Multiple nontrivial periodic solutions to a second-order partial difference equation

1.   Introduction

Denote the real number set and the integer set by $\mathbb{R}$ and $\mathbb{Z}$, respectively. For any $a, b\in\mathbb{Z}$ with $a\leq b$, define $\mathbb{Z}(a, b)$: = $\{a, a+1, \cdots, b\}$. In this paper, we focus on multiple nontrivial periodic solutions of the following second-order partial difference equation:

where $\triangle_{1}u(n, m) = u(n+1, m)-u(n, m)$ and $\triangle_{2}u(n, m) = u(n, m+1)-u(n, m)$. $f((n, m), u): \mathbb{Z}^2\times\mathbb{R}\rightarrow \mathbb{R}$ is continuous with respect to $u$. Given integers $T_1, T_2 > 0$, for any $n, m\in\mathbb{Z}$, let nonzero sequences $\{p(n, m)\}$, $\{r(n, m)\}$ and $\{q(n, m)\}$ satisfy

and

Let $\eta$ be the ratio of odd positive integers such that $(-1)^{\eta} = -1$. If a solution $u = \{u(n, m)\}$ satisfies $u(n+T_{1}, m) = u(n, m) = u(n, m+T_{2})$ for any $n, m\in\mathbb{Z}$, we call $u$ a $(T_{1}, T_{2})$-periodic solution. To help with understanding if a solution $u$ is $(T_{1}, T_{2})$-periodic, we give an example as a remark.

Remark 1.1. Consider (1.1) with $T_{1} = T_{2} = 2$. Suppose (1.1) possesses four solutions, denoted by

If they are (2, 2)-periodic, that is, $u(n+2, m) = u(n, m) = u(n, m+2)$, $n, m = 1, 2$, then

Actually, $u_{1} = u_{3}$ and $u_{2} = u_{4}$. Therefore, $u_{11} = u_{13}$ and $u_{12} = u_{14}$ ensure that the solution $u_{1} = (u_{11}, u_{12}, u_{13}, u_{14})$ is (2, 2)-periodic.

In socio-economic activities and natural science research, we often encounter variables similar to time $t$. Meanwhile, one can often only observe or record values of these variables in discrete cases. Solving this problem is inseparable from difference equations. During past decades, difference equations have been used extensively ^[1,2], and scholars have studied difference equations in many ways, including period solutions, boundary value problems, homoclinic solutions, heteroclinic solutions ^{[3,4,5,6,7,8]} and so on. It is worth mentioning that Guo and Yu ^[3] made critical point theory an effective tool to discuss periodic solutions by constructing a new variational structure for the first time. In ^[9], by critical point theory, Cai and Yu studied the existence of solutions to the following equation:

Obviously, Eq (1.2), involving only one independent variable, is a special case of (1.1). It has been studied by many authors, and certain conclusions ^[10,11,12] have been yielded.

On the other side, as modern technology advances rapidly, the use of mathematical modeling to solve problems is not only becoming more and more frequent, but also there are more and more factors needing to be considered. As a result, partial difference equations, containing multiple independent integer variables, have widespread applications in image processing, life sciences, quantum mechanics, and other fields ^[13] and capture great interest of many scholars. For example, ^[14,15,16] obtain multiple results on discrete Kirchhoff problems, and ^{[17,18,19,20]} concern second order partial difference equations via Morse theory. Very recently, ^[21] investigated periodic solutions of the equation

via critical point theory. Clearly, letting $\eta\equiv 1$, (1.1) is just (1.3), and (1.1) is more general than (1.3). Moreover, via critical point theory, ^[22,23] deal with the existence of multiple solutions for a partial discrete Dirichlet boundary value problem with mean curvature operator and homoclinic solutions for a differential inclusion system involving the $p(t)$-Laplacian, respectively. In view of the abovementioned results, we find that critical point theory serves as a robust method for studying both differential equations and difference equations. Therefore, motivated by the above obtained results, we intend to study periodic solutions to (1.1) by critical point theory. We also provide numerical stimulations to illustrate applications of our theoretical results. Our results generalize some results in ^[9] and ^[21]. The resulting problem engages two major difficulties: First, to estimate relations between norms, we need to transfer (1.1) into an equivalent form to compute its eigenvalues. Another difficulty we must overcome is verifying the link geometry and certifying boundedness of the Palais-Smale sequence.

