在共振点的一阶微分系统的周期解

The Existence of Periodic Solutions for Nonlinear Systems of First-Order Differential Equations at Resonance

考虑具偏差变元的一阶非线性微分系统 : x(t) =Bx(t) +F(x(t-τ) ) +p(t) ,其中 ,x(t)∈R2 ,τ∈R ,B∈R2×2 ,F是有界的 ,p(t)是连续的 2π_周期函数· 应用Brouwer度及Mawhin重合度理论 ,在共振的情况下 ,给出了上述方程存在 2π_周期解的充分条件及其在Duffing方程上的应用·

The nonlinear system of first_order differential equations with a deviating argument (t)=Bx(t)+F(x(t-τ))+p(t) is considered, where x(t)∈R 2,τ∈R,B∈R 2×2 ,F is bounded and p(t) is continuous and 2 π _periodic. Some sufficient conditions for the existence of 2 π _periodic solutions of the above equation, in a resonance case, by using the Brouwer degree theory and a continuation theorem based on Mawhin's coincidence degree are obtained. Some applications of the main results to Duffing's equations are also given.