Research Article Iterative Approximation to Convex Feasibility Problems in Banach Space

The convex feasibility problem (CFP) of finding a point in the nonempty intersection ∩Ni=1Ci 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 approxima-tion 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(mod N) 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)