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Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems. (English) Zbl 1053.39011

Some results are obtained for the existence and subharmonic solutions to discrete Hamiltonian systems

$\begin{array}{cc}\hfill {\Delta }{x}_{1}\left(n\right)& =-{H}_{{x}_{2}}\left(n,{x}_{1}\left(n+1\right),{x}_{2}\left(n\right)\right),\hfill \\ \hfill {\Delta }{x}_{2}\left(u\right)& ={H}_{{x}_{1}}\left(n,{x}_{1}\left(n+1\right),{x}_{2}\left(n\right)\right)\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(1\right)$

by using critical point theory, where ${x}_{1},{x}_{2}\in {ℝ}^{d}$, $H\in {C}^{1}\left(ℝ×{ℝ}^{d}×{ℝ}^{d},ℝ\right)$, ${\Delta }{x}_{i}\left(n\right)={x}_{i}\left(n+1\right)-{x}_{i}\left(n\right)$, $i=1,2$.

Denote $z={\left({x}_{1}^{T},{x}_{2}^{T}\right)}^{T}$, $H\left(t,z\right)=H\left(t,{x}_{1},{x}_{2}\right)$ and assume the following conditions

$\left({\text{H}}_{1}\right)$ $H\left(t,z\right)\in {C}^{1}\left(ℝ×{ℝ}^{2d},ℝ\right)$ and there exists a positive integer $m$ such that $\forall \left(x,z\right)\in ℝ×{ℝ}^{2d}$, $H\left(t+m,z\right)=H\left(t,z\right)$,

$\left({\text{H}}_{2}\right)$ For any $\left(t,z\right)\in ℝ×{ℝ}^{2d}$, $H\left(t,z\right)\ge 0$ and $H\left(t,z\right)=o\left(|z{|}^{2d}\right)$ (as $z\to 0$);

$\left({\text{H}}_{3}\right)$ There exist some constants $R>0$, $\beta >2$ such that for any $|z|\ge R$, $\left(z,{H}_{z}^{1}\left(t,z\right)\right)\ge \beta H\left(t,z\right)>0$;

$\left({\text{H}}_{4}\right)$ $H\left(t,z\right)$ is even for the second variable $z$, namely, $H\left(t,-z\right)=H\left(t,z\right)$, $\forall \left(t,z\right)\in ℝ×{ℝ}^{2d}$. By assumptions $\left({\text{H}}_{1}\right)$ and $\left({\text{H}}_{3}\right)$ the authors see that $H\left(t,z\right)\ge {a}_{1}{|z|}^{\beta }-{a}_{2}$, $\forall \left(t,z\right)\in ℝ×{ℝ}^{2d}$. So assumptions $\left({\text{H}}_{1}\right)$$\left({\text{H}}_{3}\right)$ imply that $H\left(t,z\right)$ grows superquadratically both at zero and at infinity. The main results of this paper are the following two theorems:

Theorem 1: Under assumption $\left({\text{H}}_{1}\right)$$\left({\text{H}}_{4}\right)$ for any given positive integer $p$, there exist at least $d\left(pm-1\right)$ geometrically distance nontrivial periodic solutions of the system (1) with period $pm$.

Without assumption $\left({\text{H}}_{4}\right)$, they have

Theorem (2): Under assumptions $\left({\text{H}}_{1}\right)$$\left({\text{H}}_{3}\right)$, for any given positive integer $p$, there exist at least two nontrivial periodic solutions of (1) with period $pm$.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 58E50 Applications of variational methods in infinite-dimensional spaces