Research Article | Open Access
Existence of Multiple Solutions for a Class of -Dimensional Discrete Boundary Value Problems
By using critical point theory, we obtain some new results on the existence of multiple solutions for a class of -dimensional discrete boundary value problems. Results obtained extend or improve existing ones.
Let , , be the set of all natural numbers, integers, and real numbers, respectively. For any , define , when .
In this paper, we consider the existence of multiple solutions for the following -dimensional discrete nonlinear boundary value problem:
where , , is the forward difference operator defined by , with , and is continuous for . with , and are constants, and are -dimensional vectors, and are real value functions defined on and , respectively, and .
Existence of solutions of discrete boundary value problems has been the subject of many investigations. Motivated by Hartman's landmark paper , Henderson  showed that the uniqueness of solutions implies the existence of solutions for some conjugate boundary value problems. In recent years, by using various methods and techniques, such as nonlinear alternative of Leray-Schauder type, the cone theoretic fixed point theorem, and the method of upper and lower solution, a series of existence results for the solutions of the BVP (1.1) in some special cases, for example, ; have been obtained in literatures. We refer to [3–7].
The critical point theory has been an important tool for investigating the periodic solutions and boundary value problems of differential equations [8–10]. In recent years, it is applied to the study of periodic solutions [11–13] and boundary value problems [14–17] of difference equations.
For the case when , the scalar BVP (1.1) was studied by Yu and Guo in . By using critical point theorem, they obtained various conditions to guarantee the existence of one solution, but they did not obtain the existence conditions of multiple solutions. In this scalar case, the BVP can be viewed as the discrete analogue of the following self-adjoint differential equation:
which is a generalization of Emden-Fowler equation:
The Emden-Fowler equation originated from earlier theories of gaseous dynamics in astrophysics , and later, found applications in the study of fluid mechanics, relative mechanics, nuclear physics, and in the study of chemical reaction system .
For the case where , , (the zero vector of ), BVP (1.1) are reduced to
which were studied by Jiang and Zhou in . They obtained the existence results of multiple solutions by using critical point theory again.
In the aforementioned references, most of the difference equations involved are scalar. The purpose of this paper is further to demonstrate the powerfulness of critical point theory in the study of existence of discrete boundary value problems and obtain various conditions for the existence of at least two nontrivial solutions for the BVP (1.1).
The remaining of this paper is organized as follows. First, in Section 2, we will establish the variational framework associated with (1.1) and transfer the problem of the existence of solutions of (1.1) into that of the existence of critical points of the corresponding functional. Some basic results will also be recalled. Then, in Section 3, we present various new conditions on the existence of at least two nontrivial solutions for the BVP (1.1). Some examples are given to illustrate the conclusions. We mention that our results generalize the ones in  and improve the ones in .
2. Preliminary and Variational Framework
Let be a real Hilbert space, , which means that is a continuously Frećhet-differentiable functional defined on . is said to satisfy Palais-Smale condition (P-S condition for short), if any sequence for which is bounded and as possesses a convergent subsequence in .
Let be the open ball in with radius and centered at and let denote its boundary. The following lemmas will be useful in the proofs of our main results.
Lemma 2.1. (Mountain Pass Lemma ). Let be a real Hilbert space, and assume that satisfies the P-S condition and the following conditions. (1)There exist constant and such that , where .(2) and there exists such that . Then possesses a critical value . Moreover, where
Lemma 2.2. Let , , , then
Without loss of generality, we assume that , where , and there exists a function , such that for any , , where is the gradient of in , and where is the zero vector in .
Let , then BVP (1.1) reduces to
where , .
Then, can be equipped with the inner product
by which the norm can be induced by
where , , and and are the norm and the inner product in , respectively.
Define a linear mapping by
Then is a linear and one to one mapping. Clearly, .
where , and
Define afunctional on as
where . Then .
For any , denote
where satisfying (2.4). Thus, there is a one to one correspondence from to
3. Main Results
In this section, we will suppose that the matrix defined in (2.11) is positive definite, , are the minimal eigenvalue and maximal eigenvalue of , respectively. It is clear that , are also the minimal eigenvalue and maximal eigenvalue of , respectively. The first result is as follows.
Proof. Since , we know that where denotes the zero element of . By (3.1), we get and . Let
Then by (3.2), we get
where . Then , we have
Since , we see that as . Thus is bounded from above on , and can achieve its maximum on . In other words, there exists , such that . So is a critical point of , and is a solution of BVP (2.3) and (2.4).
For any with , we have
So and is a nontrivial solution to BVP (2.3) and (2.4).
To obtain another nontrivial solution of BVP (2.3) and (2.4), we will use Mountain Pass Lemma. We first show that satisfies P-S condition.
In fact, for any sequence in , is bounded and , then there exists such that , and it follows from (3.5) that that is, Since , this implies that is bounded and possesses a convergent subsequence. So satisfies P-S condition on .
Choosing , then for , from (3.6), we get This shows that satisfies condition (1) of the Mountain Pass Lemma.
On the other hand, from (3.5), as , so there exists such that for . Pick such that , then , and . So the condition (2) of the Mountain Pass Lemma is satisfied. Therefore, possesses a critical value where
A critical point corresponding to is nontrivial as . Let be a critical point corresponding to the critical value of . If , then we are done. Otherwise, , which gives Pick , then , and we have Thus, there exists such that , and is also a critical point of in .
If , then Theorem 3.1 holds. Otherwise, . In this situation, we replace with in the above arguments; then possesses a critical value and where Assume that is a critical point corresponding to . If , then the proof is complete. Otherwise, . Similarly, we can find a critical point of such that holds for some . Clearly, . The proof of Theorem 3.1 is now complete.
From Theorem 3.1, we have the following corollaries.
Proof. For any , , then , . So and condition (3.1) of Theorem 3.1 is satisfied. Let Then where . So, for any , and condition (3.2) of Theorem 3.1 is satisfied. The conclusion follows from Theorem 3.1.
Remark 3.3. For the special case where , the BVP (2.3) and (2.4) was studied in . Here, the corresponding matrix becomes which is positive definite when and . So, . In this case, Corollary 3.2 reduces to Theorem of . Therefore, our results extend the ones in . Corollary 3.2 also improves the conclusion of Theorem in .
The following corollaries is obvious.
Now we will give an example to illustrate our results.
Example 3.7. Consider the discrete boundary problem:
for and , .
In this example, , , , , , , , , So, . With the Matlab software, we can get the approximate eigenvalue of matrix : Thus is positive definite and , . Take , we find that Pick , , , then for any , , we have For any , , ,
By a similar method, we can obtain the following results.
At last, we will give another example.
Example 3.13. Consider the boundary value problem (3.24) and (3.25) where is replaced by Clearly, and . Pick , , then we have , and Pick , , then for any , we can see that For any , we can get According to Corollary 3.9, we know that the given BVP in this example has at least two nontrivial solutions.
Remark 3.14. When is negative definite, we can get similar conclusions. We do not repeat here.
This work is supported by the Specialized Fund for the Doctoral Program of Higher Eduction. (no. 20071078001), by the Natural Science Foundation of GuangXi (no. 0991279), by the Foundation of Education Department of GuangXi Province (no. 200807MS121), and by the project of Scientific Research Innovation Academic Group for the Education System of Guangzhou City.
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Copyright © 2010 Weiming Tan and Zhan Zhou. 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.