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Oscillations of higher-order neutral equations. (English) Zbl 0566.34055
Consider the neutral delay differential equation of order n $(*)\quad (d\sp n/dt\sp n)[y(t)+py(t-\tau)]+qy(t-\sigma)=0,$ $t\ge t\sb 0$ 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 $(-q/(1+p))\sp{1/n}(\tau -\sigma)/n>1/e$; (ii) $p=-1$; (iii) $p>-1$ and $(q/(1+p))\sp{1/n}(\sigma -\tau)/n>1/e$; (iv) $- 1<p<0$ and $q\sp{1/n}\sigma /n>1/e.$ Theorem 3. Assume that n is even. Then each of the following two conditions implies that all solutions of (*) oscillate: (i) $p\ge 0$; (ii) $-1\le p<0$ and $(q/p)\sp{1/n}(\sigma -\tau)/n>1/e$. Theorem 4. Assume that n is even, $p<-1$, and $(q/p)\sp{1/n}(\sigma -\tau)/n>1/e$. Then every bounded solution of (*) oscillates.

34K99Functional-differential equations
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
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