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Multiple periodic solutions for discrete Hamiltonian systems. (English) Zbl 1115.39017

This work deals with the discrete Hamiltonian system \[ \begin{cases} \Delta u_1(n)=-H_{u_2}(n,u_1(n+1),u_2(n)),\\ \Delta u_2(n)=H_{u_1}(n,u_1(n+1),u_2(n)), &n\in \mathbb Z,\end{cases}\tag{1} \] where \(u_1,\,u_2\in \mathbb R^N\) and \(\Delta u_i(n)=u_i(n+1)-u_i(n)\), \(i=1,2\). The function \(H\) is \(T\)-periodic in the first variable \(n\) (\(T>0\) is a given integer), it is of class \(C^2\) in the second variable \(u_1\) and the third variable \(u_2\), and the gradient of \(H\) with respect to the last two variables, \(\nabla H\) is asymptotically linear at zero. In addition, \(\nabla H\) is asymptotically linear at infinity or \(H(n,z)\) has superquadratic growth as \(| z| \to\infty\). Under the above assumptions, the authors prove the existence of multiple \(T\)-periodic solutions for the problem (1), by using some results from Morse theory.

MSC:

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
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