Structure of the fixed point set and common fixed points of asymptoticallynonexpansive mappings

Let X be a Banach space, C a weakly compact convex subset of X and T : C -- > C an asymptotically nonexpansive mapping. Under the usual assumptions on X which assure the existence of fixed point for T, we prove that the set of fixed points is a nonexpansive retract of C. We use this result to prove t hat all known theorems about existence of fixed point for asymptotically no nexpansive mappings can be extended to obtain a common fixed point for a co mmuting family of mappings. We also derive some results about convergence o f iterates.