Fixed Point Theory and Applications
Volume 2007 (2007), Article ID 46797, 19 pages
Iterative Approximation to Convex Feasibility Problems in Banach Space
1Department of Mathematics, Yibin University, Yibin 644007, Sichuan, China
2Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
3Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
4Department of Mathematics Education, Kyungnam University, Masan 631-701, South Korea
5Department of Mathematics, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China
Received 7 November 2006; Accepted 6 February 2007
Academic Editor: Billy E. Rhoades
Copyright © 2007 Shih-Sen Chang et al. 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.
The convex feasibility problem (CFP) of finding a point in the nonempty intersection is considered, where is an integer and each is assumed to be the fixed point set of a nonexpansive mapping , where is a reflexive Banach space with a weakly sequentially continuous duality
mapping. By using viscosity approximation methods for a finite family of nonexpansive mappings, it is shown that for any given contractive mapping , where is a nonempty closed convex subset of and for any given the iterative scheme is strongly convergent to a solution of (CFP), if and only if
and satisfy certain conditions, where and is a sunny nonexpansive retraction of onto . The results presented in the paper extend and improve some recent results in Xu (2004), O'Hara et al. (2003), Song and Chen (2006), Bauschke (1996), Browder (1967), Halpern (1967), Jung (2005), Lions (1977), Moudafi (2000), Reich (1980), Wittmann (1992), Reich (1994).