A Graph Theoretic Approach to Strong Monotonicity with respect to Polyhedral Cones

A Graph Theoretic Approach to Strong Monotonicity with respect to Polyhedral Cones Consider the flow ϕt for the system of differential equations $$\dot x\left( t \right) = f\left( x \right)$$ , xεΩ, Ω⊂ $$\mathbb{R}$$ n, Ω open. Let K(t) be an expanding polyhedral cone of constant dimension, k be a unit vector in K(0), and x 0εΩ. A sufficient condition for $$\frac{{\partial \phi _t }}{{\partial k}}\left( {x_0 } \right)$$ εK(t) for t≥0 is that there exists an l so that Df(ϕt(x0))+lI leaves K(t) invariant for all t≥0. If in addition (Df(ϕt(x0))+lI)n-1 takes k into the relative interior of K(t) for all t>0 then $$\frac{{\partial \phi _t }}{{\partial k}}\left( {x_0 } \right)$$ is in the relative interior of K(t) for all t>0. The latter condition for strong monotonicity may be cumbersome to check; a graph theoretic condition which can replace it is presented in this paper. Knowledge of the facial structure of K(t) is required. The results contained in this paper are extensions of the Kamke-Müller theorem and Hirsch's theorem for strong monotone flows. Applications from chemical kinetics and epidemiology are considered. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

A Graph Theoretic Approach to Strong Monotonicity with respect to Polyhedral Cones

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2002 by Kluwer Academic Publishers
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1015290601993
Publisher site
See Article on Publisher Site

Abstract

Consider the flow ϕt for the system of differential equations $$\dot x\left( t \right) = f\left( x \right)$$ , xεΩ, Ω⊂ $$\mathbb{R}$$ n, Ω open. Let K(t) be an expanding polyhedral cone of constant dimension, k be a unit vector in K(0), and x 0εΩ. A sufficient condition for $$\frac{{\partial \phi _t }}{{\partial k}}\left( {x_0 } \right)$$ εK(t) for t≥0 is that there exists an l so that Df(ϕt(x0))+lI leaves K(t) invariant for all t≥0. If in addition (Df(ϕt(x0))+lI)n-1 takes k into the relative interior of K(t) for all t>0 then $$\frac{{\partial \phi _t }}{{\partial k}}\left( {x_0 } \right)$$ is in the relative interior of K(t) for all t>0. The latter condition for strong monotonicity may be cumbersome to check; a graph theoretic condition which can replace it is presented in this paper. Knowledge of the facial structure of K(t) is required. The results contained in this paper are extensions of the Kamke-Müller theorem and Hirsch's theorem for strong monotone flows. Applications from chemical kinetics and epidemiology are considered.

Journal

PositivitySpringer Journals

Published: Oct 14, 2004

References

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