Theτ-fixed point property for nonexpansive mappings

Abstract

LetXbe a Banach space andτa topology onX. We say thatXhas theτ-fixed point property (τ-FPP) if every nonexpansive mappingTdefined from a bounded convexτ-sequentially compact subsetCofXintoChas a fixed point. Whenτsatisfies certain regularity conditions, we show that normal structure assures theτ-FPP and Goebel-Karlovitz's Lemma still holds. We use this results to study two geometrical properties which imply theτ-FPP: theτ-GGLD andM(τ)properties. We show several examples of spaces and topologies where these results can be applied, specially the topology of convergence locally in measure in Lebesgue spaces. In the second part we study the preservence of theτ-FPP under isomorphisms. In order to do that we study some geometric constants for a Banach spaceXsuch that theτ-FPP is shared by any isomorphic Banach spaceYsatisfying that the Banach-Mazur distance betweenXandYis less than some of these constants.