Open Access

On the Nonexistence and Existence of Solutions for a Fourth-Order Discrete Boundary Value Problem

Advances in Difference Equations20092009:389624

https://doi.org/10.1155/2009/389624

Received: 16 July 2009

Accepted: 16 October 2009

Published: 20 October 2009

Abstract

By using the critical point theory, we establish various sets of sufficient conditions on the nonexistence and existence of solutions for the boundary value problems of a class of fourth-order difference equations.

1. Introduction

In this paper, we denote by the set of all natural numbers, integers, and real numbers, respectively. For , define when .

Consider the following boundary value problem (BVP):

(1.1)

Here, , is nonzero and real valued for each , is real valued for each . is real-valued for each and continuous in the second variable . is the forward difference operator defined by , and .

We may think of (1.1) as being a discrete analogue of the following boundary value problem:

(1.2)

which are used to describe the bending of an elastic beam; see, for example, [110] and references therein. Owing to its importance in physics, many methods are applied to study fourth-order boundary value problems by many authors. For example, fixed point theory [1, 3, 57], the method of upper and lower solutions [8], and critical point theory [9, 10] are widely used to deal with the existence of solutions for the boundary value problems of fourth-order differential equations.

Because of applications in many areas for difference equations, in recent years, there has been an increased interest in studying of fourth-order difference equation, which include results on periodic solutions [11], results on oscillation [1214], and results on boundary value problems and other topics [15, 16]. Recently, a few authors have gradually paid attention to applying critical point theory to deal with problems on discrete systems; for example, Yu and Guo in [17] considered the existence of solutions for the following BVP:

(1.3)

The papers [1720] show that the critical point theory is an effective approach to the study of the boundary value problems of difference equations. In this paper, we will use critical point theory to establish some sufficient conditions on the nonexistence and existence of solutions for the BVP (1.1).

Let

(1.4)

Then the BVP (1.1) becomes

(1.5)

where

(1.6)

The remaining of this paper is organized as follows. First, in Section 2, we give some preliminaries and establish the variational framework for BVP (1.5). Then, in Section 3, we present a sufficient condition on the nonexistence of nontrivial solutions of BVP (1.5). Finally, in Section 4, we provide various sets of sufficient conditions on the existence of solutions of BVP (1.5) when is superlinear, sublinear, and Lipschitz. Moreover, in a special case of we obtain a necessary and sufficient condition for the existence of unique solutions of BVP (1.5).

To conclude the introduction, we refer to [21, 22] for the general background on difference equations.

2. Preliminaries

In order to apply the critical point theory, we are going to establish the corresponding variational framework of BVP (1.5). First we give some notations.

Let be the real Euclidean space with dimension . Define the inner product on as follows:

(2.1)

by which the norm can be induced by

(2.2)

For BVP (1.5), consider the functional defined on as follows:

(2.3)

where is the transpose of a vector in :

(2.4)
(2.5)

After a careful computation, we find that the Fréchet derivative of is

(2.6)

where is defined as .

Expanding out , one can easily see that there is an one-to-one correspondence between the critical point of functional and the solution of BVP (1.5). Furthermore, is a critical point of if and only if is a solution of BVP (1.5), where .

Therefore, we have reduced the problem of finding a solution of (1.5) to that of seeking a critical point of the functional defined on .

In order to obtain the existence of critical points of on , for the convenience of readers, we cite some basic notations and some known results from critical point theory.

Let be a real Banach space, , that is, is a continuously Fréchet differentiable functional defined on , and is said to satisfy the Palais-Smale condition (P-S condition), if any sequence for which is bounded and as possesses a convergent subsequence in .

Let denote the open ball in about of radius and let denote its boundary. The following lemmas are taken from [23, 24] and will play an important role in the proofs of our main results.

Lemma 2.1 (Linking theorem).

Let be a real Banach space, where is a finite-dimensional subspace of . Assume that satisfies the P-S condition and the following.
  • (F3) There exist constants , such that .

  • (F4) There is an and a constant such that and .

Then possesses a critical value , where
(2.7)

and , where id denotes the identity operator.

Lemma 2.2 (Saddle point theorem).

Let be a real Banach space, where and is finite-dimensional. Suppose that satisfies the P-S condition and the following.
  • (F1) There exist constants , such that

  • (F2) There is and a constant such that

Then possesses a critical value , where
(2.8)

and , where id denotes the identity operator.

Lemma 2.3 (Clark theorem).

Let be a real Banach space, with being even, bounded from below, and satisfying P-S condition. Suppose , there is a set such that is homeomorphic to ( dimension unit sphere) by an odd map, and . Then has at least distinct pairs of nonzero critical points.

3. Nonexistence of Nontrivial Solutions

In this section, we give a result of nonexistence of nontrivial solutions to BVP (1.5).

Theorem 3.1.

