Existence of periodic solutions for nonautonomous secondorder discrete Hamiltonian systems
 DaBin Wang^{1},
 HuaFei Xie^{1} and
 Wen Guan^{1}Email author
https://doi.org/10.1186/s1366201610367
© The Author(s) 2016
Received: 14 July 2016
Accepted: 22 November 2016
Published: 29 November 2016
Abstract
In this paper, we consider the existence of periodic solutions for a class of nonautonomous secondorder discrete Hamiltonian systems in case the sum on the time variable of potential is periodic. The tools used in our paper are the direct variational minimizing method and Rabinowitz’s saddle point theorem.
Keywords
MSC
1 Introduction and main results
 (A)
\(F(t,x)\in C^{1}(\mathbb{R}^{N},\mathbb{R})\) for any \(t\in\mathbb {Z}\), and \(F(t+T,x)=F(t,x)\) for \((t,x)\in\mathbb{Z}\times\mathbb {R}^{N}\), where \(T>0\) is a integer.
In the last years, a great deal of work has been devoted to the study of the existence and multiplicity of periodic solutions for discrete Hamiltonian system (1.1); see [1–16] and the references therein. In particular, Guo and Yu [7] considered the existence of one periodic solution to system (1.1) in case \(\nabla F(t,x)\) is bounded. Xue and Tang [12, 13] generalized these results when the gradient of potential energy does not exceed sublinear growth.
Tang and Zhang [11] completed and extended the results obtained in [12, 13] under a more weaker assumption on \(F(t,x)\).
 (i)
\(F(t,x)\) is \(T_{i}\)periodic in \(x_{i}\), \(1\leq i\leq r\).
 (ii)$$ x^{2\alpha}\sum_{t=1}^{T}F(t,x) \rightarrow\pm\infty\quad \text{as } x\rightarrow\infty, x\in\{0\}\times \mathbb{R}^{Nr}. $$
In this paper, motivated by the results mentioned and [17], we further study the existence of periodic solutions to the discrete Hamiltonian system (1.1).
Our main results are the following theorems.
Theorem 1.1
 (H_{1}):

\(\sum_{t=1}^{T}F(t,x+T_{i}e_{i})=\sum_{t=1}^{T}F(t,x)\), \(1\leq i\leq N\), where \(T_{i}>0\), and \(\{e_{i}1\leq i\leq N\}\) is an orthogonal basis in \(\mathbb{R}^{N}\);
 (H_{2}):

there exist \(0< C_{1}<2\sin^{2}\frac{\pi}{T}\) and \(C_{2}>0\) such that$$\bigl\vert F(t,x)\bigr\vert \leq C_{1}\vert x\vert ^{2}+C_{2}. $$
Then system (1.1) has at least one Tperiodic solution.
Corollary 1.1
Remark 1.1
When \(F(t,x)=a\cos xe(t)x\) (\(a\geq0\)), system (1.1) is a discrete form of forced equations studied by Mawhin and Willem [18–20], in which they require the assumption that the forced potential is periodic on spatial variables. So, our results, Theorem 1.1 and Corollary 1.1, generalize their results in discrete situation.
Theorem 1.2
 (H_{3}):

there exist \(\mu_{1}<2\) and \(\mu_{2}\in\mathbb{R}\) such that$$\bigl(\nabla F(t,x), x\bigr)\leq\mu_{1}F(t,x)+\mu_{2}; $$
 (H_{4}):

there exists \(\delta>0\) such that, for \(t\in\mathbb{Z}\), we have$$F(t,x)>\delta,\quad x\rightarrow+\infty; $$
 (H_{5}):

there exists \(0< b<2\sin^{2}\frac{\pi}{T}\) such that$$F(t,x)\leq bx^{2}. $$
Then system (1.1) has at least one Tperiodic solution. Furthermore, system (1.1) has at least one nonconstant Tperiodic solution if \(\sum_{t=1}^{T}F(t,x)\geq0\) for all \(x\in\mathbb{R}^{N}\).
2 Some important lemmas
According to assumption (A), it is well known that Φ is continuously differentiable and the Tperiodic solutions of problem (1.1) correspond to the critical points of the functional Φ.
Definition 2.1
[21]
Lemma 2.1
Then \(C>\inf_{X_{2}}f\). Furthermore, if f satisfies \((\mathit{CPS})_{C}\) condition, then C is a critical value of f.
Lemma 2.2
[11]
3 Proof of main results
Proof of Theorem 1.1
For any \(u\in H_{T}\), there are \(\tilde{u}\in\widetilde{H}_{T}\) and \(\overline{u}\in\mathbb{R}^{N}\) such that \(u=\tilde {u}+\overline{u}\).
By (3.2) and (3.3), \(\{u_{k}\}\) is a bounded minimizing sequence of Φ in \({H}_{T}\).
Going to a subsequence if necessary, since \({H}_{T}\) is finite dimensional, we can assume that \(\{u_{k}\}\) converges to some \(u_{0}\in {H}_{T}\).
Therefore, the proof is finished. □
Proof of Theorem 1.2
From this inequality we have that \(\sum_{t=1}^{T}\bigtriangleup {u}_{k}(t)^{2}\) is bounded.
From these results we have that \(\{u_{k}\}\) is bounded.
Since \({H}_{T}\) is a finitedimensional Banach space, it is easy to see that Φ satisfies the \((\mathit{CPS})_{C}\) condition.
We now prove that the conditions of Rabinowitz’s saddle point theorem are satisfied.
From this it follows that the conditions of Rabinowitz’s saddle point theorem are all satisfied.
So, by Lemma 2.1 there exists a periodic solution of system (1.1). Furthermore, if \(\sum_{t=1}^{T}F(t,x)\geq0\), then there exists a nonconstant periodic solution u̅ of system (1.1) such that \(\Phi(\overline{u})=C>\inf_{X_{2}}\geq0\) since otherwise we would have a contradiction with the fact that \(\Phi(\overline{u})=\sum_{t=1}^{T}F(t,\overline{u}(t))\leq0\).
Therefore, the proof is finished. □
Declarations
Acknowledgements
Research was supported by NSFC (11561043).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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