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.

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.

In this paper, oscillation and asymptotic behaviour of solutions of $$y^{\text{'}\text{'}\text{'}}+a\left(t\right)y^{\text{'}\text{'}}+b\left(t\right)y^{\text{'}}+c\left(t\right)y=0$$ have been studied under suitable assumptions on the coefficient functions $a,b,c\in C\left(\right[\sigma ,\infty ),R)$, $\sigma \in R$, such that $a\left(t\right)\ge 0$, $b\left(t\right)\le 0$ and $c\left(t\right)<0$.