Suppose that matrix is negative semidefinite and for ,
(3.1)

Then BVP (1.5) has no nontrivial solutions.

Proof.

Assume, for the sake of contradiction, that BVP (1.5) has a nontrivial solution. Then has a nonzero critical point . Since
(3.2)
we get
(3.3)
On the other hand, it follows from (3.1) that
(3.4)

This contradicts with (3.3) and hence the proof is complete.

In the existing literature, results on the nonexistence of solutions of discrete boundary value problems are scarce. Hence Theorem 3.1 complements existing ones.

4. Existence of Solutions

Theorem 3.1 gives a set of sufficient conditions on the nonexistence of solutions of BVP (1.5). In this section, with part of the conditions being violated, we establish the existence of solutions of BVP (1.5) by distinguishing three cases: is superlinear, is sublinear, and is Lipschitzian.

4.1. The Superlinear Case

In this subsection, we need the following conditions.
  • (P1) For any , and , as .

  • (P2) There exist constants and such that
    (4.1)
  • (P3) Matrix exists at least one positive eigenvalue.

  • (P4) is odd for the second variable , namely,
    (4.2)

Theorem 4.1.

Suppose that satisfies . Then BVP (1.5) possesses at least one solution.

Proof.

For any , we have
(4.3)

Let . We have, for any ,

(4.4)

Since matrix is symmetric, its all eigenvalues are real. We denote by its eigenvalues. Set . Thus for any ,

(4.5)

The above inequality means that is coercive. By the continuity of , attains its maximum at some point, and we denote it by , that is, , where . Clearly, is a critical point of . This completes the proof of Theorem 4.1.

Theorem 4.2.

Suppose that satisfies the assumptions , and . Then BVP (1.5) possesses at least two nontrivial solutions.

To prove Theorem 4.2, we need the following lemma.

Lemma 4.3.

Assume that holds, then the functional satisfies the P-S condition.

Proof.

Assume that is a P-S sequence. Then there exists a constant such that for any and as . By (4.5) we have
(4.6)
and so
(4.7)

Due to , the above inequality means is bounded. Since is a finite-dimensional Hilbert space, there must exist a subsequence of which is convergent in . Therefore, P-S condition is satisfied.

Proof of Theorem 4.2.

Let , , , be the positive eigenvalues and the negative eigenvalues, where , . Let be an eigenvector of corresponding to the eigenvalue , , and let be an eigenvector of corresponding to the eigenvalue , , such that
(4.8)

Let and be subspaces of defined as follows:

(4.9)
For any , where . Then
(4.10)

Let , , then has the following decomposition of direct sum:

(4.11)

By assumption , there exists a constant , such that for any , , . So for any ,

(4.12)
Denote . Then
(4.13)

That is to say, satisfies assumption of Linking theorem.

Take . For any , let , because , where . Then

(4.14)

Set and Then . Furthermore, and are bounded from above. Accordingly, there is some , such that for any , where . By Linking theorem, possesses a critical value , where

(4.15)

Let be a critical point corresponding to the critical value of , that is, . Clearly, since . On the other hand, by Theorem 4.1, has a critical point satisfying . If , then Theorem 4.2 holds. Otherwise, . Then , which is the same as .

Choosing , we have . Because the choice of is arbitrary, we can take . Similarly, there exists a positive number , for any , where . Again, by the Linking theorem, possesses a critical value , where

(4.16)

If , then the proof is complete. Otherwise . Because and , then attains its maximum at some point in the interior of sets and . But , and for . Thus there is a critical point satisfying , .

The proof of Theorem 4.2 is now complete.

Theorem 4.4.

Suppose that satisfies the assumptions , , and . Then BVP (1.5) possesses at least distinct pairs of nontrivial solutions, where is the dimension of the space spanned by the eigenvectors corresponding to the positive eigenvalues of .

Proof.

From the proof of Theorem 4.2, it is easy to know that is bounded from above and satisfies the P-S condition. It is clear that is even and , and we should find a set and an odd map such that is homeomorphic to by an odd map.

We take where and are defined as in the proof of Theorem 4.2. It is clear that is homeomorphic to ( dimension unit sphere) by an odd map. With (4.13), we get . Thus all the conditions of Lemma 2.3 are satisfied, and has at least distinct pairs of nonzero critical points. Consequently, BVP (1.5) possesses at least distinct pairs nontrivial solutions. The proof of Theorem 4.4 is complete.

4.2. The Sublinear Case

In this subsection, we will consider the case where is sublinear. First, we assume the following.

There exist constants and such that

(4.17)

The first result is as follows.

Theorem 4.5.

Suppose that is satisfied and that matrix M is positive definite. Then BVP (1.5) possesses at least one solution.

Proof.

The proof will be finished when the existence of one critical point of functional defined as in (2.3) is proved.

