On periodic solutions of second-order partial difference equations involving p-Laplacian

1.   Introduction

Let $\mathbb{Z}$, $\mathbb{N}$, and $\mathbb{R}$ stand for the sets of integers, natural numbers and real numbers, respectively. Consider the existence and nonexistence of periodic solutions of a partial difference equation in the following form:

Here, $\triangle_{i}$ $(i = 1, 2)$ represents the forward difference operator, and $\triangle_{1}x(n-1, m) = x(n, m)-x(n-1, m)$, $\triangle_{2}x(n, m-1) = x(n, m)-x(n, m-1)$. The p-Laplacian operator is defined as $\phi_{p}(x) = |x|^{p-2}x$ for $1 < p < +\infty$ and $x\in \mathbb{R}$. Given integers $T_1, T_2 > 0$, $x = \{x(n, m)\}$ is $(T_{1}, T_{2})$-periodic, which means that $x(n+T_{1}, m) = x(n, m) = x(n, m+T_{2})$ for all $(n, m)\in\mathbb{Z}^{2}$. The nonlinearity $f \in C(\mathbb{Z}^2\times\mathbb{R}, \mathbb{R})$ is $T_{1}$-periodic in $n$ and $T_{2}$-periodic in $m$. Denote $F\left((n, m), x\right) = \int^{x}_{0}f\left((n, m), s\right)ds$ for all $(n, m)\in \mathbb{Z}^2$.

Owing to both in our real life and scientific research, many phenomena and data are recorded with discrete data; difference equations have a wide range of applications and a long research history in various fields to describe discrete phenomena ^[1,2]. With the popularization of computers and the rapid development of computer technology, the study of difference equation theory has made great progress in various aspects since Guo and Yu ^[3] first applied the variational method to difference equations. For example, the authors obtained periodic solutions ^[4], homoclinic solutions ^[5] of second-order difference equations, and standing waves solutions ^[6] for the discrete Schrödinger equations. As to difference equations involving p-Laplacians, here is a list of a few:

where $\triangle x_{n} = x_{n+1}-x_{n}$, is a special case of Equation (1.1). Results on periodic solutions and positive solutions of (1.2) were given in ^[7] and ^[8], respectively. The authors ^[9] studied periodic solutions of

As (1.3) is in higher-order, homoclinic solutions and periodic solutions were displayed in ^[10] and ^[11].

Nowadays, more and more phenomena need to be described by two or more multi-variables. Subsequently, both partial differential equations and partial difference equations, containing two or more than two variables, have caught the keen attention of many scholars, and rich results have emerged. Here mention a few; in ^[12,13,14], authors obtained a series of results for partial differential equations. Long studied discrete Kirchhoff-type problems and obtained a series of results on multiple solutions ^[15,16], least energy solutions ^[17] and infinitely many large energy solutions ^[18] (see also ^[19,20,21] and reference therein). In ^[22], the authors gave results on periodic solutions for a second- order difference equation. When partial difference equations contain $p$-Laplacian, multiple existence results were given in ^[23]. As to homoclinic solutions, Mei and Zhou ^[24] gave results for partial difference equations with mixed nonlinearities, and Long ^[25] considered nonlinear $(p, q)$-Laplacian partial difference equations with a parameter $\lambda > 0$.

Motivated by the above mentioned results, we deal with periodic solutions of (1.1) by variational techniques together with the saddle point theorem. To demonstrate the validity of our main results, we also present some examples and numerical simulations. Our results generalize and complement some known ones, as detailed in Remark 1.2.

Now we state our main results as follows:

Theorem 1.1. Assume the following suppositions are fulfilled.

$(A_1)$ There exists a constant $M_{0} > 0$ such that

$(A_2)$

Then Equation (1.1) possesses at least a $(T_{1}, T_{2})$-periodic solution.

Theorem 1.2. Let $f$ satisfy

$(A_3)$ there exist positive constants $R_{1}$ and $\alpha$ $(\frac{2}{p} < \alpha < 2)$ such that

$(A_4)$ there exist positive constants $b_{1}$, $b_{2}$, and $\beta$ $(\frac{2}{p} < \beta\leq \alpha)$ such that

Then Equation (1.1) admits at least a $(T_{1}, T_{2})$-periodic solution.

