#### Abstract

This paper concerns the existence of solutions for the Dirichlet boundary value problems of -Laplacian difference equations containing both advance and retardation depending on a parameter . Under some suitable assumptions, infinitely many solutions are obtained when lies in a given open interval. The approach is based on the critical point theory.

#### 1. Introduction

Let and be the sets of integers and real numbers, respectively. For , denotes the discrete interval if .

Assume that is a positive integer. In this paper, we consider the following boundary value problem of difference equation containing both advance and retardation:where is a positive real parameter, the forward difference operator is defined as . For every , , is a continuous function, is the -Laplacian operator, that is, , and .

As it is to know, difference equations have been widely used in various research fields such as computer science, economics, network, and control system. Relevant examples and mathematical models can be found in [1–3]. By means of critical point theory, many researchers devote themselves to the study of difference equations and achieve many excellent results, for example, the results on boundary value problems [4–16], periodic solutions [17–22], and homoclinic solutions [23–31] had been obtained.

Difference equations containing both advance and retardation have many applications in physical and biological phenomena [32]. Yu et al. [23] obtained the existence of nontrivial homoclinic orbits for the following second-order difference equation containing both advance and retardation:

The operator is a second-order difference operator given bywhere and are real valued for each and .

Mei and Zhou [21] considered the existence of the periodic and subharmonic solutions of a order -Laplacian difference equation containing both advances and retardations:

There are more about the results of the difference equations containing both advance and retardation which can be seen in [13, 19, 22, 24].

Here, we are interested in using critical point theory to investigate infinitely many solutions for (1) containing both advance and retardation.

In fact, there are some papers which studied the existence of infinitely many solutions for the boundary value problems of difference equations. Bonanno and Candito [33] in 2009 proved the existence of infinitely many solutions of the following discrete boundary value problem:

Recently, Zhou and Ling [10] studied the existence of infinitely many positive solutions for the following boundary value problem of the second order nonlinear difference equation with -Laplacian:

However, to the best of our knowledge, no similar results are obtained for problem (1) containing both advance and retardation. The main difficulty is caused by the advance and retardation. In this paper, we obtain some sufficient conditions to guarantee the existence of infinitely many solutions of (1). In fact, under some assumptions, we prove the existence of infinitely many solutions of equation (1) for each in Theorem 1. Moreover, Theorem 2 guarantees the existence of infinity positive solutions for (1) by applying a strong maximum principle. Finally, we show that (1) possesses a sequence of distinct solutions which converges to zero for each in Theorem 3.

This paper is organized as follows. In Section 2, some definitions and preliminaries on difference equations are collected. In Section 3, our main results are established. Finally, two examples are given to illustrate our main results.

#### 2. Preliminaries

Let be a reflexive real Banach space and be a function satisfying the following structure hypothesis: Assume that is a real positive parameter. , , where , is coercive, that is, .

Provided that , write

Obviously, and . When or , we agree to read or as .

Lemma 1 (see [34]). *Assume that condition holds, and one has*(a)*For every and every , the restriction of functional to admits a global minimum, which is a critical point (local minimum) of in .*(b)*If , then for each , the following alternative holds: possesses a global minimum There is a sequence of critical points (local minimum) of such that .*(c)

*If , then for each , the following alternative holds:*

*There is a global minimum of which is a local minimum of**There is a sequence of pairwise distinct critical points (local minimum) of , with , which weakly converges to a global minimum of .*Consider the -dimensional Banach space:endowed with the norm

We define another two norms on as follows:

According to Lemma 2.2 in [12], we have the following inequality:for every .

Lemma 2. *For every , one haswhere .*

*Proof. *Making use of (11), we obtainthen

For every , putwhere satisfies the following conditions: , , , for Direct computation ensures that is a functional of class on *S* withThen, is a critical point of on *S* if and only ifThat is, the function is just the variational framework of (1). For the reader’s convenience, we recall a consequence of strong maximum principle [8]. The strong maximum principle is used to obtain positive solutions to (1), that is, for every .

