## Existence solutions for second order Hamiltonian systems.(English)Zbl 1333.34061

The paper discusses the existence of periodic solutions of the non-autonomous dynamical system $-\ddot{x}(t)=B(t)x(t)+\nabla_x V(t,x(t)), \eqno(1)$ where $$x(t)=(x_1(t),\dots,x_n(t))$$, and $$V(t,x)$$ is continuous on $$\mathbb{R}^{n+1}$$ with $\nabla_x V(t,x(t))=(\partial V/\partial x_1,\dots, \partial V/\partial x_n)\in C(\mathbb{R}^{n+1}, \mathbb{R}^n).$ For each $$x\in \mathbb{R}^n$$, the function $$V(t,x)$$ is $$T$$-periodic in $$t$$. The matrix $$B(t)$$ is symmetric with $$T$$-periodic components from $$L^1(0,T)$$. The authors provide conditions which imply that that there exists a $$T$$-periodic weak solution of (1) whose weak second derivative belongs to $$L^1(0,T)$$, and conditions giving infinitely many of such solutions. Both subquadratic and superquadratic problems are considered.

### MSC:

 34C25 Periodic solutions to ordinary differential equations 37C27 Periodic orbits of vector fields and flows 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 37C60 Nonautonomous smooth dynamical systems 58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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### References:

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