Iterative approximation to convex feasibility problems in Banach space

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+1 f (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).