一类非线性微分方程空间周期解的存在性及唯一性

UNIQUENESS OF SPACE PERIODIC SOLUTION OF A KIND OF NONLINEAR DIFFERENTIAL EQUATION

利用Ｂｅｎｄｉｘｓｏｎ－Ｄｕｌａｃ定理，讨论系统在唯一空间周期解的条件，得到如下结果：当（ｉ）ｘ２＊［ｍ２－（ｋ＋ ｑ）］（ｑλ２－ ｋ）（ｍ２ｋ２λ）－１≤成立时，系统在Ω内关于正平衡点是全局稳定的；当成立时，系统在Ω内存在稳定的周期解；当（ｉｉ）和（ｉｉｉ） ｋ２（ｋ＋ ≥ｓ０δ（ｍ２－ ｋ）同时成立时，系统在Ω内存在唯一稳定的周期解．

In this paper, the following nonlinear differential equations are discussed: It is proved that the dynamical system is globally stable on Ω under the following condition: x2*[m2-(k + q)](q2-k)(m2k22)-1 Furthermore, there exists a stable periodic solution on Ω for the system with the other condition: x２＊[m２ -(k＋q)](q Moreover, if (*) and condition: are satisfied, this solution is unique and stable on Ω.