We derive Lyapunov-type inequalities for third-order half-linear difference equations. The results are significant as the existing work on Lyapunov-type inequalities for difference equations is limited only to even order cases. As the techniques for even order difference equations are not applicable to odd order difference equations, the methods used here are novel. The resulting inequalities utilize $q_{+}(t)$ and $q_{-}(t)$. New results for linear difference equations are obtained as a special case and are further extended to more general linear equations. Moreover, brief discussions are also given for third-order backward difference equations and dynamic equations on time scales.

我们推导了三阶半线性差分方程的Lyapunov型不等式。结果非常重要，因为有关差分方程的Lyapunov型不等式的现有工作仅限于偶数情况。由于用于偶数阶差分方程的技术不适用于奇数阶差分方程，因此这里使用的方法是新颖的。由此产生的不平等利用$q_{+}（Ť）$ 和 $q_{-}（Ť）$。作为特殊情况，可以获得线性差分方程的新结果，并将其进一步扩展到更通用的线性方程。此外，还简要讨论了时标上的三阶后向差分方程和动力学方程。