Remark 1.1. Substitute $(A_3)$ by

$(A_{3}^{\prime})$ there exist constants $a_{1}, a_{2} > 0$ such that

The conclusion of Theorem 1.2 is still valid.

Further, to obtain nontrivial periodic solutions, we have

Theorem 1.3. Assume the following conditions hold

$(A_5)$ $F((n, m), 0) = 0, \qquad \forall \ (n, m)\in\mathbb{Z}^{2}$;

$(A_6)$ there exists a constant $\frac{2}{p} < \alpha < 2$ such that

$(A_7)$ there exist constants $b_{3} > 0$ and $\frac{2}{p} < \beta\leq \alpha$ such that

Then Equation (1.1) has at least one nontrivial $(T_{1}, T_{2})$-periodic solution.

Theorem 1.4. Suppose $(A_1)$, $(A_2)$, and $(A_5)$ hold. Moreover,

$(A_8)$ there exist constants $b_{4} > 0$ and $0 < \gamma < 2$ such that

Then Equation (1.1) possesses at least one nontrivial $(T_{1}, T_{2})$-periodic solution.

Theorem 1.5. If for all $(n, m)\in\mathbb{Z}^{2}$ and $x \neq 0$, there holds

Then Equation (1.1) has no nontrivial $(T_{1}, T_{2})$-periodic solution.

Remark 1.2. Our Theorems 1.1, 1.2, 1.3 and 1.4 are generalizations of Theorems 1.1, 1.2, 1.3, and 1.4 in ^[9], respectively. Moreover, Theorem 1.5 supplements the nonexistence of periodic solutions of Equations (1.1) and (1.2).

The rest of this paper is organized as follows. In Section 2, we establish the variational framework corresponding to Equation (1.1) and give some basic lemmas that play a vital role in proving our main results. Section 3 presents detailed proofs of our main results. Finally, three examples and numerical simulations are provided in Section 4.

2.   Preliminaries

For convenience, we give some notations. Denote $\mathbb{Z}(t, s): = \{t, t+1, \cdots, s\}$ with integers $t \leq s$ and $\Omega: = \mathbb{Z}(1, T_1)\times\mathbb{Z}(1, T_2)$. Let

Define a $T_{1}T_{2}$-dimensional subspace $E$ of vector space $S = \{x = \{x(n, m)\}|x(n, m)\in\mathbb{R}, n, m\in\mathbb{Z}\}$ by

which is endowed with the inner product

Thus, the induced norm $\|\cdot\|$ is

and $E$ is isomorphic to $\mathbb{R}^{T_{1}T_{2}}$.

Write

It follows that $\|x\|_{2} = \|x\|$ and there exist positive constants $\zeta_{p}$ and $\xi_{p}$ with $\frac{\zeta_{p}}{\xi_{p}} = (T_{1}T_{2})^{\frac{-\mid 2-p\mid}{2p}}$ such that

Further, we have, for any $x\in E$, there exist positive constants $C_{1}$, $C_{2}$, $C_{3}$ such that

Consider the associated functional $I: E\rightarrow \mathbb{R}$ in the form as

Then $I$ is $C^{1}$. Using periodic conditions, simple calculation yields that

which means that Equation (1.1) is the corresponding Euler-Lagrange equation for $I$. Consequently, we transform the problem to find $(T_{1}, T_{2})$-periodic solutions of Equation (1.1) to the problem to seek critical points of $I$ in $E$.

Identify $x = \{x(n, m)\}_{n, m\in\mathbb{Z}} \in E$ with

and write

where

Then $\|x\|_{s} = \|x^{\prime}\|_{s}$ for all $s > 1$.

