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Existence of periodic solutions for a class of second order discrete Hamiltonian systems
Advances in Difference Equations volume 2016, Article number: 68 (2016)
Abstract
By using the variational minimizing method and the saddle point theorem, the periodic solutions for non-autonomous second-order discrete Hamiltonian systems are considered. The results obtained in this paper complete and extend previous results.
1 Introduction and main results
Consider the second-order discrete Hamiltonian system
where \(\Delta^{2}u(n)=\Delta(\Delta u(n))\) and \(\nabla F(n,x)\) denotes the gradient of F with respect to the second variable. F satisfies the following assumption:
-
(A)
\(F(n,x)\in C^{1}(\mathbb{R}^{N},\mathbb{R})\) for any \(n\in \mathbb{Z}\), \(F(n+T,x)=F(n,x)\) for \((n,x)\in\mathbb{Z}\times\mathbb {R}^{N}\), T is a positive integer.
Since Guo and Yu developed a new method to study the existence and multiplicity of periodic solutions of difference equations by using critical point theory (see [1–4]), the existence and multiplicity of periodic solutions for system (1.1) have been extensively studied and lots of interesting results have been worked out; see [5–16] and the references therein. System (1.1) is a discrete form of classical second-order Hamiltonian systems, which has been paid much attention to by many mathematicians in the past 30 years; see [17–24] for example.
In particular, when the nonlinearity \(\nabla F(n,x)\) is bounded, Guo and Yu [3] obtained one periodic solution to system (1.1). When the gradient of the potential energy does not exceed sublinear growth, i.e. there exist \(M_{1}>0\), \(M_{2}>0\), and \(\alpha\in [0,1)\), such that
where \(\mathbb{Z}[a,b]:=\mathbb{Z}\cap[a,b]\) for every \(a,b\in\mathbb {Z}\) with \(a\leq b\), Xue and Tang [12, 13] considered the periodic solutions of system (1.1), which completed and extended the results in [3] under the condition where
or
Under weaker conditions on \(\nabla F(n,x)\), i.e.,
or
Tang and Zhang [11] considered the periodic solutions of system (1.1), which completed and extended the results in [12, 13].
In this paper, we will further investigate periodic solutions to the system (1.1) under the conditions of (1.5) or (1.6). Our main results are the following theorems.
Theorem 1.1
Suppose that \(F(n,x)=F_{1}(n,x)+F_{2}(x)\), where \(F_{1}\) and \(F_{2}\) satisfy (A) and the following conditions:
-
(1)
there exist \(f,g:\mathbb{Z}[1,T]\rightarrow\mathbb{R^{+}}\) and \(\alpha\in[0,1)\) such that
$$ \bigl|\nabla F_{1}(n,x)\bigr|\leq f(n)|x|^{\alpha}+g(n),\quad \textit{for all } (n,x)\in\mathbb{Z}[1,T]\times\mathbb{R}^{N}; $$ -
(2)
there exist constants \(r>0\) and \(\gamma\in[0,2)\) such that
$$ \bigl(\nabla F_{2}(x)-\nabla F_{2}(y),x-y\bigr) \geq-r|x-y|^{\gamma}, \quad\textit{for all } x,y\in\mathbb{R}^{N}; $$ -
(3)
$$ \liminf_{|x|\rightarrow+\infty}|x|^{-2\alpha}\sum _{n=1}^{T}F(n,x)>\frac {1}{8\sin^{2}\frac{\pi}{T}}\sum _{n=1}^{T}f^{2}(n). $$
Then system (1.1) has at least one T-periodic solution.
Theorem 1.2
Suppose that \(F(n,x)=F_{1}(n,x)+F_{2}(x)\), where \(F_{1}\) and \(F_{2}\) satisfy (A), (1), (2), and the following conditions:
-
(4)
there exist \(\delta\in[0,2)\) and \(C>0\) such that
$$ \bigl(\nabla F_{2}(x)-\nabla F_{2}(y),x-y\bigr)\leq C|x-y|^{\delta}, \quad\textit{for all } x,y\in\mathbb{R}^{N}; $$ -
(5)
$$ \limsup_{|x|\rightarrow+\infty}|x|^{-2\alpha}\sum _{n=1}^{T}F(n,x)< -\frac {3}{8\sin^{2}\frac{\pi}{T}}\sum _{n=1}^{T}f^{2}(n). $$
Then system (1.1) has at least one T-periodic solution.
