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

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.