Oscillation of Third-order Nonlinear Delay Difference Equation

Abstract

Third-order nonlinear difference equation of the form $\Delta \left(c_{n} \Delta \left(d_{n} \Delta x_{n} \right)\right)+p_{n} \Delta x_{n+1} +q_{n} f\left(x_{n-\sigma } \right)=0,n\ge n_{0} $ are considered. Here, $\left\{c_{n} \right\}$, $\left\{d_{n} \right\}$, $\left\{p_{n} \right\}$, and $\left\{a_{n} \right\}$ are sequence of positive real number for $n_{0} \in N,f$ is a continuous function such that ${f\left(u\right) \mathord{\left/{\vphantom{f\left(u\right) u}}\right.\kern-\nulldelimiterspace} u} \ge k>0$ for $u\ne 0$. By means of a Riccati transformation technique we obtain some new oscillation criteria. Examples are given to illustrate the importance of the results.