Oscillation of 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 second-order differential equations, $y''\left ( t \right ) + Q\left ( t \right )y\left ( t \right ) = 0$, $t \in [a,\infty )$; $Q$ is a continuous matrix-valued function on $[a,\infty )$ whose values are real symmetric matrices of order $n$; it is assumed that the largest eigenvalue of the matrix $\int _a^t {Q\left ( s \right )ds}$ tends to infinity as $t \to \infty$. Various sufficient conditions are given which guarantee oscillatory behavior at infinity; these conditions generalize those of Mingarelli [C.R. Math. Rep. Acad. Sci. Canada 2 (1980), 287-290, and Proc. Amer. Math. Soc. 82 (1981), 593-598].