Let be a Banach space and let be a nonempty closed convex subset of . Let be a subset of and let be a mapping of into . Then is said to be sunny if whenever for and . A mapping of into is said to be a retraction if . A subset of is said to be sunny nonexpansive retract on if there exists a sunny and nonexpansive retraction of onto .
In the present paper the authors establish, among others, the following theorem: Let be a uniformly convex Banach space with a Fréchet differentiable norm and let be a nonempty closed convex subset of . Let be sunny nonexpansive retracts of such that . Let be an operator on given by , , , , such that for each , , , where is a sunny nonexpansive retraction of onto . Then and further, for each , converges weakly to an element of .
Finally the authors prove a common fixed point theorem for a finite commuting family of nonexpansive mappings in a uniformly convex and uniformly smooth Banach space.