For the rest of this paper, we organize in the following way. In Section 2, we give a variational structure and look for the corresponding functional to (1.1). Moreover, some definitions and lemmas are recalled. Our main results and detailed proofs are provided in Section 3. Finally, Section 4 presents three examples to demonstrate the application of our main results.

2.   Preliminaries and notations

In this section, we establish the corresponding variational framework to (1.1) and state some preliminaries and notations to make preparation for our main results.

Write

and let

be a vector space which is composed of all $u = \{u(n, m)\}_{n, m\in\mathbb{Z}}$. Define

as a subset of $S$. Define an inner product $\langle\cdot, \cdot\rangle$ on $E$ as

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

Clearly, the dimension of the Hilbert space $E$ is $T_{1}T_{2}$-dimensional. Thus, $E$ is homeomorphic to $\mathbb{R}^{T_{1}T_{2}}$.

For $s > 1$, define another norm $\|\cdot\|_s$ on $E$ as

Then, $\|u\|_{2} = \|u\|$, and for $\beta > \eta+1$ there exist constants $C_{2}\geq C_{1} > 0$, $C_{4}\geq C_{3} > 0$ such that

Moreover, there holds

Then,

which means that we can choose $C_1 = (T_{1}T_{2})^{-\frac{\eta-1}{2(\eta+1)}}$.

Consider a functional $I: E \rightarrow \mathbb{R}$ in the following form:

where $F\left((n, m), u\right) = \int^{u}_{0}f\left((n, m), s\right)ds$. Since $f((\cdot, \cdot), u)$ is continuous with respect to $u$, it follows that $I\in C^{1}(E, \mathbb{R})$. Moreover, for any $u\in E$, using the periodic condition, direct computation yields

Hence, $u\in E$ being a critical point for $I$ is equivalent to

which is just (1.1). Therefore, we transform the problem to find $(T_{1}, T_{2})$-periodic solutions to (1.1) to the problem to seek critical points of $I$ on $E$.

For convenience, write $u\in E$ as

where $\cdot^{T}$ denotes the transpose of vector $\cdot$. Let matrices $A$ and $B$ be defined by

By matrix theory, we find that the matrix $B$ is semi-definite positive, and its eigenvalues are

Then, $\lambda_1 = 0 < \lambda_2\leq\lambda_3\leq\cdots\leq\lambda_{T_{1}}$, and

Moreover, matrices $A$ and $B$ have the same eigenvalues $\lambda_1 = 0 < \lambda_2\leq\lambda_3\leq\cdots\leq\lambda_{T_{1}}$, and the multiplicity of each eigenvalue $\lambda_k$ of matrix $A$ is $T_{2}$. Direct computation gives

and

Define an orthogonal matrix

such that

Then, the matrix $P$ is a rearrangement transformation of $u$, and $\|u\|_{s} = \|v\|_{s}$ for any $s > 0$.

Similarly, given matrices $C$ and $D$ as

it follows that eigenvalues of matrix $C$ are $\mu_{1} = 0 < \mu_{2}\leq\mu_{3}\leq\cdots\leq\mu_{T_{2}}$, and

In the same manner, we have that eigenvalues of matrix $D$ are also $\mu_{1} = 0 < \mu_{2}\leq\mu_{3}\leq\cdots\leq\mu_{T_{2}}$, and each eigenvalue $\mu_\kappa$, $1\leq \kappa \leq T_2$ is $T_{1}$-multiple. Further,

and

Set $W = \left\{w\in E|w = \{c\}, c\in\mathbb{R}\right\}$ and $Y = W^{\perp}$. Then, $E = Y\oplus W$. Thus, for any $u\in Y$, we have

Thus, for any $w\in W$, we get

In the following, we recall some definitions and lemmas which are useful to our main results.

Definition 2.1. Let $I\in C^{1}\left(E, \mathbb{R} \right)$. If any sequence $\{u_k\}\subset E$ such that $\{I(u_k)\}$ is bounded and $I'\left(u_{k}\right)\rightarrow 0$ as $k\to\infty$ possesses a convergent subsequence, then $I$ satisfies the Palais-Smale ($P. S.$ for short) condition.

