Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings

Let X X be a Banach space, C C a weakly compact convex subset of X X and T : C → C T:C\to C an asymptotically nonexpansive mapping. Under the usual assumptions on X X which assure the existence of fixed point for T T , we prove that the set of fixed points is a nonexpansive retract of C C . We use this result to prove that all known theorems about existence of fixed point for asymptotically nonexpansive mappings can be extended to obtain a common fixed point for a commuting family of mappings. We also derive some results about convergence of iterates.