H\"older-contractive mappings, nonlinear extension problem and fixed point free results

TL;DR

This paper investigates the fixed point property for H"older nonexpansive maps in various Banach spaces, establishing conditions for existence or absence of fixed points, especially in infinite-dimensional spaces.

Contribution

It characterizes when H"older maps have fixed points in Banach spaces, highlighting the role of space dimensionality, reflexivity, and weak continuity, and constructs fixed point free examples.

Findings

Only finite-dimensional spaces have the H"older-FPP.

Infinite-dimensional unit balls lack the FPP for H"older maps with positive minimal displacement.

Reflexivity and weak sequential continuity ensure fixed points for bounded orbits.

Abstract

For a bounded closed convex set $K$, in this note, we study the FPP for $\alpha$-H\"older nonexpansive maps, i.e. mappings $T\colon K\to K$ for which $\|T x -Ty\| \leq\| x - y\|^\alpha$ for all $x, y\in K$, $\alpha\in (0,1)$. First, we note that only finite-dimensional spaces have the H\"older-FPP. Moreover, the unit ball $B_X$ of any infinite-dimensional space fails the FPP for H\"older maps with $\mathrm{d}(T, B_X)>0$, where $\mathrm{d}(T, K)$ denotes the minimal displacement of $T$. We further show that reflexivity and weak sequential continuity are sufficient conditions to capture fixed points of H\"older-Lipschitz maps with bounded orbits. Next we focus on the existence of fixed point free $\alpha$-H\"older maps $T\colon K\to K$ with $\mathrm{d}(T, K)\leq \varphi(\alpha)$ where either $\varphi(\alpha)=0$ or $\varphi(\alpha)\to 0$ as $\alpha \to 1$. Interesting results are obtained for the spaces $\mathrm{c}$, $\co$, $\ell_1$ and $\ell_2$, and also for $L_p$-spaces with $p\in[ 1, \infty]$. We also study the problem in spaces containing copies of $\co$ and $\ell_1$. Some questions are left open.