Oscillation and Nonoscillation of Solutions of a Second-Order Nonlinear Ordinary Differential Equation

Abstract

We consider the Emden–Fowler nonlinear differential equation

$$\begin{aligned} x'' + a(t)|x|^{\gamma }\mathrm {sgn}\,x = 0, \quad t \ge t_{0},\quad \quad \quad \quad (1.1) \end{aligned}$$

and discuss the problem of oscillation and nonoscillation of solutions of (1.1). The results in this paper are described by means of the function

$$\begin{aligned} B(t) = \lim _{\tau \rightarrow \infty }\frac{1}{\tau }\int _{t}^{\tau }\left( \int _{t}^{s}a(r)\,dr\right) ds, \quad t \ge t_{0}. \end{aligned}$$

In the case that $B(t) \ge 0$ for $t \ge t_{0}$, a necessary and sufficient condition for oscillation of all solutions of (1.1) can be established.