By means of critical point theory, we study the existence and multiplicity of homoclinic solutions of the damped second-order difference equation$$\begin{aligned} \Delta ^{2}u(n-1)-c\Delta u(n-1)-a(n)u(n)+f(n,u(n))=0 ,\quad n\in {\mathbb {Z}}, \end{aligned}$$where \(c>-1\) is a constant, \(a: {\mathbb {Z}}\rightarrow (0,+\infty )\) and \(f: {\mathbb {Z}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is continuous with respect to the second variable and satisfies some additional assumptions. The proofs of our results are based on variational methods in some weighted Hilbert space of sequences. Some recent results in the literature are extended even in the case of \(c=0\).

中文翻译：

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二阶阻尼差分方程的快速同宿解

利用临界点理论，研究了阻尼二阶差分方程$$ \ begin {aligned} \ Delta ^ {2} u（n-1）-c \ Delta u（n -1）-a（n）u（n）+ f（n，u（n））= 0，\ quad n \ in {\ mathbb {Z}}，\ end {aligned} $$其中\（c> -1 \）是常数，\（a：{\ mathbb {Z}} \ rightarrow（0，+ \ infty）\）和\（f：{\ mathbb {Z}} \ times {\ mathbb {R} } \ rightarrow {\ mathbb {R}} \）关于第二个变量是连续的，并且满足一些其他假设。我们的结果证明基于序列的某些加权希尔伯特空间中的变分方法。即使在\（c = 0 \）的情况下，文献中的一些最新结果也得到扩展。