In this paper, we study the oscillatory behavior of the solutions of the delay differential equation of the form $${\left(\frac{1}{r\left(t\right)}{y}^{\text{'}}\left(t\right)\right)}^{\text{'}}+p\left(t\right)y\left(\tau \left(t\right)\right)=0.$$ The obtained results are applied to n-th order delay differential equation with quasi-derivatives of the form $$L_{n}u\left(t\right)+p\left(t\right)u\left(\tau \left(t\right)\right)=0.$$

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 $Chanturiya$.

We offer sufficient conditions for the oscillation of all solutions of the perturbed difference equation $$|{\Delta }^m{y\left(k\right)|}^{\alpha -1}{\Delta }^{m}y\left(k\right)+Q(k,y(k-{\sigma }_k),\Delta y(k-{\sigma }_k),\cdots ,{\Delta }^{m-2}y(k-{\sigma }_k))$$ $=P(k,y(k-{\sigma }_k),\Delta y(k-{\sigma }_k),\cdots ,{\Delta }^{m-1}y(k-{\sigma }_k)),k\ge k_0$ where $\alpha >0.$ Examples which dwell upon the importance of our results are also included.