Hölder-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 $\alpha$-Hölder nonexpansive maps, i.e. mappings $T\text{\hspace{0.17em}⁣}:K\to K$ for which $\mathrm{\parallel }Tx-Ty\mathrm{\parallel }\le \mathrm{\parallel }x-y{\mathrm{\parallel }}^{\alpha }$ for all $x,y\in K$, $\alpha \in (0,1)$. First, we note that only finite-dimensional spaces have the Hölder-FPP. Moreover, the unit ball $B_X$ of any infinite-dimensional space fails the FPP for Hölder 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 Hölder-Lipschitz maps with bounded orbits. Next we focus on the existence of fixed point free $\alpha$-Hölder maps $T\text{\hspace{0.17em}⁣}:K\to K$ with $\mathrm{d}(T,K)\le \phi (\alpha )$ where either $\phi (\alpha )=0$ or $\phi (\alpha )\to 0$ as $\alpha \to 1$. Interesting results are obtained for the spaces $\mathrm{c}$, $\text{co}$, ${\mathrm{\ell }}_1$ and ${\mathrm{\ell }}_2$, and also for $L_p$-spaces with $p\in [1,\mathrm{\infty }]$. We also study the problem in spaces containing copies of $\text{co}$ and ${\mathrm{\ell }}_1$. Some questions are left open.