非自治二阶微分方程周期解的存在唯一性

Existence and uniqueness of periodic solutions of nonautonomous differential equations of second order

在不要求ｆ（ｘ）→±∞（ｘ→±∞）的条件下，利用Ｂｒｏｕｗｅｒ不动点定理，得到了方程组ｘ＝φ（ｙ）－ｆ（ｘ），ｙ＝－ｇ（ｘ）＋ｅ（ｔ）周期解的存在性，并给出了此方程组的一个唯一性结果．另外，通过构造Ｌｙａｐｕｎｏｖ泛函，推广了另一类方程前人的相关结果．

By using Brouwers fixed point theorem, several sufficient conditions for existence and uniqueness of periodic solutions of differential equation x′(t)=φ(y(t))-f(x(t)),y′(t)=-g(x(t))+e(t ) are obtained without the assumption f(x )→±∞ as x →±∞. In addition, the related results of another class of differential equations are given by constructing Lyapunov functions, which extends the known results.