Theorem 1.3
Suppose that \(F(n,x)=F_{1}(n,x)+F_{2}(x)\), where \(F_{1}\) and \(F_{2}\) satisfy (A), (1), and the following conditions:
-
(6)
there exists a constant \(0< r<\frac{6}{T^{2}-1}\), such that
$$ \bigl(\nabla F_{2}(x)-\nabla F_{2}(y),x-y\bigr) \geq-r|x-y|^{2},\quad \textit{for all } x,y\in\mathbb{R}^{N}; $$ -
(7)
$$ \liminf_{|x|\rightarrow+\infty}|x|^{-2\alpha}\sum _{n=1}^{T}F(n,x)>\frac {3}{ (24-4(T^{2}-1)r )\sin^{2}\frac{\pi}{T}}\sum _{n=1}^{T}f^{2}(n). $$
Then system (1.1) has at least one T-periodic solution.
Theorem 1.4
Suppose that \(F=F_{1}+F_{2}\), where \(F_{1}\) and \(F_{2}\) satisfy (A), (1), and the following conditions:
-
(8)
there exist \(h:\mathbb{Z}[1,T]\rightarrow\mathbb{R}^{+}\) and \((\lambda,u)\)-subconvex potential \(G:\mathbb{R}^{N}\rightarrow\mathbb {R}\) with λ>1/2 and \(1/2<\mu<2\lambda^{2}\), such that
$$ \bigl(\nabla F_{2}(n,x),y\bigr)\geq-h(n)G(x-y), \quad\textit{for all } x,y\in\mathbb {R}^{N} \textit{ and } n\in\mathbb{Z}[1,T]; $$ -
(9)
$$\begin{aligned}& \limsup_{|x|\rightarrow+\infty}|x|^{-2\alpha}\sum _{n=1}^{T}F_{1}(n,x)< -\frac{3}{8\sin^{2}\frac{\pi}{T}}\sum _{n=1}^{T}f^{2}(n), \\& \limsup_{|x|\rightarrow+\infty}|x|^{-\beta}\sum _{n=1}^{T}F_{2}(n,x)< -8\mu\max _{|s|\leq1}G(s)\sum_{n=1}^{T}h(n), \end{aligned}$$
where \(\beta=\log_{2\lambda}(2\mu)\).
Then system (1.1) has at least one T-periodic solution.
Remark 1.5
Theorems 1.1-1.3 extend some existing results. On the one hand, we decomposed the potential F into \(F_{1}\) and \(F_{2}\). On the other hand, if \(F_{2}=0\), the theorems in [11], Theorems 1 and 2, are special cases of Theorem 1.1 and Theorem 1.2, respectively. Some examples of F are given in Section 4, which are not covered in the references. Moreover, our Theorem 1.4 is a new result.
2 Some important lemmas
\(H_{T}\) can be equipped with the inner product
by which the norm \(\|\cdot\|\) can be induced by
Define
and
for \(u,v\in H_{T}\).
By (A), it is easy to see that Φ is continuously differentiable, and the critical points of Φ are the T-periodic solutions of system (1.1).
The following lemma is a discrete form of Wirtinger’s inequality and Sobolev’s inequality (see [19]).
Lemma 2.1
[11]
If \(u\in H_{T}\) and \(\sum_{t=1}^{T}u(t)=0\), then
Lemma 2.2
[25]
Let \(E=V\oplus X\), where E is a real Banach space and \(V\neq\{0\}\) and is finite dimensional. Suppose \(I\in C^{1}(E, \mathbb{R})\), it satisfies (PS), and
-
(i)
there is a constant α and a bounded neighborhood D of 0 in V such that \(I\mid_{\partial D}\leq\gamma\), and
-
(ii)
there is a constant \(\beta>\gamma\) such that \(I\mid_{X}\geq\beta\).
