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Our aim in this paper is to present criteria for oscillation of the nonlinear differential equation $$u^{\text{'}\text{'}}\left(t\right)+p\left(t\right)f\left(u\left(g\left(t\right)\right)\right)=0.$$ The obtained oscillatory criteria improve existing ones.

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 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.

In the paper we offer criteria for oscillation of the third order Euler differential equation with delay $$y^{\text{'}\text{'}\text{'}}\left(t\right)+\frac{k^2}{t^3}y\left(ct\right)=0.$$ We provide detail analysis of the properties of this equation, we fill the gap in the oscillation theory and provide necessary and sufficient conditions for oscillation of equation considered.