On never nonexpansive mappings in reflexive Banach spaces

Abstract

We give conditions under which a self-mapping on a bounded closed convex subset, containing zero, of a reflexive Banach space is never nonexpansive, i.e., there is no renorming with respect to which the mapping is nonexpansive. This provides with a unifying procedure to prove that the classical mappings of Kakutani, Nirenberg and P. K. Lin are not uniformly Lipschitzian in any bounded closed convex subset of $$\ell _2$$ which they leave invariant. In an analogous way, the well known fixed point free mappings of Goebel-Kirk-Thele and Baillon, although they are uniformly Lipschitzian, are also seen to be never nonexpansive in any subset of $$\ell _2$$ where they are self-mappings.