Fixed Point Theory and Applications
Volume 2007 (2007), Article ID 46797, 19 pages
doi:10.1155/2007/46797

Iterative approximation to convex feasibility problems in Banach space

Shih-Sen Chang1 , Jen-Chih Yao2 , Jong Kyu Kim3 and Li Yang4

1Department of Mathematics, Yibin University, Yibin 644007, Sichuan, China
 2Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
 3Department of Mathematics Education, Kyungnam University, Masan 631-701, South Korea
 4Department of Mathematics, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China

Abstract

The convex feasibility problem (CFP) of finding a point in the nonempty intersection ∩i=1NCi is considered, where N≥1 is an integer and each Ci is assumed to be the fixed point set of a nonexpansive mapping Ti:E→E, 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:C→C, where C is a nonempty closed convex subset of E and for any given x0∈C the iterative scheme xn+1=P[αn+1f(xn)+(1−αn+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).