A Necessary and Sufficient Condition for a Linear Differential System to be Strongly Monotone |
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Authors: | Andersen Kurt Munk; Sandqvist Allan |
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Institution: | Department of Mathematics, Technical University of Denmark Building 303, DK2800 Lyngby, Denmark |
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Abstract: | In order to present the results of this note, we begin withsome definitions. Consider a differential system formula] where IR is an open interval, and f(t, x), (t, x)IxRn, is acontinuous vector function with continuous first derivativesfr/xs, r, s=1, 2, ..., n. Let Dxf(t, x), (t, x)IxRn, 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)IxRn. For two vectors x, yRn, we use the notations x>y and x>>yaccording to the following definitions: formula] An nxn matrix A=(ars) is called reducible if n2 and there existsa partition formula] (p1, q1, p+q=n) such that formula] The matrix A is called irreducible if n=1, or if n2 and A isnot reducible. The system (1) is called strongly monotone if for any t0I, x1,x2Rn 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 forall (t, x)IxRn the off-diagonal elements of the nxn matrix Dxf(t,x) are nonnegative. 1991 Mathematics Subject Classification34A30, 34C99. |
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