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

Let $X$ be a Banach space, $C$ a weakly compact convex subset of $X$and $T:C\to C$ 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 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.