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

Allen Devinatz

Vol. 15 (1965), No. 1, 75–83

DOI: 10.2140/pjm.1965.15.75

Abstract

Suppose $y^{\prime }\left(t\right)=\left[A+V\left(t\right)+R\left(t\right)\right]y\left(t\right)$ is a system of differential equations defined on $\left[0,\infty \right)$, where $A$ is a constant matrix, $V\left(t\right)\to 0$ as $t\to \infty$ and the norms of the matrices $V^{\prime }\left(t\right)$ and $R\left(t\right)$ 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 $A+V\left(t\right)$ a theorem of N. Levinson gives an asymptotic estimate of the behavior of the solutions of the differential system as $t\to \infty$. In this paper Levinson’s theorem is improved by removing the condition that the characteristic roots of $A$ are simple. Under suitable conditions on $V\left(t\right)$ and $R\left(t\right)$ and the characteristic roots of $A+V\left(t\right)$, 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

Milestones

Received: 3 February 1964

Published: 1 March 1965