Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Eigenvectors of homogeneous order-bounded order-preserving maps

Pages: 1073 - 1097, Volume 22, Issue 3, May 2017      doi:10.3934/dcdsb.2017053

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Horst R. Thieme - Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804, United States (email)

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.

Keywords:  Ordered normed vector space, homogeneous map, cone spectral radius, Collatz-Wielandt numbers, Krein-Rutman theorem, nonlinear eigenvectors, power compactness, population models with mating.
Mathematics Subject Classification:  Primary: 47H07, 47J10; Secondary: 47N60.

Received: September 2015;      Revised: May 2016;      Available Online: January 2017.