(TP)jyn)
for all n≥0, where x,x0,y,y0∈C, {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.