具临界常系数线性中立型方程的周期解

Periodic Solution of Linear Neutral Differential Equations with Critial Coefficient

利用Ｆｏｕｒｉｅｒ级数理论研究了单时滞具临界常系数线性中立型方程ｄ＾２ｄｔ＾２ｘ（ｔ）＋ａ１ｄｄｔｘ（ｔ）＋ａ２ｘ（ｔ）＋ｃｄ＾２ｄｔ＾２ｘ（ｔ－ｈ）＋ｃ１ｄｄｔｘ（ｔ－ｈ）＋ｃ２ｘ（ｔ－ｈ）＝ｆ（ｔ）的周期解的存在性、唯一性问题，其中ｈ≥０，│ｃ│＝１，ａｉ，ｃｉ（ｉ＝１，２）为常数，ｆ（ｔ）是连续可微的以２π为周期的函数。在一定条件下，如果ｆ（ｔ）足够光滑，那么上述方程的周期解存在且唯一。

Using the theory of Fourier series,the existence and uniqueness of the continuous differential periodic solution of linear autonomous neutral differential equations with delays are as following d 2 d t 2x(t)+a 1 d d tx(t)+a 2x(t)+c d 2 d t 2x(t-h)+c 1 d d tx(t-h)+c 2x(t-h)=f(t)(*)where h≥0,｜c｜=1,a i,c i(i=1,2) are constants, f(t) is the continuous differential function,and its period is 2π .If f(t) is smooth enough,the periodic solution of () is existing and unique.