2n阶常微分方程周期边值问题解的存在唯一性

Existence and uniqueness for periodic boundary value problems of 2nth-order ordinary differential equations

研究2n阶非线性常微分方程周期边值问题{u(2n)(t)+au(t)=f(t,u(t),u′(t),…,u(2n-1)(t)),t∈I,u(i)(0)=u(i)(2π),i=0,1,…,2n-1解的存在唯一性,其中n≥1是整数,I=[0,2π],(-1)na>0,f:I×R2n—→R连续且关于t以2π为周期.运用Fourier分析法和Leray-Schauder不动点定理,获得了当非线性项f满足适当增长条件时,该问题解的存在唯一性结果.

This paper deals with the existence and uniqueness of solutions for 2nth-order ordinary differential equation with periodic boundary value conditionu{（2n）（t）＋au（t）=f（t,u（t）,u′（t）,…,u（2n-1）（t））,t∈I,u（i）（0）=u（i）（2π）,i=0,1,…,2n-1.Where n≥1is a integer,I=[0,2π],（-1）na0,f：I×R2n →Ris continuous and 2π-peridoic with respect to t.By applying the Fourier analysis method and Leray-Schauder fixed point theorem,the results of existence and uniqueness are obtained when the nonlinearity fsatisifies proper growth conditions.