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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
Abstract
In order to present the results of this note, we begin with some definitions. Consider a differential system
formula here
where I⊆ℝ is an open interval, and f(t, x), (t, x)∈I×ℝn, is a continuous vector function with continuous first derivatives δfr/δxs, 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), t∈I(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 t0∈I, 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.
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- © The London Mathematical Society 1998