International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 16470, 12 pages
doi:10.1155/IJMMS/2006/16470
Strong convergence of approximation fixed points for nonexpansive
nonself-mapping
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
Received 17 May 2006; Accepted 22 June 2006
Copyright © 2006 Rudong Chen and Zhichuan Zhu. 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 C be a closed convex subset of a uniformly smooth Banach
space E, and T:C→E a nonexpansive nonself-mapping satisfying the weakly inwardness condition such that F(T)≠∅, and f:C→C a fixed contractive mapping. For t∈(0,1), the implicit iterative sequence {xt} is defined by xt=P(tf(xt)+(1−t)Txt), the explicit iterative sequence {xn} is given by xn+1=P(αnf(xn)+(1−αn)Txn), where αn∈(0,1) and P is a sunny nonexpansive retraction of E onto C.
We prove that {xt} strongly converges to a fixed point of T
as t→0, and {xn} strongly converges to a fixed point of T as αn satisfying appropriate conditions. The results presented extend and improve the corresponding results of Hong-Kun Xu (2004) and Yisheng Song and Rudong Chen (2006).