Nonlinear Perron-Frobenius theory in finite dimensions

A nonlinear Perron-Frobenius theory in finite dimensions is described which applies to positively homogeneous and increasing maps. Sufficient conditions for the existence and uniqueness of eigenvectors in the interior of a cone are developed even when eigenvectors at the boundary of the cone exist. Several ways to benefit from nonlinearities are pointed out in order to show that the classical, linear, theory truly deserves nonlinear generalization. The main technical novelties are: a link between Collatz-Wielandt numbers and Gâteaux derivatives, and the almost one-dimensional dynamics of the nonlinear map.