## Periodic solutions with prescribed minimal period for the second order Hamiltonian systems with even potentials.(English)Zbl 1211.34051

The article deals with $$T$$-periodic solutions of the autonomous superquadratic second order Hamiltonian system
$\ddot{x} + V'(x) = 0,\tag{1}$
where $$V: {\mathbb R}^n \to {\mathbb R}$$ is an even smooth function.
The main result is the following one: if $$V$$ is even and there exists $$\theta > 1$$ such that $$0 < \theta V'(x)x \leq V''(x)[x]^2$$, $$x \neq 0$$, then, for every $$T > 0$$, system (1) has at least one $$T$$-periodic solution with $$T$$ as its minimal period.

### MSC:

 34C25 Periodic solutions to ordinary differential equations 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 58E30 Variational principles in infinite-dimensional spaces
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### References:

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