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This paper deals with oscillatory and asymptotic behaviour of solutions of second order quasilinear difference equation of the form $$\Delta (a_{n-1}|\Delta y_{n-1}{|}^{\alpha -1}\Delta y_{n-1})+F(n,y_n)=G(n,y_n,\Delta y_n),n\in N\left(n_0\right)\left(\mathrm{E}\right)$$ where $\alpha >0$. Some sufficient conditions for all solutions of (E) to be oscillatory are obtained. Asymptotic behaviour of nonoscillatory solutions of (E) are also considered.

In this paper the authors give necessary and sufficient conditions for the oscillation of solutions of nonlinear delay difference equations of Emden– Fowler type in the form ${\Delta }^2{y}_{n-1}+q_n{y}_{\sigma \left(n\right)}^{\gamma }=g_n$, where $\gamma$ is a quotient of odd positive integers, in the superlinear case $(\gamma >1)$ and in the sublinear case $(\gamma <1)$.