- Research Article
- Open Access
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.
- Periodic Solution
- Boundary Value Problem
- Difference Equation
- Nontrivial Solution
- Real Banach Space
Consider the following boundary value problem (BVP):
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.
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 .
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).
Lemma 2.2 (Saddle point theorem).
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.
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.
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
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.
To prove Theorem 4.2, we need the following lemma.
Proof of Theorem 4.2.
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 .
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
The first result is as follows.
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.
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.
Consider the following special case
Then BVP (4.33) possesses at least one solution.
4.3. The Lipschitz Case
In this subsection, we suppose the following.
then condition (4.36) is satisfied.
In view of (4.36), we have
By using the theory of linear algebra, we have the next necessary and sufficient conditions.
This work is supported by the Specialized Fund for the Doctoral Program of Higher Eduction (no. 20071078001).
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