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→2X 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. © 2005 Elsevier Ltd. All rights reserved.