In this paper we have considered completely the equation $$y^{\text{'}\text{'}\text{'}}+a\left(t\right)y^{\text{'}\text{'}}+b\left(t\right)y^{\text{'}}+c\left(t\right)y=0,(*)$$ where $a\in C^2([\sigma ,\infty ),R)$, $b\in C^1([\sigma ,\infty ),R)$, $c\in C\left(\right[\sigma ,\infty ),R)$ and $\sigma \in R$ such that $a\left(t\right)\le 0$, $b\left(t\right)\le 0$ and $c\left(t\right)\le 0$. It has been shown that the set of all oscillatory solutions of (*) forms a two-dimensional subspace of the solution space of (*) provided that (*) has an oscillatory solution. This answers a question raised by S. Ahmad and A.  C. Lazer earlier.

Qualitative comparison of the nonoscillatory behavior of the equations $$L_{n}y\left(t\right)+H(t,y\left(t\right))=0$$ and $$L_{n}y\left(t\right)+H(t,y\left(g\left(t\right)\right))=0$$ is sought by way of finding different nonoscillation criteria for the above equations. $L_n$ is a disconjugate operator of the form $$L_n=\frac{1}{p_n\left(t\right)}\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{p_{n-1}\left(t\right)}\frac{\mathrm{d}}{\mathrm{d}t}...\frac{\mathrm{d}}{\mathrm{d}t}\frac{·}{p_0\left(t\right)}.$$ Both canonical and noncanonical forms of $L_n$ have been studied.

In this paper we consider the equation $$y^{\text{'}\text{'}\text{'}}+q\left(t\right){y^{\text{'}}}^{\alpha }+p\left(t\right)h\left(y\right)=0,$$ where $p,q$ are real valued continuous functions on $[0,\infty )$ such that $q\left(t\right)\ge 0$, $p\left(t\right)\ge 0$ and $h\left(y\right)$ is continuous in $(-\infty ,\infty )$ such that $h\left(y\right)y>0$ for $y\ne 0$. We obtain sufficient conditions for solutions of the considered equation to be nonoscillatory. Furthermore, the asymptotic behaviour of these nonoscillatory solutions is studied.