Weak fixed point property for nonexpansive mappings with respect to orbits in Banach spaces

Abstract

In this paper, we first show that a Banach space X has weak normal structure if and only if X has the weak fixed point property for nonexpansive mappings with respect to (wrt) orbits. Then, we give a counterexample to show that the Goebel–Karlovitz lemma does not hold for minimal invariant sets of nonexpansive mappings wrt orbits, and we present a modified version of the Goebel–Karlovitz lemma. © 2016, Springer International Publishing.