Iterative Approximation to Convex Feasibility Problems in Banach Space

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).

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