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Periodic solutions with prescribed minimal period for convex autonomous Hamiltonian systems. (English) Zbl 0594.58035
In 1977, Rabinowitz proved the existence of a nonconstant periodic solution of any prescribed period \(T>0\) for the Hamiltonian systems under some reasonable conditions. However, he does not claim that the T- periodic solution he finds actually has minimal period T. Therefore, it challenges one’s attention to prove minimality of the period for that T- periodic solution.
The first researchers to make progress on this question were A. Ambrosetti and G. Mancini, and then M. Girardi and M. Matzeu. This paper is a similar one devoted to the same question for convex autonomous Hamiltonian systems with some (more or less) weaker conditions. The authors combine two new tools: An index theory for periodic solutions of Hamiltonian systems, and a complete description of \(C^ 2\) functionals near critical points of mountain pass type. The main result is that the Ambrosetti-Rabinowitz theorem, when applied to the dual action functional, will actually yield solutions with minimal period T.
Reviewer: Ding Tongren

MSC:
37G99 Local and nonlocal bifurcation theory for dynamical systems
34C25 Periodic solutions to ordinary differential equations
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References:
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