Discrete Dynamics in Nature and Society

Volume 2018, Article ID 6703503, 9 pages

https://doi.org/10.1155/2018/6703503

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

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

Correspondence should be addressed to Xia Liu; moc.361@200199aix

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 equationsatisfying the boundary value conditionswhere , 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 equationfor 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 conditionsSome 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 equationLeng [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 conditionsThey 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 thatfor any .

Letand

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 byand for any , defineand

*Remark 1. *For any , it is easy to see thatAs 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 byThen 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 , letwhere .

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 thatThere 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*

*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 .

DenoteandLet . Clearly, is an invariant subspace of . The space is defined byLet be such that is bounded and as Thus, for any , there exists a positive constant such thatFor any , it comes from (27) and thatTherefore,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 . DenoteAs a consequence, on the one hand, there is a sequence in such thatBy (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 . LetBy 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 satisfiesTakeThenIn other words, there exist two positive constants and such that . Hence, satisfies of Lemma 2.

For any , . Combining with , we haveTherefore, 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 haveThereby, there exists a constant such thatwhere . According to Lemma 2, has a critical value , whereand .

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.

SetIt 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 . Sincewe haveOn the one hand, it follows from and thatOn the other hand, by , we havewhich is a contradiction with (52).

*5. Some Examples*

*5. Some Examples**In this section, we shall provide three examples to illustrate our main results.*

*Example 1. *For , assume thatsatisfying the boundary value conditionsWe havewithandBesides,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 thatsatisfying the boundary value conditionsWe havewithandBesides,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 thatsatisfying the boundary value conditionsWe havewithandIt is easy to verify that all the suppositions of Theorem 9 are satisfied and then BVP (66), (67) has no nontrivial solutions.

