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A NECESSARY AND SUFFICIENT CONDITION FOR A LINEAR DIFFERENTIAL SYSTEM TO BE STRONGLY MONOTONE

Published online by Cambridge University Press:  01 November 1998

KURT MUNK ANDERSEN
Affiliation:
Department of Mathematics, Technical University of Denmark, Building 303, DK–2800 Lyngby, Denmark
ALLAN SANDQVIST
Affiliation:
Department of Mathematics, Technical University of Denmark, Building 303, DK–2800 Lyngby, Denmark
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Abstract

In order to present the results of this note, we begin with some definitions. Consider a differential system

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where I⊆ℝ is an open interval, and f(t, x), (t, x)∈I×ℝn, is a continuous vector function with continuous first derivatives δfrxs, r, s=1, 2, …, n.

Let Dxf(t, x), (t, x)∈I×ℝn, denote the Jacobi matrix of f(t, x), with respect to the variables x1, …, xn. Let x(t, t0, x0), tI(t0, x0) denote the maximal solution of the system (1) through the point (t0, x0)∈I×ℝn.

For two vectors x, y∈ℝn, we use the notations x>y and x[Gt ]y according to the following definitions:

formula here

An n×n matrix A=(ars) is called reducible if n[ges ]2 and there exists a partition

formula here

(p[ges ]1, q[ges ]1, p+q=n) such that

formula here

The matrix A is called irreducible if n=1, or if n[ges ]2 and A is not reducible.

The system (1) is called strongly monotone if for any t0I, x1, x2∈ℝn

formula here

holds for all t>t0 as long as both solutions x(t, t0, xi), i=1, 2, are defined. The system is called cooperative if for all (t, x)∈I×ℝn the off-diagonal elements of the n×n matrix Dxf(t, x) are nonnegative.

Type
Notes and Papers
Copyright
© The London Mathematical Society 1998

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