Advanced Search

Our aim in this paper is to present the relationship between property (B) of the third order equation with delay argument y'''(t) - q(t)y(τ(t)) = 0 and the oscillation of the second order delay equation of the form y''(t) + p(t)y(τ(t)) = 0.

We establish some new oscillation criteria for the second order neutral delay differential equation $${\left[r\left(t\right)\right|[x\left(t\right)+p\left(t\right)x\left[\tau \left(t\right)\right]]}^{\text{'}}{{|}^{\alpha -1}{[x\left(t\right)+p\left(t\right)x\left[\tau \left(t\right)\right]]}^{\text{'}}]}^{\text{'}}+q\left(t\right)f\left(x\left[\sigma \left(t\right)\right]\right)=0.$$ The obtained results supplement those of Dzurina and Stavroulakis, Sun and Meng, Xu and Meng, Baculíková and Lacková. We also make a slight improvement of one assumption in the paper of Xu and Meng.

In this paper we consider the third-order nonlinear delay differential equation (*) $${\left(a\left(t\right){\left(x^{\text{'}\text{'}}\left(t\right)\right)}^{\gamma }\right)}^{\text{'}}+q\left(t\right)x^{\gamma }\left(\tau \left(t\right)\right)=0,t\ge t_0,$$ where $a\left(t\right)$, $q\left(t\right)$ are positive functions, $\gamma >0$ is a quotient of odd positive integers and the delay function $\tau \left(t\right)\le t$ satisfies ${lim}_{t\to infty}\tau \left(t\right)=infty$. We establish some sufficient conditions which ensure that (*) is oscillatory or the solutions converge to zero. Our results in the nondelay case extend and improve some known results and in the delay case the results can be applied to new classes of equations which are not covered by the known criteria....