Sun and Ma (J. Differ. Equ. 255:2534–2563, 2013) proved the existence of a nonzero T-periodic solution for a class of one-dimensional lattice dynamical systems, $$\begin{aligned} \ddot{q_{i}}=\varPhi _{i-1}'(q_{i-1}-q_{i})- \varPhi _{i}'(q_{i}-q_{i+1}),\quad i\in \mathbb{Z}, \end{aligned}$$ where $q_{i}$ denotes the co-ordinate of the ith particle and $\varPhi _{i}$ denotes the potential of the interaction between the ith and the $(i+1)$ th particle. We extend their results to the case of the least energy of nonzero T-periodic solution under general conditions. Of particular interest is a new and quite general approach. To the best of our knowledge, there is no result for the ground states for one-dimensional lattice dynamical systems.

中文翻译：

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具有最邻近相互作用的无限晶格的基态

Sun和Ma（J. Differ。Equ。255：2534–2563，2013）证明了一类一维晶格动力学系统存在非零T周期解，$$ \ begin {aligned} \ ddot {q_ {i}} = \ varPhi _ {i-1}'（q_ {i-1} -q_ {i}）-\ varPhi _ {i}'（q_ {i} -q_ {i + 1}），\ Quad i \ in \ mathbb {Z}，\ end {aligned} $$，其中$ q_ {i} $表示第i个粒子的坐标，$ \ varPhi _ {i} $表示ith和第（i + 1）$个粒子。我们将其结果推广到一般条件下非零T周期解能量最小的情况。特别令人感兴趣的是一种新的且相当通用的方法。据我们所知，一维晶格动力学系统的基态没有结果。