A geometrical coefficient implying the fixed point property and stability results

Abstract

In this paper we define a new geometric constant M(X) in Banach spaces such that X has the fixed point property for nonexpansive mappings if M(X) > 1. We prove that M(X) •_ WCS(X), the inequality being strict in many important classes of Banach spaces and we obtain lower bounds for M(X) based upon either the modulus of near uniform smoothness or the modulus of the Opia] property of the conjugated space. We show that this new constant gives us stability results for the fixed point property with respect to œp-spaces which improve all previous results.