## 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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### References:

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