AbstractThe convex feasibility problem (CFP) of finding a point in the nonempty intersection "Equation missing" is considered, where "Equation missing" is an integer and each "Equation missing" is assumed to be the fixed point set of a nonexpansive mapping "Equation missing", where "Equation missing" 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 "Equation missing", where "Equation missing" is a nonempty closed convex subset of "Equation missing" and for any given "Equation missing" the iterative scheme "Equation missing" is strongly convergent to a solution of (CFP), if and only if "Equation missing" and "Equation missing" satisfy certain conditions, where "Equation missing" and "Equation missing" is a sunny nonexpansive retraction of "Equation missing" onto "Equation missing". 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).