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.