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Existence of periodic solutions for nonautonomous secondorder discrete Hamiltonian systems
Advances in Difference Equations volume 2016, Article number: 309 (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.
1 Introduction and main results
Consider the following discrete Hamiltonian system:
where \(\bigtriangleup^{2}u(t)=\bigtriangleup(\bigtriangleup u(t))\), and \(\nabla F(t,x)\) denotes the gradient of \(F(t,x)\) in x. In this paper, we always suppose that the following condition is satisfied:

(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 \(F(t,x)\).
Recently, Yan et al. [15] obtained multiple periodic solutions for system (1.1) when the growth of \(\nabla F(t,x)\) is sublinear and there exists an integer \(r\in[0,N]\) such that:

(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
Suppose that (A) holds and
 (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
Let \(F(t,x)=a\cos xe(t)x\). If \(e(t)\) satisfies
then system (1.1) has at least one Tperiodic solution.
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
Suppose that (A) and (H_{1}) hold and
 (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
Let
with norm
Set
and
for \(u,v\in H_{T}\).
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]
Assume that X is a Banach space and \(f\in C^{1}(X, \mathbb{R})\). If \(\{u_{n}\}\subset X\) satisfies
then we say that \(\{u_{n}\}\) is a \((\mathit{CPS})_{C}\) sequence of f. For any \((\mathit{CPS})_{C}\) sequence \(\{u_{n}\}\), if there exists a subsequence of \(\{ u_{n}\}\) convergent in X, then we say that f satisfies \((\mathit{CPS})_{C}\) condition.
Lemma 2.1
Assume that X is a Banach space and \(f\in C^{1}(X, \mathbb{R})\). Let \(X=X_{1}\oplus X_{2}\) and
where \(S^{1}_{R}=\{u\in X_{1}\mid u=R\}\).
Set \(B^{1}_{R}=\{u\in X_{1}, u\leq R\}\), \(M=\{g\in C(B^{1}_{R},X)\mid g(s)=s, s\in S^{1}_{R}\}\),
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]
If \(u\in H_{T}\) and \(\sum_{t=1}^{T}u(t)=0\), then
and
3 Proof of main results
Proof of Theorem 1.1
Let
where \(\widetilde{H}_{T}=\{u\in H_{T}:\overline{u}=\frac{1}{T}\sum_{t=1}^{T}u(t)=0\}\).
For any \(u\in H_{T}\), there are \(\tilde{u}\in\widetilde{H}_{T}\) and \(\overline{u}\in\mathbb{R}^{N}\) such that \(u=\tilde {u}+\overline{u}\).
According to (H_{2}), we have that
So,
Suppose that \(\{u_{k}\}\) is a minimizing sequence for Φ, that is,
Then \(u_{k}=\tilde{u}_{k}+\overline{u}_{k}\), where \(\tilde {u}_{k}\in\widetilde{H}_{T}\), \(\overline{u}_{k}\in\mathbb{R}^{N}\). By (3.1) there exists \(c>0\) such that
By (H_{1}) we have that
Hence, if \(\{u_{k}\}\) is a minimizing sequence for Φ, then
is also a minimizing sequence of Φ.
Therefore, we can assume that
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 \(u_{0}\in {H}_{T}\).
Since Φ is continuously differentiable, we have
Therefore, the proof is finished. □
Proof of Theorem 1.2
For the proof, we will apply Rabinowitz’s saddle point theorem. First, to prove that Φ satisfies the \((\mathit{CPS})_{C}\) condition. Suppose that for C, a sequence \(\{u_{k}\}\in{H}_{T}\) satisfies
Since \(\Phi(u_{k})\rightarrow C\), we have that
From (H_{3}) we have that
By (3.4) we have that
So, we have
Therefore,
From this inequality we have that \(\sum_{t=1}^{T}\bigtriangleup {u}_{k}(t)^{2}\) is bounded.
By (H_{1}) we have that
Therefore, if \(\{u_{k}\}\) is a \((\mathit{CPS})_{C}\) sequence of Φ, then
is also a \((\mathit{CPS})_{C}\) sequence of Φ.
So, we can assume that
that is, \(\overline{u}_{k}\) 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.
Let
For any \(u\in X_{2}\), by (H_{5}) and Lemma 2.2 we have that
On the other hand, for any \(u\in X_{1}\), by (H_{4}) we have that
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. □
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Research was supported by NSFC (11561043).
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Wang, DB., Xie, HF. & Guan, W. Existence of periodic solutions for nonautonomous secondorder discrete Hamiltonian systems. Adv Differ Equ 2016, 309 (2016). https://doi.org/10.1186/s1366201610367
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DOI: https://doi.org/10.1186/s1366201610367