一类渐近线性差分方程的周期解

Periodic Solution of a Kind of Non-autonomous Second Order Equation

考虑一类二阶差分方程Δ2xn-1+f(n,xn)=0,其中,f∈C(Z×Rm,Rm)为梯度算子,即存在连续可微函数F(n,z)满足F(n,z)=f(n,z),且存在正整数M使得对于任何的(n,z)∈Z×Rm,有f(n+M,z)=f(n,z)。使用临界点理论得到方程存在三周期解的一个充分条件,改进了已有文献中的结果。

Consider the second order discrete system Δ2xn-1＋f（n,xn）=0 , where f∈C（Z×Rm,Rm） is gradient operator,i.e., there is a continuous differential functional F（ n,z ） satisfies △↓ F（ n,z ） =f（ n,z ）, there is a positive integer M satisfies （n,z）∈Z×Rm ,and we can have f（ n ＋ M,z ） =f（ n,z ）. By using critical point theory, some sufficient conditions are obtained, and the results in existed literature are improved.