# zbMATH — the first resource for mathematics

On periodic solutions of nonlinear second order vector differential equations. (English) Zbl 0626.34040
Consider the vector differential equation $(1)\quad x''(t)+d/dt[\text{grad} f(x(t))]+g(t,x(t)),$ together with periodic boundary conditions (ii) $$x(0)=x(2\pi)$$, $$x'(0)=x'(2\pi)$$. Let $$f\in C^ 2({\mathbb{R}}^ n,{\mathbb{R}})$$ and let the function $$g: [0,2\pi]\times {\mathbb{R}}^ n\to {\mathbb{R}}^ n$$ satisfy the Carathéodory conditions. If $$g(t,x)=Q(t,x)x+h(t,x),$$ where Q is a symmetric matrix, then under some assumptions on the eigenvalues of Q, the problem (i)-(ii) has at least one solution. (The class of all possible Q is described.) Some other results refer to the same problem for approximate equations, especially for the vector Duffing equation.
Reviewer: D.Bobrowski

##### MSC:
 34C25 Periodic solutions to ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems
Full Text: