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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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