Fixed Point Theory and Applications
Volume 2007 (2007), Article ID 46797, 19 pages
Research Article

Iterative Approximation to Convex Feasibility Problems in Banach Space

Shih-Sen Chang,1,2 Jen-Chih Yao,3 Jong Kyu Kim,4 and Li Yang5

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 i=1NCi is considered, where N1 is an integer and each Ci is assumed to be the fixed point set of a nonexpansive mapping Ti:EE, where E 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 f:CC, where C is a nonempty closed convex subset of E and for any given x0C the iterative scheme xn+1=P[αn+1f(xn)+(1αn+1)Tn+1xn] is strongly convergent to a solution of (CFP), if and only if {αn} and {xn} satisfy certain conditions, where αn(0,1),Tn=Tn(modN) and P is a sunny nonexpansive retraction of E onto C. 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).