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Periodic solutions for second-order difference equations with quadratic–supquadratic condition

Abstract

In this paper, we consider the existence of multiple periodic solutions for a class of second-order difference equations with quadratic–supquadratic growth condition at infinity. Moreover, we give three examples to illustrate our main result.

Introduction and main result

Discrete equations have been widely employed as mathematical models depicting the nature phenomena in many practical problems including computer sciences, life sciences, mathematical biology, and so on; see [15]. Among these discrete equations, discrete nonlinear Schrödinger (DNLS) equations are very important nonlinear lattice models in the nonlinear science, ranging from condensed matter physics to biology [611]. Let N, Z, and R stand for the sets of all natural numbers, integers, and real numbers, respectively. For c,dZ with \(c\leq d\), we define Z[c]={c,c+1,} and Z[c,d]={c,c+1,,d}. It is well known that by the standing wave assumptions DNLS equations can change into the following nonlinear second-order difference equation [1218]:

{ Δ 2 x n 1 + f ( n , x n ) = 0 , n Z , f ( n , x n ) = x n F ( n , x n ) , n Z ,
(1.1)

where \(\Delta x_{n-1}=x_{n}-x_{n-1}\), \(\Delta ^{2}=\Delta (\Delta )\), F C 1 (R×R), and \(F(n+M,\cdot )=F(n,\cdot )\) for some MN.

As is known, the critical-point theory is an important tool when dealing with the existence of solutions of differential equations (see [19]), and for discrete system (1.1), there are some results on the existence of periodic solutions in the last few years: especially, for \(F(n,X)\) with supquadratic growth condition with respect to X at infinity,

$$\begin{aligned} \lim_{ \vert X \vert \rightarrow +\infty }\frac{F(n,X)}{ \vert X \vert ^{2}}=+\infty , \end{aligned}$$
(1.2)

Guo and Yu [18] developed a new approach to obtain the existence and multiplicity of periodic solutions to discrete system (1.1). Later, for \(F(n,X)\) with subquadratic growth condition with respect to X at infinity, Guo and Yu [20] proved the existence of nontrivial periodic solutions. For the case of \(F(n,X)\) with quadratic–supquadratic growth condition in X at infinity,

$$ \liminf_{ \vert X \vert \rightarrow +\infty } \frac{ \vert F(n,X) \vert }{ \vert X \vert ^{2}}=\kappa >0, $$

in 2004, under the assumption that κ depends on M (especially, \(\kappa (M)>2\) for even M), Zhou, Yu, and Guo [21] improved the Guo–Yu method of [18] and obtained the existence of two nontrivial M-periodic solutions for discrete system (1.1); for more details on the existence of multiple nontrivial M-periodic solutions with quadratic–supquadratic condition, we refer to [22, 23]. Moreover, the existence of one nontrivial solution for general nonlinear difference equations, that is, discrete ϕ-Laplacian equations with quadratic–supquadratic condition, is considered in [13, 15, 24]. For other related works, we refer to [14, 16, 17, 2531].

Note that in [21], for F with quadratic–supquadratic condition with respect to X at infinity, by introducing the smallest and largest eigenvalues of the matrix

$$\begin{aligned} \begin{pmatrix} 2 & -1 & 0 & \cdots & 0 & -1 \\ -1 & 2 & -1 & \cdots & 0 & 0 \\ 0 & -1 & 2 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ -1 & 0 & 0 & \cdots & -1 & 2 \end{pmatrix}_{M \times M}, \end{aligned}$$
(1.3)

that is,

$$\begin{aligned} \lambda _{\min } =2 \biggl(1-\cos \frac{2\pi }{M} \biggr), \quad\quad \lambda _{\max } = \textstyle\begin{cases} 4 &\text{when } M \text{ is even}, \\ 2(1+\cos {\frac{\pi }{M}}) & \text{when } M \text{ is odd,} \end{cases}\displaystyle \end{aligned}$$
(1.4)

Zhou, Yu, and Guo obtained the existence of two nontrivial M-periodic solutions for system (1.1). Four years later, Xue and Tang obtained the following more general result for F with quadratic–supquadratic condition with respect to X at infinity.

