Necessary and sufficient conditions for oscillation of second order neutral differential equations (Q1840802)

The author deals with the second-order nonlinear neutral differential equation with delays: \[ \frac{d^2}{dt^2}[y(t)-py(t-\tau)]+q(t)f(y(t-\sigma))=0, \quad \text{for }t\in [0,\infty),\tag{E} \] where \(q(t), f(y)\) are continuous functions, \(q(t)\geq 0, yf(y)>0 (y\not =0)\), and \(0<p<1, \tau >0, \sigma >0\). Necessary and sufficient conditions are established for the oscillation of all continuable solutions when \(f(y)\) satisfies either superlinear or sublinear conditions. This paper extends the well known theorems of Atkinson and Belohorec which deal with the Emden-Fowler equation: \(y''(t)+q(t) y |y|^{\gamma -1} =0\).