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.

We investigate two boundary value problems for the second order differential equation with $p$-Laplacian $${\left(a\left(t\right){\Phi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}=b\left(t\right)F\left(x\right),t\in I=[0,\infty ),$$ where $a$, $b$ are continuous positive functions on $I$. We give necessary and sufficient conditions which guarantee the existence of a unique (or at least one) positive solution, satisfying one of the following two boundary conditions: $$\mathrm{i})x\left(0\right)=c>0,\underset{t\to \infty }{lim}x\left(t\right)=0;\mathrm{ii})x^{\text{'}}\left(0\right)=d<0,\underset{t\to \infty }{lim}x\left(t\right)=0.$$

In the paper it is shown that each solution $u(r,\alpha )$ ot the initial value problem (2), (3) has a finite limit for $r\to \infty$, and an asymptotic formula for the nontrivial solution $u(r,\alpha )$ tending to 0 is given. Further, the existence of such a solutions is established by examining the number of zeros of two different solutions $u(r,\overline{\alpha })$, $u(r,\widehat{\alpha })$.