Abstract
Sufficient conditions are obtained for the third order nonlinear delay difference equation of the form $$ \Delta (a_n(\Delta (b_n(\Delta y_n)^{\alpha })))+q_nf(y_{\sigma (n)})=0 $$ to have property ${(\rm A)}$ or to be oscillatory. These conditions improve and complement many known results reported in the literature. Examples are provided to illustrate the importance of the main results.
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