On the Nonexistence and Existence of Solutions for a Fourth-Order Discrete Boundary Value Problem
© S. Huang and Z. Zhou 2009
Received: 16 July 2009
Accepted: 16 October 2009
Published: 20 October 2009
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.
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):
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:
which are used to describe the bending of an elastic beam; see, for example, [1–10] 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, 5–7], the method of upper and lower solutions , 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 , results on oscillation [12–14], 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  considered the existence of solutions for the following BVP:
The papers [17–20] 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).
Then the BVP (1.1) becomes
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).
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:
by which the norm can be induced by
For BVP (1.5), consider the functional defined on as follows:
where is the transpose of a vector in :
After a careful computation, we find that the Fréchet derivative of is
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 .
Lemma 2.1 (Linking theorem).
(F3) There exist constants , such that .
(F4) There is an and a constant such that and .
and , where id denotes the identity operator.
Lemma 2.2 (Saddle point theorem).
(F1) There exist constants , such that
(F2) There is and a constant such that
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).
Then BVP (1.5) has no nontrivial solutions.
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
(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)
Suppose that satisfies . Then BVP (1.5) possesses at least one solution.
Let . We have, for any ,
Since matrix is symmetric, its all eigenvalues are real. We denote by its eigenvalues. Set . Thus for any ,
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.
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.
Assume that holds, then the functional satisfies the P-S condition.
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 and be subspaces of defined as follows:
Let , , then has the following decomposition of direct sum:
By assumption , there exists a constant , such that for any , , . So for any ,
That is to say, satisfies assumption of Linking theorem.
Take . For any , let , because , where . Then
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
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
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.
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 .
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
The first result is as follows.
Suppose that is satisfied and that matrix M is positive definite. Then BVP (1.5) possesses at least one solution.
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
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 .
Then BVP (1.5) possesses at least one solution.
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.
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.
Suppose that is nonsingular, satisfies . Then BVP (1.5) possesses at least one solution.
Let and be subspaces of defined as follows:
Then has the following decomposition of direct sum:
Let be defined as in (2.3). Then . By (4.20),
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 ,
which implies that is bounded.
Now we are going to prove that is also bounded. By (4.25),
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
This implies that is true.
For any , ,
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
It can be verified that is positive definite, then we have the following corollaries.
Then BVP (4.33) possesses at least one solution.
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 ,
When is Lipschitzian in the second variable , namely, there exists a constant such that for any , , ,
then condition (4.36) is satisfied.
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.
In view of (4.36), we have
It follows, by using the inequality for and Hölder's inequality, that
By a similar argument to the proof of Theorem 4.8, we can decompose into the following form of direct sum:
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 ,
Thus, for any , ,
for some positive constant .
For any , , we have
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
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:
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:
By using the theory of linear algebra, we have the next necessary and sufficient conditions.
- (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)
BVP (4.48) has a unique solution if and only if .
This work is supported by the Specialized Fund for the Doctoral Program of Higher Eduction (no. 20071078001).
