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

Consider the second order nonlinear neutral differential equation with dela ys: (E) d(2)/dt(2)[y(t) - py(t - tau)] + q(t)f(y(t - sigma)) = 0, far t is an element of [0, infinity), where q(t),f(x) are continuous functions, q(t) greater than or equal to 0, yf(y) > 0 if y not equal 0, and 0 < p < 1, pi > 0, sigma > 0. When f(y) satisfies either the superlinear or sublinear con ditions which include the special case f(y) = y/y/(y-1) of gamma > 1 and 0 < <gamma> < 1, respectively, we give necessary and sufficient conditions fo r the oscillation of all continuable solutions of (E). When p = <tau> = sig ma = 0 in (E), these results reduce to the well known theorems of Atkinson and Belohorec in the special case when f(y)= y/y/(y-1), gamma not equal 1. (C) 2000 Academic Press.