On a conjecture for oscillation of second-order ordinary differential systems

Abstract:

We present here some results pertaining to the oscillatory behavior at infinity of the vector differential equation \[ y'' + Q(t)y = 0,\quad t \in [0,\infty )\] , where $Q(t)$ is a real continuous $n \times n$ symmetric matrix function. It has been conjectured (cf., e.g. [6]) that the criterion \[ \lim \limits _{t \to \infty } {\lambda _1}\left \{ {\int _0^t {Q(s)\;ds} } \right \} = \infty \] where ${\lambda _1}( \cdot )$ denotes the maximum eigenvalue of the matrix concerned, implies oscillation. We show that this is so under the tacit assumption \[ \lim \inf \limits _{t \to \infty } {t^{ - 1}}{\text {tr}}\left \{ {\int _0^t {Q(s)\;ds} } \right \} > - \infty \] where ${\text {tr}}( \cdot )$ represents the trace of the matrix under consideration.