Journal of Applied Mathematics

Volume 2014, Article ID 343129, 6 pages

http://dx.doi.org/10.1155/2014/343129

## Boundary Value Problems for Fourth Order Nonlinear -Laplacian Difference Equations

^{1}School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong 510006, China^{2}Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, Guangdong 510006, China

Received 6 October 2013; Accepted 26 December 2013; Published 30 January 2014

Academic Editor: Shih-sen Chang

Copyright © 2014 Qinqin Zhang. 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

We consider the boundary value problem for a fourth order nonlinear *p*-Laplacian difference equation containing both advance and retardation. By using Mountain pass lemma and some established inequalities, sufficient conditions of the existence of solutions of the boundary value problem are obtained. And an illustrative example is given in the last part of the paper.

#### 1. Introduction

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

Consider the following fourth order nonlinear difference equation: with boundary value conditions where , is a positive number for , is the forward difference operator defined by , , and is the -Laplacian operator; that is, , .

In the last decade, by using various techniques such as critical point theory, fix point theory, topological degree theory, and coincidence degree theory, a great deal of works have been done on the existence of solutions to boundary value problems of difference equations (see [1–7] and references therein). Among these approaches, the critical point theory seems to be a powerful tool to deal with this problem (see [5, 7–9]). However, compared to the boundary value problems of lower order difference equations ([6, 8, 10–13]), the study of boundary value problems of higher order difference equations is relatively rare (see [9, 14, 15]), especially the works by using the critical point theory [16]. For the background on difference equations, we refer to [17].

In this paper, we will consider the existence of solutions of the boundary value problem of (1) with (2). First, we will construct a functional such that solutions of the boundary value problem (1) with (2) correspond to critical points of . Then, by using Mountain pass lemma, we obtain the existence of critical points of . We mention that (1) is a kind of difference equation containing both advance and retardation. This kind of difference equation has many applications both in theory and practice. For example, in [17], Agarwal considered the following difference equation: with the boundary value conditions as an example. It represents the amplitude of the motion of every particle in the string. And in [7], the authors considered the following second order functional difference equation: with different boundary value conditions where the operator is the Jacobi operator given by In [18], the authors considered the second order -Laplacian difference equation: with boundary value conditions As for the periodic and subharmonic solutions of -Laplacian difference equations containing both advance and retardation, we refer to [19]. And for the periodic solutions of -Laplacian difference equations, we refer to [20].

Throughout this paper, we assume that there exists a function which is differentiable in and for each , satisfying for .

#### 2. Preliminaries and Main Results

Lemma 1. *Let ; then there exist two positive sequences and such that
**
holds for any with , where , for and , for .*

*Proof. *If , by Hölder’s inequality, we have
which implies that
and is obvious. If , then we have
which implies that
and is obvious. Now the proof is complete.

Lemma 2. *There exist two positive sequences and such that
**
holds for any with and , where
*

*Proof. *There is no harm in assuming that , . Then
where means the transpose of , and is the matrix given by
We will calculate the eigenvalues of . Similar to [21], assume that is an eigenvalue of . Since is positive-definite for and negative-definite for , where is the identity matrix, we see that . Assume that is an eigenvector associated to and define the sequence as
Then satisfies
Since the roots of the equation are
set
Then
for some constants and . implies that , and implies that . Therefore, for . By (23), we have which implies that the eigenvalues of are
The maximum eigenvalue of is , and the minimal eigenvalue of is . Equation (16) follows from (18) and the fact that and .

*Before we apply the critical point theory, we will establish the corresponding variational framework for (1) with (2).*

*Let
Then is a -dimensional Hilbert space.*

*Obviously, is isomorphic to . In fact, we can find a map defined by
*

*Define the inner product on as
The corresponding norm can be induced by
*

*For all , define the functional on as follows:
Clearly, . We can compute the partial derivative as
for , . Therefore, is a critical point of if and only if is a solution of (1) with (2).*

*Definition 3. *Let be a real Banach space; the functional is said to satisfy the Palais-Smale (P.S. for short) condition if any sequence in such that is bounded and as contains a convergent subsequence.

