Oscillations in neutral equations with periodic coefficients

Abstract:

We obtain a necessary and sufficient condition for the oscillation of all solutions of the neutral delay differential equation: (1) \[ \tfrac {d}{{dt}}[x(t) + px(t - \tau )] + Q(t)x(t - \sigma ) = 0,\] where $p \in {\mathbf {R}},Q \in C[[0,\infty ),{{\mathbf {R}}^ + }],Q$ is $\omega$-periodic with $\omega > 0,Q(t)[unk]0$ for $t \geqq 0$, and there exist positive integers ${n_1}$ and ${n_2}$ such that $\tau = {n_1}\omega$ and $\sigma = {n_2}\omega$. More precisely we show that every solution of (1) oscillates if and only if every solution of an associated neutral equation with constant coefficients oscillates.