## 1 Introduction and main results

Consider the following discrete Hamiltonian system:

$$\bigtriangleup^{2}u(t-1)+\nabla F\bigl(t,u(t)\bigr)=0,\quad t\in \mathbb{Z},$$
(1.1)

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:

1. (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 [116] 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:

1. (i)

$$F(t,x)$$ is $$T_{i}$$-periodic in $$x_{i}$$, $$1\leq i\leq r$$.

2. (ii)
$$|x|^{-2\alpha}\sum_{t=1}^{T}F(t,x) \rightarrow\pm\infty\quad \text{as } |x|\rightarrow\infty, x\in\{0\}\times \mathbb{R}^{N-r}.$$

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

(H1):

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

(H2):

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 T-periodic solution.

### Corollary 1.1

Let $$F(t,x)=-a\cos x-e(t)x$$. If $$e(t)$$ satisfies

$$e(t+T)=e(t),\qquad \sum_{t=1}^{T}e(t)=0,$$

then system (1.1) has at least one T-periodic solution.

### Remark 1.1

When $$F(t,x)=-a\cos x-e(t)x$$ ($$a\geq0$$), system (1.1) is a discrete form of forced equations studied by Mawhin and Willem [1820], 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 (H1) hold and

(H3):

there exist $$\mu_{1}<2$$ and $$\mu_{2}\in\mathbb{R}$$ such that

$$\bigl(\nabla F(t,x), x\bigr)\leq\mu_{1}F(t,x)+\mu_{2};$$
(H4):

there exists $$\delta>0$$ such that, for $$t\in\mathbb{Z}$$, we have

$$F(t,x)>\delta,\quad |x|\rightarrow+\infty;$$
(H5):

there exists $$0< b<2\sin^{2}\frac{\pi}{T}$$ such that

$$F(t,x)\leq b|x|^{2}.$$

Then system (1.1) has at least one T-periodic solution. Furthermore, system (1.1) has at least one nonconstant T-periodic solution if $$\sum_{t=1}^{T}F(t,x)\geq0$$ for all $$x\in\mathbb{R}^{N}$$.

## 2 Some important lemmas

Let

$$H_{T}=\bigl\{ u:\mathbb{Z}\rightarrow\mathbb{R}^{N}\mid u(t)=u(t+T) \text{ for all } t\in\mathbb{Z}\bigr\}$$

with norm

$$\|u\|= \Biggl(\sum_{t=1}^{T}\bigl\vert \bigtriangleup u(t)\bigr\vert ^{2} \Biggr)^{\frac {1}{2}}+\Biggl\vert \sum_{t=1}^{T}u(t)\Biggr\vert .$$

Set

$$\Phi(u)=\frac{1}{2}\sum_{t=1}^{T} \bigl\vert \bigtriangleup u(t)\bigr\vert ^{2}-\sum _{t=1}^{T}F\bigl(t,u(t)\bigr)$$

and

$$\bigl\langle \Phi'(u), v\bigr\rangle =\sum _{t=1}^{T}\bigl(\bigtriangleup u(t),\bigtriangleup v(t)\bigr)-\sum_{t=1}^{T}\bigl(\nabla F \bigl(t,u(t)\bigr),v(t)\bigr)$$

for $$u,v\in H_{T}$$.

According to assumption (A), it is well known that Φ is continuously differentiable and the T-periodic 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

$$f(u_{n})\rightarrow C, \qquad \bigl(1+\Vert u_{n}\Vert \bigr)f'(u_{n})\rightarrow0,$$

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

[19, 22]

Assume that X is a Banach space and $$f\in C^{1}(X, \mathbb{R})$$. Let $$X=X_{1}\oplus X_{2}$$ and

$$\dim X_{1}< +\infty,\qquad \sup_{S^{1}_{R}} f< \inf _{X_{2}}f,$$

where $$S^{1}_{R}=\{u\in X_{1}\mid |u|=R\}$$.