Then I possesses a critical value \(c\geq\beta\). Moreover, c can be characterized as
where
3 Proof of theorems
For convenience, we denote
Proof of Theorem 1.1
According to (3), there exists \(a_{1}>\frac{1}{4\sin^{2}\frac{\pi}{T}}\) satisfying
From (1) and Lemma 2.1, for any \(u\in H_{T}\), one has
From (2) and Lemma 2.1, for any \(u\in H_{T}\), we have
Combining (3.1) with (3.2), for all \(u\in H_{T}^{1}\) one has
Hence, \(\varphi(u)\rightarrow\infty\) as \(\|u\|\rightarrow\infty\). From this result, if \(\{u_{k}\}\subset H_{T}\) is a minimizing sequence for φ, i.e., \(\varphi(u_{k})\rightarrow\inf\varphi \), \(k\rightarrow\infty\), then \(\{u_{k}\}\) is bounded. Since \(H_{T}\) is finite dimensional, going if necessary to a subsequence, we can assume that \(\{u_{k}\}\) converges to some \(u_{0}\in H_{T}\). Because of φ is continuously differentiable on \(H_{T}\), one has
Obviously, \(u_{0}\in H_{T}\) is a T-periodic solution of system (1.1). □
Proof of Theorem 1.2
Step 1. To prove φ satisfies the (PS) condition. Suppose that \({u_{k}}\) is a (PS) sequence, that is, \(\varphi'(u_{k})\rightarrow0\) as \(k\rightarrow\infty \) and \({\varphi(u_{k})}\) is bounded. According to (5), there exists \(a_{2}>\frac{1}{4\sin^{2}\frac{\pi}{T}}\) satisfying
In the same way as (3.1), for any \(u\in H_{T}\), one has
and
Hence, we have
for all large k.
By Lemma 2.1, one has
By (3.4) and (3.5), for all \(u\in H_{T}^{1}\) one has
where
By the choice of \(a_{2}>\frac{1}{4\sin^{2}\frac{\pi}{T}}\), \(-\infty < C_{1}<0\). Hence
and then
where \(0< C_{2}<+\infty\).
From Theorem 1.1, one has
By (4), we obtain
Combining the boundedness of \(\{\varphi(u_{k})\}\) and (3.7)-(3.9), one has
for large k. By the choice of \(a_{2}\), \(\{\bar{u}_{k}\}\) is bounded. From (3.7), \(\{u_{k}\}\) is bounded. In view of \(H_{T}\) is finite dimensional Hilbert space, φ satisfies the (PS) condition.
Step 2. Let \(\tilde{H}_{T}=\{u\in H_{T}:\bar{u}=0 \}\). We show that, for \(u\in\tilde{H}_{T}\),
From (1) and Lemma 2.1, one has
for all \(u\in\tilde{H}_{T}\). It follows from (2) that
Hence, we have
In view of Lemma 2.1, \(\|u\|\rightarrow+\infty\) in \(\tilde{H}_{T}\) if and only if \((\sum_{n=1}^{T}|\Delta u(n)|^{2})^{1/2}\rightarrow\infty\). Hence (3.10) is satisfied.
Step 3. By (5), for all \(u\in(\tilde{H}_{T})^{\bot }=\mathbb{R}^{N}\), one has
Above all, all conditions of Lemma 2.2 are satisfied. So, by Lemma 2.2, system (1.1) has at least one T-periodic solution. □
Proof of Theorem 1.3
By (7), there exists \(a_{3}>\frac {3}{(12-2(T^{2}-1)r)\sin^{2}\frac{\pi}{T}}\) satisfying
Similar to (3.1), we have
By (6) and Lemma 2.1, one has
So, for any \(u\in H_{T}\), we have
Therefore, \(\varphi(u)\rightarrow+\infty\) as \(\|u\|\rightarrow+\infty\) due to the choice of \(a_{3}\) and \(r<\frac{6}{T^{2}-1}\). The rest is similar to the proof of Theorem 1.1. □
Proof of Theorem 1.4
First, we prove that φ satisfies the (PS) condition. Suppose that \(\{u_{k}\}\subset H_{T}\) is a (PS) sequence of φ, that is, \(\varphi'(u_{k})\rightarrow0\) as \(k\rightarrow\infty\) and \(\{\varphi(u_{k})\}\) is bounded. By (9), there exists \(a_{4}>\frac{1}{4\sin^{2}\frac{\pi}{T}}\) satisfying
By the \((\lambda,\mu)\)-subconvexity of \(G(x)\), we have
for all \(x\in\mathbb{R}^{N}\), where \(G_{0}=\max_{|s|\leq1}G(s)\), \(\beta =\log_{2\lambda}(2\mu)<2\).
