二阶非线性微分方程+f(t,x)=0的2π-周期解

2π-Periodic Solutions of Second-order Nonlinear Differential Equation x" + f(t,x) =0

本文讨论二阶非线性微分方程χ+(t,χ)=0的2π-周期解,在不要求超线性条件(|χ|→∞时对t一致地x^(-1),f(t,χ)→+∞)或次线性条件(|χ|→0时对t一致地x^(-1)f(t,x)→+∞)的情况下,利用推广的Poincare-Birkhoff定理,给出了存在多个2π-周期解的条件.同时,在不要求超线性或次线性条件的情况下给出了存在无穷多个2π-周期解的一组充分条件,这组条件是就函数,f(t,χ)本身直接提出的.对于χ+f(t,χ)=O (1)以后恒假定:f∈C°(R × R,R),f(t+2π,)≡f(t,·),并保证方程(1)的初值解存在唯一及解对初值连续依赖.

Without super linear or sub linear assumptions, in this paper sufficient criteria for the existence of multiple harmonic solutions, multiple subharmoaic solutions and infinite harmonic solutions of the second-order nonlinear differential equation X + f(t,x) = 0 are given respectively.