The Perron-Frobenius Theorem for Homogeneous, Monotone Functions

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dc.creatorGaubert, Stephane
dc.creatorGunawardena, Jeremy
dc.date2001-05-11
dc.date2003-08-18
dc.date.accessioned2026-07-07T04:41:39Z
dc.date.available2026-07-07T04:41:39Z
dc.descriptionIf A is a nonnegative matrix whose associated directed graph is strongly connected, the Perron-Frobenius theorem asserts that A has an eigenvector in the positive cone, (R^+)^n. We associate a directed graph to any homogeneous, monotone function, f: (R^+)^n -> (R^+)^n, and show that if the graph is strongly connected then f has a (nonlinear) eigenvector in (R^+)^n. Several results in the literature emerge as corollaries. Our methods show that the Perron-Frobenius theorem is ``really'' about the boundedness of invariant subsets in the Hilbert projective metric. They lead to further existence results and open problems.
dc.description20 pages, 3 Postscript figures, v2 (minor revision)
dc.identifierhttps://arxiv.org/abs/math/0105091
dc.identifierhttp://arxiv.org/abs/math/0105091
dc.identifierTrans. Amer. Math. Soc. 356 (2004), 4931-4950.
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/61449
dc.subjectFunctional Analysis
dc.subject47J10 (Primary) 47H09, 47H07, 15A48 (Secondary)
dc.titleThe Perron-Frobenius Theorem for Homogeneous, Monotone Functions
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