A unified approach to nonlinear Perron-Frobenius theory

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.