Upper and lower bounds for the iterates of order-preserving homogeneous maps on cones

2012/05/31 by Philip S. Chodrow, Philip Chodrow, Chodrow, Philip +4
Computer Science · Immunology and Microbiology · Mathematics · #47H07 (Primary) 15B48 (Secondary) #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Macrophage Migration Inhibitory Factor #Matrix Theory and Algorithms #math.DS #math.FA #msc:15B48 #msc:47H07

paper · pdf · doi:10.48550/arxiv.1205.7003

arxiv created 2012/05/31 · openalex publication_date 2012/05/31 · arxiv updated 2012/06/01 · openalex created_date 2022/09/19 · openalex updated_date 2026/07/28

Abstract

We define upper bound and lower bounds for order-preserving homogeneous of degree one maps on a proper closed cone in \Rⁿ in terms of the cone spectral radius. We also define weak upper and lower bounds for these maps. For a proper closed cone C ⊂ \Rⁿ, we prove that any order-preserving homogeneous of degree one map f: \inter C → \inter C has a lower bound. If C is polyhedral, we prove that the map f has a weak upper bound. We give examples of weak upper bounds for certain order-preserving homogeneous of degree one maps defined on the interior of \Rⁿ₊.

Related

• The Berger-Wang formula for order-preserving homogeneous maps on cones
• A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones
• Eigenvectors and eigenfunctionals of homogeneous order-preserving maps
• Continuity of the cone spectral radius
• Comparison of spectral radii and Collatz-Wielandt numbers for homogeneous maps, and other applications of the monotone companion norm on ordered normed vector spaces