Infinitely Many Solutions for the Discrete Boundary Value Problems of the Kirchhoff Type

In this paper, we study the existence and multiplicity of solutions for the discrete Dirichlet boundary value problem of the Kirchhoff type, which has a symmetric structure. By using the critical point theory, we establish the existence of infinitely many solutions under appropriate assumptions on the nonlinear term. Moreover, we obtain the existence of infinitely many positive solutions via the strong maximum principle. Finally, we take two examples to verify our results.


Introduction
Let N be a positive integer and denote with [1, N] the discrete set {1, . . . , N}. In this paper, we consider the following discrete boundary value problem of the Kirchhoff type: where a, b are two positive constants, and ∆ is the forward difference operator defined by ∆u k = u k+1 − u k . ∆ 2 = ∆(∆) and f (k, ×) ∈ C(R, R) for any k ∈ [1, N] and λ ∈ R + . Problem (1) has a symmetric structure in the variable u k ; that is, if we replace u k−1 with u k+1 , and replace u k+1 with u k−1 in (1), then (1) is invariant since ∆ 2 u k−1 = u k+1 + u k−1 − 2u k .
In the past two decades, there has been a lot of interest in the study of difference equations, such as in biology, economics, and other research fields [1][2][3][4][5]. Most results about the boundary value problems of difference equations are proved by using the method of upper and lower solutions as well as fixed-point methods; see [6][7][8][9][10] for more details. In 2003, Guo and Yu [11] discussed the second-order difference equation by using critical point theory, and they obtained the existence of periodic and subharmonic solutions. Since then, many researchers have studied difference equations via critical point theory, including boundary value problems [12][13][14][15][16][17][18], periodic solutions [19,20] as well as homoclinic solutions [21][22][23][24] and heteroclinic solutions [25].
As to problem (2), Zou and He [26] established the existence of infinitely many positive solutions by using variational methods. In the case of λ = 1 in problem (2), Cheng and Wu [27] studied the two existence results, including at least one or no positive solution via variational methods. In 2016, Tang and Cheng [28] studied the existence of ground state sign-changing solutions when λ = 1 in problem (2) by applying the non-Nehari manifold method. As for Kirchhoff's changes and related applications, we refer the reader to [29,30] and the references therein. Problem (2) is related to the stationary case of a nonlinear wave equation such as which was proposed by Kirchhoff [31] as an extension of the classical D'Alembert's wave equation by considering the effects of the changes in the length of the string during the vibrations.
As for the discrete case, when the parameter λ = 1 in problem (1) and f satisfies various assumptions, Yang and Liu [32] studied the existence of at least one nontrivial solution via variational methods and critical groups. A class of partial discrete Kirchhofftype problems was discussed by Long and Deng [33] via invariant sets of descending flow and minimax methods, and some results on the existence of sign-changing solutions, positive solutions, and negative solutions were obtained.
To the best of our knowledge, although most of the previous works have been dedicated to boundary value problems, few have been studied in the discrete problems of the Kirchhoff type. Inspired by the above results, we intend to investigate the multiplicity of solutions for the discrete Kirchhoff-type problem with a Dirichlet boundary value condition by applying critical point theory.

Preliminaries
Let X be a reflexive real Banach space and I λ : X → R be a function satisfying the following structure hypothesis: where Φ, Ψ : X → R are two functions of class C 1 on X, and Φ is coercive, i.e., lim u →∞ Φ(u) = +∞ and λ ∈ R + .
(b) If δ < +∞, then for each λ ∈ (0, 1 δ ), the following alternatives hold: (β 1 ) T is a global minimum of Φ, which is a local minimum of I λ ; (β 2 ) There is a sequence {u n } of pairwise distinct critical points (local minima) of I λ , with lim n→+∞ Φ(u n ) = inf X Φ, which weakly converges to a global minimum of Φ. Now we consider the N-dimensional Banach space S ={u : [0, N + 1] → R : u 0 = u N+1 = 0} and define the norm as follows: From ( [35], Lemma 2.2), we have the following inequality: Let , I λ is also a class of C 1 (S, R). Using the summation by parts method and the boundary condition, one has Thus, u is a critical point of I on S if and only if u is a solution of problem (1). Now we have reduced the existence of a solution for problem (1) to the existence of a critical point of I on S.
Finally, we point out the following two lemmas used to obtain positive solutions for our problem. The first is the following strong maximum principle.

