Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 6703503, 9 pages
https://doi.org/10.1155/2018/6703503

Research Article

Existence and Nonexistence of Solutions for Fourth-Order Nonlinear Difference Boundary Value Problems via Variational Methods

Xia Liu,1,2 Tao Zhou,³ and Haiping Shi4,5

¹College of Continuing Education and Open College, Guangdong University of Foreign Studies, Guangzhou 510420, China
²Science College, Hunan Agricultural University, Changsha 410128, China
³School of Economics and Management, South China Normal University, Guangzhou 510006, China
⁴Modern Business and Management Department, Guangdong Construction Polytechnic, Guangzhou 510440, China
⁵School of Mathematics and Statistics, Central South University, Changsha 410083, China

Correspondence should be addressed to Xia Liu; xia991002@163.com

Received 3 June 2018; Accepted 2 September 2018; Published 16 September 2018

Academic Editor: Pasquale Candito

Copyright © 2018 Xia Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is concerned with boundary value problems for a fourth-order nonlinear difference equation. Via variational methods and critical point theory, sufficient conditions are obtained for the existence of at least two nontrivial solutions, the existence of distinct pairs of nontrivial solutions, and nonexistence of solutions. Some examples are provided to show the effectiveness of the main results.

1. Introduction

Throughout this paper, we denote by , , the sets of all natural numbers, integers, and real numbers, respectively. Let the symbol denote the transpose of a vector. For any integers and with ,   is defined by the discrete interval .

Now, we are concerned with the existence and nonexistence solutions to the fourth-order nonlinear difference equation

satisfying the boundary value conditions

where ,   is the forward difference operator defined by ,  ,  ,   with ,  ,  ,  .

As usual, a solution of (1), (2), in other words, a function , satisfies both (1) and (2).

We may think of boundary value problem (BVP) (1), (2) as being a discrete analogue of the following fourth-order nonlinear differential equation:

with boundary value conditions

(3) includes the following differential equation:

which is used to describe the bending of an elastic beam [1]. Equations similar in structure to (3) have been studied by many researchers using a variety of methods; see, for example, [2–12].

It is well-known that the study of nonlinear difference equations [13–29] has long been an important one as a result of the fact that they arise in numerical solutions of both ordinary and partial differential equations as well as in applications to different areas of applied mathematics and physics.

Domshlak and Matakaev [17] in 2001 investigated the oscillation properties of the delay difference equation

for and near the 2-periodic critical states with respect to its oscillation properties. By making use of “the Sturmian comparison method: discrete version”, they obtained some conditions for the existence and for the nonexistence of eventually positive solution.

Using the critical point theory, Yang [27] studied the following higher order nonlinear difference equation:

with boundary value conditions

Some sufficient conditions for the existence of the solution to the boundary value problem (7), (8) are obtained.

In 2010, He, Yang and Yang [18] considered the following second-order three-point discrete boundary value problem:

By using the topological degree theory and the fixed point index theory, they provided sufficient conditions for the existence of sign-changing solutions, positive solutions, and negative solutions.

Investigating the high order difference equation

Leng [22] established some new criteria for the existence and multiplicity of periodic and subharmonic solutions of (10) based on the linking theorem in combination with variational technique.

Leszczyński [23] considered the difference equation with both advance and retardation,

with boundary value conditions

They applied the direct method of the calculus of variations and the mountain pass technique to prove the existence of at least one and at least two solutions. Nonexistence of nontrivial solutions is also undertaken.

In this paper, we shall study the boundary value problem for a fourth-order nonlinear difference equation (1), (2). Via variational methods and critical point theory, sufficient conditions are obtained for the existence of at least two nontrivial solutions, the existence of distinct pairs of nontrivial solutions, and nonexistence of solutions. The motivation for the present work stems from the recent papers [3, 5].

Throughout the whole paper, we suppose that there exists a function such that

for any .

Let

and

The remainder of this article is organized as follows. In Section 2, we shall give some preliminary lemmas and establish the variational structure of BVP (1), (2). In Section 3, we shall give sufficient conditions to the existence and nonexistence solutions. In Section 4, we shall complete the proofs of the main results. Some examples illustrating our main results are given in Section 5.

For the basic knowledge of variational methods, the reader is referred to [30–32].

2. Preliminary Lemmas

Assume that is a real Banach space and is a continuously Fréchet differentiable functional defined on . As usual, is said to satisfy the Palais-Smale condition if any sequence for which is bounded and as possesses a convergent subsequence. Here, the sequence is called a Palais-Smale sequence.

Let be a real Banach space. We denote by the symbol the open ball in about 0 of radius , its boundary, and its closure.

In the present article, we define a vector space by

and for any , define

and

Remark 1. For any , it is easy to see that

As the case stands, is isomorphic to . In the following and in the sequel, when we write , we always imply that can be extended to a vector in so that (19) is satisfied.

For any , let the functional be denoted by

Then and

Thus, if and only if

Thereupon a function is a critical point of the functional on if and only if is a solution of BVP (1), (2).

Let be the matrix. If , let

where .

If , let

If , let

If , let

If , let .

Then the functional can be rewritten as

Lemma 2 (linking theorem [30]). Let be a real Banach space, , where is finite dimensional. Assume that satisfies the Palais-Smale condition and the following:

There are positive constants and such that There is a and a positive constant such that , where . Then possesses a critical value , where

 and , where denotes the identity operator.

