International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 16470, 12 pages

Strong convergence of approximation fixed points for nonexpansive nonself-mapping

Rudong Chen and Zhichuan Zhu

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.


Let C be a closed convex subset of a uniformly smooth Banach space E, and T:CE a nonexpansive nonself-mapping satisfying the weakly inwardness condition such that F(T), and f:CC a fixed contractive mapping. For t(0,1), the implicit iterative sequence {xt} is defined by xt=P(tf(xt)+(1t)Txt), the explicit iterative sequence {xn} is given by xn+1=P(αnf(xn)+(1αn)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 t0, 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).