A discrete second-order Hamiltonian system with asymptotically linear conditions

1.   Introduction and main results

We are interested in the following discrete second-order Hamiltonian system (SOHS in short):

where $u(n)\in{\mathbb R^{N}}$, $\Delta ^{2}{u(n)} = \Delta(\Delta {u(n)})$, $\Delta {u(n)} = u(n+1)-u(n),$ and $F(n, x)$ is continuously differentiable about the first variable. $F(n+T, x) = F(n, x), (n, x)\in {\mathbb Z\times \mathbb R^{N}}$, with the positive integer $T\geq2$.

The study of nonlinear Hamiltonian systems (HS) is one of the important research directions in mathematics, and it is related to many mathematical physics fields. Many scholars are committed to studying the continuous HS and have obtained a lot of results on nonlinear HS in studies of past decades via different critical point theories (see ^[1,2] and references therein). The critical point theories also have important applications in nonlocal elliptic problems ^[3].

Guo and Yu ^[4,5,6] first studied the discrete SOHS by using variational method and aroused much research enthusiasm in this topic. Specially, in ^[4] the authors obtained the existence of periodic solutions with the help of the saddle point theorem with $F(n, x)$ satisfying superlinear conditions at the origin and infinity.

In ^[7] Tang and Zhang investigated a discrete SOHS under sublinear conditions. Wang, Zhang and Chen ^[8] introduced a control function $h(s)$ and discussed a class of non-autonomous SOHS via the least action principle.

Tang and Xiao ^[9] obtained the existence of a nontrivial homoclinic solution for continuous HS via the mountain pass theorem. Tang and Xue ^[10] investigated the multiplicity of periodic solutions for a discrete SOHS under superquadratic conditions via the operator theory. Gradually, more and more scholars have devoted themselves to studying the non-autonomous SOHS under quadratic conditions. Some solvable results have been for the non-autonomous SOHS by the minimax methods, such as in the papers^[11,12,13]. Among them, ^[11,13] considered the cases with subquadratic conditions, while in ^[12], it was treated with superquadratic conditions. However, Xie, Li and Luo in ^[14] studied a continuous SOHS with the help of the linking theorem.

Zhao, Yang and Chen considered an asymptotically linear case for the SOHS in ^[15], that is, $F(t, x) = W(t, x)-K(t, x)$, where $W$ satisfies the asymptotically linear condition at infinity, and $K$ satisfies the coupling condition. Chen, Guo and Liu ^[16] demonstrated continuous HS with asymptotically linear terms, which further extended the previous results under coupling conditions.

HS with asymptotically linear terms have been extensively studied in the continuous case, e.g., ^[17,18,19], whereas few results have been obtained in the discrete case. Inspired by the above literature, we will discuss a non-autonomous SOHS with asymptotically linear terms in discrete cases. The difference with ^[16] is that we construct a new workspace to estimate the minimax level associated with the energy functional.

We first write the nonlinear term $F(n, x)$ in (1.1) in the form $F(n, x) = -G(n, x)+H(n, x)$ with the following conditions:

$(G_1):$ for any $(n, x)\in{\mathbb Z[1, T]}\times\mathbb {R}^{N}$, there exist $b > 0$ and $g_{1}(n)\in \mathbb {R}$ satisfying

$(G_2):$ for any $n\in{\mathbb Z[1, T]}$, there exists $K_{1} > 0$ such that

$(H_1):$ for any $(n, x)\in{\mathbb Z[1, T]}\times\mathbb {R}^{N}$, there exist $g_{2}(n)\in{\mathbb R}$ and

such that

$(H_2):$ $(\nabla{H(n, x)}, x)-2H(n, x)\rightarrow {+\infty}$ uniformly for $n\in{\mathbb Z[1, T]}$, when $|x|\rightarrow {+\infty}$;

$(W_1):$ for all $n\in{\mathbb Z[1, T]}$, there exists $K_{2} > 0$, such that

$(W_2):$ for any ${n\in{\mathbb Z[1, T]}}$, there exists a constant $K_{3} > 0$, such that

Theorem 1.1. If $(G_1), (G_2), (H_1), (H_2), (W_1)$ and $(W_2)$ hold, then system $(1.1)$ has a nontrivial T-periodic solution.

2.   Preliminaries

Define the Hilbert space

with its inner product

and the corresponding norm

The corresponding functional of the equation is

which is continuously differentiable. So,

Clearly, the critical point of functional $(2.1)$ is the T-periodic solution of problem $(1.1)$.