Then
where \(R_{4}=G_{0}\sum_{n=1}^{T}h(n)\). For large k, according to (3.3) and (3.13) we have
where
By the choice of \(a_{4}\), \(-\infty< C_{8}<0\). By (3.15), we have
and then
where \(C_{9}>0\). By (8) and (3.12), for any \(u\in H_{T}\), we get
Combining the boundedness of \(\{\varphi(u_{k})\}\) and (3.16)-(3.18), one has
Combining (3.11) and the above inequality, we see that \(\{|\bar {u}|\}\) is bounded. By (3.16), \(\{u_{k}\}\) is bounded. Since \(H_{T}\) is a finite dimensional Hilbert space, φ satisfies the (PS) condition.
Similar to the proof of Theorem 1.2, all conditions of Lemma 2.2 are satisfied. So, the proof of Theorem 1.4 is completed. □
4 Examples
In this section, we give some examples to illustrate our results.
Example 4.1
Let \(F=F_{1}+F_{2}\), with
where \(k:\mathbb{Z}[1,T]\longrightarrow\mathbb{R}\) and \(k(n+T)=k(n)\), for all \(n\in\mathbb{Z}\), \(r>0\), \(C(x)=\frac{3r}{4} (|x_{1}|^{4/3}+|x_{2}|^{4/3}+\cdots+|x_{N}|^{4/3})\). It is easy to see that
For all \((n,x)\in\mathbb{Z}[1,T]\times\mathbb{R}^{N}\), where \(\varepsilon>0\),
Thus, (1), (2) hold with \(\alpha=3/4\), \(\gamma=4/3\), and
So, we have
On the other hand, one has
If \(T\in\{2,3,4,5,6,7\}\), we can choose \(\varepsilon>0\) such that
So, (3) holds. By Theorem 1.1, system (1.1) has at least one T-periodic solution.
Example 4.2
Let \(F=F_{1}+F_{2}\), with
where \(k:\mathbb{Z}[1,T]\longrightarrow\mathbb{R}^{N}\) and \(k(n+T)=k(n)\) for all \(n\in\mathbb{Z}\), \(r>0\).
In a way similar to Example 4.1, it is easy to see that condition (1) and (4) are satisfied with \(\alpha=3/4\). So,
If \(T\in\{2,3,4,5\}\), we can choose \(\varepsilon>0\) small enough such that
which implies that (5) holds. By Theorem 1.2, system (1.1) has at least one T-periodic solution.
Example 4.3
Let \(F=F_{1}+F_{2}\), with
where \(k:\mathbb{Z}[1,T]\longrightarrow\mathbb{R}\) and \(k(n+T)=k(n)\) for all \(n\in\mathbb{Z}\), \(r>0\), \(C(x)=\frac{r}{2} (|x_{1}|^{4}+|x_{2}|^{2}+\cdots+|x_{N}|^{2})\), \(0< r<\frac {6}{T^{2}-1}\).
In a way similar to Example 4.1, it is easy to see that conditions (1) and (6) are satisfied with \(\alpha=3/4\). So
If \(T\in\{2,3\}\), we choose \(\varepsilon>0\), such that
which implies that (7) holds. By Theorem 1.3, system (1.1) has at least one T-periodic solution.
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Acknowledgements
Research was supported by the Postdoctoral fund in China (Grant No. 2013M531717) and NSFC (11561043).
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The main idea of this paper was proposed by KY and WG, WG prepared the manuscript initially and KY performed a part of the steps of the proofs in this research. All authors read and approved the final manuscript.
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Guan, W., Yang, K. Existence of periodic solutions for a class of second order discrete Hamiltonian systems. Adv Differ Equ 2016, 68 (2016). https://doi.org/10.1186/s13662-016-0787-5
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DOI: https://doi.org/10.1186/s13662-016-0787-5
Keywords
- periodic solutions
- second-order discrete Hamiltonian systems
- saddle point theorem
- least action principle