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Oscillatory and asymptotic behaviour of n order neutral functional differential equations. (English) Zbl 0693.34073
The author studies the neutral functional differential equation: $$ \frac{d\sp n}{dt\sp n}[x(t)-cx(t-\tau)]+(-1)\sp{n-1}\int\sp{0}\sb{- \tau\sp*}x(t+\theta)d\eta (\theta)=0, $$ where $\tau >0$, $\tau\sp*>0$, 1-c$\ge 0$, $\eta$ ($\theta)$ is a nondecreasing bounded variational function on $[-\tau\sp*,0]$. The main results are: 1) some sufficient conditions for all solutions to be oscillatory when n is odd are otained. 2) Some sufficient conditions for all bounded solutions to be oscillating when n is even are obtained. Let $\{t\sb k,k=1,2,...,m\}$, $0>t\sb 1>t\sb 2>...>t\sb m\ge -\tau\sp*$ be a sequence in $[-\tau\sp*,0]$ and $\eta$ ($\theta)$ have positive damp on $\{t\sb k\}$, consider the equation $$ \frac{d\sp n}{dt\sp n}[x(t)-cx(t-\tau)]+(-1)\sp{n-1}\int\sp{t\sb n}\sb{t\sb 1}x(t+\theta)d\eta (\theta)=0. $$ The main results are: 1) sufficient conditions for the existence of bounded nonoscillatory solutions for all n are obtained. 2) A sufficient condition for all solutions to be oscillatory for all n is obtained.
Reviewer: L.Sheng

34K99Functional-differential equations
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34E05Asymptotic expansions (ODE)