In this paper we establish some oscillation or nonoscillation criteria for the second order half-linear differential equation $${\left(r\left(t\right)\Phi \left(u^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}+c\left(t\right)\Phi \left(u\left(t\right)\right)=0,$$ where (i) $r,c\in C([t_0,\infty )$, $ℝ:=(-\infty ,\infty ))$ and $r\left(t\right)>0$ on $[t_0,\infty )$ for some $t_0\ge 0$; (ii) $\Phi \left(u\right)={\left|u\right|}^{p-2}u$ for some fixed number $p>1$. We also generalize some results of Hille-Wintner, Leighton and Willet.

In this paper there are generalized some results on oscillatory properties of the binomial linear differential equations of order $n(\ge 3$) for perturbed iterative linear differential equations of the same order.

Consider the third order differential operator $L$ given by $$L(·)\equiv \frac{1}{a_3\left(t\right)}\frac{\text{d}}{\text{d}t}\frac{1}{a_2\left(t\right)}\frac{\text{d}}{\text{d}t}\frac{1}{a_1\left(t\right)}\frac{\text{d}}{\text{d}t}(·)$$ and the related linear differential equation $L\left(x\right)\left(t\right)+x\left(t\right)=0$. We study the relations between $L$, its adjoint operator, the canonical representation of $L$, the operator obtained by a cyclic permutation of coefficients $a_i$, $i=1,2,3$, in $L$ and the relations between the corresponding equations. We give the commutative diagrams for such equations and show some applications (oscillation, property A).

Necessary and sufficient conditions for discreteness and boundedness below of the spectrum of the singular differential operator $\ell \left(y\right)\equiv \frac{1}{w\left(t\right)}\left(r\right(t\left)y\right)$, $t\in [a,\infty )$ are established. These conditions are based on a recently proved relationship between spectral properties of $\ell$ and oscillation of a certain associated second order differential equation.