Ramı́rez, P.: Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings

Abstract. 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 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 One of the most celebrated result in Metric Fixed Point Theory was given by R. Bruck [1] who proved that the fixed point set of a nonexpansive mapping T: C → C is a nonexpansive retract of C when C is a weakly compact convex subset of a Banach space X which has ”nice ” properties with respect to the existence of fixe