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Periodic solution of second-order Hamiltonian systems with a change sign potential on time scales. (English) Zbl 1182.34118

Summary: This paper is concerned with the second-order Hamiltonian system on time scales \(\mathbb T\) of the form
\[ u^\Delta\Delta(\rho(t))+\mu b(t)|u(t)|\mu-2u(t)+\overline\nabla H(t,u(t))=0,\quad \Delta\text{-a.e. }t\in [0,T]_{\mathbb T}, \]
\[ u(0)-u(T)=u\Delta (\rho(0))-u\Delta(\rho(T))=0, \]
where \(0,T\in\mathbb T\). By using the minimax methods in critical theory, an existence theorem of periodic solution for the above system is established. As an application, an example is given to illustrate the result. This is probably the first time the existence of periodic solutions for second-order Hamiltonian system on time scales has been studied by critical theory.

MSC:

34N05 Dynamic equations on time scales or measure chains
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)
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References:

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