*Let denote the open ball in with radius and center 0, and let denote its boundary.*

*In order to obtain the existence of critical points of on , we need to use the following basic lemma, which is important in the proof of our main results.*

*Lemma 4 (Mountain pass lemma [22]). Let be a real Hilbert space and satisfies the P.S. condition, if and the following conditions hold.There exist constants and such that .There exists such that .Then possesses a critical value given by
where
*

*Let
Then, for ,
For ,
*

*Now we state our main results.*

*Theorem 5. Assume that satisfies the following conditions.There exist constants and such that
There exist constants and such that
Then (1) with (2) possesses at least two nontrivial solutions.*

*Remark 6. *Comparing our results with the results of the boundary value problems of second order -Laplacian difference equations in [18], we find that our results are more precisely.

*In view of (37) and (38), it is easy to obtain the following corollary.*

*Corollary 7. Assume that satisfies
Then (1) with (2) possesses at least two nontrivial solutions.*

*For the case when , we have the following corollary for the boundary value problems of the fourth order nonlinear difference equations.*

*Corollary 8. Assume that satisfies the following conditions.There exist constants and such that
There exist constants and such that
Then the following fourth order nonlinear difference equation
with the boundary value conditions (2) possesses at least two nontrivial solutions.*

*3. Proof of Theorem 5*

*3. Proof of Theorem 5*

*In order to prove Theorem 5, we first establish the following lemma.*

*Lemma 9. Assume that satisfies ; then the functional satisfies the P.S. condition.*

*Proof. *Let be a sequence in such that is bounded and as . Then there exists a positive constant such that for .

By (11) and (16), we have
And by , (11), and (16), we have
Therefore, by (30), we obtain
Noticing that and , by (45), we have
Since is a finite-dimensional space, (46) implies that is bounded and has a convergent subsequence. Thus . condition is verified.

*Now we give the proof of Theorem 5.*

*Proof. *For any with , according to (11) and (16), we have
By , (11), and (16), we have
So, by (30), we get
Since , we let and . Then by (49),
which means that satisfies the condition of the Mountain pass lemma.

By our assumptions, it is clear that . In order to use Mountain pass lemma, it suffices to verify that condition holds. In fact, similar to the proof of (45), we have
for any . Since , it is easy to see that there exists an with such that . Thus holds.

According to Mountain pass lemma, possesses a critical value given by
where

Let be a critical point of corresponding to the critical value ; then is nontrivial and .

On the other hand, by (51), we have
Since is a -dimensional space, by the continuity of on , we see that there exists such that
Clearly, is a nonzero critical point of , and .

If , then the proof is finished. Otherwise, . Since and , then by Mountain pass lemma again, possesses a critical value given by
where

Let be a critical point of corresponding to the critical value . If , then the proof is finished. Otherwise . Let for ; then . By the definition of , we see that there exists such that . Thus is a critical point of . Similar, let for ; then . By the definition of , we see that there exists such that . And is a critical point of . Clearly . The proof is now completed.

*In the last part of this paper, we give an example to illustrate our results.*

*Example 10. *Consider (1) with (2), where is defined by
for . Here for . Define
Then for and (10) holds. Moreover, it is easy to see that satisfies (39) for . By Corollary 7, we see that (1) with (2) when is defined by (58) has at least two nontrivial solutions.

*Conflict of Interests *

*The author declares that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgments*

*The author would like to thank the anonymous referee for his/her valuable suggestions. This work is supported by Program for Changjiang Scholars and Innovative Research Team in University (no. IRT1226), the National Natural Science Foundation of China (no. 11171078), and the Specialized Fund for the Doctoral Program of Higher Education of China (no. 20104410110001).*