Theorem 1.1

([23, Theorem 2])

Suppose that \(F(n,X)\) satisfies

\((F_{1})\):

There are constants \(\delta >0\) and k[0,[ M 2 ]1]Z such that for any \(\vert X \vert \leq \delta \) and tZ[1,M],

$$\begin{aligned} \frac{1}{2}\mu _{k} \vert X \vert ^{2}\leq F(n,X)\leq \frac{1}{2}\mu _{k+1} \vert X \vert ^{2}, \end{aligned}$$

where \(\mu _{k}=2-2\cos kw\), \(w=\frac{2\pi }{M}\), \(M>2\), and \([\cdot ]\) denotes the Gauss function;

\((F_{2})\):

For tZ[1,M], there exists a constant \(\beta \in (\frac{\lambda _{\max }}{2}, +\infty )\) such that

$$\begin{aligned} \lim_{ \vert X \vert \rightarrow +\infty }\inf \frac{F(n,X)}{ \vert X \vert ^{2}}\geq \beta . \end{aligned}$$
Then system (1.1) has at least two nontrivial M-periodic solutions.

In condition \((F_{2})\), \(\lambda _{\max }\) is dependent on M; when M is even, \(\beta >2\), and when \(M\geq 3\) is odd, \(\beta \geq 1+\cos {\pi /M}\geq 3/2\). Therefore the constant β is at least greater than \(3/2\). Moreover, in condition \((F_{1})\), if \(k=0\), then when \(M=4\), \(0\leq F(n,X)\leq 1-\cos (2\pi /4)=1\), and when \(M>4\), \(0\leq F(n,X)\leq 1-\cos (2\pi /M)<1\). Clearly, the range of parameter values in \((F_{1})\) and \((F_{2})\) will play a critical role in discrete model (1.1) when proving the existence of periodic solutions. However, all parameters in Theorem 1.1 are limited. In this paper, we want to establish an existence result for periodic solutions without this limitation. The main result of this paper is the following:

Theorem 1.2

Let \(M\geq 4\), and let \(F(n,X)\) satisfy the following conditions:

\((F_{1}^{\prime })\):

There exist constants \(\delta >0\) and \(0<\alpha <1\) such that

0F(n,X)α|X | 2 for nN,XR and |X|δ.
\((F_{2}^{\prime })\):

For tZ[1,M], there exists a constant \(\beta \in (1, +\infty )\) such that

$$\begin{aligned} \lim_{ \vert X \vert \rightarrow +\infty }\inf \frac{F(n,X)}{ \vert X \vert ^{2}}\geq \beta . \end{aligned}$$
Then system (1.1) has at least two nontrivial M-periodic solutions.

Remark 1.1

(i) Since \(M\geq 4\), the parameter β in Theorem 1.1 can only take values in \((1+\cos (\pi /5),+\infty )\), but it can take any value in \((1,+\infty )\) in Theorem 1.2. Moreover, if \(M>4\) and \(k=0\), then the parameter α in Theorem 1.1 can only take values in \((0,1-\cos (2\pi /M))\), but in the present paper, we prove that this parameter can take any value in \((0,1)\). In this sense, we extend the ranges of parameters.

(ii) In Theorem 1.1, if \(k\neq 0\), then \(F(n,X)=O( \vert X \vert ^{2})\) as \(\vert X \vert \rightarrow 0\), but in Theorem 1.2, as \(\vert X \vert \rightarrow 0\), both \(F(n,X)=O( \vert X \vert ^{2})\) and \(F(n,X)=o( \vert X \vert ^{2})\) are admissible.

Remark 1.2

(i) In 2020, under the assumptions that \(M\geq 5\) and \(F(n,X)\rightarrow -\infty \) as \(\vert X \vert \rightarrow +\infty \), by using an extended mountain pass theorem, we obtained the existence of two nontrivial M-periodic solutions for quadratic–supquadratic vector field \(F(n, X)\) (see [22, Theorem 1.1]) in X at infinity. Different from the method in [22, Theorem 1.1], now by constructing a new functional \(J_{1}(x)\), two new orthogonal direct sum decompositions, and Linking theorem [19, Theorem 5.3], under the assumptions that \(M\geq 4\) and \(F(n,X)\rightarrow +\infty \) as \(\vert X \vert \rightarrow +\infty \), we also obtain the existence of two nontrivial M-periodic solutions for quadratic–supquadratic condition in X at infinity.

(ii) The method improved here may be applied to the general difference equations [13, 15, 21, 24, 28, 29], and under general quadratic–supquadratic growth conditions at infinity, we may also obtain the existence of multiple periodic solutions.

Now we give three examples to explain Theorem 1.2. First, we give an example for completely quadratic condition with respect to X at infinity.

Example 1

Let F be given by

$$\begin{aligned} F({n},X)=a_{1} \vert X \vert ^{2} \bigl(\phi ({n})+D \bigr), \end{aligned}$$

where \(a_{1}\) is an arbitrary constant that belongs to \((0, 1)\), a constant \(D>0\), and \(\phi ({n})\) is a continuous M-periodic function satisfying \(\vert \phi ({n}) \vert < D\) for every n. Now \(F({n}, X)\) satisfies all assumptions in Theorem 1.2. Thus (1.1) has at least two nontrivial M-periodic solutions.

