一类具有偏差变元的二阶泛函微分方程周期解

利用重合度理论,研究一类具有偏差变元的二阶微分方程x″+f(t,x′(t))+g(t,x(t-τ(t)))=p(t)的周期解的存在性问题.其中,f,g∈C(R×R,R),且对任意的x∈R,g(t+ω,x)=g(t,x),p∈C(R,R),τ∈C(R,R)是ω-周期的.在不要求对所有的y∈R,函数f(t,y)≤0(f(t,y)≥0),t∈R的情况下,得到该类方程至少存在一个ω-周期解的充分条件.

Periodic Solutions for a Class of Second Order Functional Differential Equations with a Deviating Argument

In this paper,by means of Mawhin's continuation theorem,we study the problem on the existence of periodic solutions for the second order differential equations with a deviating argument x″+f(t,x′(t))+g(t,x(t-τ(t)))=p(t),where f,g∈C(R×R,R),and for any x∈R,g(t+ω,x)=g(t,x),p∈C(R,R);and τ∈C(R,R) arecω-periodic.Without the condition f(t,y)≤0(f(t,y)≥0) for all y∈R and t∈R,we obtain some sufficient conditions on the existence of at least one periodic solution for this equation.