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The aim of this paper is to present sufficient conditions for all bounded solutions of the second order neutral differential equation $${(x\left(t\right)-px(t-\tau ))}^{\text{'}\text{'}}-q\left(t\right)x\left(\sigma \left(t\right)\right)=0$$ to be oscillatory and to improve some existing results. The main results are based on the comparison principles.

In this paper property (A) of the linear delay differential equation $$L_{n}u\left(t\right)+p\left(t\right)u\left(\tau \left(t\right)\right)=0,$$ is to deduce from the oscillation of a set of the first order delay differential equations.

In this paper, we study the oscillatory behavior of the solutions of the delay differential equation of the form $${\left(\frac{1}{r\left(t\right)}{y}^{\text{'}}\left(t\right)\right)}^{\text{'}}+p\left(t\right)y\left(\tau \left(t\right)\right)=0.$$ The obtained results are applied to n-th order delay differential equation with quasi-derivatives of the form $$L_{n}u\left(t\right)+p\left(t\right)u\left(\tau \left(t\right)\right)=0.$$

The aim of this paper is to deduce oscillatory and asymptotic behavior of delay differential equation $$L_{n}u\left(t\right)-p\left(t\right)u\left(\tau \left(t\right)\right)=0,$$ from the oscillation of a set of the first order delay equations.

In this paper we study asymptotic behavior of solutions of second order neutral functional differential equation of the form $${\left(x\left(t\right)+px(t-\tau )\right)}^{\text{'}\text{'}}+f(t,x\left(t\right))=0.$$ We present conditions under which all nonoscillatory solutions are asymptotic to $at+b$ as $t\to \infty$, with $a,b\in R$. The obtained results extend those that are known for equation $$u^{\text{'}\text{'}}+f(t,u)=0.$$

In this paper we compare the asymptotic behaviour of the advanced functional equation $$L_{n}u\left(t\right)-F(t,u\left[g\left(t\right)\right])=0$$ with the asymptotic behaviour of the set of ordinary functional equations $${\alpha }_{i}u\left(t\right)-F(t,u\left(t\right))=0.$$ On the basis of this comparison principle the sufficient conditions for property (B) of equation (*) are deduced.

In this paper the oscillatory and asymptotic properties of the solutions of the functional differential equation $L_{n}u\left(t\right)+p\left(t\right)f\left(u\left[g\left(t\right)\right]\right)=0$ are compared with those of the functional differential equation ${\alpha }_{n}u\left(t\right)+q\left(t\right)h\left(u\left[w\left(t\right)\right]\right)=0$.

In this paper we present some new oscillatory criteria for the $n$-th order neutral differential equations of the form $${(x\left(t\right)\pm p\left(t\right)x\left[\tau \left(t\right)\right])}^{\left(n\right)}+q\left(t\right)x\left[\sigma \left(t\right)\right]=0.$$ The results obtained extend and improve a number of existing criteria.