The Domínguez–Lorenzo condition and multivalued nonexpansive mappings

Abstract

Let E be a nonempty bounded closed convex separable subset of a reflexive Banach space X which satisfies the Domínguez–Lorenzo condition, i.e., an inequality concerning the asymptotic radius of a sequence and the Chebyshev radius of its asymptotic center. We prove that a multivalued nonexpansive mapping T : E → 2 X which is compact convex valued and such that T ( E ) is bounded and satisfies an inwardness condition has a fixed point. As a consequence, we obtain a fixed-point theorem for multivalued nonexpansive mappings in uniformly nonsquare Banach spaces which satisfy the property WORTH, extending a known result for the case of nonexpansive single-valued mappings. We also prove a common fixed point theorem for two nonexpansive commuting mappings t : E → E and T : E → KC ( E ) (where KC ( E ) denotes the class of all compact convex subsets of E) when X is a uniformly convex Banach space.