Description

• Title:
  Fixed point theorems and Denjoy-Wolff theorems for Hilbert's projective metric in infinite dimensions
• Creator:
  Nussbaum, Roger D.
• Description:
  Let $K$ be a closed, normal cone with nonempty interior $\inta(K)$ in a Banach space $X$. Let $\Sigma = \{x\in\inta(K) : q(x) = 1\}$ where $q \colon \inta(K)\rightarrow (0,\infty)$ is continuous and homogeneous of degree $1$ and it is usually assumed that $\Sigma$ is bounded in norm. In this framework there is a complete metric $d$, {\it Hilbert's projective metric}, defined on $\Sigma$ and a complete metric $\overline d$, {\it Thompson's metric}, defined on ${\rm \int}(K)$. We study primarily maps $f\colon \Sigma\rightarrow\Sigma$ which are nonexpansive with respect to $d$, but also maps $g \colon {\rm \int}(K)\rightarrow {\rm \int}(K)$ which are nonexpansive with respect to $\overline{d}$. We prove under essentially minimal compactness assumptions, fixed point theorems for $f$ and $g$. We generalize to infinite dimensions results of A. F. Beardon (see also A. Karlsson and G. Noskov) concerning the behaviour of Hilbert's projective metric near $\partial\Sigma := \overline\Sigma \setminus \Sigma$. If $x \in \Sigma$, $f \colon \Sigma\rightarrow \Sigma$ is nonexpansive with respect to Hilbert's projective metric, $f$ has no fixed points on $\Sigma$ and $f$ satisfies certain mild compactness assumptions, we prove that $\omega (x;f)$, the omega limit set of $x$ under $f$ in the norm topology, is contained in $\partial\Sigma$; and there exists $\eta\in\partial\Sigma$, $\eta$ independent of $x$, such that $(1 - t) y + t\eta \in\partial K$ for $0 \leq t \leq 1$ and all $y\in \omega (x;f)$. This generalizes results of Beardon and of Karlsson and Noskov. We give some evidence for the conjecture that $\text{\rm co}(\omega(x;f))$, the convex hull of $\omega(x;f)$, is contained in $\partial K$.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2007.06.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 29, No 2 (June 2007); 199 - 249;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2007.009/8831
• Subject:
  Fixed point theorems;
  Hilbert's projective metric;
  Denjoy-Wolff theorems;
  47H07;
  47H09;
  47H10
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2007.009
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other