Let

with

By ^[9], the eigenvalues of matrix $A_{T_{1}T_{2}}$ are $\lambda_{i} = 2(1-\cos\frac{2i\pi}{T_{1}})$, $i = 0, 1, 2, \cdots, T_{1}-1$. Thus $\lambda_{0} = 0$ and $\lambda_{i} > 0$ for $1\leq i\leq T_{1}-1$. Further, each $\lambda_{i}$ is $T_{2}$-multiple and

Similarly, $A_{T_{2}T_{1}}$ has eigenvalues $\mu_{j}$ $(0\leq j \leq T_{2}-1)$ and

We split $E$ as $E = V\oplus Y$ with $Y = \left\{y\in E|y = \{c, c, \cdots, c\}, c\in\mathbb{R}\right\}$. It follows that

and

Now, we state some basic definitions. Let $X$ be a real Banach space. $I\in C^{1}\left(X, \mathbb{R}\right)$ satisfies the Palais-Smale ($P. S.$ for short) condition, which states that any sequence $\{x_n\}\subset X$ such that $\{I(x_n)\}$ is bounded and $\lim\limits_{n\to\infty}I^{\prime}\left(x_{n}\right)\rightarrow 0$ possesses a convergent subsequence.

We denote by $B_\rho$, the open ball with center $0$ and radius $\rho$ in $X$, and $\partial B_\rho$ its boundary. Recall the Saddle Point Theorem, introduced in ^[26], which plays a crucial role in proofs of our main results.

Lemma 2.1. (Saddle Point Theorem ^[26]) Let $X = X_{1}\oplus X_{2}$ be a real Banach space with finite-dimensional subspace $X_{1}\ne\{0\}$. Suppose $I\in C^{1}\left(X, \mathbb{R}\right)$ fulfills the $P. S.$ condition and

$(J_{1})$ $I \mid_{\partial B_{\rho}\cap X_{1}}\leq\sigma$ for constants $\sigma$ and $\rho > 0$;

$(J_{2})$ $I \mid_{e+ X_{2}}\geq\omega$ for constants $e\in B_{\rho}\cap X_{1}$ and $\omega > \sigma$.

Then $I$ admits a critical value $c\geq\omega$ with

3.   Proofs of main results

In this section, we present detailed proofs of our main results.

Proof of Theorem 1.1 We complete the proof by Lemma 2.1 in three steps.

Step 1 $I$ satisfies the $P.S.$ condition on $E$.

Assume that $\{x_{k}\}\subset E$ is a $P. S.$ sequence, that is, $\lim\limits_{k\rightarrow \infty}I^{\prime}(x_{k}) = 0$ and there exists a constant $M_{1} > 0$ such that $\mid I(x_{k})\mid\leq M_{1}$. Then for $k$ large enough and any $x\in E$, we have

Take $x_{k} = v_{k}+y_{k}\in V\oplus Y$, it follows that

Together with $(A_1)$ and (3.1), we deduce that

By (2.8), we have

Thus, combining (3.2) with (3.3), we obtain

Since $p > 1$, (3.4) ensures that $\|v_{k}\|$ has a maximum value. Thus, $\{v_{k}\}$ is a bounded sequence.

Next, we show that $\{y_{k}\}$ is also a bounded sequence. Owing to $(A_1)$, (2.7) and

we attain that, for $\theta\in(0, 1)$, there holds

Notice that $\{v_{k}\}$ is bounded, then $\left\{\sum\limits_{n = 1}^{T_{1}}\sum\limits_{m = 1}^{T_{2}} F\left((n, m), y_{k}(n, m)\right)\right\}$ is bounded. We claim that $\{y_{k}\}$ is bounded. Otherwise, we assume that $\lim\limits_{k\rightarrow \infty}\|y_{k}\| = \infty$. Let $y_{k} = (c_{k}, c_{k}, \cdots, c_{k})^{T}\in Y$ where $c_{k}\in\mathbb{R}$, $k\in\mathbb{N}$, then

In view of $(A_2)$,

Thus, $\left\{\sum\limits_{n = 1}^{T_{1}}\sum\limits_{m = 1}^{T_{2}} F\left((n, m), y_{k}(n, m)\right)\right\}\rightarrow +\infty$, which is a contradiction. Therefore, $\{y_{k}\}$ is bounded. Consequently, $\{x_{k}\}\subset E$ is a bounded sequence on the finite-dimensional space $E$, and the $P. S.$ condition is verified.

