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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 641617, 5 pages
http://dx.doi.org/10.1155/2013/641617
Research Article

Existence Results for a Fully Fourth-Order Boundary Value Problem

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 21 December 2012; Accepted 27 June 2013

Academic Editor: To Ma

Copyright © 2013 Yongxiang Li and Qiuyan Liang. 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.

Abstract

We discuss the existence of solution for the fully fourth-order boundary value problem , , . A growth condition on guaranteeing the existence of solution is presented. The discussion is based on the Fourier analysis method and Leray-Schauder fixed point theorem.

1. Introduction and Main Results

In this paper we deal with the existence of solution for the fully fourth-order ordinary differential equation boundary value problem (BVP) where is continuous. This problem models deformations of an elastic beam whose two ends are simply supported in equilibrium state, and its research has important significance in mechanics.

For the special case of BVP(1) that does not contain derivative terms and , namely, simply fourth-order boundary value problem

the existence of solution has been studied by many authors; see [18]. In [1], Aftabizadeh showed the existence of a solution to PBV(2) under the restriction that is a bounded function. In [2, Theorem 1], Yang extended Aftabizadeh’s result and showed the existence for BVP(2) under the growth condition of the form where , , and are positive constants such that

In [3], under a more general linear growth condition of two-parameter nonresonance, del Pino and Manásevich also discussed the existence of BVP(2) and the result of Yang was further extended. For more results involving two-parameter nonresonance condition see [4, 7]. All these works are based on Leray-Schauder degree theory. In [5, 6], the upper and lower solutions method is applied to discuss the existence of BVP(2). Recently, in [8] the fixed point index theory in cones is employed to BVP(2) and some existence results of positive are obtained, where may be super-linear growth.

For the more simple case of BVP(1) that does not contain any derivative terms, the following fourth-order boundary value problem

has been studied by more researchers, and various theorems and methods of nonlinear analysis have been applied; see [913] and reference therein.

However, few researchers consider the fully fourth-order boundary value problem BVP(1). The purpose of this paper is to discuss the existence of solution of BVP(1). We will extend the Yang’s result previously mentioned from BVP(2) to the general BVP(1). Our results are as follows.

Theorem 1. Assume that and it satisfies the growth condition
for all and , where and are constants and satisfy the restriction
Then the BVP(1) possesses at least one solution.

Theorem 1 is a directly extension of Yang’s result previously mentioned. In Theorem 1, the condition (7) is optimal. If the condition (7) does not hold, the existence of solution of BVP(1) cannot be guaranteed. Strengthening the condition (6) of Theorem 1, we can obtain the following uniqueness result.

Theorem 2. Assume that and it satisfies the Lipschitz-type condition
for any and , where , , , are constants and satisfy (7). Then BVP(1) has a unique solution.

If the partial derivatives , , , and exist, then from Theorem 2 and the theorem of differential mean value, we have the following.

Corollary 3. Let and the partial derivatives , , , and exist. If there exist positive constants such that
and the constants satisfy (7), then BVP(1) has one unique solution.

The proofs of Theorems 1 and 2 are based on the Fourier analysis method and Leray-Schauder fixed point theorem, which will be given in Section 2.

2. Proof of the Main Results

Let and be the usual Hilbert space with the interior product and the norm . For , let be the usual Sobolev space with the norm . means that , is absolutely continuous on and .

Given , we consider the linear fourth-order boundary value problem (LBVP)

Let be the Green’s function to the second-order linear boundary value problem

which is explicitly expressed by

For every given , it is easy to verify that the LBVP(10) has a unique solution in Carathéodory sense, which is given by

If , the solution is in and is a classical solution. Moreover, the solution operator of LBVP(10), is a linearly bounded operator. By the compactness of the Sobolev embedding and the continuity of embedding , we see that maps into and is a completely continuous operator.

Choose a subspace of by

Clearly, is a closed subspace, and hence is a Banach space by the norm of . Define another norm on by

One easily verifies that is equivalent to . Hereafter, we use to denote the Banach space endowed the norm , namely,

By the boundary condition of LBVP(10), the solution operator maps into . Hence is completely continuous.

Lemma 4. For LBVP(10), the following two conclusions hold.(a)The norm of the solution operator of LBVP(10) satisfies .(b)For every , the unique solution of LBVP(10) satisfies the inequalities

Proof. Since sine system is a complete orthogonal system of , every can be expressed by the Fourier series expansion where , , and the Parseval equality
holds. Let ; then is the unique solution of LBVP(10), and , , and can be expressed by the Fourier series expansion of the sine system. Since , by the integral formula of Fourier coefficient, we obtain that
On the other hand, since cosine system is another complete orthogonal system of , every can be expressed by the cosine series expansion where , . For the above , by the integral formula of the coefficient of cosine series, we obtain the cosine series expansions of and :
Now from (23), (19), and Parseval equality, it follows that
This means that , namely, (a) holds.
By (20)–(22) and Paserval equality, we have that

This shows that the conclusion (b) holds.

Proof of Theorem 1. We define a mapping by
From the assumption (6) and the property of Carathéodory mapping it follows that is continuous and it maps every bounded set of into a bounded set of . Hence, the composite mapping is completely continuous. We use the Leray-Schauder fixed-point theorem to show that has at least one fixed-point. For this, we consider the homotopic family of the operator equations:
We need to prove that the set of the solutions of (29) is bounded in . See [14].
Let be a solution of an equation of (29) for . Set ; then by the definition of , is the unique solution of LBVP(10). By (a) of Lemma 4, we have
From (28), (6), and (b) of Lemma 4, it follows that
Combining this inequality with (30), we obtain that
This means that the set of the solutions for (29) is bounded in . Therefore, by the Leray-Schauder fixed-point theorem [14], has a fixed-point . Let . By the definition of , is a solution of LBVP(10) for . Since , from (28) it follows that . Hence is a classical solution of LBVP(10), and by (28) is also a solution of BVP(1).
The proof of Theorem 1 is completed.

Proof of Theorem 2. Let . From condition (8) of Theorem 2 we easily see that Condition (6) of Theorem 1 holds. By Theorem 1, the BVP(1) has at least one solution.
Now, let be two solutions of BVP(1); then , . From (8) and (28), we obtain that
for . Since is the solution of LBVP(10) for , by (33) and (b) of Lemma 4, we have
From this and (a) of Lemma 4, it follows that
Since , from (35) we see that , that is . Therefore, BVP(1) has only one solution.
The proof of Theorem 2 is completed.

Example 5. Consider the following fully linear fourth-order boundary value problem where the coefficient functions and the inhomogeneous term . All the known results of [113] are not applicable to this equation. Let
It is easy to see that the partial derivatives , , , and exist and
Assume that the constants satisfy (7). Then by Corollary 3, (36) has a unique solution.

Example 6. Consider the following nonlinear fourth-order boundary value problem where , . Let
Then and it satisfies that
From this one easily proves that there exists a positive constant such that
Since (7) holds for the constants , by (42) satisfies the conditions of Theorem 1. Hence by Theorem 1, (39) has at least one solution. This conclusion cannot be obtained from the results in [113].

Acknowledgments

This research is supported by NNSFs of China (11261053, 11061031) and the NFS of Gansu province (1208RJZA129).

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