Eigenvectors of homogeneous order-bounded order-preserving maps

Abstract

The existence of eigenvectors associated with the cone spectral radius is shown for homogenous, order-preserving, continuous maps that have compact and order-bounded powers (iterates). The order-boundedness makes it possible to show the existence of eigenvectors for perturbations of the maps using Hilbert's projective metric, while the power compactness or similar compactness properties together with a uniform continuity condition let the eigenvectors of the perturbations converge to an eigenvector of the original map.