Integral Averages and the Oscillation of Second Order Ordinary Differential Equations

Abstract

Some of the more important and useful tests for the oscillation of the second order scalar linear differential equation $y'' + qy = 0$ are given by the classical Fite–Wintner theorem and its generalizations by Wintner and by Hartman. These tests involve the behavior of the integral of q or, more generally, the average behavior of the integral. Several years ago, Waltman extended the Fite-Wintner theorem to nonlinear equations. We show that the Wintner and Hartman theorems also extend to a large class of nonlinear equations which includes the Emden–Fowler equation. Further generalizations of the averaging technique for the linear equation due to Coles and to Willett are also shown to extend to some degree to nonlinear equations.