Hölder-contractive mappings, nonlinear extension problem and fixed point free results

Abstract

For a bounded closed convex set K, in this note, we study the FPP for α-Hölder nonexpansive maps, i.e. mappings T:K→K for which ‖Tx−Ty‖≤‖x−y‖α for all x,y∈K, α∈(0,1). First, we note that only finite-dimensional spaces have the Hölder-FPP. Moreover, the unit ball BX of any infinite-dimensional space fails the FPP for Hölder maps with d(T,BX)>0, where d(T,K) denotes the minimal displacement of T. We further show that reflexivity and weak sequential continuity are sufficient conditions to capture fixed points of Hölder-Lipschitz maps with bounded orbits. Next we focus on the existence of fixed point free α-Hölder maps T:K→K with d(T,K)≤φ(α) where either φ(α)=0 or φ(α)→0 as α→1. Interesting results are obtained for the spaces c, c0 and ℓ1, and also for Lp-spaces with p∈{1,∞}. We also study the problem in spaces containing copies of c0 and ℓ1. Some questions are left open.