Let $B_{\rho}$ denote an open ball whose center is $0$ and radius is $\rho$ in $E$. Let $\partial B_\rho$ stand for the boundary of $B_{\rho}$. The following Lemmas 2.1–2.3 are main tools to prove our results, and we can refer to ^[24] for detail.

Lemma 2.1. (Mountain Pass Lemma^[24]) Let $X$ be a real Banach space and $I\in C^{1}(X, \mathbb{R})$ satisfy the $P. S.$ condition. Moreover, $I(0) = 0$. Suppose

$(f_{1})$ there exist constants $\rho$, $a > 0$ such that $I|_{\partial B_{\rho}}\geq a$;

$(f_{2})$ there exists $e\in X\backslash B_{\rho}$ such that $I(e)\leq 0$.

Then, $I$ admits a critical value $c\geq a$ given by

where

Lemma 2.2. (Linking theorem^[24]) Let $X = X_{1}\oplus X_{2}$ be a real Banach space, where $X_{1}$ is a finite-dimensional subspace of $X$. Suppose that $I\in C^{1}\left(E, \mathbb{R}\right)$ satisfies the $P. S.$ condition. If

$(f_{3})$ there exist constants $\rho$, $a > 0$ such that $I|_{\partial B_{\rho}\cap X_{2}}\geq a$, and

$(f_{4})$ there exist constants $e\in \partial B_{1}\cap X_{2}$, $R_0 > \rho$ such that $I|_{\partial Q}\leq 0$, where $Q = \left(\bar{B}_{R_{0}}\cap X_{1}\right)\oplus\left\{re|0 < r < R_{0}\right\}$,

then $I$ has a critical value $c\geq a$, and

where

Lemma 2.3. (Saddle point theorem^[24]) Let $X = X_{1}\oplus X_{2}$ be a real Banach space and $X_{1}\ne\{0\}$ be a finite-dimensional subspace of $X$. Suppose $I\in C^{1}\left(E, \mathbb{R}\right)$ satisfies the $P. S.$ condition. If

$(f_{5})$ there exist constants $\sigma$ and $\rho > 0$ such that $I|_{\partial B_{\rho}\cap X_{1}}\leq\sigma$, and

$(f_{6})$ there exist constants $e\in B_{\rho}\cap X_{1}$, $\omega > \sigma$ such that $I|_{e+ X_{2}}\geq\omega$,

then I has a critical value $c\geq\omega$, and

where

3.   Main results and proofs

For convenience, we give some notations first. Write $\Omega: = \mathbb{Z}(1, T_1)\times \mathbb{Z}(1, T_2)$, and

To study (1.1), the following assumptions are needed:

$(F_1) \quad \lim\limits_{u\rightarrow 0}\cfrac{f((n, m), u)}{u^\eta} = 0$, $\qquad \forall((n, m), u)\in\Omega\times\mathbb{R}$.

$(F_2) \quad$ There exist constants $a_1 > 0$, $a_2 > 0$ and $\beta > \eta+1$ such that

Remark 3.1. By $(F_2)$, we have

$(F'_{2}) \quad \lim\limits_{|u|\rightarrow +\infty}\cfrac{F((n, m), u)}{u^{\eta+1}} = +\infty, \quad \forall((n, m), u)\in\Omega\times\mathbb{R}$.

Thus, $(F_1)$ and $(F'_2)$ mean that $f((n, m), u)$ is superlinearly increasing at both $0$ and $\infty$.

Now, we are in the position to present our main results.

Theorem 3.1. Let $(F_1)$ and $(F_2)$ hold. Moreover,

$(q) \quad$ for any $(n, m)\in\Omega$, $q(n, m) < 0$.

Then, (1.1) possesses at least two nontrivial $(T_{1}, T_{2})$-periodic solutions.

Theorem 3.2. Suppose $(F_1)$ and $(F_2)$ are satisfied. If $\mathop{\sum}\limits_{n = 1}^{T_1}\mathop{\sum}\limits_{m = 1}^{T_2}F((n, m), u)\geq 0$ and

$(q') \quad$ $q(n, m)\equiv 0$, $\forall (n, m)\in\Omega$,

then (1.1) admits at least two nontrivial $(T_{1}, T_{2})$-periodic solutions.

