Oscillation of second-order sublinear neutral delay difference equations. (Q1412548)

Consider the second-order sublinear delay difference equation \[ \Delta (a_n\Delta (x_n+p_nx_{n-\tau }))+q_nx_{n-\sigma }^\gamma =0,\tag{*} \] where \(0<\gamma <1\) is a quotient of odd positive integers, \(\Delta u_n=u_{n+1}-u_n\) for any sequence \(\{u_n\},\) \(\tau, \sigma\) are fixed nonnegative integers, \(\{a_n\}, \{p_n\}\) and \(\{q_n\}\) are sequences of real numbers such that \(a_{n+1}\geq a_n>0,\) \(\sum^\infty_{n=0} \frac 1{a_n}=\infty,\) \(0\leq p_n<1\) for all \(n\geq 0\) and \(q_n\geq 0\) and \(q_n\) is not identically zero for large \(n.\) The authors establish some sufficient conditions so that every solution of (*) oscillates.