Second, we give an example for quadratic–supquadratic condition with respect to X at infinity.

Example 2

Let F be given by

$$\begin{aligned} F({n},X)= \bigl(a_{4} \vert X \vert ^{2}-a_{5} \vert X \vert ^{4}+a_{6} \vert X \vert ^{6} \bigr) \bigl(\phi ({ n})+D \bigr), \end{aligned}$$

where \(a_{4}\), \(a_{5}\), \(a_{6}\) are arbitrary constants that belong to \((0, 1)\), a constant \(D>0\), and \(\phi ({n})\) is as in Example 1. Then \(F({n},X)\) satisfies all assumptions in Theorem 1.2. Thus (1.1) has at least two nontrivial M-periodic solutions.

Remark 1.3

Since the constants \(a_{1}\) and \(a_{4}\) in Examples 1 and 2 are arbitrary, these examples cannot be solved by Theorem 1.1.

Finally, we give an example for completely supquadratic condition with respect to X at infinity.

Example 3

Let F be given by

$$\begin{aligned} F({n},X)= \bigl(a_{2} \vert X \vert ^{4}+a_{3} \vert X \vert ^{6} \bigr) \bigl(\phi ({n})+D \bigr), \end{aligned}$$

where \(a_{2}\) and \(a_{3}\) are arbitrary constants that belong to \((0, 1)\), a constant \(D>0\), and \(\phi ({n})\) is as in Example 1. Then \(F({n},X)\) satsifies all assumptions in Theorem 1.2. Thus (1.1) has at least two nontrivial M-periodic solutions.

Some useful lemmas

To use the critical point theory to study the existence of periodic solutions to (1.1), we introduce some notions and notations.

  • Let S be the set of sequences, that is, S={s={ s j }= ( , s j , , s 0 , , s j , ), s j R,jZ}. When \(x,y \in S\) and a,bR, \(ax+by\) is given by \(ax+by = \{ax_{j}+by_{j}\}_{j=-\infty }^{+\infty }\). For any given positive integer M, \(E_{M}\) is a subspace of S defined as

    E M = { x = { x j } S x j + M = x j , j Z } .

    Then, with the common Euclid inner product (\(\Vert x \Vert =(\sum_{j=1}^{M}(x_{j})^{2})^{ \frac{1}{2}}\)), \(E_{M}\) is an M-dimensional Hilbert space. Let

    $$\begin{aligned} \Vert x \Vert _{\alpha } = \Biggl(\sum_{k=1}^{M} \vert x_{k} \vert ^{\alpha } \Biggr)^{ \frac{1}{\alpha }}, \quad \alpha \in (1,\infty ). \end{aligned}$$

    Then

    $$\begin{aligned} \frac{1}{M} \Vert x \Vert _{4} \leq \Vert x \Vert \leq M \Vert x \Vert _{4}, \quad\quad \frac{1}{M} \Vert x \Vert _{\frac{3}{2}} \leq \Vert x \Vert \leq M \Vert x \Vert _{\frac{3}{2}}, \quad \forall x\in E_{M}. \end{aligned}$$
  • Let H be a real Hilbert space. \(J \in {\mathcal{C}}^{1}(H)\) is said to satisfy the PS condition if any sequence \(\{x^{(j)}\} \subset H\) for which \(\{J(x^{(j)})\}\) is bounded and \(J'(x^{(j)}) \rightarrow 0\) as \(j\rightarrow \infty \) possesses a convergent subsequence in H.

  • Different from the known literature used to study the existence of periodic solutions for discrete system (1.1), the result of this paper is not related to the complicated smallest and largest eigenvalues of matrix (1.3), that is, \(\lambda _{\min }\) and \(\lambda _{\max }\) in (1.4), and now for any \(x\in E_{M}\) and Δx= ( Δ x 1 , Δ x 2 , , Δ x M ) R M , we let

    $$ \sum_{j=1}^{M}(\Delta x_{j})^{2}+2 \Delta x_{n-1} \Delta x_{n}=( \Delta x_{1}, \Delta x_{2},\ldots, \Delta x_{M})^{\top }P(\Delta x_{1}, \Delta x_{2},\ldots, \Delta x_{M}), $$

