Oscillation criteria for higher-order neutral type difference equations

Authors: TURHAN KÖPRÜBAŞI, ZAFER ÜNAL, YAŞAR BOLAT

Abstract: In this paper, oscillation criteria are obtained for higher-order neutral-type nonlinear delay difference equations of the form% \begin{equation} \Delta (r_{n}(\Delta ^{k-1}(y_{n}+p_{n}y_{\tau _{n}}))+q_{n}f(y_{\sigma _{n}})=0\text{, }n\geq n_{0}\text{,} \tag{0.1} \end{equation}% where $r_{n},p_{n},q_{n}\in \lbrack n_{0},\infty ),$ $r_{n}>0$, $q_{n}>0$; $% 0\leq p_{n}\leq p_{0}<\infty $; $\lim\limits_{n\rightarrow \infty }\tau _{n}=\infty $, $\lim\limits_{n\rightarrow \infty }\sigma _{n}=\infty $; $% \sigma _{n}\leq n$, $\sigma _{n}$ is nondecreasing; $\Delta \tau _{n}\geq \tau _{0}>0$; $\tau _{\sigma }=\sigma _{\tau }$; $\frac{f(u)}{u}\geq m>0$ for $u\neq 0$. Moreover, we provide some examples to illustrate our main results.

Keywords: Oscillation, oscillatory, difference equations

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