Infinitely Many Positive Solutions for a Coupled Discrete Boundary Value Problem

Abstract

In this paper, we obtain some results for the existence of infinitely many positive solutions for a coupled discrete boundary value problem. The approach is based on variational methods.

1. Introduction

Let and denote the sets of integers and real numbers, respectively. For , define , when .

In this paper, we consider the following coupled discrete boundary value problem:for all , , where , and is a positive parameter, is the forward difference operator, for all , and for all , is a function in satisfying =0 for every , and denotes the partial derivative of with respect to for .

As we know, results for existence of solutions for difference equations have been widely studied because of their applications to various fields of applied sciences, like mechanical engineering, control systems, computer science, economics, artificial or biological neural networks, and many others. Many scholars have studied such problems and main tools are fixed point methods, Brouwer degree theory, and upper and lower solution techniques; see [1–4] and references therein. In recent years, variational methods have been employed to study difference equation and various results have been obtained. See, for instance, [5–22].

More recently, especially, in [23–31], by starting from the seminal papers [32, 33], many results for the existence and multiplicity of solutions for discrete boundary value problems have been obtained also by adopting variational methods.

However, these papers only deal with a single equation. For instance, in [23], by studying the Dirichlet discrete boundary value problemthe author obtained the existence of two positive solutions of the problem through appropriate variational methods. In this paper, we consider system (1) with difference equations by using Ricceri’s variational principle proposed in [33]. In Theorem 4 and Remark 5, we prove the existence of an interval such that, for each , problem (1) admits a sequence of positive solutions which is unbounded in . To the best of our knowledge, this is the first time to deal with coupled discrete boundary value problems. This method has already been used for the continuous counterparts [34–36]. The monographs [37–39] are related books of the critical point theory and difference equations.

The rest of the paper is organized as follows: Section 1 consists of some definitions and mathematical symbols. In Section 2, we emphasize that a strong maximum principle (Lemma 1) is presented so that if , for all and , our results guarantee the existence of infinitely many positive solutions (Remark 5 in Section 3). Section 3 contains a more precise version (Lemma 3) of Ricceri’s variational principle, the statements and proofs of the main results (Theorem 4 and Remark 5), two corollaries (Corollaries 6 and 7), and an example (Example 10).

Throughout this paper, we let and be the Cartesian product of Banach spaces ; i.e., endowed with the normwhere which is a norm in .

Put where for every , and where for all .

2. Preliminaries

First, we establish a strong maximum principle.

Lemma 1. Fix such that either orThen either in or

Proof. Let such thatIf , then it is clear that in .
If , then by (9), we havethat is On the other hand, by the definition of , we see thatThus, by (12), we obtain that
By similar arguments applied to and and continuing in this way, we have

Remark 2. Let be such that for all and .
Put Clearly, is continuous in for each . Owing to Lemma 1, all solutions of problemare either zero or positive and hence are also solutions for problem (1). Hence we emphasize that when (15) admits nontrivial solutions, then problem (1) admits positive solutions, independently of the sign of .

3. Main results

Let be a reflexive real Banach space and let be a function satisfying the following structure hypothesis:  for all , where are two functions of class on with coercive; i.e., , and is a real positive parameter.

Provided that , put and Clearly, . When , in the sequel, we agree to read as .

For the readers’ convenience, we recall a more precise version of Theorem 2.1 of [32] (see also Theorem 2.5 of [33]) which is the main tool used to investigate problem (1).

Lemma 3. Assume that condition holds. If , then, for each , the following alternative holds: either  possesses a global minimum, or there is a sequence of critical points (local minima) of such that

Put Our main result is the following theorem.

Theorem 4. Assume that  (i) is nonnegative in  (ii) , where and are given by (6) and (8), respectively Then, for each , system (1) admits an unbounded sequence of solutions.

Proof. For each , put and Standard arguments show that and that critical points of are exactly the solutions of problem (1). In fact, ; that is, and are continuously Fréchet differentiable in . Using the summation by parts formula and the fact that for any , we get and for any .
Moreover, we obtain for all .
Now we verify that . Let be a real sequence such that , andIt follows from [24] thatfor . And put for all . Hence a computation ensures that whenever .
Taking into account the fact that , one has Therefore, since from assumption (ii) one has , we obtainNow fix . We claim that is unbounded from below. Let be positive real sequences such that , andfor all .
For each , let for all .
Clearly, , and Therefore, we have for all .
If , let . By (29) there exists such that for all . Moreover, for all . Taking into account the choice of , we have If , let us consider . By (29) there exists such that for all . Moreover, for all . Taking into account the choice of , in this case we also have Due to Lemma 3, for each , the functional admits an unbounded sequence of critical points, and the conclusion is proven.

Remark 5. When for all and , owing to Remark 2, the solutions in the conclusion of Theorem 4 are positive.
It is interesting to list some special cases of the above results.

Corollary 6. Assume that  () is nonnegative in  () Then, for each , the system for , admits an unbounded sequence of solutions.

Corollary 7. Let be a -function and assume that  () is nonnegative in  (),where and Then, for each , the system for , admits an unbounded sequence of solutions.

Remark 8. When for all and , owing to Remark 2, the solutions in the conclusion of Corollary 6 are positive.

Remark 9. When for , owing to Remark 2, the solutions in the conclusion of Corollary 7 are positive.
Now we give an example to illustrate our results.

Example 10. Let and consider the increasing sequence of positive real numbers given by for every .
Define the function as follows: If for some positive integer , then otherwise, where denotes the open unit ball of center .
By the definition of , we see that it is nonnegative in and . Further it is a simple matter to verify that . We will denote by and , respectively, the partial derivative of with respect to and . Now, for every , the restriction attains its maximum in and one has . Obviously,owing to the fact that On the other hand, by setting for every , one has Thenand henceFinallyThe previous observations and computations ensure that all the hypotheses of Corollary 7 are satisfied. Then, for each , the problemadmits an unbounded sequence of solutions.
Taking partial derivative to with respect to gives if for some positive integer ; otherwise, .
It is easy to see thatIn a similar way, we obtainConsequently, according to Remark 9, problem (51) admits an unbounded sequence of positive solutions.

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

Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11571084) and the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT-16R16).