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Multiple periodic solutions for discrete Hamiltonian systems. (English) Zbl 1115.39017
This work deals with the discrete Hamiltonian system $$\cases \Delta u_1(n)=-H_{u_2}(n,u_1(n+1),u_2(n)),\\ \Delta u_2(n)=H_{u_1}(n,u_1(n+1),u_2(n)), &n\in \Bbb Z,\endcases\tag1$$ where $u_1,\,u_2\in \Bbb R^N$ 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 $\vert z\vert \to\infty$. 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
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