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 $(\gamma <1)$.