论文标题
Hölder取消映射，非线性扩展问题和固定点的无效结果
Hölder-contractive mappings, nonlinear extension problem and fixed point free results
论文作者
论文摘要
对于有限的封闭凸套$ k $，在本说明中，我们以$α$-Hölderinonexpansive Maps的形式研究FPP，即映射$ t \ colon k \ colon k \ to k $ to $ \ | t x -ty \ | \ leq \ | x -y \ |^α$用于所有$ x，y \ in K $，$α\ in（0,1）$。首先，我们注意到只有有限维空间才具有Hölder-fpp。此外，任何无限维空间的单位球$ b_x $均在Hölder映射的FPP上，用$ \ mathrm {d}（t，b_x）> 0 $，其中$ \ mathrm {d}（t，k）$表示最小位移$ t $。我们进一步表明，反身性和弱连续性是用有界轨道捕获Hölder-Lipschitz地图的固定点的足够条件。接下来，我们将重点介绍固定点的存在$α$-Hölder映射$ t \ colon k \ to k $ to k $，使用$ \ mathrm {d}（t，k）\leqφ（α）$，其中$φ（α）= 0 $或$φ（α）\ 0 $或$φ（α）可为空格$ \ mathrm {c} $，$ \ co $，$ \ ell_1 $和$ \ ell_2 $获得有趣的结果，以及$ l_p $ -spaces，in [1，\ infty] $。我们还研究了包含$ \ co $和$ \ ell_1 $的副本的空间中的问题。一些问题是打开的。
For a bounded closed convex set $K$, in this note, we study the FPP for $α$-Hölder nonexpansive maps, i.e. mappings $T\colon K\to K$ for which $\|T x -Ty\| \leq\| x - y\|^α$ for all $x, y\in K$, $α\in (0,1)$. 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 $α$-Hölder maps $T\colon K\to K$ with $\mathrm{d}(T, K)\leq φ(α)$ where either $φ(α)=0$ or $φ(α)\to 0$ as $α\to 1$. 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.