Recall $C_{1} = (T_{1}T_{2})^{-\frac{\eta-1}{2(\eta+1)}}$, $\lambda_{2} = 2(1-\cos\frac{2\pi}{T_{1}})$ and $\mu_{2} = 2(1-\cos\frac{2\pi}{T_{2}})$. We have the following.

Theorem 3.3. If $(q)$ and

$(F_3)$

hold, then equation

has at least a $(T_{1}, T_{2})$-periodic solution.

Remark 3.2. In fact, (1.2) and (1.3) are special cases of (1.1). Consider (1.1) with $r(n, m)\equiv0$, and (1.1) can be written in the form of $(1.2)$. Meanwhile, if $q(n, m) = 0$ and $\eta = 1$, then (1.1) changes to (1.3). Moreover, our Theorems 3.1–3.3 are able to override Theorem 3.2 of ^[9], Theorem 3.1 of ^[21] and Theorem 3.3 of ^[9]. Consequently, (1.1) is a generalization of both (1.2) and (1.3), and our results are more universal.

Before stating proofs of Theorems 3.1–3.3, we need to prove the compactness of $I$ first.

Lemma 3.1. Assume $(F_2)$ holds. Then, $I$ satisfies the $P. S.$ condition on $E$.

Proof. Assume that for any $\{u_k\}\subset E$ there exists a constant $M > 0$ such that

As $E$ is a finite-dimensional space, we only need to prove that $\{u_k\}$ is bounded. By $(F_2)$, we have

Moreover, due to (2.1) and (2.3), it follows that

Therefore, combining (3.2) with (3.3), it yields that

That is,

Since $\beta > \eta+1$, (3.4) ensures that $\{u_k\}\subset E$ is a bounded sequence. Consequently, $I$ satisfies the $P. S.$ condition, and the proof is finished.

Proof of Theorem 3.1 By Lemma 3.1, $I$ satisfies the $P. S.$ condition on $E$. In the following, we verify conditions $(f_1)$ and $(f_2)$ of Lemma 2.1 to complete the proof.

In fact, from $(F_1)$, it follows that

Then, there is a $\rho > 0$ such that

Hence, for any $u\in E$ with $\|u\|\leq\rho$, we obtain

Take $a = -\cfrac{q_{\max}C_{1}^{\eta+1}}{2(\eta+1)}\rho^{\eta+1} > 0$, and then (3.5) ensures $I(u) |_{\partial B_{\rho}}\geq a > 0$. Thus, $(f_1)$ of Lemma 2.1 is fulfilled.

Given $\omega\in E$ with $\|\omega\| = 1$ and $\alpha > 0$, we have

which means that there exists $\alpha > \rho$ large enough such that $I(u_0) < 0$, where $u_{0} = \alpha\omega\in E \backslash B_{\rho}$. Moreover, $I(0) = 0$. Thus, Lemma 2.1 guarantees that there is a critical value $c\ge a > 0$. Assume $\bar{u}$ is a critical point, namely, $I(\bar{u}) = c$ and $I'(\bar{u}) = 0$.

In the following, we look for another critical point $\tilde{u}$ for $I$. By (3.4), we get

which indicates $I$ is bounded from above. Denote the supremum of ${\{I(u)\}}_{u\in E}$ by $c_{\max}$, and then $\tilde{u}\in E$ and $I(\tilde{u}) = c_{\max}$. Obviously, $\tilde{u}\ne 0$. If $\bar{u}\ne\tilde{u}$, the proof is finished. Else, $c = c_{\max}$. Lemma 2.1 means that

where

Hence, for any $h\in\Gamma$, $c_{\max} = \mathop{\max}\limits_{t\in [0, 1]}I(h(t))$. In view of $I(h(t))$ being continuous with respect to $t$, $I(0)\leq 0$ and $I(u_0) < 0$, it follows that there exists a $t_0\in(0, 1)$ such that $I(h(t_0)) = c_{\max}$. Choose $h_1, h_2$ such that $\{h_{1}(t)|t\in(0, 1)\}\cap\{h_{2}(t)|t\in(0, 1)\} = \emptyset$, and then there exist $t_1, t_2\in (0, 1)$ such that $I(h_{1}(t_1)) = I(h_{2}(t_2)) = c_{\max}$. Thus, we obtain two different critical points $u_{1} = h_{1}(t_{1})$ and $u_{2} = h_{2}(t_{2})$. Consequently, there exist at least two nontrivial critical points which correspond to the critical value $c_{\max}$. This completes the proof.

