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Periodic solutions for second order Hamiltonian systems. (English) Zbl 1265.34154

The authors consider the periodic boundary value problem for the Hamiltonian system \[ \begin{aligned} &\ddot {u}(t)=\nabla F(t,u(t))\; \; \text{for a.e. } t\in [0,T], \\ &u(0)-u(T)=\dot {u}(0)-\dot {u}(T=0)\end{aligned} \] and establish some existence theorems for periodic solutions. They use the technique of the least-action principle and the minimax method in critical point theory. The results extend those of Z. Wang and J. Zhang [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 12, 4480–4487 (2010; Zbl 1206.34060)].

MSC:

34C25 Periodic solutions to ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34B15 Nonlinear boundary value problems for ordinary differential equations
58E30 Variational principles in infinite-dimensional spaces

Citations:

Zbl 1206.34060

References:

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