n维二阶系统周期边值问题解的存在性

EXISTENCE OF SOLUTIONS TO SYSTEMS OF SECOND ORDER PERIODIC BOUNDARY VALUE PROBLEMS

应用上、下解的方法研究二阶方程的周期边值问题(PBVP)解的存在性目前已有许多成果(见[1—3]及其参考文献).与两点边值问题的研究类似,PBVP解的存在性研究一般要先对方程的右端函数关于一阶导数项的增长阶作些限制,如满足Nagumo 条件等,再将问题转化成某种修正方程边值问题解的存在性问题.对于二阶系统(即向量二阶方程)PBVP解的存在性也已有了一定的研究(见[4—7]).但已有的这些研究对上、下解都要作较强的限制。

In this paper we use the method of upper and lower solutions to discuss the periodicboundary value problem of the general second order differential system-x=f(t,x,x), Thedefinition of upper and lower solutions in our paper is an immediate extension of that ofscalar equations. Using our results, we can prove the existence of periodic solutions for somesecond order systems which do not satisfy Nagumo condition.And we can solve the existenceproblem of periodic solutions for a class of perturbed linear systems.