Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

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

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 [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, 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≫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 t0∈I, x1, x2∈Rn [formula] 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. 1991 Mathematics Subject Classification 34A30, 34C99. © London Mathematical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

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

Download PDF
4 pages

Loading next page...
 
/lp/oxford-university-press/a-necessary-and-sufficient-condition-for-a-linear-differential-system-wYO5hy7gU8

References (5)

Publisher
Oxford University Press
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/S0024609398004561
Publisher site
See Article on Publisher Site

Abstract

Abstract 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, 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≫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 t0∈I, x1, x2∈Rn [formula] 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. 1991 Mathematics Subject Classification 34A30, 34C99. © London Mathematical Society

Journal

Bulletin of the London Mathematical Society – Oxford University Press

Published: Nov 1, 1998

There are no references for this article.