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Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems. (English) Zbl 1043.34045

The authors obtain some existence theorems for periodic solutions of the class of unbounded nonautonomous nonconvex subquadratic second-order Hamiltonian systems \[ \ddot u(t)+A(t)u(t)+\nabla F(t,u(t))= 0, \quad\text{a.e. }t\in [0,T], \qquad u(0)- u(T)= \dot u(0)- \dot u(T)= 0, \] where \(T>0\), \(A(t)\) is a continuous symmetric matrix of order \(N\). \(F:[0,T]\times \mathbb{R}^N\to \mathbb{R}\) and satisfies the following condition: \(F(t,x)\) is measurable in \(t\) for every \(x\in \mathbb{R}^N\) and continuously differentiable in \(x\) for a.e. \(t\in [0,T]\), and there exist \(a\in \mathbb{C}(\mathbb{R}^+, \mathbb{R}^+)\), \(b\in L^1(0,T;\mathbb{R}^+)\) such that \(| F(t,x)|\leq a(| x|)b(t)\), \(|\nabla F(t,x)|\leq a(| x|) b(t)\) for all \(x\in \mathbb{R}^N\) and a.e. \(t\in [0,T]\), where \(\mathbb{R}^+\) is the set of all nonnegative real numbers. The results are obtained by the minimax methods in critical point theory.

MSC:

34C25 Periodic solutions to ordinary differential equations
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