## Infinitely many periodic solutions for a class of second-order Hamiltonian systems.(English)Zbl 1347.37110

In this paper, the authors study the following second-order Hamiltonian systems $\begin{cases} \ddot{u}(t)+A(t)u(t)+\nabla F(t,u(t))=0,\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0, \end{cases}\tag{PS}$ where $$A \in C ([0, T],\mathbb{R}^{N^2})$$ is a symmetric matrix valued function, $$F : [0, T ]\times \mathbb{R}^N \rightarrow \mathbb{R}$$ is subquadratic, even in $$u$$ and satisfies the Ahmad-Lazer-Paul condition. By using the dual fountain theorem due to T. Bartsch and M. Willem [Proc. Am. Math. Soc. 123, No. 11, 3555–3561 (1995; Zbl 0848.35039)], the authors obtain the existence of infinitely many periodic solutions for systems (PS).

### MSC:

 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 34C25 Periodic solutions to ordinary differential equations 70H05 Hamilton’s equations

Zbl 0848.35039
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### References:

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