An oscillation criterion for linear second-order differential systems

Abstract:

This article is concerned with the oscillatory behavior at infinity of the solution $y:[a,\infty ) \to {{\mathbf {R}}^n}$ of a system of $n$ second-order differential equations, $y''(t)y(t) = 0,\;t \in [a,\infty );\;Q$ is a continuous matrix-valued function on $[a,\infty )$ whose values are real symmetric matrices of order $n$. It is shown that the solution is oscillatory at infinity if (at least) $n - 1$ eigenvalues of the matrix $\smallint _a^tQ(t)\;dt$ dt end to infinity as $t \to \infty$.