By using the variational approach and critical point theory, we establish several criteria for the existence of at least three anti-periodic solutions for a higher order difference equation with p-Laplacian and which contains both advance and retardation. As special cases, we obtain results when the nonlinear function is free of advance or retardation. To illustrate our results, we discuss some consequences of our main results when the function is specified. Our work complements the existing results in the literature. Finally, we present an example to show the applicability of the results.

中文翻译：

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一类高阶非线性差分方程的多个反周期解的存在性

通过使用变分方法和临界点理论，我们为存在p -Laplacian的高阶差分方程建立了至少三个反周期解的存在标准，该方程包含超前和滞后。作为特殊情况，当非线性函数没有提前或滞后时，我们可以获得结果。为了说明我们的结果，我们讨论了指定功能时主要结果的一些后果。我们的工作是对文献中现有结果的补充。最后，我们提供一个示例来说明结果的适用性。