    where

    $$\begin{aligned} P= \begin{pmatrix} 1 & 0 & \cdots & \cdots &\cdots & \cdots & \cdots &\cdots &\cdots &0 \\ 0 & 1 & 0 & \cdots & \cdots & \cdots &\cdots &\cdots &\cdots &0 \\ \vdots & \ddots & \ddots & \ddots & \cdots & \cdots &\cdots &\cdots &\cdots &\vdots \\ \vdots & \cdots & \ddots & \ddots & 0 & 0 & \cdots &\cdots &\cdots &\vdots \\ \vdots & \cdots & \cdots & 0 & 1 & 1 & 0&\cdots &\cdots &\vdots \\ \vdots & \cdots & \cdots & 0 & 1 & 1 & 0&\cdots &\cdots &\vdots \\ \vdots & \cdots & \cdots & \vdots & 0 & 0 & \ddots &0&\cdots &\vdots \\ \vdots & \cdots & \cdots & \vdots & \vdots & \vdots & \ddots &\ddots &\ddots &\vdots \\ 0 & \cdots & \cdots & \cdots &\cdots & \cdots & \cdots &0&1&0 \\ 0 & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots &\cdots &0&1 \end{pmatrix}_{M \times M}. \end{aligned}$$

Remark 2.1

Obviously, the matrix P is different from (1.3), and it is easy to get that the eigenvalues of P are \(\underbrace{1,\ldots,1}_{M-2}\), 0, 2. Besides, we compute that the matrix P has M linearly independent eigenvectors.

  • To obtain the general existence of periodic solutions for second-order difference equation (1.1) with new quadratic–supquadratic condition, we also let H 1 ={ ( x 1 , x 2 , , x M ) E M Δ x 1 ==Δ x n 2 =Δ x n + 1 ==Δ x M =0,Δ x n 1 =Δ x n =wR,M4}, and \(H_{2}=H_{1}^{\bot }\). Then we construct the first new orthogonal direct sum decomposition, \(E_{M} = H_{1} \oplus H_{2}\).

Remark 2.2

Most literatures used the method in [18, 21] and \(H_{1}\) is trivial, that is, H 1 ={ ( v , v , , v ) M vR}, but in this paper, when \(M\geq 4\), \(H_{1}\) is nontrivial, and { ( v , v , , v ) M vR} H 1 , but { ( v , v , , v ) M vR} H 1 .

Let \(M\geq 4\) be a integer, and let nZ[1,M]. To prove Theorem 1.2, we define the new functional \(J_{1}(x)\), related to the matrix P, on \(E_{M}\) as follows:

$$\begin{aligned} J_{1}(x) =&\frac{1}{2} \Vert \Delta x \Vert ^{2}-\sum_{j=1}^{M}F(j, x_{j})-M^{4} { \biggl[}\sum_{j\neq n} \vert x_{j} \vert ^{4}+\frac{1}{16} \vert x_{n+1}+x_{n-1} \vert ^{4}{ \biggr]} \\ & {} + M^{-\frac{3}{2}}{ \biggl[}\sum_{j\neq n} \vert x_{j} \vert ^{\frac{3}{2}}+ \frac{1}{2\sqrt{2}} \vert x_{n+1}+x_{n-1} \vert ^{\frac{3}{2}}{ \biggr]}- \frac{1}{2}(\Delta x)^{\top }P(\Delta x). \end{aligned}$$
(2.1)

Remark 2.3

We have the following identity:

$$\begin{aligned} \frac{\partial [(\Delta x)^{\top }P(\Delta x)]}{\partial x_{n}} =& \frac{\partial { [}\sum_{j\neq n} \vert x_{j} \vert ^{4}+\frac{1}{16} \vert x_{n+1} + x_{n-1} \vert ^{4}{ ]}}{\partial x_{n}} \\ =& \frac{\partial { [} \sum_{j\neq n} \vert x_{j} \vert ^{\frac{3}{2}} + \frac{1}{2\sqrt{2}} \vert x_{n+1} + x_{n-1} \vert ^{\frac{3}{2}}{ ]}}{\partial x_{n}}=0. \end{aligned}$$

Clearly, \(J_{1}\in \mathcal{C}^{1} (E_{M})\). For any x= { x j } j Z E M , according to Remark 2.3, we compute that

J 1 x n = [ Δ 2 x n 1 + f ( n , x n ) ] ,nZ[1,M].

So, the existence of critical points of \(J_{1}\) on \(E_{M}\) may imply the existence of periodic solutions of system (1.1).

We now give some useful lemmas, which will serve us later.

Lemma 2.1

Let \((F_{2}^{\prime })\) be valid. Then there exists a constant γ R such that \(F(n,X)\geq \beta \vert X \vert ^{2}-\gamma ^{\prime }\) for nZ, XR.

Proof

By \((F_{2}^{\prime })\), for \(\beta >1\), there exist constants \(\rho >0\) and \(\gamma >0\) such that

F(n,X)β|X | 2 γfor nZ,|X|ρ.

