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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 $(\gamma <1)$.

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.