Abstract
In local terms on finite and infinite intervals we obtain necessary and sufficient conditions for the conjugacy and disconjugacy of the following second order half-linear difference equation \begin{equation*} \Delta(\rho_i|\Delta y_i|^{p-2}\Delta y_i)+v_i|y_{i+1}|^{p-2}y_{i+1}=0, \qquad i=0, 1, 2,\dots, \end{equation*} where $1<p<\infty$, $\Delta y_i=y_{i+1}-y_i$, $\{\rho_i\}$ and $\{v_i\}$ are sequences of positive and non-negative real numbers, respectively. Moreover, we study oscillation and non-oscillation properties of this equation.