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

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 δf_r/δx_s, r, s=1, 2, …, n.

Let D_xf(t, x), (t, x)∈I×ℝⁿ, denote the Jacobi matrix of f(t, x), with respect to the variables x₁, …, x_n. 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:

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

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

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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

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holds for all t>t₀ as long as both solutions x(t, t₀, x_i), 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 D_xf(t, x) are nonnegative.