2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61449If 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.20 pages, 3 Postscript figures, v2 (minor revision)Functional Analysis47J10 (Primary) 47H09, 47H07, 15A48 (Secondary)text