Proof of Theorem 3.2 Let $W = \{w\in E|w = \{c\}, c\in\mathbb{R}\}$, $Y = W^\perp$, and then $E = W\oplus Y$. By $(F_1)$, there exist some $\rho > 0$ and $u\in B_{\rho}$ such that

Then, for every $u\in(\partial B_{\rho})\cap Y$, one obtains

Set $a = \cfrac{\left(p_{\min}\lambda_{2}^{\frac{\eta+1}{2}}+r_{\min}\mu_{2}^{\frac{\eta+1}{2}}\right)C_{1}^{\eta+1}\rho^{\eta+1}}{2(\eta+1)}$, and then (3.6) implies $I(u)\geq a$, $u\in(\partial B_{\rho})\cap Y$. Thus, $(f_3)$ of Lemma 2.2 is valid.

Let $e\in\partial B_{1}\cap Y$, and for every $w\in W$ and $s\in\mathbb{R}$, set $u = se+w$. From (2.4) together with $(F_2)$, it follows that $\triangle_{1}w = \triangle_{2}w = 0$, $\|e\| = 1$, and

Write

Then, both $g_{1}(s)$ and $g_{2}(\tau)$ are bounded from above. Moreover, $\beta > \eta+1$ leads to $\lim\limits_{s\rightarrow +\infty}g_{1}(s) = -\infty$ and $\lim\limits_{\tau\rightarrow +\infty}g_{2}(\tau) = -\infty$. Thus, there exists a positive constant $R_{0} > \rho$ such that $I(u)\leq 0$ holds for any $u\in\partial Q$ and $Q = \left(\bar{B}_{R_{0}}\cap W\right)\oplus\left\{se|0 < s < R_{0}\right\}$.

Notice that Lemma 3.1 shows $I$ satisfies the $P. S.$ condition on $E$. Therefore, Lemma 2.2 ensures that $I$ admits a critical value $c\geq a$, and

Take $\bar{u}\in E$ to be a critical point which corresponds to $c$, that is, $I(\bar{u}) = c$. By (3.4), $I$ is bounded from above. Hence, there will be a $\tilde{u}\in E$ such that

Then, $\bar{u}$ and $\tilde{u}$ are nontrivial $(T_{1}, T_{2})$-periodic solutions of (1.1). If $\bar{u}\ne\tilde{u}$, then Theorem 3.2 holds. Otherwise, $\bar{u} = \tilde{u}$, and then $c = c_{\max}$, that is,

Choose $h = id$, and we get $\mathop{\sup}\limits_{u\in\bar{Q}}I(u) = c_{\max}$. Consider $-e\in\partial B_\rho\cap Y$. With the arbitrariness of $e$, it follows that there is an $R_{1} > \rho$ such that $I|_{\partial Q_{1}}\leq 0$, where $Q_{1} = \left(\bar{B}_{R_{1}}\cap W\right)\oplus\left\{-se|0 < s < R_{1}\right\}$. In the same way, by Lemma 2.2, $I$ possesses a new critical value $c'\geq a > 0$, and

If $c'\ne c_{\max}$, then this proof is done. If $c' = c_{\max}$, then for any $h\in\Gamma$, we have $\mathop{\max}\limits_{u\in\bar{Q}_{1}}I(h(u)) = c_{\max}$. Specifically, take $h = id$, and then $\mathop{\max}\limits_{u\in\bar{Q}_{1}}I(u) = c_{\max}$. Since $I|_{\partial Q}\leq 0$, $I|_{\partial Q_{1}}\leq 0$ and $c_{\max} > 0$, the maximum value of $I$ is given at a point inside $Q$ and $Q_1$, respectively. In addition, $Q\cap Q_{1}\subset W$, and for every $w\in W$, we have

Therefore, a point $\hat{u}$ which is different from $\tilde{u}$ must exist in $E$ such that $I(\hat{u}) = c' = c_{\max}$. In summary, if $c < c_{\max}$, then (1.1) admits at least two nontrivial $(T_{1}, T_{2})$-periodic solutions; if $c = c_{\max}$, then (1.1) admits an infinite number of nontrivial $(T_{1}, T_{2})$-periodic solutions. This completes the proof.