Then letting γ 1 =max{|F(n,X)β|X | 2 +γ|:nZ,|X|ρ} and \(\gamma ^{\prime }=\max \{\gamma , \gamma _{1}+\gamma \}\), we obtain

F(n,X)β|X | 2 γ for nZ,XR.

The proof is complete. □

Lemma 2.2

Let \((F_{2}^{\prime })\) be in force. Then \(J_{1}(x)\) is bounded from above on \(E_{M}\).

Proof

Let \(J_{1}\) be given by (2.1). By Lemma 2.1, for all \(x \in E_{M}\), we have

$$\begin{aligned} J_{1}(x) \leq & \frac{1}{2} \Vert \Delta x \Vert ^{2}-\sum_{j=1}^{M} \bigl( \beta (x_{j})^{2}- \gamma ^{\prime } \bigr)-M^{4}{ \biggl[}\sum_{j\neq n} \vert x_{j} \vert ^{4}+ \frac{1}{16} \vert x_{n+1}+x_{n-1} \vert ^{4}{ \biggr]} \\ & {} + M^{-\frac{3}{2}}{ \biggl[} \sum_{j\neq n} \vert x_{j} \vert ^{\frac{3}{2}}+ \frac{1}{2\sqrt{2}} \vert x_{n+1}+x_{n-1} \vert ^{\frac{3}{2}}{ \biggr]} - \frac{1}{2}(\Delta x)^{\top }P(\Delta x) \\ =&\frac{1}{2} \Vert \Delta x \Vert ^{2}-\beta \Vert x \Vert ^{2}-M^{4}{ \biggl[}\sum_{j \neq n} \vert x_{j} \vert ^{4}+\frac{1}{16} \vert x_{n+1}+x_{n-1} \vert ^{4}{ \biggr]} \\ & {} +M^{-\frac{3}{2}}{ \biggl[} \sum_{j\neq n} \vert x_{j} \vert ^{\frac{3}{2}}+ \frac{1}{2\sqrt{2}} \vert x_{n+1}+x_{n-1} \vert ^{\frac{3}{2}}{ \biggr]}+ M\gamma ^{ \prime } \\ & {} -\frac{1}{2}(\Delta x)^{\top }P(\Delta x). \end{aligned}$$
(2.2)

The eigenvalues of P are \(\underbrace{1,\ldots,1}_{M-2}\), 0, 2, and the matrix P has M linearly independent eigenvectors. So we can construct the second new orthogonal direct sum decomposition R M = L 0 L 1 L 2 , where

L 0 =span { Δ x R M P Δ x = 0 } , L 1 =span { Δ x R M P Δ x = Δ x } ,

and

L 2 =span { Δ x R M P Δ x = 2 Δ x } .

For difference cases, we have the following discussions.

• Case 1: \(\Delta x\in L_{0}\). In this case, \((\Delta x)^{\top }P(\Delta x)=0\) and \(\Delta x_{n-1}=-\Delta x_{n}\). Then \(x_{1}=_{\cdots }=x_{n-1}=x_{n+1}=_{\cdots }=x_{M}\). Thus by (2.2) we have

$$\begin{aligned} J_{1}(x) \leq & x_{n}^{2}+x^{2}_{n-1}+2x_{n-1}x_{n} -\beta \Vert x \Vert ^{2}-M^{4}(M-1) \bigl( \vert x_{1} \vert ^{4}- \vert x_{1} \vert ^{\frac{3}{2}} \bigr) \\ & {} + M\gamma ^{\prime }. \end{aligned}$$
(2.3)

By the Cauchy–Schwarz inequality, for \(\beta >1\), from (2.3) we have

$$\begin{aligned} J_{1}(x) \leq &x_{n}^{2}+x^{2}_{1}+ \frac{2}{\beta -1}x^{2}_{1}+ \frac{\beta -1}{2}x_{n}^{2}- \beta \Vert x \Vert ^{2}-M^{4}(M-1) \bigl[ \vert x_{1} \vert ^{4} \\ & {} - \vert x_{1} \vert ^{\frac{3}{2}} \bigr]+ M\gamma ^{\prime } \\ \leq &\frac{\beta +1}{\beta -1}x^{2}_{1} -\frac{\beta -1}{2} \Vert x \Vert ^{2}-M^{4}(M-1) \bigl[ \vert x_{1} \vert ^{4}- \vert x_{1} \vert ^{\frac{3}{2}} \bigr]+ M\gamma ^{\prime }. \end{aligned}$$
(2.4)

Since

$$ \lim_{x_{1}\rightarrow \infty } \biggl[\frac{\beta +1}{\beta -1}x^{2}_{1} -M^{4}(M-1) \vert x_{1} \vert ^{4}+ M^{4}(M-1) \vert x_{1} \vert ^{\frac{3}{2}} \biggr]=- \infty , $$

there exists a constant \(M_{1}\) such that \(J_{1}(x)\leq M_{1}\). So, when the eigenvalue of P is 0, the functional \(J_{1}(x)\) is bounded from above on \(E_{M}\).