Step 2 $(J_1)$ of Lemma 2.1 is fulfilled.

From $(A_{1})$, there exists a constant $M_{0}^{\prime} > 0$ such that

Utilizing (2.8), for any $v \in V$, it follows that

Therefore, $(J_1)$ holds.

Step 3 $(J_2)$ of Lemma 2.1 is satisfied.

For any $y\in Y$, there is $c\in\mathbb{R}$ such that

Taking account of $(A_2)$, one gets that there exists a constant $R_{0} > 0$ such that $F((n, m), x) > 0$ for any $(n, m)\in\mathbb{Z}^{2}$ and $\mid x\mid > R_{0}$. Let

Then

Hence, we have

which indicates that $I$ satisfies $(J_2)$ of Lemma 2.1 with $e = 0$. Thus, the desired result follows.

Proof of Theorem 1.2 To prove Theorem 1.2 by Lemma 2.1, it is necessary to verify that $I$ satisfies the $P.S.$ condition on $E$ and the geometry conditions $(J_1)$ and $(J_2)$ of Lemma 2.1.

First, we testify that the $P.S.$ condition is satisfied. Suppose $\{x_{k}\}\subset E$ and there is a constant $M_{3} > 0$ such that

Then for $k$ is large enough, there holds

Recall

In combination with (3.5) and (3.6), we have

Write $\Omega = \Omega_1\cup \Omega_2$, where

Then (3.7) and $(A_3)$ imply that

Moreover, $\frac{\alpha}{2} p F((n, m), z)-f\left((n, m), z\right)\cdot z$ is continuous with respect to $z$, which means that there is a constant $M_{4} > 0$ such that

Thus,

By $(A_4)$ and (3.9), we achieve

where $M_{5} = (1-\frac{\alpha}{2})b_{2}T_{1} T_{2}+\frac{1}{p} T_{1} T_{2} M_{4}$. Joint with (2.3), we obtain

that is,

Remind $\frac{2}{p} < \beta\leq \alpha < 2$, (3.10) guarantees $\{x_{k}\}$ is bounded. Since $E$ is finite dimensional, the $P.S.$ condition is satisfied.

Second, we complete the proof by verifying that $I$ satisfies the geometry conditions $(J_1)$ and $(J_2)$ of Lemma 2.1. For any $y = (c, c, \cdots, c)\in Y$ with $c\in \mathbb{R}$, $(A_4)$ means that

For any $v\in V$, $(A_{3}^{\prime})$, (2.2) and (2.8) yield

Notice that $\frac{2}{p} < \alpha < 2$, (3.11) indicates that there is a constant $\rho_{0} > 0$ large enough such that

Thus, both $(J_1)$ and $(J_2)$ are satisfied. Therefore, Lemma 2.1 ensures that Equation (1.1) possesses at least a $(T_{1}, T_{2})$-periodic solution. The proof is completed.

Proof of Theorem 1.3 To obtain nontrivial solutions, we divide the proof of Theorem 1.3 in four steps.

Step 1 $I$ satisfies the $P.S.$ condition on $E$.

Let sequence $\{x_{k}\}\subset E$ such that

where $M_{6} > 0$ is a constant. For $k$ large enough, one obtains

Moreover,

Together with (2.3), $(A_6)$ and $(A_7)$, it follows that

Namely,

Recall $\frac{2}{p} < \beta\leq \alpha < 2$, (3.12) implies that $\{x_{k}\}$ is a bounded sequence. Due to the fact that $E$ is a finite-dimensional space, then $I$ satisfies the $P.S.$ condition.

Step 2 $I$ meets $(J_1)$ of Lemma 2.1.

For any $v\in V$, (3.11) gives

Therefore, $(J_1)$ of Lemma 1 is satisfied.

Step 3 $I$ fulfills $(J_2)$ of Lemma 2.1.

Given $x = v_{0}+y$ with $v_{0}\in V$ and $y\in Y$, from $(A_7)$, (2.3), and (2.7), we have that

which means that there is a sufficiently small positive constant $\delta_{1}$ satisfying

for $v_{0}\in \partial B_{\delta_{1}}\cap V$ and $y\in Y$. Then $(J_2)$ is valid.

