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The Perron-Frobenius Theorem for Homogeneous, Monotone Functions

Gaubert, Stephane; Gunawardena, Jeremy


Keyword(s): Collatz-Wielandt property; Hilbert projective metric; nonexpansive function; nonlinear eigenvalue; Perron- Frobenius theorem; strongly connected graph; supereigenspace; topical function

Abstract:If 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 are based on the boundedness of invariant subsets in the Hilbert projective metric and lead to further existence results and several open problems. Notes: Stephane Gaubert, INRIA, Domaine De Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France

19 Pages

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