In this paper we offer criteria for property (B) and additional asymptotic behavior of solutions of the $n$-th order delay differential equations $${\left(r\left(t\right){\left[x^{(n-1)}\left(t\right)\right]}^{\gamma }\right)}^{\text{'}}=q\left(t\right)f\left(x\left(\tau \left(t\right)\right)\right).$$ Obtained results essentially use new comparison theorems, that permit to reduce the problem of the oscillation of the n-th order equation to the the oscillation of a set of certain the first order equations. So that established comparison principles essentially simplify the examination of studied equations. Both cases ${\int }^{\infty }{r}^{-1/\gamma }\left(t\right)t=\infty$ and ${\int }^{\infty }{r}^{-1/\gamma }\left(t\right)t<\infty$ are discussed.

The aim of our paper is to study oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with deviating argument. In particular, we prove a comparison theorem for properties A and B as well as a comparison result on property A between nonlinear equations with and without deviating arguments. Our assumptions on nonlinearity f are related to its behavior only in a neighbourhood of zero and/or of infinity.

The objective of this paper is to study asymptotic properties of the third-order neutral differential equation $${\left[a\left(t\right){\left({\left[x\left(t\right)+p\left(t\right)x\left(\sigma \left(t\right)\right)\right]}^{\text{'}\text{'}}\right)}^{\gamma }\right]}^{\text{'}}+q\left(t\right)f\left(x\left[\tau \left(t\right)\right]\right)=0,t⩾t_0.\left(E\right)$$ . We will establish two kinds of sufficient conditions which ensure that either all nonoscillatory solutions of (E) converge to zero or all solutions of (E) are oscillatory. Some examples are considered to illustrate the main results.

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....

The aim of this paper is to study asymptotic properties of the third-order quasi-linear neutral functional differential equation $${\left[a\left(t\right){\left({[x\left(t\right)+p\left(t\right)x\left(\delta \left(t\right)\right)]}^{\text{'}\text{'}}\right)}^{\alpha }\right]}^{\text{'}}+q\left(t\right)x^{\alpha }\left(\tau \left(t\right)\right)=0,E$$ where $\alpha >0$, $0\le p\left(t\right)\le p_0<\infty$ and $\delta \left(t\right)\le t$. By using Riccati transformation, we establish some sufficient conditions which ensure that every solution of () is either oscillatory or converges to zero. These results improve some known results in the literature. Two examples are given to illustrate the main results.

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.