A unified approach to nonlinear Perron-Frobenius theory

TL;DR

This paper presents a unified framework for analyzing the existence and uniqueness of eigenvectors of order-preserving, homogeneous functions on positive reals, extending previous hypergraph-based tests with new computable conditions.

Contribution

It introduces a new condition involving Collatz-Wielandt numbers that generalizes existing hypergraph tests for eigenvector existence.

Findings

Eigenvectors exist and are bounded under the new condition.

The set of eigenvectors is unique up to scaling for real analytic functions.

The approach combines the new condition with hypergraph tests for systematic analysis.

Abstract

Let $f:\mathbb{R}^n_{> 0} \rightarrow \mathbb{R}^n_{>0}$ be an order-preserving and homogeneous function. We show that the set of eigenvectors of $f$ in $\mathbb{R}^n_{>0}$ is nonempty and bounded in Hilbert's projective metric if and only if $f$ satisfies a condition involving upper and lower Collatz-Wielandt numbers of readily computed auxiliary functions. This condition generalizes a test for the existence of eigenvectors using hypergraphs that was proved by Akian, Gaubert, and Hochart. We include several examples to show how the new condition can be combined with the hypergraph test to give a systematic approach to determine when homogeneous and order-preserving functions have eigenvectors in $\mathbb{R}^n_{>0}$. We also observe that if the entries of $f$ are real analytic functions on $\mathbb{R}^n_{>0}$, then the set of eigenvectors of $f$ in $\mathbb{R}^n_{>0}$ is nonempty and bounded in Hilbert's projective metric if and only if the eigenvector is unique, up to scaling.