On the location of zeros of second-order differential equations

Abstract:

The paper considers the location of zeros of the equation $(\alpha (t)x’)’ + \gamma (t)x = 0,t \in [{t_0},{t_1}]$. The following theorem is proved. Let $[a,a + T],T = na$ (n a positive integer), be a subset of $[{t_0},{t_1}]$. Denote $\omega = \pi /T$. Let the coefficient functions obey the inequality $\smallint _a^{a + T}\{ \gamma (t) - {\omega ^2}\alpha (t){\sin ^2}(\omega t)\} dt > {\omega ^2}\smallint _a^{a + T}\{ \alpha \cos 2\omega t\} dt$. Then every solution of this equation will have a zero on $[a,a + T]$. A more general form of this theorem is also proved.