Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 268270, 9 pages
doi:10.1155/2010/268270

The Szlenk index and the fixed point property under renorming

Abstract

Assume that X is a Banach space such that its Szlenk index Sz(X) is less than or equal to the first infinite ordinal ω. We prove that X can be renormed in such a way that X with the resultant norm satisfies R(X)<2, where R(⋅) is the García-Falset coefficient. This leads us to prove that if X is a Banach space which can be continuously embedded in a Banach space Y with Sz(Y)≤ω, then, X can be renormed to satisfy the w-FPP. This result can be applied to Banach spaces which can be embedded in C(K), where K is a scattered compact topological space such that K(ω)=∅. Furthermore, for a Banach space (X,‖⋅‖), we consider a distance in the space 𝒫 of all norms in X which are equivalent to ‖⋅‖ (for which 𝒫 becomes a Baire space). If Sz(X)≤ω, we show that for almost all norms (in the sense of porosity) in 𝒫, X satisfies the w-FPP. For general reflexive spaces (independently of the Szlenk index), we prove another strong generic result in the sense of Baire category.