Some oscillation results for second order neutral type difference equations (Q2857744)

This paper deals with the second-order neutral difference equation NEWLINE\[NEWLINE \Delta\big(a_n\,[\Delta(x_n+p_n\,x_{\tau(n)})]^\alpha \big) + q_n\,f(x_{\sigma(n+1)}) =0, NEWLINE\]NEWLINE where \(n\geq n_0\in{\mathbb N}\) and \(\{\tau(n)\}_{n=n_0}^\infty\) and \(\{\sigma(n)\}_{n=n_0}^\infty\) are increasing integer sequences. Four special cases are considered: (i) \(\tau(n)\geq n\) and \(\sigma(n+1)\leq n\), (ii) \(\sigma(n)\geq\tau(n)\geq n\), (iii) \(\sigma(n+1)\leq\tau(n)\leq n\), and (iv) \(\tau(n)\leq n\) and \(\sigma(n+1)\geq n\). For each case, the authors prove an oscillation criterion for solutions satisfying \(\sup\{|x_n|,\;n\geq N\}>0\) for all \(N\geq n_0\). These criteria improve the results of \textit{E. Thandapani} and \textit{S. Selvarangam} [``Oscillation theorems of second-order quasilinear neutral difference equations'', J.~Math. Comput. Sci. 2, No. 4, 866--879 (2012)], where an additional condition (that \(\sigma\) and \(\tau\) commute) was assumed.