Step 4 $I$ has a nontrivial critical point.

Applying the saddle point theorem, we find a critical value $c\geq \omega_{1} > 0$ of $I$. Let $\bar{x}\in E$ be the corresponding critical point, that is,

Further, $\bar{x}$ is a nontrivial critical point, that is, $\bar{x}\neq 0$. Or else, if $\bar{x} = 0$, by $(A_5)$, we have

This contradicts (3.13). Hence, $\bar{x}\neq 0$ and the proof is completed.

Proof of Theorem 1.4 From the proof of Theorem 1.1, we know that $(A_1)$ and $(A_2)$ ensure that the $P. S.$ condition and $(J_1)$ of Lemma 2.1 are valid. Then we only need to prove that $(J_2)$ of Lemma 2.1 also holds.

Taking $x = v_{0}+y$, where $v_{0}\in V$ and $y\in Y$, by $(A_8)$, (2.3) and (2.8), we obtain

which means that, for $v_{0}\in \partial B_{\delta_{2}}\cap V$ and any $y\in Y$, there exists a sufficiently small constant $\delta_{2} > 0$ such that

Thus $(J_2)$ is verified. Therefore, Lemma 2.1 means that $I$ admits a critical value $c\geq \omega_{2} > 0$. Denote the corresponding critical point by $\bar{x}$, that is, $I(\bar{x}) = c > 0$. With $(A_5)$, we get

which implies that $\bar{x}\neq 0$. Thus our proof is done.

Proof of Theorem 1.5 For the sake of contradiction, we assume that $x^{*}$ is a nontrivial $(T_{1}, T_{2})$-periodic solution of Equation $(1.1)$, which is equivalent to the fact that $x^{*}$ is a nontrivial critical point of $I$ on $E$. Hence, $I^{\prime}(x^{*}) = 0$ with $x^{*}\neq 0$. Direct computation gives that

Therefore,

On the other hand, $(A_9)$ gives

This is in conflict with (3.14). Consequently, the proof is finished.

4.   Examples

We give three examples to illustrate applications of our main results. Write

To facilitate the presentation of numerical simulations, we abbreviate $x\in \bar{E}$ as

Example 4.1. Take $T_{1} = T_{2} = 2$ and $p = 3$. Consider

Here

By integration, it yields

Direct calculations give

Thus, Equation (4.1) fulfills all the assumptions of Theorem 1.1, which guarantees that Equation (4.1) has at least a $(2, 2)$-periodic solution. Use Matlab; a solution $x\in\bar{E}$ of Equation (4.1) is presented as

Example 4.2. Take $p = 3$ and $T_{1} = T_{2} = 2$. Consider

Here

then

Take $\alpha = \frac{4}{3}$ and $\beta = \frac{4}{3}$, then $(A_3)$ and $(A_4)$ of Theorem 1.2 hold. Thus, Equation (4.2) has at least a $(2, 2)$-periodic solution.

Further, we know that $F((n, m), 0) = 0$ for any $n, m\in\mathbb{Z}$. Thus, $(A_5)$, $(A_6)$, and $(A_7)$ of Theorem 1.3 are also true. Thereby, it can be further confirmed that Equation (4.2) holds at least one nontrivial $(2, 2)$-periodic solution. We list a solution $x\in\bar{E}$ of Equation (4.2) as follows:

Example 4.3. Consider Equation (1.1) with $f((n, m), x) = -x$. Then Equation (1.1) turns into

It is clear that the condition $(A_9)$ of Theorem 1.5 is valid. Therefore, Equation (4.3) has no nontrivial $(T_{1}, T_{2})$-periodic solution.

Author contributions

Dan Li: Writing-original draft, formal analysis; Yuhua Long: Methodology, writing-review editing.

Use of AI tools declaration

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

Acknowledgments

We wish to thank the handling editor and the referees for their valuable comments and suggestions. We would also like to thank the National Natural Science Foundation of China 12471177.

Conflict of interest

The authors declare there is no conflict of interest.