Existence of Solutions to Boundary Value Problems for a Fourth-Order Difference Equation

Abstract

In this article, some new existence criteria of at least one solution to boundary value problems for a fourth-order difference equation are obtained by using the critical point theory. In a special case, a necessary and sufficient condition for the existence and uniqueness of solution is also established. An example of the main result is given.

1. Introduction

Throughout this article, the sets of all natural numbers, integers, and real numbers are defined as , , and , respectively. The transpose of a vector is defined as . For with , the discrete interval is denoted as .

Consider the existence of solutions to boundary value problem (BVP) for a fourth-order difference equation:satisfying the boundary value conditionswhere and , is the forward difference operator denoted as , , , with , , with , , .

And (1), (2) can be regarded as a discrete analogue ofwith boundary value conditions Equations similar in structure to (3) arise in the study of the existence of solutions to differential equations [1–11].

Difference equations are widely found in mathematics itself and in its applications to statistics, computing, electrical circuit analysis, dynamical systems, economics, biology, and so on. Many authors were interested in difference equations and obtained some significant results [12–29].

Consider the fourth-order nonlinear difference equation Thandapani and Arockiasamy [24] in 2001 established some new criteria for the oscillation and nonoscillation of solutions.

Xia [26] in 2017 considered the following second-order nonlinear difference equation with Jacobi operators containing both many advances and retardations. By using variational methods and critical point theory, some new criteria are obtained for the existence of a nontrivial homoclinic solution.

By Krasnoselskii’s fixed point theorems in cones, Cabada and Dimitrov [13] obtained some existence, multiplicity, and nonexistence results for the nonlinear singular and nonsingular fourth-order equation depending on a real parameter In [18], a higher order nonlinear difference equation is studied. By using critical point theory, sufficient conditions for the existence of periodic solutions are established.

In [23], Raafat determined the forbidden set, introduced an explicit formula for the solutions, and discussed the global behavior of solutions of the difference equation where are positive real numbers and the initial conditions are real numbers.

By using the symmetric mountain pass theorem, Chen and Tang [15] established some existence criteria to guarantee the fourth-order difference equation of the form having infinitely many homoclinic orbits.

Using the direct method of the calculus of variations and the mountain pass technique, Leszczyński [20] obtained the existence of at least one and at least two solutions of the difference equation: with boundary value conditions

Our purpose in this article is to use the critical point theory to explore some existence criteria of solutions to the boundary value problem (1), (2) for a fourth-order nonlinear difference equation. In a special case, a necessary and sufficient condition for the existence and uniqueness of solution is also established. Results obtained here are motivated by the recent papers [18, 25].

Define

The rest of this article is organized as follows. In Section 2, we shall introduce some basic notations, set up the variational framework of BVP (1), (2), and give a lemma which will be useful in the proofs of our theorems. Section 3 contains our results of at least one solution. In Section 4, we shall finish proving the main results. In Section 5, we shall provide an example to illustrate our main theorem.

2. Preliminaries

Let be a vector space: For any , denote

Remark 1. It is obvious that In reality, is isomorphic to . In the following and in the sequel, when we write , we always mean that can be extended to a vector in so that (16) is satisfied.

For any , define the functional as It is easy to see that and

For this reason, when and only when

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

Let and be the matrices as follows.

If , let .

If , let

If , let

If , let

If , let where , .

If , letLet . Therefore, we rewrite by

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 the open ball in about 0 of radius , be its boundary, and be its closure.

Lemma 2 (saddle point theorem [30]). Suppose that is a real Banach space, , where and is finite dimensional. Suppose that satisfies the Palais-Smale condition and
there are constants such that
there is and a constant such that
Then admits a critical value , where and is defined as the identity operator.

3. Main Theorems

Our main theorems are as follows.

Theorem 3. Assume that the following assumptions are satisfied.
For any , .
For any , .
For any , .
, , .
For any , where can be referred to (32).
Then the BVP (1), (2) has at least one solution.

If , we obtain the theorem as follows.

Theorem 4. Assume that , , and the following assumptions are satisfied.
For any , .
, where can be referred to (33).
For any , Then the BVP (1), (2) has at least one solution.

If , consider the following equation:with boundary value condition (2).

By a similar argument to that in Section 2, we define the functional as where . Hence, a function is a critical point of the functional on when and only when is a solution of the BVP (29), (2).

It is obvious to find that the critical point of is just the solution of the following linear equation:

Let . Making use of the linear algebraic theory, we can obtain the following necessary and sufficient conditions.

Theorem 5. The BVP (29), (2) has at least one solution when and only when , where defines the rank of the matrix .

Theorem 6. The BVP (29), (2) has a unique solution when and only when .

4. Proofs of the Theorems

In this section, we shall finish proofs of the theorems by using critical point theory.

By , it is evident that is positive semidefinite. is an eigenvalue of and is an eigenvector corresponding to 0. Let be the other eigenvalues of .

Define

Let the space and such that

Proof of Theorem 3. Let be such that is bounded and as Thus, for any , there exists a positive constant such that It comes from Hölder inequality thatBy (25) and (36), we have That is,From the assumption , (38) implies that is bounded in . Since the dimension of is finite, has a convergent subsequence. As a result, satisfies the Palais-Smale condition. Hence, it is sufficient to prove that satisfies the conditions and of Lemma 2.
For any , we have Choose Therefore, Let Thus, The condition of Lemma 2 is proved.
Then, we prove the condition of Lemma 2. For any , by Hölder inequality, we have as one finds by minimization with respect to . In other words, Let It follows from that All the assumptions of Lemma 2 are proved. According to saddle point theorem, the proof of Theorem 3 is complete.

Proof of Theorem 4. Let be such that is bounded and as Thus, for any , there exists a positive constant such that By (25), (33) and Hölder inequality, we have That is,By condition , (51) implies that is a bounded in . Since the dimension of is finite, has a convergent subsequence. Consequently, satisfies the Palais-Smale condition. Hence, it is sufficient to prove that satisfies conditions and of Lemma 2.
For any , let and similar to the proof of we have Condition of Lemma 2 is proved.
Then, we prove the condition of Lemma 2. For any , by Hölder inequality, we have as one finds by maximization with respect to . That is to say, Let By , we have Therefore, satisfies the condition of saddle point theorem. By Lemma 2, the proof of Theorem 4 is finished.

5. Examples

In this section, we shall provide two examples to illustrate our main theorem.

Example 1. For , assume that satisfying the boundary value conditions We have with and the eigenvalues of are , and . Thus, and From the above argument, we see that all the suppositions of Theorem 3 are satisfied; then the BVP (58), (59) has at least one solution.

Example 2. For , assume that satisfying the boundary value conditions We have with and the eigenvalues of are and . Thus, and From the above argument, we see that all the suppositions of Theorem 3 are satisfied; then the BVP (63), (64) has at least one solution.

6. Conclusions

Difference equations, the discrete analogue of differential equations, occur widely in numerous settings and forms, both in mathematics itself and in its applications in theory and practice. The boundary value problem discussed in this paper has important analogue in the continuous case of the fourth-order differential equation. Such problem is of special significance for the study of beam equations which are used to describe the bending of an elastic beam. The problem discussed in this paper can be extended to th-order difference boundary value problem.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contributions

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

Acknowledgments

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