A Nonlinear Perron-Frobenius Theorem

Abstract

If $T$ is a nonexpansive map on a domain in a finite-dimensional sup norm space then there is a universal bound on the periods of periodic points. This yields the same result for $T$ nonexpansive on a domain in a finite-dimensional Banach space which has a polyhedral unit ball. Similar results are obtained for certain nonexpansive maps defined on all of an infinite-dimensional ${L_p}$ space with $1 < p < \infty$.