### Fixed point theorems and Denjoy-Wolff theorems for Hilbert's projective metric in infinite dimensions

DOI: http://dx.doi.org/10.12775/TMNA.2007.009

#### Abstract

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 $\partial K$.

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 $\partial K$.

#### Keywords

Fixed point theorems; Hilbert's projective metric; Denjoy-Wolff theorems

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