*Data Availability*

*Data Availability**No data were used to support this study.*

*Conflicts of Interest*

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

*Authors’ Contributions*

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

*Acknowledgments*

*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.*

*References*

*References*

- C. P. Gupta, “Existence and uniqueness theorems for the bending of an elastic beam equation,”
*Applicable Analysis: An International Journal*, vol. 26, no. 4, pp. 289–304, 1988. View at Publisher · View at Google Scholar · View at MathSciNet - P. Chen and C. Tian, “Infinitely many solutions for Schrödinger-Maxwell equations with indefinite sign subquadratic potentials,”
*Applied Mathematics and Computation*, vol. 226, pp. 492–502, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - P. Chen, X. He, and X. H. Tang, “Infinitely many solutions for a class of fractional Hamiltonian systems via critical point theory,”
*Mathematical Methods in the Applied Sciences*, vol. 39, no. 5, pp. 1005–1019, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - C. Guo, R. P. Agarwal, C. Wang, and D. O’Regan, “The existence of homoclinic orbits for a class of first-order superquadratic Hamiltonian systems,”
*Memoirs on Differential Equations and Mathematical Physics*, vol. 61, pp. 83–102, 2014. View at Google Scholar · View at MathSciNet - C. Guo, D. O’Regan, Y. Xu, and R. P. Agarwal, “Existence of homoclinic orbits for a class of first-order differential difference equations,”
*Acta Mathematica Scientia B*, vol. 35, no. 5, pp. 1077–1094, 2015. View at Publisher · View at Google Scholar · View at MathSciNet - C. J. Guo, D. O’Regan, Y. T. Xu, and R. P. Agarwal, “Existence and multiplicity of homoclinic orbits of a second-order differential difference equation via variational methods,”
*Applied Mathematics and Mechanics*, vol. 4, no. 1, pp. 1–15, 2012. View at Google Scholar - C. Guo, D. O’Regan, Y. Xu, and R. P. Agarwal, “Existence of subharmonic solutions and homoclinic orbits for a class of even higher order differential equations,”
*Applicable Analysis*, vol. 90, no. 7, pp. 1169–1183, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - T. He, C. Chen, Y. Huang, and C. Hou, “Infinitely many sign-changing solutions for
*p*-Laplacian Neumann problems with indefinite weight,”*Applied Mathematics Letters*, vol. 39, pp. 73–79, 2015. View at Publisher · View at Google Scholar · View at MathSciNet - Yanyan Li and Yuhua Long, “Periodic solutions of a class of fourth-order superlinear differential equations,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 469863, 8 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - X. Tang and S. Chen, “Ground state solutions of Nehari-Pohozaev type for Schrödinger-POIsson problems with general potentials,”
*Discrete and Continuous Dynamical Systems - Series A*, vol. 37, no. 9, pp. 4973–5002, 2017. View at Publisher · View at Google Scholar · View at MathSciNet - X. H. Tang and S. Chen, “Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials,”
*Calculus of Variations and Partial Differential Equations*, vol. 56, no. 4, pp. 110–134, 2017. View at Publisher · View at Google Scholar · View at MathSciNet - H. Zhou, L. Yang, and P. Agarwal, “Solvability for fractional p-Laplacian differential equations with multipoint boundary conditions at resonance on infinite interval,”
*Applied Mathematics and Computation*, vol. 53, no. 1-2, pp. 51–76, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - R. P. Agarwal and P. J. Y. Wong,
*Advanced Topics in Difference Equations*, vol. 404 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. View at Publisher · View at Google Scholar · View at MathSciNet - Z. AlSharawi, J. M. Cushing, and S. Elaydi,
*Theory and Applications of Difference Equations and Discrete Dynamical Systems*, Springer, New York, NY, USA, 2014. - P. Chen, “Existence of homoclinic orbits in discrete Hamiltonian systems without Palais-Smale condition,”
*Journal of Difference Equations and Applications*, vol. 19, no. 11, pp. 1781–1794, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - P. Chen and X. H. Tang, “Infinitely many homoclinic solutions for the second-order discrete
*p*-Laplacian systems,”*Bulletin of the Belgian Mathematical Society - Simon Stevin*, vol. 20, no. 2, pp. 193–212, 2013. View at Google Scholar · View at MathSciNet - Y. Domshlak and A. Matakaev, “The third and fourth order linear difference equations near periodic critical states: sharp results in oscillation theory,”
*Journal of Difference Equations and Applications*, vol. 7, no. 4, pp. 551–575, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - T. He, W. Yang, and F. Yang, “Sign-changing solutions for discrete second-order three-point boundary value problems,”
*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 705387, 14 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - T. He, W. Yang, H. Zhao, and Y. Lei, “Positive periodic solutions for higher order functional difference systems depending on a parameter,”
*Mediterranean Journal of Mathematics*, vol. 10, no. 2, pp. 791–806, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Huang and Z. Zhou, “Ground state solutions of the periodic discrete coupled nonlinear Schrödinger equations,”
*Mathematical Methods in the Applied Sciences*, vol. 38, no. 8, pp. 1682–1695, 2015. View at Publisher · View at Google Scholar · View at MathSciNet - Meihua Huang and Zhan Zhou, “On the existence of ground state solutions of the periodic discrete coupled nonlinear schrödinger lattice,”
*Journal of Applied Mathematics*, vol. 2013, Article ID 404369, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - J. Leng, “Periodic and subharmonic solutions for 2
*n*th-order -Laplacian difference equations containing both advance and retardation,”*Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae. New Series*, vol. 27, no. 4, pp. 902–913, 2016. View at Publisher · View at Google Scholar · View at MathSciNet - M. Leszczyński, “Fourth-order discrete anisotropic boundary-value problems,”
*Electronic Journal of Differential Equations*, vol. 2015, no. 238, pp. 1–14, 2015. View at Google Scholar - G. Lin and Z. Zhou, “Homoclinic solutions of discrete -Laplacian equations with mixed nonlinearities,”
*Communications on Pure and Applied Analysis*, vol. 17, no. 5, pp. 1723–1747, 2018. View at Publisher · View at Google Scholar · View at MathSciNet - X. Liu, T. Zhou, and H. Shi, “Existence of homoclinic orbits for a higher order difference system,”
*Journal of Nonlinear Sciences and Applications. JNSA*, vol. 10, no. 4, pp. 1842–1853, 2017. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Long and B. Zeng, “Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence,”
*Open Mathematics*, vol. 15, pp. 1549–1557, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - L. Yang, “Boundary value problems of a higher order nonlinear difference equation,”
*Politehnica University of Bucharest. Scientific Bulletin. Series A. Applied Mathematics and Physics*, vol. 79, no. 4, pp. 93–102, 2017. View at Google Scholar · View at MathSciNet - Z. Zhou and D. Ma, “Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials,”
*Science China Mathematics*, vol. 58, no. 4, pp. 781–790, 2015. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Zhou, J. Yu, and Y. Chen, “Homoclinic solutions in periodic difference equations with saturable nonlinearity,”
*Science China Mathematics*, vol. 54, no. 1, pp. 83–93, 2011. View at Publisher · View at Google Scholar · View at Scopus - J. Mawhin and M. Willem,
*Critical Point Theory and Hamiltonian Systems*, vol. 74, Springer, New York, NY. USA, 1989. View at Publisher · View at Google Scholar · View at MathSciNet - P. H. Rabinowitz,
*Minimax Methods in Critical Point Theory with Applications to Differential Equations*, vol. 65, American Mathematical Society, Providence, RI, USA, 1986. View at Publisher · View at Google Scholar · View at MathSciNet - M. Struwe,
*Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems*, vol. 34 of*Modern Surveys in Mathematics*, Springer, Berlin, Germany, 4th edition, 2008. View at MathSciNet

*
*