Proof of Theorem 3.3 Similar to (2.2), the variational functional associated to (3.1) is expressed by

In the following, we utilize Lemma 2.3 to finish the proof. To begin with, it is to show that $\hat{I}$ meets the $P. S.$ condition on $E$. Suppose $\{u_k\}\subset E$, and there is a constant $\hat{M} > 0$ such that

Since the dimension of $E$ is $T_1T_2$, it is necessary for us to show $\{u_k\}$ is bounded in $E$. In view of (3.7) and the oddness of $\eta$, there holds

Recall $\eta+1 > 1$, and then (3.8) implies that $\{u_{k}\}$ is bounded in $E$. Therefore, $\hat{I}$ satisfies the $P. S.$ condition on $E$.

Next, we show $(f_5)$ and $(f_6)$ of Lemma 2.3 are met. For any $w = (z, \cdots, z)^{T}\in W$, one has

Take

Then

Thus,

which means that $(f_{5})$ of Lemma 2.3 holds.

For $y\in Y$, one has

and the last inequality is obtained by minimization with respect to $\|y\|$. Set

Together with $(F_3)$, this yields that

So, $(f_6)$ of Lemma 2.3 holds by taking $e = 0$. Thus, all conditions of Lemma 2.3 are satisfied, and (3.1) admits at least a $(T_{1}, T_{2})$-periodic solution.

4.   Examples

Finally, we give three examples to demonstrate the validity of our results. Let

Example 4.1. Take $\eta = 3$. Consider Eq (1.1) with $p(n, m) > 0$, $r(n, m)\geq 0$ and $q(n, m) < 0$ for any $n, m\in \mathbb{Z}$, and

Then,

Obviously, $f((n, m), u)$ satisfies all conditions of Theorem 3.1. Then, (1.1) with $f((n, m), u) = 6u^5$ admits at least two nontrivial $(T_{1}, T_{2})$-periodic solutions.

Specially, let $p(n, m) = r(n, m) = 1$, $q(n, m) = -2$ and $T_{1} = T_{2} = 2$, and then (1.1) can be rewritten in the form of

Using MATLAB, we find that (4.1) has at least two nontrivial solutions $u_{1} = \{u_{1}(n, m)\}\in\tilde{E}$ and $u_{2} = \{u_{2}(n, m)\}\in\tilde{E}$, where

By Remark 1.1, $u_{1}$ and $u_{2}$ are two different nontrivial (2, 2)-periodic solutions to (4.1).

Example 4.2. For every $n, m\in \mathbb{Z}$, consider (1.1) with $p(n, m) > 0$, $r(n, m)\geq 0$ and $q(n, m) = 0$. Set $\eta = 3$, and

Then, all conditions of Theorem 3.2 are satisfied, and (1.1) has at least two nontrivial $(T_{1}, T_{2})$-periodic solutions.

Take $p(n, m) = r(n, m) = 1$ and $T_{1} = T_{2} = 2$, and then (1.1) becomes

Utilizing MATLAB, (4.2) has at least two nontrivial solutions

From Remark 1.1, (4.2) has at least two different nontrivial (2, 2)-periodic solutions $u_{1}$ and $u_{2}$.

Example 4.3. Set $T_{1} = T_{2} = 2$, $\eta = 3$ and $f(n, m) = 2$. Consider (3.1) with $p(n, m) = 1$, $r(n, m) = 1$ and $q(n, m) = -2$. Clearly, $p(n, m)$, $r(n, m)$, $q(n, m)$ and $f(n, m)$ are all (2, 2)-periodic, and (3.1) is in the following form:

Obviously, we only need to verify $(F_3)$. Simple calculation gives

Thus, Theorem 3.3 ensures that (4.3) possesses at least one nontrivial (2, 2)-periodic solution. By MATLAB and Remark 1.1, (4.3) admits at least a (2, 2)-periodic solution

Acknowledgments

This work is supported by the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT_16R16).

Conflict of interest

All authors declare no conflicts of interest in this paper.