一类二阶时滞微分方程周期解的存在性

Existence of periodic solutions to a second-order delay differential equation

二阶时滞泛函微分方程周期解问题的主要研究方法是利用度理论得到方程的先验界,再运用不动点或重合度定理得出周期解的存在性结果。文章尝试运用上、下解方法和紧向量场方程的解集连通理论研究了一类时滞泛函微分方程x″(t)=f(t,x(t),x(t-τ))周期解的存在性,在较弱的条件下,得出此类二阶时滞泛函微分方程周期解存在的充分条件。

In this paper, the existence of periodic solutions for a kind of second-order differential equation with a deviating argument, x″（t）=f（t,x（t）,x（t-τ）） , is studied based on the methods of lower and upper solutions and the connectivity properties of the solution set of parameterized families of compact vector fields. The sufficient conditions for the existence of the periodic solutions are obtained under weak conditions.