二阶微分方程组正周期解的存在性

摘要

考虑二阶微分方程组{x”+H(t)x'+A(t)x=F(t,x),0＜t＜T,x(0)=x(T),x'(0)=x'(T),正周期解的存在性.利用Krasnoselskii不动点定理以及所对应的格林函数正性,证明上述二阶微分方程组新的正周期解的存在定理,将所得结论进行推广,得到上述二阶微分方程组多个正周期解的存在定理.将结论应用在经典实例上,验证了所得结果的正确性.%Abstract:The existence of positive periodic solutions for the following second order differential equations is considered.{x”+H(t)x' +A(t)x=F(t,x),0＜t＜T,x(0)=x(T),x'(0) =x'(T).First of all,by using the Krasnoselskii fixed point theorem and the positivity of the Green's function,the new existence theorem of positive periodic solutions of the equations is obtained; Then,the above main conclusions are generalized to existence theorem of multiple positive periodic solutions; Finally,a classic example is tested to present the value of the main results.