A comparison theorem

Abstract:

In this paper the authors consider a pair of differential equations ${y''_1} + {p_1}(x){y_1} = 0,{y''_2} + {p_2}(x){y_2} = 0$, where ${p_i}(x)$ are positive and continuous, and where solutions ${y_1}(x)$ and ${y_2}(x)$ have common consecutive zeros at $x = a$ and $x = b$. They show that if the curves $y = {p_1}(x)$ and $y = {p_2}(x)$ have a single intersection (possibly a closed subinterval) and if ${p_1}(a) > {p_2}(a),{p_2}(b) > {p_1}(b)$, the first conjugate point of $a + {\varepsilon }$ (${\varepsilon } > 0$ and small) for the second equation precedes that of the first.