*(English)*Zbl 0566.34055

Consider the neutral delay differential equation of order n $(*)\phantom{\rule{1.em}{0ex}}({d}^{n}/d{t}^{n})[y\left(t\right)+py(t-\tau )]+qy(t-\sigma )=0,$ $t\ge {t}_{0}$ where q is a positive constant, the delays $\tau $ and $\sigma $ are nonnegative constants and the coefficient p is a real number. Theorem 1. (a) Assume that n is odd and that $p<-1$. Then every nonoscillatory solution of (*) tends to $+\infty $ or -$\infty $ as $t\to \infty $. (b) Assume that n is odd or even and that $p>-1$. Then every nonoscillatory solution of (*) tends to zero as $t\to \infty $. Theorem 2. Assume that n is odd. Then each of the following four conditions implies that every solution of (*) oscillates: (i) $p<-1$ and ${(-q/(1+p))}^{1/n}(\tau -\sigma )/n>1/e$; (ii) $p=-1$; (iii) $p>-1$ and ${(q/(1+p))}^{1/n}(\sigma -\tau )/n>1/e$; (iv) $-1<p<0$ and ${q}^{1/n}\sigma /n>1/e\xb7$

Theorem 3. Assume that n is even. Then each of the following two conditions implies that all solutions of (*) oscillate: (i) $p\ge 0$; (ii) $-1\le p<0$ and ${(q/p)}^{1/n}(\sigma -\tau )/n>1/e$. Theorem 4. Assume that n is even, $p<-1$, and ${(q/p)}^{1/n}(\sigma -\tau )/n>1/e$. Then every bounded solution of (*) oscillates.

##### MSC:

34K99 | Functional-differential equations |

34C10 | Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory |