Oscillatory and asymptotic behaviour of n order neutral functional differential equations
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.
neutral functional differential equation
nondecreasing bounded variational function
scientific article; zbMATH DE number 4134519