# Article

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Keywords:
half-linear difference equation; Picone identity; Reid Roundabout Theorem; oscillation criteria
Summary:
We study oscillatory properties of the second order half-linear difference equation $\Delta (r_k|\Delta y_k|^{\alpha -2}\Delta y_k)-p_k|y_{k+1}|^{\alpha -2}y_{k+1}=0, \quad \alpha >1. \qquad \mathrm{(HL)}$ It will be shown that the basic facts of oscillation theory for this equation are essentially the same as those for the linear equation $\Delta (r_k\Delta y_k)-p_ky_{k+1}=0.$ We present here the Picone type identity, Reid Roundabout Theorem and Sturmian theory for equation (HL). Some oscillation criteria are also given.
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