• Case 2: \(\Delta x\in L_{2}\).

Now \(P(\Delta x)=\Delta x\) and \(\Delta x_{n-1}=\Delta x_{n}=0\). Therefore \(x_{n}=x_{n+1}=x_{n-1}\), and

$$\begin{aligned} J_{1}(x) \leq & {-}\beta \Vert x \Vert ^{2}-M^{4} \Vert x \Vert ^{4}_{4} +M^{-\frac{3}{2}} \Vert x \Vert ^{\frac{3}{2}}_{\frac{3}{2}}+ M\gamma ^{\prime } \\ \leq &{-}\beta \Vert x \Vert ^{2}- \Vert x \Vert ^{4} + \Vert x \Vert ^{\frac{3}{2}}+ M\gamma ^{ \prime }. \end{aligned}$$
(2.5)

So there exists a constant \(M_{2}\) such that \(J_{1}(x)\leq M_{2}\).

• Case 3: \(\Delta x\in L_{3}\).

Now \(P(\Delta x)=2\Delta x\), so \(\Delta x_{1}=_{\cdots }=\Delta x_{n-2}=\Delta x_{n+2}=_{\cdots }= \Delta x_{M}=0\), \(\Delta x_{n-1}=\Delta x_{n}\). Thus \(2x_{n}=x_{n+1}+x_{n-1}\), and

$$\begin{aligned} J_{1}(x)\leq -\frac{1}{2} \Vert \Delta x \Vert ^{2} -\beta \Vert x \Vert ^{2}- \Vert x \Vert ^{4} + \Vert x \Vert ^{\frac{3}{2}}+ N\gamma ^{\prime }\leq M_{2}. \end{aligned}$$
(2.6)

Take \(M_{3}=\max \{M_{1}, M_{2}\}\). Then by Cases 1–3 we get \(J_{1}(x)\leq M_{3}\). Now the proof of Lemma 2.2 is complete. □

Lemma 2.3

Let hypothesis \((F_{2}^{\prime })\) be in force. Then \(J_{1}\) satisfies the PS condition.

Proof

Let \(\{J_{1}(x^{(j)})\}\) be a bounded sequence from bellow, that is, there exists a positive constant \(M_{4}\) such that

$$\begin{aligned} {J_{1} \bigl(x^{(j)} \bigr) \geq -M_{4},}\quad \forall j \in N. \end{aligned}$$

From (2.4)–(2.6) we have the following inequality:

$$\begin{aligned} J_{1} \bigl(x^{(j)} \bigr)\leq \textstyle\begin{cases} \frac{\beta +1}{\beta -1} \vert x^{(j)}_{1} \vert ^{2} -\frac{\beta -1}{2} \Vert x^{(j)} \Vert ^{2}-M^{4}(M-1)[ \vert x^{(j)}_{1} \vert ^{4}- \vert x^{(j)}_{1} \vert ^{\frac{3}{2}}]\\ \quad {}+ M\gamma ^{\prime } & \text{when } \Delta x^{(j)}\in L_{0}, \\ {-}\beta \Vert x^{(j)} \Vert ^{2}- \Vert x^{(j)} \Vert ^{4} + \Vert x^{(j)} \Vert ^{\frac{3}{2}}+ M\gamma ^{\prime }& \text{otherwise}, \end{cases}\displaystyle \end{aligned}$$

which implies

$$\begin{aligned} \textstyle\begin{cases} \frac{\beta -1}{2} \Vert x^{(j)} \Vert ^{2}+M^{4}(M-1)[ \vert x^{(j)}_{1} \vert ^{4}- \vert x^{(j)}_{1} \vert ^{\frac{3}{2}}]-\frac{\beta +1}{\beta -1} \vert x^{(j)}_{1} \vert ^{2}\\ \quad \leq M\gamma ^{\prime }+M_{4} & \text{when } \Delta x^{(j)}\in L_{0},\\ \beta \Vert x^{(j)} \Vert ^{2}+ \Vert x^{(j)} \Vert ^{4} - \Vert x^{(j)} \Vert ^{\frac{3}{2}} \leq M\gamma ^{\prime }+M_{4} & \text{otherwise}. \end{cases}\displaystyle \end{aligned}$$
(2.7)

