在共振点的一阶微分系统的周期解

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

The nonlinear system of first-order differential equations with a deviating argument x>(t)=Bx(t)+F(x(t-τ))+p(t),is considered,where x(t)∈R2,τ∈R,B∈R2×2 F is bounded and p(t) is continuous and 2π-periodic.Some sufficient conditions for the existence of 2π-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.