Lemma 2.
Fix u ∈ S, such that either Proof. Let u j = min k∈ [1,N] u k . If u j > 0, then u k > 0 for each k ∈ [1, N], and the conclusion follows. If u j ≤ 0, then we have Owing to a, b > 0, one has ∆ 2 u j−1 ≤ 0. Considering the fact that u j is the minimum, Replacing j with j + 1, we get u j+2 = u j+1 . Continuing this process N + 1 − j times, we have u j = u j+1 = · · · = u N = u N+1 = 0. In the same way, we also get u j = u j−1 = · · · = u 1 = u 0 = 0. Thus, we prove that u ≡ 0, and the proof is complete.
is defined as before. Similarly, the critical points of I λ + are the solutions of the following problem: Proof. From Lemma 2, it follows that all solutions of problem (5) are either zero or positive. Then, problem (1) admits positive solutions when problem (5) admits non-zero solutions. Therefore, the conclusion holds.

Main Results
Let Our main results are the following theorems. and 2b n 2 a(N + 1) + 2b × b n 2 − a n 2 (N + 1) 2 (a + b × a n 2 ) Then, for each λ ∈ b H ∞ , (1) admits an unbounded sequence of solutions.
Next, assuming that H ∞ = +∞, and taking L > 0 such that L > b λ , we also put a real sequence {c n } with lim n→+∞ c n = +∞, such that N ∑ k=1 F(k, c n ) > L × c n 4 , ∀n ∈ N.
Therefore, we prove that γ < +∞ and I λ is unbounded from below in both cases. Bearing in mind Lemma 1 part (α), the proof is complete.

Theorem 2.
Assume that there exist two real sequences {d n } and {e n }, with e n > 0 and lim n→+∞ e n = 0, such that and Then, for each λ ∈ a H 0 , 1 (N+1) 2 G 0 , problem (1) admits a sequence of non-zero solutions that converge to zero.
Our aim is to verify if the global minimum of Φ is different from the local minimum of I λ . As a matter of fact, it is easy to see that the global minimum of Φ is 0, and Φ = 0 if and only if u k = 0 for every k ∈ [1, N]. Therefore, our task is reduced to proving that 0 is not a local minimum of I λ .
Using the same argument as in the proof of Theorem 1, we firstly assume that H 0 < +∞. Since λ > a H 0 , we fix ε > 0, such that H 0 − a λ > ε. Thus, we can take a real sequence {r n } with lim n→+∞ r n = 0, such that Moreover, by taking in S the sequence {µ n } that, for each n ∈ N, is defined by (µ n ) k := r n for every k ∈ [1, N], we have Thus, I λ (µ n ) < 0. Next, assuming that H 0 = +∞, we fix M > 0, such that M > a λ ; we also put a real sequence {r n } with lim n→+∞ r n = 0, such that N ∑ k=1 F(k, r n ) > M × r n 2 , ∀n ∈ N.
Choosing a real sequence {µ n } from S in the same way as mentioned above, we have Therefore, I λ (µ n ) < 0. Hence, the conclusion follows from part (β) of Lemma 1.
By setting particular conditions, we obtain the following consequences. Let

Proposition 1. Assume that
Conditions (6) and (7) of Theorem 1 follow when we take sequence a n = 0 for each n ∈ N. Let for each k ∈ [1, N]. From Lemma 3, our proof is complete.
If h : [0, +∞) → R is a continuous function with h(0) = 0, and σ : [1, N] → R is a non-negative and non-zero function. Then, for each admits an unbounded sequence of positive solutions.
Proof. Let for each k ∈ [1, N] and t ∈ R. Therefore, we have f (k, 0) ≥ 0 for each k ∈ [1, N], and the conclusion follows from Proposition 1.
Then, the conclusion follows from Proposition 1.
Then, the solutions are also positive from Proposition 2.

Remark 3.
If we replace t → +∞ with t → 0 + , we can also obtain the similar propositions and remarks in Theorem 2 in the same way.

Examples
In this section, we present the following examples to illustrate our results.
It is easy to see that h(s) ≥ 0, and when ε is sufficiently small, Hence, condition (13) holds.
Then, from Remark 2, for each λ ∈ 1 admits an unbounded sequence of positive solutions.
Example 2. Let a, b, N be such that Then, for each λ ∈ 1 By applying Remark 3, our aim is achieved and the conclusion holds.

Conclusions
In recent years, Kirchhoff-type problems have been widely studied in the continuous case, while few have been discussed in the discrete case. In this paper, we considered the multiplicity of solutions for the discrete Kirchhoff-type problem with a Dirichlet boundary value condition. In Section 2, we recalled critical point theory and showed some basic lemmas. In Section 3, we proved the existence of infinitely many solutions for problem (1) by using critical point theory. Moreover, we obtained the existence of infinitely many positive solutions by means of the strong maximum principle.