Research Article | Open Access
Existence of Positive Solutions for Fourth-Order Boundary Value Problems with Sign-Changing Nonlinear Terms
the existence of positive solutions for a fourth-order boundary value problem with a sign-changing nonlinear term is investigated. By using Krasnoselskii’s fixed point theorem, sufficient conditions that guarantee the existence of at least one positive solution are obtained. An example is presented to illustrate the application of our main results.
In this paper, we consider the existence of positive solutions to the following fourth-order boundary value problem (BVP): where is a positive parameter, is continuous and may be singular at , and is Lebesgue integrable and has finitely many singularities in .
Boundary value problems for ordinary differential equations play a very important role in both theory and applications. They are used to describe a large number of physical, biological, and chemical phenomena. The work of Timoshenko  on elasticity, the monograph by Soedel  on deformation of structures, and the work of Dulcska  on the effects of soil settlement are rich sources of such applications. There has been a great deal of research work on BVPs for second and higher order differential equations, and we cite as recent contributions the papers of Anderson and Davis , Baxley and Haywood , and Hao et al. . For surveys of known results and additional references, we refer the readers to the monographs by Agarwal et al. [7, 8].
Many authors have studied the existence of positive solutions for fourth-order boundary value problems where the nonlinearity takes nonnegative values, see [9–13]. However, for problems with sign-changing nonlinearities, only a few studies have been reported.
Owing to the importance of high order differential equations in physics, the existence and multiplicity of the solutions to such problems have been studied by many authors, see [9, 12–17]. They obtained the existence of positive solutions provided is superlinear or sublinear in by employing the cone expansion-compression fixed point theorem.
In , by using the strongly monotone operator principle and the critical point theory to discuss BVP the authors established some sufficient conditions for to guarantee that the problem has a unique solution, at least one nonzero solution, or infinitely many solutions.
In , Feng and Ge considered the fourth-order singular differential equation subject to one of the following boundary conditions: where . By using a fixed point index theorem in cones and the upper and lower solutions method, the authors discussed the existence of positive solutions for the above BVP.
However, most papers only focus on attention to the case where the nonlinearity has no singularities or/and takes nonnegative values on and . Inspired by the work of the above papers, our aim in the present paper is to investigated the existence of positive solutions to BVP (1) by employing the fixed point theorem of cone expansion and compression of norm type. Some well-known results in the literature are generalized and improved.
By singularity we mean that the function in BVP (1) are allowed to be unbounded at some point. In the paper, BVP (1) is allowed to have finitely many singularities in . In BVP(1), are allowed to change sign and tend to negative infinity. An element for a.e. is called a positive solution of BVP (1) if it satisfies BVP (1) and for any .
2. Preliminaries and Several Important Lemmas
Let be equipped with norm , then is a real Banach space.
Definition 1. We define a ordering in by for if and only if
In the following, let us define a cone in by
is the Green’s function of the following BVP
For convenience, we list the following assumptions: is continuous and there exist constants , such that for any , is Lebesgue integrable such that where , .
Remark 2. The inequality (10) is equivalent to the following inequality:
For any , let us define a function by
Let , . Obviously is continuous on . By , we obtain so is well defined in . By direct computation, we have which imply that is a positive solutions of the following BVP:
Now, we consider the following BVP: It is well known that for a.e. is a solution of BVP (17) if and only if is a solution of the following nonlinear integral equation:
Define an operator as Obviously, the existence of solutions of the BVP (17) is equivalent to the existence of fixed points of the operator in the real Banach space .
Lemma 3. Suppose that holds, then is nondecreasing in in , for any fixed .
Proof. For any fixed and for any , without the loss of the generality, let . If , obviously the equation holds. If , let , then we obtain . It follows from (10) that that is, is nondecreasing in in .
Proof. Assume that is a positive solution of BVP (17) such that , then from (17) and the definition of , we have Let , then for a.e. , which imply that Thus, (21) becomes Notice that and (23), we know that is a positive solution of BVP (1), that is, is a positive solution of BVP (1).
Lemma 5. Assume that and hold. Then, is well defined and is a completely continuous operator.
Proof. For any , choose such that , then we obtain . Thus, by (10), (12), and Lemma 3, we have
Hence, for any , we get
where . Thus, is well defined.
Next, for any , let . Then, there exists such that . Since we obtain Hence, we have So, we conclude that .
Let be any bounded set, then there exists a constant such that for any , we have By (12), (29) and Lemma 3, for any , we have From (9), (30), and Lemma 3, for any , we have where . Therefore, is uniformly bounded.
Since is continuous in , is uniformly continuous. Hence, for any , there exists such that , for any we have from (30), (32) and , for any , we obtain Therefore, is equicontinuous on . According to the Ascoli-Arzela Theorem, is a relatively compact set.
At the end, Let . Then, is bounded, let , for any , we have By (12), (34) and Lemma 3, we get From (35), the continuity of and Lebesgue dominated convergence theorem, we have Therefore, is continuous. So is a completely continuous operator.
The proof of our main result is based upon an application of the following fixed point theorem in a cone.
Theorem 6 (see ). Let be a Banach space, and be a cone. Assume are open bounded subsets of E with , and let be a completely continuous operator such that(i), and , ; or(ii), and , .Then, has at least one fixed point in .
3. The Main Results and Proofs
Theorem 7. Suppose that and hold. Assume that there exist constants and such that where Then, for sufficiently small, BVP (1) has at least one positive solution for a.e. in P.
where are defined in . For any , we have , since
for any and , we have
From Lemma 3, we get
Then, for any and , we have
Thus, we obtain that
Let . For and , we have Hence, by (37) and (47), for any and , we have Thus, we get
By Theorem 6, we know that has at least a fixed point with .
Thus, for any , we have It follows from Lemma 4 that is a positive solution of BVP (1).
Corollary 8. Suppose that and hold. Assume that there exist constants and such that Then, for adequately small, BVP (1) has at least one positive solution for a.e. in P.
4. An Example
Now, we present an example to illustrate the main result.
Example 1. Consider the following BVP where is positive parameter, clearly Let then holds. By calculating, it is easy to obtain that thus holds. Obviously, for any fixed , we have Let then for any . By Corollary 8, we know that BVP (52) has at least one positive solution for a.e. in .
This paper is supported by the NNSF of China (10971045) and HEBNSF of China (A2012506010).
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