From (2.7) it is not difficult to deduce that there exists a constant \(M_{5}\) such that \(\Vert x^{(j)} \Vert \leq M_{5}\), that is, \(\{x^{(j)}\}\) is bounded in \(E_{M}\). Since \(E_{M}\) is finite-dimensional, there exists a subsequence of \(\{ x^{(j)} \}\) (not labeled) convergent in \(E_{M}\), so the PS condition is satisfied. □

Lemma 2.4

([19, Theorem 5.3]; Linking theorem)

Let H be a real Hilbert space, \(H = H_{1}\oplus H_{2}\), where \(H_{2}\) is a finite-dimensional subspace of H. Assume that \(J \in \mathcal{C}^{1}(H)\) satisfies the PS condition and

\({(\mathrm{A}_{1})}\):

there exist constants \(\sigma > 0\) and \(\rho > 0\) such that \(J| _{\partial B_{\rho } \cap H_{1}} \geq \sigma \);

\({(\mathrm{A}_{2})}\):

there are \(e \in \partial B_{1} \cap H_{1}\) and a constant \(R_{1} > \rho \) such that \(J| _{\partial Q}\leq 0\),

where \(Q=(\overline{B}_{R_{1}} \cap H_{2})\oplus \{re\mid 0< r< R_{1}\}\), \(B_{\rho }\) denotes the open ball in H with radius ρ and centered at 0, and \(\partial B_{\rho }\) is its boundary. Then J possesses a critical value \(c \geq \sigma \), where

$$ c = \inf_{h \in \Gamma }\max_{u \in Q}J \bigl(h(u) \bigr), \quad\quad \Gamma = \bigl\{ h \in C(\overline{Q}, H) \mid h| _{\partial Q} = \mathit{id} \bigr\} , $$

and id denotes the identity operator.

Proof of Theorem 1.2

Based upon Lemmas 2.22.4, we divide the proof into four steps.

Step 1. We show that \((A_{1})\) in the Linking theorem holds.

When \(x \in H_{1}\) and \(\Delta x\in L_{0}\), then by \((F_{1}^{\prime })\)

$$\begin{aligned} J_{1}(x) \geq & \frac{1}{2} \Vert \Delta x \Vert ^{2}-\alpha \Vert x \Vert ^{2} -M^{4}{ \biggl[}\sum_{j\neq n} \vert x_{j} \vert ^{4}+\frac{1}{16} \vert x_{n+1}+x_{n-1} \vert ^{4}{ \biggr]} \\ & {} + M^{-\frac{3}{2}}{ \biggl[} \sum_{j\neq n} \vert x_{j} \vert ^{\frac{3}{2}}+ \frac{1}{2\sqrt{2}} \vert x_{n+1}+x_{n-1} \vert ^{\frac{3}{2}}{ \biggr]} \\ =& \bigl(x_{n}^{2}+x^{2}_{n-1}+2x_{n-1}x_{n} \bigr)-\alpha \Vert x \Vert ^{2}-M^{4}{ \biggl[}\sum _{j\neq n} \vert x_{j} \vert ^{4}+\frac{1}{16} \vert x_{n+1} \\ & {} +x_{n-1} \vert ^{4}{ \biggr]}+M^{-\frac{3}{2}}{ \biggl[} \sum_{j\neq n} \vert x_{j} \vert ^{ \frac{3}{2}}+\frac{1}{2\sqrt{2}} \vert x_{n+1}+x_{n-1} \vert ^{\frac{3}{2}}{ \biggr]}. \end{aligned}$$
(3.1)

Using the Cauchy–Schwarz inequality \(2x_{n-1}x_{n}\leq \frac{1-\alpha }{2}x^{2}_{n-1}+\frac{2}{1-\alpha }x^{2}_{n}\) and \(0<\alpha <1\), from (3.1) we arrive at

$$\begin{aligned} J_{1}(x)\geq \frac{1-\alpha }{2}x_{n}^{2}- \frac{1+\alpha }{1-\alpha }x^{2}_{1} -(M-1)\alpha x_{1}^{2}-M^{5}x_{1}^{4} +x_{1}^{\frac{3}{2}}/\sqrt{M}. \end{aligned}$$
(3.2)

On the other hand, when \(\Vert x \Vert \leq \delta \), \((\sum_{j\neq n}x_{j}^{2})^{1/2}\leq \delta \), so if we choose δ sufficiently small, then from (3.2) we conclude

$$\begin{aligned} J_{1}(x) \geq &\frac{1-\alpha }{2}x_{n}^{2} +\frac{1}{2\sqrt{M}}x_{1}^{ \frac{3}{2}} \\ \geq & \frac{1-\alpha }{2}x_{n}^{2} +\frac{(1-\alpha )(M-1)}{2}x_{1}^{2}=(1- \alpha ) \Vert x \Vert ^{2}. \end{aligned}$$
(3.3)

Taking \(\sigma =(1-\alpha )\delta ^{2}\), we have

$$\begin{aligned} J_{1}(x)\geq \sigma > 0, \quad \forall x \in H_{1}\cap \partial B_{\delta }. \end{aligned}$$
(3.4)

Thus condition \((A_{1})\) in Lemma 2.4 is satisfied.