Set $$B^{1}_{R}=\{u\in X_{1}, |u|\leq R\}$$, $$M=\{g\in C(B^{1}_{R},X)\mid g(s)=s, s\in S^{1}_{R}\}$$,

$$C=\inf_{g\in M}\max_{s\in B^{1}_{R}}f\bigl(g(s)\bigr).$$

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

$$\sum_{t=1}^{T}\bigl\vert u(t)\bigr\vert ^{2}\leq\frac{1}{4\sin^{2}\frac{\pi}{T}}\sum_{t=1}^{T} \bigl\vert \bigtriangleup u(t)\bigr\vert ^{2}$$

and

$$\|u\|^{2}_{\infty}:= \Bigl(\max_{t\in\mathbb{Z}[1,T]}\bigl\vert u(t)\bigr\vert \Bigr)^{2}\leq\frac{T^{2}-1}{6T}\sum _{t=1}^{T}\bigl\vert \bigtriangleup u(t)\bigr\vert ^{2}.$$

## 3 Proof of main results

### Proof of Theorem 1.1

Let

$$H_{T}=\mathbb{R}^{N}\oplus\widetilde{H}_{T},$$

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 (H2), we have that

\begin{aligned} \Phi(\tilde{u})&=\frac{1}{2}\sum_{t=1}^{T} \bigl\vert \bigtriangleup{\tilde {u}}(t)\bigr\vert ^{2}-\sum _{t=1}^{T}F\bigl(t,\tilde{u}(t)\bigr) \\ &\geq \frac{1}{2}\sum_{t=1}^{T}\bigl\vert \bigtriangleup{\tilde {u}}(t)\bigr\vert ^{2}-C_{1} \sum_{t=1}^{T}\bigl\vert \tilde{u}(t)\bigr\vert ^{2}-TC_{2} \\ &\geq\frac{1}{2}\sum_{t=1}^{T}\bigl\vert \bigtriangleup{\tilde {u}}(t)\bigr\vert ^{2}-\frac{C_{1}}{4\sin^{2}\frac{\pi}{T}} \sum_{t=1}^{T}\bigl\vert \bigtriangleup{ \tilde{u}}(t)\bigr\vert ^{2}-TC_{2} \\ &= \biggl(\frac{1}{2}-\frac{C_{1}}{4\sin^{2}\frac{\pi}{T}} \biggr)\sum _{t=1}^{T}\bigl\vert \bigtriangleup{\tilde{u}}(t) \bigr\vert ^{2}-TC_{2}. \end{aligned}

So,

$$\Phi(\tilde{u})\rightarrow+\infty\quad \text{as } \|\tilde{u}\| \rightarrow \infty.$$
(3.1)

Suppose that $$\{u_{k}\}$$ is a minimizing sequence for Φ, that is,

$$\Phi(u_{k})\rightarrow\inf\Phi,\quad k \rightarrow\infty.$$

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

$$\|\tilde{u}_{k}\|\leq c.$$
(3.2)

By (H1) we have that

$$\Phi(u+T_{i}e_{i})=\Phi(u),\quad u\in{H}_{T}, 1 \leq i\leq N.$$

Hence, if $$\{u_{k}\}$$ is a minimizing sequence for Φ, then

$$(\tilde{u}_{k}\cdot e_{1}+\overline{u}_{k}\cdot e_{1}+k_{1}T_{1}, \ldots, \tilde{u}_{k} \cdot e_{N}+\overline{u}_{k}\cdot e_{N}+k_{N}T_{N})$$

is also a minimizing sequence of Φ.

Therefore, we can assume that

$$0\leq\overline{u}_{k}\cdot e_{i}\leq T_{i},\quad 1\leq i\leq N.$$
(3.3)

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

$$\Phi(u_{0})=\inf\Phi(u),\qquad \Phi'(u_{0})=0.$$

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

$$\Phi(u_{k})\rightarrow C, \qquad \bigl(1+\Vert u_{k} \Vert \bigr)\Phi'(u_{k})\rightarrow0.$$

Since $$\Phi(u_{k})\rightarrow C$$, we have that

$$\frac{1}{2}\sum_{t=1}^{T}\bigl\vert \bigtriangleup{u}_{k}(t)\bigr\vert ^{2}-\sum _{t=1}^{T}F\bigl(t,u_{k}(t)\bigr) \rightarrow C.$$
(3.4)

From (H3) we have that

\begin{aligned} \bigl\langle \Phi'({u}_{k}), {u}_{k}\bigr\rangle &=\sum_{t=1}^{T}\bigl\vert \bigtriangleup {u}_{k}(t)\bigr\vert ^{2}-\sum _{t=1}^{T}\bigl(\nabla F\bigl(t,{u}_{k}(t) \bigr),{u}_{k}(t)\bigr) \\ &\geq \sum_{t=1}^{T}\bigl\vert \bigtriangleup{u}_{k}(t)\bigr\vert ^{2}-\mu_{1} \sum_{t=1}^{T}F\bigl(t,{u}_{k}(t) \bigr)-\mu_{2}T. \end{aligned}