Assume that matrix is positive definite. We denote by its eigenvalues, where . Then for any , followed by we have

(4.18)

By the continuity of on , the above inequality means that there exists a lower bound of values of functional . Classical calculus shows that attains its minimal value at some point, and then there exist such that . Clearly, is a critical point of the functional .

Corollary 4.6.

Suppose that matrix M is positive definite, and satisfies that there exist constants , and such that
(4.19)

Then BVP (1.5) possesses at least one solution.

Corollary 4.7.

Suppose that matrix M is positive definite, and satisfies the following.

There exists a constant such that for any , .

Then BVP (1.5) possesses at least one solution.

Proof.

Assume that matrix is positive definite. In this case, for any ,
(4.20)

Since the rest of the proof is similar to Theorem 4.5, we do not repeat them here.

When is neither positive definite nor negative definite, we now assume that is nonsingular, and we have the following result.

Theorem 4.8.

Suppose that is nonsingular, satisfies . Then BVP (1.5) possesses at least one solution.

Proof.

We may assume that is neither positive definite nor negative definite. Let , denote all eigenvalues of , where and . For any , let be an eigenvector of corresponding to the eigenvalue , , such that
(4.21)

Let and be subspaces of defined as follows:

(4.22)

Then has the following decomposition of direct sum:

(4.23)

Let be defined as in (2.3). Then . By (4.20),

(4.24)

Suppose that is a P-S sequence. Then there exists a constant such that for any , and as . Thus, for sufficiently large and for any , we have .

Let . We have, by (2.6), for any ,

(4.25)
Thus for sufficiently large , we get
(4.26)
Thus,
(4.27)

which implies that is bounded.

Now we are going to prove that is also bounded. By (4.25),

(4.28)
Thus we have
(4.29)
And so
(4.30)

Due to is bounded. Then is bounded. Since is a finite-dimensional Hilbert space, there must exist a subsequence of which is convergent in . Therefore, P-S condition is satisfied.

In order to apply the saddle point theorem to prove the conclusion, we consider the functional and verify the conditions of Lemma 2.2.

For any , , we have

(4.31)

This implies that is true.

For any , ,

(4.32)

This implies that is true.

So far we have verified all the assumptions of Lemma 2.2 and hence has at least a critical point in . This completes the proof.

Consider the following special case

(4.33)

Here,

(4.34)

It can be verified that is positive definite, then we have the following corollaries.

Corollary 4.9.

Suppose that there exist constants and such that
(4.35)

Then BVP (4.33) possesses at least one solution.

Corollary 4.10.

Suppose that satisfies . Then BVP (4.33) possesses at least one solution.

4.3. The Lipschitz Case

In this subsection, we suppose the following.

Assume that there exist positive constants , such that for any ,

(4.36)

When is Lipschitzian in the second variable , namely, there exists a constant such that for any , , ,

(4.37)

then condition (4.36) is satisfied.

Theorem 4.11.

Suppose that is satisfied and M is nonsingular. If , where and are the minimal positive eigenvalue and maximal negative eigenvalue of , respectively, then BVP (1.5) possesses at least one solution.

Proof.

Assume that is a P-S sequence. Then as . Thus for sufficiently large , we get . Since , then for sufficiently large ,
(4.38)

In view of (4.36), we have

(4.39)

It follows, by using the inequality for and Hölder's inequality, that

(4.40)

By a similar argument to the proof of Theorem 4.8, we can decompose into the following form of direct sum:

(4.41)
where and can be referred to (4.22). Thus can be expressed by
(4.42)
and , where . Therefore,
(4.43)

Hence,

(4.44)

By the fact that , we know that the sequence is bounded and therefore the P-S condition is verified.

Now we are going to check conditions and for functional . In fact, by (4.36), we have for any ,

(4.45)

Thus, for any , ,

(4.46)

for some positive constant .

For any , , we have

(4.47)

Until now, we have verified all the assumptions of Lemma 2.2 and hence has at least a critical point in . This completes the proof.

Finally, we consider the special case that is independent of the second variable ; that is, for any , the BVP (1.1) becomes

(4.48)

As in Section 2, we reduce the existence of solutions of BVP (4.48) to the existence of critical points of a functional defined on as follows:

(4.49)

where is defined as in (2.4), and . Then we can see that the critical point of is just the solution to the following system of linear algebraic equations:

(4.50)

By using the theory of linear algebra, we have the next necessary and sufficient conditions.

Theorem 4.12.
  1. (i)
    BVP (4.48) has at least one solution if and only if , where denotes the rank of matrix and is the augmented matrix defined as follows:
    (4.51)
     
  2. (ii)

    BVP (4.48) has a unique solution if and only if .

     

Declarations

Acknowledgment

This work is supported by the Specialized Fund for the Doctoral Program of Higher Eduction (no. 20071078001).

Authors’ Affiliations

(1)
School of Mathematics and Information Science, Guangzhou University

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© S. Huang and Z. Zhou 2009

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