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Zbl 0693.34073
Ruan, Jiong
Oscillatory and asymptotic behaviour of n order neutral functional differential equations.
(English)
[J] Chin. Ann. Math., Ser. B 10, No.2, 143-153 (1989). ISSN 0252-9599; ISSN 1860-6261/e

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.
[L.Sheng]
MSC 2000:
*34K99 Functional-differential equations
34C10 Qualitative theory of oscillations of ODE: Zeros, etc.
34E05 Asymptotic expansions (ODE)

Keywords: neutral functional differential equation; nondecreasing bounded variational function

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