Proposition 2.1. ^[16] Suppose that $H(n, x)$ satisfies $(H_2)$, $G(n, x)$ satisfies $(G_2)$, for any $n\in{\mathbb Z[1, T]}$. Then, there is a sufficiently large $M > 0$, such that

Recall that $(PS)$ condition in ^[8], a sequence ${\{u_{p}\}\subset{H_{T}}}$ has a convergent sequence when $\varphi{(u_{p})}$ is bounded, and $\|\varphi^{'}(u_{p})\|\rightarrow {0}, p\rightarrow {+\infty}$. Similarly, we can see $(C)$ condition in ^[16], if $\varphi{(u_{p})}$ is bounded, and $\|\varphi^{'}(u_{p})\|(1+\|u_{p}\|)\rightarrow {0}, p\rightarrow {+\infty},$ then the sequence ${\{u_{p}\}\subset{H_{T}}}$ has a convergent sequence.

Lemma 2.2. If $G$ satisfies $(G_1)$ and $(G_2)$, $H$ satisfies $(H_1)$ and $(H_2)$, then the functional $\varphi$ fulfills the $(C)$ condition.

Proof. Assume that $\{u_p\}\subset{H_T}$ is a $(C)$ sequence, i.e.,

In addition, there exists $M_1 > 0$ sufficiently large, such that

We claim that $\{u_p\}$ is bounded. Conversely, we obtain $\|u_{p_k}\| \rightarrow {+\infty}$, as $k \rightarrow {+\infty}$. We replace $\{u_{p_k}\}$ with $\{u_p\}$.

Let $s_p = u_p/{\|u_p\|}$, and obviously, we know that $\|s_p\| = 1$. There exists $s\in{H_T}$, such that $s_p\rightarrow {s}$ in $H_T$.

As

we have $\|s\| = \|s_p\| = 1$. Therefore, $s\not\equiv{0}.$

As $s\not\equiv{0}$, we set

From the previously described results, we obtain

When $p\in{\mathbb{N^*}}, \lambda > \max\{K_1, M\}$, set

by $(G_2)$, we have

We set

By $(H_2)$, we obtain

Combining $(H_2)$ with $(2.5)$, we obtain

As $|\varphi(u_p)|\leq{M_1}$, one has

Therefore, we have

Combining (2.1) and (2.2), we have

To be convenient, we denote

by $I_1$ and

by $I_2$. By (2.6), (2.7) and (2.8), we see that

By (2.10)-(2.12), we have

which contradicts the boundedness of $M_1$.

Therefore $\{u_p\}$ is bounded in $H_T$, and then we have $u_p\rightarrow {u}$ in $H_T$, which demonstrates that $\varphi$ fulfills $(C)$ condition. □

3.   Proof of Theorem 1.1

Set

Let

and therefore

Just as in ^[7], we have

Next, we will prove Theorem 1.1 in two steps.

Step 1 Set $\mathcal{W} = X\bigoplus{Y}$ as $\mathcal{H} = V\bigoplus{X}$ in [16, Lemma 4], where $X = \mathbb{R}^N$ and $Y = \widetilde{H}_T$. As is discussed in ^[16], $(PS)$ condition can be replaced by $(C)$ condition, and [16, Lemma 4] holds under $(C)$ condition. Now, we will prove that (i) holds in [16, Lemma 4]. By $(W_1)$ and Proposition 2.1, let $m_0\in{\mathbb{R}^N}$, and $|m_0| = 1$. If $z\geq\max\{M, K_2\}$, we obtain

There exist $r > 0$ large enough and a constant

such that

Step 2 We will show that (ii) holds in [16, Lemma 4]. By $(G_1), (H_1),$ and (3.1), for all $u\in\widetilde{H}_{T}$, we get

Let constant

and then $\varphi|_Y\geq\beta$. From [16, Lemma 4], we know that $\varphi$ has a critical value

Therefore, there is a $u\in{H_T}$ such that

Finally, we can get our conclusion according to $(W_2)$.

4.   Example

We provide an example to demonstrate the rationality of Theorem 1.1.

Example 1 Suppose $T = 2$ and set

with

where

Let

where $n\in\mathbb Z[1, T]$ and $x\in\mathbb R^{N}$.

First of all, set

Then, we have

and

Then, $G$ satisfies $(G_1)$ and $(G_2)$ conditions.

In the same way, let

Then, we obtain

and

Obviously, $H(n, x)$ satisfies $(H_1)$ and$(H_2)$ conditions. Also,

for any $a\in \mathbb R$ and $n\in\mathbb Z[1, T]$, so that $(W_1)$ holds. In addition,

Therefore,

which means that $(W_2)$ holds.

According to Theorem 1.1, we can get the existence of a nontrivial 2-periodic solution for system (1.1).

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Chungen Liu is supported by the National Natural Science Foundation of China (Grant No.11790271, 12171108), Guangdong Basic and Applied basic Research Foundation (Grant No. 2020A1515011019), Innovation and development project of Guangzhou University. Jiabin Zuo is supported by the Guangdong Basic and Applied Basic Research Foundation (2022A1515110907) and the Project funded by China Postdoctoral Science Foundation (2023M730767).

Conflict of interest

The authors declare there is no conflict of interest.