*References*

- J. Henderson and H. B. Thompson, “Existence of multiple solutions for second-order discrete boundary value problems,”
*Computers & Mathematics with Applications*, vol. 43, no. 10-11, pp. 1239–1248, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Li and J. Shu, “Solvability of boundary value problems with Riemann-Stieltjes $\Delta $-integral conditions for second-order dynamic equations on time scales at resonance,”
*Advances in Difference Equations*, vol. 42, pp. 8–18, 2011. View at Google Scholar · View at MathSciNet - R. Ma and C. Gao, “Bifurcation of positive solutions of a nonlinear discrete fourth-order boundary value problem,”
*Zeitschrift für Angewandte Mathematik und Physik*, vol. 64, no. 3, pp. 493–506, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Yuan, “Positive solutions of a singular positone and semipositone boundary value problems for fourth-order difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 312864, 16 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Gao, “Existence of multiple solutions for a second-order difference equation with a parameter,”
*Applied Mathematics and Computation*, vol. 216, no. 5, pp. 1592–1598, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Liang and P. Weng, “Existence and multiple solutions for a second-order difference boundary value problem via critical point theory,”
*Journal of Mathematical Analysis and Applications*, vol. 326, no. 1, pp. 511–520, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Deng and H. Shi, “On boundary value problems for second order nonlinear functional difference equations,”
*Acta Applicandae Mathematicae*, vol. 110, no. 3, pp. 1277–1287, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Liu, S. Wang, and J. Zhang, “Multiple solutions for boundary value problems of second-order difference equations with resonance,”
*Journal of Mathematical Analysis and Applications*, vol. 374, no. 1, pp. 187–196, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. R. Zou and P. X. Weng, “Solutions of 2nth-order boundary value problem for difference equation via variational method,”
*Advances in Difference Equations*, vol. 2009, Article ID 730484, 10 pages, 2009. View at Publisher · View at Google Scholar - D. Bai and Y. Xu, “Nontrivial solutions of boundary value problems of second-order difference equations,”
*Journal of Mathematical Analysis and Applications*, vol. 326, no. 1, pp. 297–302, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Zhang, “Positive solutions of BVPs for third-order discrete nonlinear difference systems,”
*Journal of Applied Mathematics and Computing*, vol. 35, no. 1-2, pp. 551–575, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Yu and Z. Guo, “Boundary value problems of discrete generalized Emden-Fowler equation,”
*Science in China A*, vol. 49, no. 10, pp. 1303–1314, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. He and Y. Xu, “Positive solutions for nonlinear discrete second-order boundary value problems with parameter dependence,”
*Journal of Mathematical Analysis and Applications*, vol. 379, no. 2, pp. 627–636, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Berger, “Existence of nontrivial solutions of a two point boundary value problem for a $2n$th order nonlinear difference equation,”
*Advances in Dynamical Systems and Applications*, vol. 3, no. 1, pp. 131–146, 2008. View at Google Scholar · View at MathSciNet - S. Xie and J. Zhu, “Positive solutions of the system for $n$th-order singular nonlocal boundary value problems,”
*Journal of Applied Mathematics and Computing*, vol. 37, no. 1-2, pp. 119–132, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - S. Huang and Z. Zhou, “On the nonexistence and existence of solutions for a fourth-order discrete boundary value problem,”
*Advances in Difference Equations*, vol. 2009, Article ID 389624, 18 pages, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Agarwal,
*Difference Equations and Inequalities: Theory, Methods, and Applications*, vol. 228, Marcel Dekker, New York, NY, USA, 2000. View at MathSciNet - H. Shi, Z. Liu, and Z. Wang, “Dirichlet boundary value problems for second order $p$-Laplacian difference equations,”
*Rendiconti dell'Istituto di Matematica dell'Università di Trieste*, vol. 42, pp. 19–29, 2010. View at Google Scholar · View at MathSciNet - X. Liu, Y. Zhang, B. Zheng, and H. Shi, “Periodic and subharmonic solutions for second order $p$-Laplacian difference equations,”
*Indian Academy of Sciences*, vol. 121, no. 4, pp. 457–468, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - J. Mawhin, “Periodic solutions of second order nonlinear difference systems with
*p*-Laplacian: a variational approach,”*Nonlinear Analysis: Theory, Methods & Applications*, vol. 75, no. 12, pp. 4672–4687, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Zhou, J. Yu, and Z. Guo, “Periodic solutions of higher-dimensional discrete systems,”
*Proceedings of the Royal Society of Edinburgh A*, vol. 134, no. 5, pp. 1013–1022, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · 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 MathSciNet

*
*