Vol. 15, No. 1, 1965

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The asymptotic nature of the solutions of certain linear systems of differential equations

Allen Devinatz

Vol. 15 (1965), No. 1, 75–83

Suppose y(t) = [A + V (t) + R(t)]y(t) is a system of differential equations defined on [0,), where A is a constant matrix, V (t) 0 as t and the norms of the matrices V (t) and R(t) are summable. If the roots of the characteristic polynomial of A are simple, then under suitable conditions on the real parts of the roots of the characteristic polynomials of A + V (t) a theorem of N. Levinson gives an asymptotic estimate of the behavior of the solutions of the differential system as t . In this paper Levinson’s theorem is improved by removing the condition that the characteristic roots of A are simple. Under suitable conditions on V (t) and R(t) and the characteristic roots of A + V (t), which reduce to Levinson’s conditions when the characteristic roots of A are simple, asymptotic estimates are obtained for the solutions of the given system.

Mathematical Subject Classification
Primary: 34.50
Received: 3 February 1964
Published: 1 March 1965
Allen Devinatz