Lemma 3 (Clark theorem [30]). Let be a real Banach space, , with being even, bounded from below, and satisfying Palais-Smale condition. Suppose , there is a set such that is homeomorphic to (-1 dimension unit sphere) by an odd map, and . Then has at least distinct pairs of nonzero critical points.

3. Main Results

We now state our main theorems in this paper.

Theorem 4. Suppose that the function and the following conditions are satisfied:

For any ,  .For any ,  ,   and .,  ,  .There are constants and such that

There are constants and such that

 Here and are constants which can be referred to (32) and (33). Then BVP (1), (2) has at least three solutions.

Corollary 5. Suppose that the function and the conditions , , , , and are satisfied. Then BVP (1), (2) has at least two nontrivial solutions.

Theorem 6. Suppose that the function ,  ,  ,   and the following conditions are satisfied:

For any , .For any , there are constants , and such that

 Then BVP (1), (2) has at least three solutions.

Corollary 7. Suppose that the function , , , and the conditions and are satisfied. Then BVP (1), (2) has at least two nontrivial solutions.

Theorem 8. Suppose that the function , , , , , and the following condition are satisfied:

. Then BVP (1), (2) has at least distinct pairs of nontrivial solutions, where is the dimension of which can be referred to (34).

Theorem 9. Suppose that the following conditions are satisfied:

For any ,  .For any ,  .For any ,  . Then BVP (1), (2) has no nontrivial solutions.

4. Proofs of the Main Results

In this section, we shall finish proofs of the main results via variational methods.

Proof of Theorem 4. The matrix satisfies that is positive semidefinite. In fact, from , it is obvious that 0 is an eigenvalue of with an eigenvector . Define the eigenvalues of by .
Denote

and

Let . Clearly, is an invariant subspace of . The space is defined by

Let be such that is bounded and as Thus, for any , there exists a positive constant such that

For any , it comes from (27) and that

Therefore,

From , we have . (37) implies that is a bounded in . Since the dimension of is finite, possesses a convergent subsequence. Consequently, the functional satisfies the Palais-Smale condition. Therefore, it suffices to prove that satisfies the conditions and of Lemma 2.
For any , it follows from , , and that and . Hence, is a trivial solution of BVP (1), (2).
From the proof of (36), we have that is bounded from above in . Denote

As a consequence, on the one hand, there is a sequence in such that

By (27), on the other hand, the functional satisfies

(40) means that which implies that is bounded. As a result, has a convergent subsequence in denoted by . Let

By the reason of the continuity of in , it is easy to see that . That is, is a critical point of .
For any , via (27) and , the functional satisfies

Take

Then

In other words, there exist two positive constants and such that . Hence, satisfies of Lemma 2.
For any ,  . Combining with , we have

Therefore, and the critical point of corresponding to the critical value is a nontrivial solution of BVP (1), (2).
Next, we shall prove of Lemma 2.
Take ; for any and , let . By , we have

Thereby, there exists a constant such that

where . According to Lemma 2, has a critical value , where

and .
Similar to the proof of Theorem 1.1 in [22], we can prove that BVP (1), (2) has at least three solutions. For simplicity, its proof is omitted.

Remark 10. From the course of the proof of Theorem 4, the conclusion of Corollary 5 is evidently correct.

Remark 11. The techniques of the proof of Theorem 6 are just the same as those carried out in the proof of Theorem 4. We do not repeat them here.

Remark 12. According to Theorem 6, it is easy to see that the conclusion of Corollary 7 is true.

Proof of Theorem 8. For any , by the continuity of in , can be viewed as a continuously differentiable functional defined on . It comes from and that . Owing to the condition , is even. From the process of proof of Theorem 4, is bounded from below and satisfies the Palais-Smale condition. Next, in the light of Clark Theorem, we shall find a set and an odd map such that is homeomorphic to by an odd map.
Set

It is obvious that is homeomorphic to by an odd map. (44) implies that . As a result of Clark Theorem, has at least distinct pairs of nonzero critical points. As a consequence, BVP (1), (2) has at least distinct pairs of nontrivial solutions. The desired result is obtained.

Proof of Theorem 9. On the contrary, we suppose that BVP (1), (2) has a nontrivial solution. Therefore, has a nonzero critical point . Since

we have

On the one hand, it follows from and that

On the other hand, by , we have

which is a contradiction with (52).

5. Some Examples

In this section, we shall provide three examples to illustrate our main results.

Example 1. For , assume that

satisfying the boundary value conditions

We have

with

and

Besides,

and the eigenvalues of are , and . It is easy to verify that all the suppositions of Theorem 4 are satisfied and then BVP (54), (55) has at least three solutions.

Example 2. For , assume that

satisfying the boundary value conditions

We have

with

and

Besides,

and the eigenvalues of are , and . It is easy to verify that all the suppositions of Theorem 8 are satisfied and then BVP (60), (61) has at least 3 distinct pairs of nontrivial solutions.

Example 3. For , assume that

satisfying the boundary value conditions

We have

with

and

It is easy to verify that all the suppositions of Theorem 9 are satisfied and then BVP (66), (67) has no nontrivial 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

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

This project is supported by the National Natural Science Foundation of China (No. 11501194). This work was carried out while visiting Central South University. The author Haiping Shi wishes to thank Professor Xianhua Tang for his invitation.

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