一类积分微分方程周期解的存在性和唯一性

本文考虑具连续时滞和离散时滞的非线性积分微分方程x'(t)=A(t,x(t))x(t)+∫-∞tC(t,s)x(s)ds+∑i=1i gi(t,x(t—τi(t)))+b(t)和x’(t)=f(t,x(t))+∫-∞tC(t,s)x(s)ds+∑i=1igi(t,x(t-τi(t)))+b(t)周期解的存在性和唯一性问题,这里t∈R,x∈Rn;A(t,x),C(t,s)为n×n阶连续的函数矩阵; f(t,x),gi(t,x)(i=1,2,…,l),b(t)是n维连续向量.通过利用线性系统指数型二分性理论和泛函分析方法研究上述系统,获得了保证其周期解存在性、唯一性的充分性条件.我们除了实质性的推广和改进了已有的结果外,还得到三个新的定理,这是用已有的方法无法获得的(见文1-30]).

On the Existence and Uniqueness of Periodic Solutions of a Kind of Integro-Differential Equations

In this paper, nonlinear integro- differential equations with both continuous delay and discrete delays of the form x'(t) = A(t,x(t))x(t) + f-∞t C(t,s)x(s)ds + + b(t) are considered, where the n × n matrix function A(t,x) and the n-vectors f(t,x),gi(t,x) (i = 1,2, ...,l) are continuous in (t,x) ∑ R × Rn, C(t,s) ∑ C(R ××R, Rn×n). By combining the theory of exponential dichotomies of linear system and the method of functional analysis, some sufficient conditions that guarantee the existence and uniqueness of periodic solution of the systems are obtained. We not only make an essential improvement and extension of the main results but also obtain some new criteria which guarantee the existence and uniqueness of peroidic solution of the sytems; these criteria can not be obtained by the existing methods (see 1-30]).