Iterative approximation to convex feasibility problems in Banach space

The convex feasibility problem (CFP) of finding a point in the nonempty intersection boolean AND C-N(i=1)i is considered, where N >= 1 is an integer and each C-i is assumed to be the fixed point set of a nonexpansive mapping T-i : 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 x(0) is an element of C the iterative scheme x(n+1) = P[alpha(n+1) f (x(n)) + (1- alpha(n+1)) T(n+1)x(n)] is strongly convergent to a solution of ( CFP), if and only if {an} and {xn} satisfy certain conditions, where alpha(n) is an element of (0,1), T-n = T-n(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), 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).