### Nonexpansive mappings on Hilbert's metric spaces

#### Abstract

This paper deals with the iterative behavior of nonexpansive mappings

on Hilbert's metric spaces $(X,d_X)$. We show that if $(X,d_X)$

is strictly convex and does not contain a hyperbolic plane, then

for each nonexpansive mapping, with a fixed point in $X$, all orbits

converge to periodic orbits. In addition, we prove that if $X$ is

an open $2$-simplex, then the optimal upper bound for the periods

of periodic points of nonexpansive mappings on $(X,d_X)$ is $6$.

The results have applications in the analysis of nonlinear mappings

on cones, and extend work by Nussbaum and others.

on Hilbert's metric spaces $(X,d_X)$. We show that if $(X,d_X)$

is strictly convex and does not contain a hyperbolic plane, then

for each nonexpansive mapping, with a fixed point in $X$, all orbits

converge to periodic orbits. In addition, we prove that if $X$ is

an open $2$-simplex, then the optimal upper bound for the periods

of periodic points of nonexpansive mappings on $(X,d_X)$ is $6$.

The results have applications in the analysis of nonlinear mappings

on cones, and extend work by Nussbaum and others.

#### Keywords

Nonexpansive mappings; Hilbert's metric; periodic orbits; hyperbolic plane; simplices; nonlinear mappings on cones

#### Full Text:

FULL TEXT### Refbacks

- There are currently no refbacks.