By (3.4) we have that

$$-\sum_{t=1}^{T}F\bigl(t,{u}_{k}(t) \bigr)=C-\frac{1}{2}\sum_{t=1}^{T}\bigl\vert \bigtriangleup{u}_{k}(t)\bigr\vert ^{2}+ \varepsilon.$$

So, we have

\begin{aligned} \bigl\langle \Phi'({u}_{k}), {u}_{k}\bigr\rangle &=\sum_{t=1}^{T}\bigl\vert \bigtriangleup {u}_{k}(t)\bigr\vert ^{2}-\sum _{t=1}^{T}\bigl(\nabla F\bigl(t,{u}_{k}(t) \bigr),{u}_{k}(t)\bigr) \\ &\geq \biggl(1-\frac{\mu_{1}}{2} \biggr)\sum_{t=1}^{T} \bigl\vert \bigtriangleup {u}_{k}(t)\bigr\vert ^{2}+C \mu_{1}-\mu_{2}T+\varepsilon. \end{aligned}

Therefore,

$$\biggl(1-\frac{\mu_{1}}{2} \biggr)\sum_{t=1}^{T} \bigl\vert \bigtriangleup {u}_{k}(t)\bigr\vert ^{2}+C \mu_{1}-\mu_{2}T\leq0.$$

From this inequality we have that $$\sum_{t=1}^{T}|\bigtriangleup {u}_{k}(t)|^{2}$$ is bounded.

By (H1) we have that

$$\Phi(u+T_{i}e_{i})=\Phi(u),\quad u\in{H}_{T}, 1 \leq i\leq N.$$

Therefore, if $$\{u_{k}\}$$ is a $$(\mathit{CPS})_{C}$$ sequence of Φ, then

$$(\tilde{u}_{k}\cdot e_{1}+\overline{u}_{k}\cdot e_{1}+k_{1}T_{1}, \ldots, \tilde{u}_{k} \cdot e_{N}+\overline{u}_{k}\cdot e_{N}+k_{N}T_{N})$$

is also a $$(\mathit{CPS})_{C}$$ sequence of Φ.

So, we can assume that

$$0\leq\overline{u}_{k}\cdot e_{i}\leq T_{i},\quad 1\leq i\leq N,$$

that is, $$|\overline{u}_{k}|$$ is bounded.

From these results we have that $$\{u_{k}\}$$ is bounded.

Since $${H}_{T}$$ is a finite-dimensional 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

$$X_{1}=\mathbb{R}^{N}, \qquad X_{2}=\Biggl\{ u \in{H}_{T}:\sum_{t=1}^{T}u(t)=0 \Biggr\} .$$

For any $$u\in X_{2}$$, by (H5) and Lemma 2.2 we have that

\begin{aligned} \Phi(u)&=\frac{1}{2}\sum_{t=1}^{T} \bigl\vert \bigtriangleup u(t)\bigr\vert ^{2}-\sum _{t=1}^{T}F\bigl(t,u(t)\bigr) \\ &\geq \frac{1}{2}\sum_{t=1}^{T}\bigl\vert \bigtriangleup u(t)\bigr\vert ^{2}-b\sum _{t=1}^{T}\bigl\vert u(t)\bigr\vert ^{2} \\ &\geq \frac{1}{2}\sum_{t=1}^{T}\bigl\vert \bigtriangleup u(t)\bigr\vert ^{2}-\frac{b}{4\sin ^{2}\frac{\pi}{T}}\sum _{t=1}^{T}\bigl\vert \bigtriangleup u(t) \bigr\vert ^{2} \\ &= \biggl(\frac{1}{2}- \frac{b}{4\sin^{2}\frac{\pi}{T}} \biggr)\sum _{t=1}^{T}\bigl\vert \bigtriangleup u(t)\bigr\vert ^{2} \\ &\geq 0. \end{aligned}

On the other hand, for any $$u\in X_{1}$$, by (H4) we have that

$$\Phi(u)=-\sum_{t=1}^{T}F\bigl(t,u(t)\bigr) \leq-\delta, \quad |u|\rightarrow+\infty.$$

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 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. □