Step 2. We show that \((A_{2})\) holds.

By Lemma 2.3, \(J_{1}(x)\) meets the PS condition. Taking \(e \in \partial B_{1} \cap H_{1}\), for any \(z \in H_{2}\), rR. Let \(x = re+z\). By (2.4)–(2.6)

$$\begin{aligned} J_{1}(x)\leq \textstyle\begin{cases}\frac{\beta +1}{\beta -1} \vert x_{1} \vert ^{2} -\frac{\beta -1}{2} \Vert x \Vert ^{2}-M^{4}(M-1)[ \vert x_{1} \vert ^{4}- \vert x_{1} \vert ^{\frac{3}{2}}]+ M\gamma ^{\prime } & \text{when } x\in H_{1},\\ {-}\beta \Vert x \Vert ^{2}- \Vert x \Vert ^{4} + \Vert x \Vert ^{\frac{3}{2}}+ M\gamma ^{\prime } & \text{when } x\in H_{2}. \end{cases}\displaystyle \end{aligned}$$
(3.5)

Observing that as \(\Vert x \Vert \rightarrow \infty \), the right-hands in (3.5) are approaching to negative infinity, and thus there exists a big enough constant \(R_{2}>0\) such that \(J_{1}(x) \leq 0\) for all \(x \in \partial Q\), where

$$\begin{aligned} Q = (\overline{B}_{R_{2}} \cap H_{2}) \oplus \{re\mid 0 < r < R_{2}\}. \end{aligned}$$

Step 3. Existence of the first nontrivial N-periodic solution.

By Linking theorem (Lemma 2.4), \(J_{1}\) has a critical value \(c \geq \sigma > 0\), where

$$\begin{aligned} c = \inf_{h \in \Gamma }\max_{x \in Q}J_{1} \bigl(h(x) \bigr), \quad\quad \Gamma = \bigl\{ h \in C(\overline{Q}, E_{M}) \vert h \vert _{\partial Q} = \mathit{id} \bigr\} . \end{aligned}$$

Step 4. Existence of the second nontrivial N-periodic solution.

Inequalities (2.4)–(2.6) imply \(\lim_{ \Vert x \Vert \rightarrow \infty }J_{1}(x)=-\infty \). Let \(c_{0} = \sup_{x \in E_{M}}J_{1}(x)\). By the continuity of \(J_{1}\) on \(E_{M}\) there exists \(\bar{x} \in E_{M}\) such that \(J_{1}(\bar{x}) = c_{0}\) and is a critical point of \(J_{1}\). Combining (3.4), we have \(J_{1}(\bar{x})=c_{0}>0\). Note that when \(x_{1}=\cdots =x_{M}=0\), by (2.1) and condition \((F_{1}^{\prime })\) we get \(J_{1}(x)= -\sum_{j=1}^{M}F(j, x_{j})\leq 0\). So the critical point associated with the critical value \(c_{0}\) of \(J_{1}\), is a nontrivial M-periodic solution of system (1.1).

The rest of the proof of the other nontrivial M-periodic solution is similar to that of [18, Theorem 1.1], and we omit it.

By now the proof of Theorem 1.2 is complete, which means that discrete system (1.1) has at least two nontrivial M-periodic solutions.

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Acknowledgements

The authors are thankful to the editor and anonymous referees for their valuable comments and suggestions.

Funding

The first author was partially supported by the research funding project of Guizhou Minzu University (GZMU[2019]QN04) and the Guizhou Province Science and Technology Department (ZK[2021]003). The second author was partially supported by the National Natural Science Foundation of China (No. 11901442). The third author was partially supported by National Science Foundation of China (No. 11501577).

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Ding, L., Tian, R. & Wei, J. Periodic solutions for second-order difference equations with quadratic–supquadratic condition. Adv Differ Equ 2021, 406 (2021). https://doi.org/10.1186/s13662-021-03560-0

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MSC

  • 49J35
  • 34K13
  • 34G20

Keywords

  • Critical points
  • Linking theorem
  • Nontrivial periodic solutions
  • Discrete systems