  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 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.
