A Necessary and Sufficient Condition for a Linear Differential System to be Strongly Monotone

In order to present the results of this note, we begin with some definitions. Consider a differential system [formula] 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, t, x), t∈I(t, x) denote the maximal solution of the system (1) through the point (t, x)∈I×Rn. For two vectors x, y∈Rn, we use the notations x>y and x≫y according to the following definitions: [formula] An n×n matrix A=(ars) is called reducible if n≥2 and there exists a partition [formula] (p≥1, q≥1, p+q=n) such that [formula] The matrix A is called irreducible if n=1, or if n≥2 and A is not reducible.

The system (1) is called strongly monotone if for any t∈I, x1, x2∈Rn [formula] holds for all t>t as long as both solutions x(t, t, 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. 1991 Mathematics Subject Classification 34A30, 34C99.