Inspired by Chang (1981) and Long (1990), we consider the existence of periodic solutions for second-order difference equations with asymptotically linear nonlinearity via Conley index theory. Moreover, when we study the following one-dimensional autonomous difference equations with form ${\Delta }^2{x}_{n-1}+f\left(x_n\right)=0,n\in \mathbb{Z},x_n\in \mathbb{R},$we can derive at least three periodic solutions under suitable assumptions.

中文翻译：

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Conley指数理论在非共振差分方程中的应用

受Chang（1981）和Long（1990）的启发，我们通过Conley指数理论考虑了渐近线性非线性的二阶差分方程周期解的存在性。此外，当我们研究以下形式的一维自治差分方程时${\Delta }^2{X}_{ñ-\mathrm{1个}}+F（X_{ñ}）=0，ñ\in \mathbb{ž}，X_{ñ}\in \mathbb{[R}，$我们可以在适当的假设下得出至少三个周期解。