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.
