二阶跨共振Duffing方程奇调和解的存在性和唯一性

The Existence and Uniqueness of Odd-harmonic Solutions for Second Order Crossing Resonance Duffing Equations

讨论了二阶Duffing方程x″+cx′+g(t,x)=e(t)的奇调和解,利用Leray Schauder度理论,在可跨0特征值的渐近非一致条件γ(t)≤gx(t,x)≤Γ(t)下,得到了所讨论方程奇调和解的存在唯一性定理;基于此,还讨论了方程x″+g(t,x)=e(t)具有对称性的奇调和解的存在性和唯一性.

In this paper, the existence and uniqueness of odd-harmonic solutions for second order Duffing equation x″+cx′+g(t,x)=e(t) are studied using Leray-Schauder degree theory under asymptotic nonuniform condition γ(t)≤g_x(t,x)≤Γ(t), such that a crossing of the zero eigenvalue is allowed. Based on this, the existence and uniqueness of symmetric odd-harmonic solutions for equation x″+g(t,x)=e(t) are studied with the same theory.