Oscillatory and asymptotic behaviour of n order neutral functional differential equations
type: Item
instance of: scholarly article
title: Oscillatory and asymptotic behaviour of n order neutral functional differential equations
zbMATH Open document ID: 0693.34073
author: Jiong Ruan
published in: Chinese Annals of Mathematics. Series B
publication date: 1989-01-01T00:00:00Z
review text: The author studies the neutral functional differential equation: \[ \frac{d^ n}{dt^ n}[x(t)-cx(t-\tau)]+(-1)^{n-1}\int^{0}_{- \tau^*}x(t+\theta)d\eta (\theta)=0, \] where \(\tau >0\), \(\tau^*>0\), 1-c\(\geq 0\), \(\eta\) (\(\theta)\) is a nondecreasing bounded variational function on \([-\tau^*,0]\). 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 \(\{t_ k,k=1,2,...,m\}\), \(0>t_ 1>t_ 2>...>t_ m\geq -\tau^*\) be a sequence in \([-\tau^*,0]\) and \(\eta\) (\(\theta)\) have positive damp on \(\{t_ k\}\), consider the equation \[ \frac{d^ n}{dt^ n}[x(t)-cx(t-\tau)]+(-1)^{n-1}\int^{t_ n}_{t_ 1}x(t+\theta)d\eta (\theta)=0. \] 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.
Mathematics Subject Classification ID: 34K99
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34E05
zbMATH DE Number: 4134519
zbMATH Keywords: neutral functional differential equation
zbMATH Keywords: nondecreasing bounded variational function
reviewed by: Li-Ren Sheng
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4134519