- Aftabizadeh AR: Existence and uniqueness theorems for fourth-order boundary value problems. Journal of Mathematical Analysis and Applications 1986,116(2):415-426. 10.1016/S0022-247X(86)80006-3MATHMathSciNetView ArticleGoogle Scholar
- Gupta CP: Existence and uniqueness theorems for the bending of an elastic beam equation. Applicable Analysis 1988,26(4):289-304. 10.1080/00036818808839715MATHMathSciNetView ArticleGoogle Scholar
- Agarwal RP: On fourth order boundary value problems arising in beam analysis. Differential and Integral Equations 1989,2(1):91-110.MATHMathSciNetGoogle Scholar
- Gupta CP: Existence and uniqueness theorems for some fourth order fully quasilinear boundary value problems. Applicable Analysis 1990,36(3-4):157-169. 10.1080/00036819008839930MATHMathSciNetView ArticleGoogle Scholar
- Ma R, Wang H: On the existence of positive solutions of fourth-order ordinary differential equations. Applicable Analysis 1995,59(1–4):225-231.MATHMathSciNetGoogle Scholar
- Bai Z, Wang H: On positive solutions of some nonlinear fourth-order beam equations. Journal of Mathematical Analysis and Applications 2002,270(2):357-368. 10.1016/S0022-247X(02)00071-9MATHMathSciNetView ArticleGoogle Scholar
- Yao Q: Positive solutions of a nonlinear elastic beam equation rigidly fastened on the left and simply supported on the right. Nonlinear Analysis: Theory, Methods & Applications 2008,69(5-6):1570-1580. 10.1016/j.na.2007.07.002MATHView ArticleGoogle Scholar
- Ruyun M, Jihui Z, Shengmao F: The method of lower and upper solutions for fourth-order two-point boundary value problems. Journal of Mathematical Analysis and Applications 1997,215(2):415-422. 10.1006/jmaa.1997.5639MATHMathSciNetView ArticleGoogle Scholar
- Li F, Zhang Q, Liang Z: Existence and multiplicity of solutions of a kind of fourth-order boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 2005,62(5):803-816. 10.1016/j.na.2005.03.054MATHMathSciNetView ArticleGoogle Scholar
- Han G, Xu Z: Multiple solutions of some nonlinear fourth-order beam equations. Nonlinear Analysis: Theory, Methods & Applications 2008,68(12):3646-3656. 10.1016/j.na.2007.04.007MATHMathSciNetView ArticleGoogle Scholar
- Cai X, Yu J, Guo Z: Existence of periodic solutions for fourth-order difference equations. Computers & Mathematics with Applications 2005,50(1-2):49-55. 10.1016/j.camwa.2005.03.004MATHMathSciNetView ArticleGoogle Scholar
- Agarwal RP, Grace SR, Manojlovic JV: On the oscillatory properties of certain fourth order nonlinear difference equations. Journal of Mathematical Analysis and Applications 2006,322(2):930-956. 10.1016/j.jmaa.2005.09.059MATHMathSciNetView ArticleGoogle Scholar
- Thandapani E, Arockiasamy IM: Fourth-order nonlinear oscillations of difference equations. Computers & Mathematics with Applications 2001,42(3–5):357-368.MATHMathSciNetView ArticleGoogle Scholar
- Schmeidel E, Szmanda B: Oscillatory and asymptotic behavior of certain difference equation. Nonlinear Analysis: Theory, Methods & Applications 2001,47(7):4731-4742. 10.1016/S0362-546X(01)00585-5MATHMathSciNetView ArticleGoogle Scholar
- He Z, Yu J: On the existence of positive solutions of fourth-order difference equations. Applied Mathematics and Computation 2005,161(1):139-148. 10.1016/j.amc.2003.12.016MATHMathSciNetView ArticleGoogle Scholar
- Zhang B, Kong L, Sun Y, Deng X: Existence of positive solutions for BVPs of fourth-order difference equations. Applied Mathematics and Computation 2002,131(2-3):583-591. 10.1016/S0096-3003(01)00171-0MATHMathSciNetView ArticleGoogle Scholar
- Yu J, Guo Z: On boundary value problems for a discrete generalized Emden-Fowler equation. Journal of Differential Equations 2006,231(1):18-31. 10.1016/j.jde.2006.08.011MATHMathSciNetView ArticleGoogle Scholar
- Jiang L, Zhou Z:Three solutions to Dirichlet boundary value problems for -Laplacian difference equations. Advances in Difference Equations 2008, 2008:-10.Google Scholar
- Jiang L, Zhou Z: Multiple nontrivial solutions for a class of higher dimensional discrete boundary value problems. Applied Mathematics and Computation 2008,203(1):30-38. 10.1016/j.amc.2007.12.014MATHMathSciNetView ArticleGoogle Scholar
- Bai D, Xu Y: Nontrivial solutions of boundary value problems of second-order difference equations. Journal of Mathematical Analysis and Applications 2007,326(1):297-302. 10.1016/j.jmaa.2006.02.091MATHMathSciNetView ArticleGoogle Scholar
- Agarwal RP: Difference Equations and Inequalities: Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics. Volume 228. 2nd edition. Marcel Dekker, New York, NY, USA; 2000:xvi+971.Google Scholar
- Elaydi SN: An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, NY, USA; 1996:xiv+389.View ArticleGoogle Scholar
- Rabinowitz PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics. Volume 65. American Mathematical Society, Providence, RI, USA; 1986:viii+100.Google Scholar
- Chang KC: Critical Point Theory and Its Applications. Science and Technical Press, Shanghai, China; 1986:iv+316.MATHGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.