一类非自治三阶常微分方程周期解的存在性

Existence of periodic solutions for a class of nonautonomous third-order ordinary differential equations

本文研究了一类非自治三阶常微分方程x■-a(t)x+b(t)x 2-c(t)x 3=0正周期解的存在性,其中a(t),b(t),c(t)是连续的T-周期函数,满足0<a≤a(t)≤A,0<b≤b(t)≤B,0<c≤c(t)≤C,a,A,b,B,c,C是正常数.运用Mawhin延拓定理,本文证明了方程至少存在两个正T-周期解.

In this paper,we study the existence of positive periodic solutions for a class of nonautonomous third-order ordinary differential equations x■-a(t)x+b(t)x 2-c(t)x 3=0,where a(t),b(t),c(t)are continuous T-periodic functions satisfying 0<a≤a(t)≤A,0<b≤b(t)≤B,0<c≤c(t)≤C,a,A,b,B,c,C are positive constants.Using Mawhin′s continuation theorem,we prove that there exist at least two positive T-periodic solutions for this equation.