Iterative approaches to convex feasibility problems in Banach spaces

The convex feasibility problem (CFP) of finding a point in the nonempty intersection ⋂i=1NCi is considered, where N⩾1N⩾1 is an integer and each CiCi is assumed to be the fixed point set of a nonexpansive mapping Ti:X→XTi:X→X with X   a Banach space. It is shown that the iterative scheme xn+1=λn+1y+(1-λn+1)Tn+1xnxn+1=λn+1y+(1-λn+1)Tn+1xn is strongly convergent to a solution of (CFP) provided the Banach space X   either is uniformly smooth or is reflexive and has a weakly continuous duality map, and provided the sequence {λn}{λn} satisfies certain conditions. The limit of {xn}{xn} is located as Q(y)Q(y), where Q is the sunny nonexpansive retraction from X   onto the common fixed point set of the Ti′s.