Lemma 3 (see [8]). *Fix such that**Then, either or .*

*Remark 1. *Assume is continuous for each and for all . Putthen for each .

Consider the following boundary value problem:From Lemma 3, all solutions of (19) are either zero or positive; hence, they are also solutions for (1). When (19) possesses nontrivial solutions, (1) possesses positive solutions, independently of the sign of .

#### 3. Main Results

Put

Now, we consider the suitable oscillating behavior of when goes to . We have the following theorem.

Theorem 1. *Assume that are satisfied and there exist two real sequences , with , such that*

Then, problem (1) admits an unbounded sequence of solutions for each .

*Proof. *Our aim is to apply Lemma 1(b) to prove our conclusion. Fix , which clearly holds. Our conclusion needs to provide that . Put for all . Owing to (12), if , then for every , and we obtainLet for every , and . Clearly, and . From (21), we obtain , and one hasHence, .

Now, we verify that is unbounded from below. By (20), let be a positive real sequence with andFirst, we assume that , and then we can fix , and there exists such thatWe take a sequence in such that for every , , and one hasfor all . That is, .

Next, we assume that . Since , from (25), we can fix such that , arguing as before, there is a such thatBy choosing in as above, we obtain thatfor all . Hence, one has .

We have verified all assumptions of Lemma 1(b); then, there is a sequence of critical points (local minima) of such that .

According to Theorem 1, it is easy to obtain the following corollary.

Corollary 1. *Let be satisfied, and assume that*

Then, problem (1) admits an unbounded sequence of solutions for each .

*Proof. *Let be a real sequence with such thatWe take for all in (21); combining with , (30) and (31), we can apply Theorem 1 to reach the conclusion.

From the argument of Remark 1, we have the following theorem and corollary.

Theorem 2. *If for every and the hypotheses of Theorem 1 hold. Then, problem (1) admits an unbounded sequence of positive solutions for each .*

Corollary 2. *If for every and the hypotheses of Corollary 1 are satisfied, then problem (1) admits an unbounded sequence of positive solutions for each .*

Next, we consider the oscillating behavior of when goes to 0. We obtain the following theorem.

Theorem 3. *Assume that are satisfied andwhere and . Then, problem (1) possesses a sequence of pairwise distinct solutions, which converges to zero for each .*

*Proof. *We will check the part of Lemma 1. Fix , let be a positive real sequence such that andAs before, we let for each . In view of (12), note that implies for every , by the definition of , we havethen .

Clearly, 0 is a global minimum of in , and by .

Moreover, we can verify that 0 is not a local minimum of . Given a positive real sequence with such thatif , fix , and there exists such thatLet be a sequence satisfying for every , . Since as , we havefor all . Thus, 0 is not a local minimum of .

If , since , we can also find a positive real sequence with such that (35) holds. Fix , and there exists , such thatArguing as before, there exist a sequence , such thatfor all . Obviously, 0 is not a local minimum of , and of Lemma 1(c) is true.

A similar result to Theorem 2 is obtained as follows.

Theorem 4. *If for every and the hypotheses of Theorem 3 hold, then problem (1) admits a sequence of pairwise distinct positive solutions, which converges to zero for each .*

#### 4. Examples

Finally, we give two examples to illustrate our results.

*Example 1. *Consider the boundary value problem (1) withfor . PutObviously, satisfies conditions and . It is easy to see thatClearly,andLet be sufficiently small, then . Hence, for each , (1) admits an unbounded sequence of solutions by Corollary 1. Moreover, we have for all ; according to Corollary 2, (1) admits an unbounded sequence of positive solutions.

*Example 2. *Consider the boundary value problem (1) withif and for . Letthen holds. Clearly, holds andBy computation, we obtainObviously, we have , when be sufficiently small. On the contrary, for all .

The computations ensure that all the assumptions of Theorem 4 are satisfied. Then, for each , (1) admits a sequence of pairwise distinct positive solutions, which converges to zero.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Authors’ Contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11971126), Program for Changjiang Scholars and Innovative Research Team in University (Grant no. IRT_16R16), and Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi (Grant no. 2019L0955).