Abstract
It is well known that an ordered normed vector space X with normal cone X+ has an order-preserving norm that is equivalent to the original norm. Such an equivalent order-preserving norm is given by
This paper explores the properties of this norm and of the half-norm ψ(x) = d(x,–X+) independently of whether or not the cone is normal. We use ψ to derive comparison principles for the solutions of abstract integral equations and compare Collatz–Wielandt numbers, bounds, and order-spectral radii for order-preserving homogeneous maps and give conditions for a local upper Collatz–Wielandt radius to have a lower positive eigenvector. For illustration, we consider a rank-structured population with mating.
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© 2016 Springer International Publishing Switzerland
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Thieme, H.R. (2016). Spectral Radii and Collatz–Wielandt Numbers for Homogeneous Order-preserving Maps and the Monotone Companion Norm. In: de Jeu, M., de Pagter, B., van Gaans, O., Veraar, M. (eds) Ordered Structures and Applications. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-27842-1_26
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DOI: https://doi.org/10.1007/978-3-319-27842-1_26
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-27840-7
Online ISBN: 978-3-319-27842-1
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