Necessary and Sufficient Conditions for Oscillation of Second Order Neutral Differential Equations

Consider the second order nonlinear neutral differential equation with delays: ŽE. d2 dt2 yŽt. pyŽt . qŽt.fŽyŽt .. 0, for t 0, ., where qŽt., fŽx. are continuous functions, qŽt. 0, yfŽ y. 0 if y 0, and 0 p 1, 0, 0. When fŽy. satisfies either the superlinear or sublinear conditions which include the special case fŽy. y y 1 of 1 and 0 1, respectively, we give necessary and sufficient conditions for the oscillation of all continuable solutions of ŽE.. When p 0 in ŽE., these results reduce to the well known theorems of Atkinson and Belohorec in the special case when fŽy. y y 1, 1.