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

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

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).