## Oscillations of higher-order neutral equations.(English)Zbl 0566.34055

Consider the neutral delay differential equation of order n $$(*)\quad (d^ n/dt^ n)[y(t)+py(t-\tau)]+qy(t-\sigma)=0,$$ $$t\geq t_ 0$$ 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 $$q^{1/n}\sigma /n>1/e.$$
Theorem 3. Assume that n is even. Then each of the following two conditions implies that all solutions of (*) oscillate: (i) $$p\geq 0$$; (ii) $$-1\leq p<0$$ 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 (including equations with delayed, advanced or state-dependent argument) 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

### Keywords:

neutral delay differential equation
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