We characterize the family of nonexpansive mappings which are invariant under renormings and we also compare the families of nonexpansive mappings under two equivalent norms.

<abstract><p>In this paper we present the following two results: 1.- A characterization of the renorming invariant family of asymptotically nonexpansive mappings defined on a convex, closed and bounded set of a Banach space; 2.- A comparison of the renorming invariant family of asymptotically nonexpansive mappings with the renorming invariant family of nonexpansive mappings. Additionally, a series of examples are shown for general and particular cases.</p></abstract>

Given a Banach space (X, • ) and a subset C of X, we consider the family of bounded Lipschitzian mappings BLip(C, X). This family is endowed with a norm and a topology that does not depend on renormings. With this topology we prove that it is not enough to consider the family of mappings that are nonexpansive with respect to finitely many renormings, to get the family of mappings that are nonexpansive w.r.t. all renormings.