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

Loading next page...
 
/lp/springer_journal/a-graph-theoretic-approach-to-strong-monotonicity-with-respect-to-3fnGCtqUPc
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

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off