Upper and lower bounds for the iterates of order-preserving homogeneous maps on cones

TL;DR

This paper investigates bounds for order-preserving homogeneous maps on cones, establishing the existence of lower bounds generally and weak upper bounds for polyhedral cones, with examples illustrating these bounds.

Contribution

It introduces new concepts of upper and lower bounds for such maps and proves their existence under various conditions, advancing understanding of their iterative behavior.

Findings

Existence of lower bounds for all such maps on the interior of cones.

Existence of weak upper bounds for maps on polyhedral cones.

Examples of weak upper bounds on the interior of the positive orthant.

Abstract

We define upper bound and lower bounds for order-preserving homogeneous of degree one maps on a proper closed cone in $\R^n$ in terms of the cone spectral radius. We also define weak upper and lower bounds for these maps. For a proper closed cone $C \subset \R^n$, we prove that any order-preserving homogeneous of degree one map $f: \inter C \rightarrow \inter C$ has a lower bound. If $C$ is polyhedral, we prove that the map $f$ has a weak upper bound. We give examples of weak upper bounds for certain order-preserving homogeneous of degree one maps defined on the interior of $\R^n_+$.