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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

34C25 Periodic solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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