Our aim in this paper is to obtain a new oscillation criterion for equation $$y^{\text{'}\text{'}\text{'}}+p\left(t\right)y^{\text{'}}+q\left(t\right)y=0$$ with a nonnegative coefficients which extends and improves some oscillation criteria for this equation. In the special case of equation (*), namely, for equation $y^{\text{'}\text{'}\text{'}}+q\left(t\right)y=0$, our results solve the open question of $Chanturiya$.

Oscillation and nonoscillation criteria for the higher order self-adjoint differential equation $${(-1)}^n{\left(t^{\alpha }{y}^{\left(n\right)}\right)}^{\left(n\right)}+q\left(t\right)y=0(*)$$ are established. In these criteria, equation $(*)$ is viewed as a perturbation of the conditionally oscillatory equation $${(-1)}^n{\left(t^{\alpha }{y}^{\left(n\right)}\right)}^{\left(n\right)}-\frac{{\mu }_{n,\alpha }}{t^{2n-\alpha }}y=0,$$ where ${\mu }_{n,\alpha }$ is the critical constant in conditional oscillation. Some open problems in the theory of conditionally oscillatory, even order, self-adjoint equations are also discussed.

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.

Sufficient conditions are presented for all bounded solutions of the linear system of delay differential equations $${(-1)}^{m+1}\frac{d^m{y}_i\left(t\right)}{d{t}^m}+\sum _{j=1}^n{q}_{ij}{y}_j(t-h_{jj})=0,m\ge 1,i=1,2,...,n,$$ to be oscillatory, where $q_{ij}\epsilon (-\infty ,\infty )$, $h_{jj}\in (0,\infty )$, $i,j=1,2,...,n$. Also, we study the oscillatory behavior of all bounded solutions of the linear system of neutral differential equations $${(-1)}^{m+1}\frac{d^m}{d{t}^m}(y_i\left(t\right)+c{y}_i(t-g))+\sum _{j=1}^n{q}_{ij}{y}_j(t-h)=0,$$ where $c$, $g$ and $h$ are real constants and $i=1,2,...,n$.