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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 572172, 11 pages
Positive Solutions for Sturm-Liouville Boundary Value Problems in a Banach Space
1School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China
2School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
3Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia
Received 20 February 2012; Accepted 13 June 2012
Academic Editor: Shaoyong Lai
Copyright © 2012 Hua Su et al. 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.
We consider the existence of single and multiple positive solutions for a second-order Sturm-Liouville boundary value problem in a Banach space. The sufficient condition for the existence of positive solution is obtained by the fixed point theorem of strict set contraction operators in the frame of the ODE technique. Our results significantly extend and improve many known results including singular and nonsingular cases.
Boundary value problems for ordinary differential equations play a very important role in both theoretical study and practical application in many fields. They are used to describe a large number of physical, biological, and chemical phenomena. In this paper, we study the existence of positive solutions for the following second-order nonlinear Sturm-Liouville boundary value problem (BVP) in a Banach Space where are constants such that , and . Moreover, may be singular at and/or .
The BVP (1.1) is often referred to as a model for the deformation of an elastic beam under a variety of boundary conditions [1–12]. We notice that previous work is limited to use the completely continuous operators and the function is required to satisfy some growth condition or assumptions of monotonicity.
The aim of this paper is to consider the existence of positive solutions for the more general Sturm-Liouville boundary value problem (1.1) by using the fixed point theorem of strict set contraction operators. Here we allow to have singularity at . The results obtained in this paper improve and generalize many well-known results.
The rest of the paper is organized as follows. In Section 2, we first present some properties of Green's functions to be used to define a positive operator. Then we approximate the singular second-order boundary value problem by constructing an integral operator. In Section 3, the sufficient condition for the existence of single and multiple positive solutions for the BVP (1.1) is established. In Section 4, we give a example to demonstrate the application of our results.
2. Preliminaries and Lemmas
In this paper, we denote by a real Banach space. A nonempty closed convex subset in is said to be a cone if for and , where denotes the zero element of . The cone defines a partial ordering in by if and only if . Recall that the cone is said to be normal if there exists a positive constant such that implies .
In this paper, we assume is normal, and without loss of generality, we may assume that the normal constant of is 1. Let , and For , let , then is a Banach space with the norm .
We notice that if is a positive solution of the BVP (1.1) and , then .
Now we denote by the Green's functions for the following boundary value problem: It is well known that can be written by where .
It is easy to verify the following properties of : (I); (II), for any , where
Throughout this paper, we adopt the following assumptions:() and satisfies () is a uniformly continuous function and there exists such that for any bounded set , we have where denotes the Kuratowski measure of noncompactness in .
The following Lemmas play an important role in this paper (see ).
Lemma 2.2. Let be bounded and equicontinuous on , then .
Lemma 2.3. Let be bounded and equicontinuous on , then is continuous on and
Lemma 2.4. Let be a bounded set on . Then .
Now, for the given and the as in (II), we introduce It is easy to check that is a cone in and for , we have .
Next, we define an operator given by Clearly, is a solution of the BVP (1.1) if and only if is a fixed point of the operator .
Through direct calculation, by (II) and for , we have So, this implies that .
Lemma 2.5. Assume that, hold. Then is a strict set contraction operator.
Proof. Firstly, The continuity of is easily obtained. In fact, if and in the norm in , then for any , we get
so, by the uniformly continuity of , we have
This implies that in , that is, is continuous.
Now, let be a bounded set. It follows from that there exists a positive number such that for any . Then, we can get where is as defined in. So, is a bounded set in .
For any , by (), there exists a such that Let . It follows from the uniform continuity of on that there exists such that Consequently, when , we have This implies that is an equicontinuous set on . Therefore, by Lemma 2.2, we have
Without loss of generality, by condition, we may assume that is singular at . So, there exist with being a strictly increasing sequence and such that Next, we let Then, from the above discussion we know that is continuous on for every and
For any , by (2.19) and, there exists an such that for any , we have Therefore, for any bounded set , by Lemmas 2.3 and 2.4,, the above discussion and noting that , as , we have that As is arbitrarily, we get So, it follows from (2.17) and (2.24) that for any bounded set , we have And note that , we have is a strict set contraction operator. The proof is completed.
Remark 2.6. When , (2.6) naturally holds. In this case, we may take as 0, consequently, is a completely continuous operator. Furthermore, if , Dalmasso  used the following condition: Clearly, our condition is weaker than (2.26).
Our main tool used in this paper is the following fixed point index theorem of cone.
Theorem 2.7 (see ). Suppose that is a Banach space, is a cone, and let the be two bounded open sets of such that . Let operator be strict set contraction. Suppose that one of the following two conditions holds:(i), for all , for all ; (ii), for all , for all . Then has at least one fixed point in .
Theorem 2.8 (see ). Suppose is a real Banach space, is a cone. Let , and let the operator be completely continuous and satisfy , for all . Then(i)If , for all , then ;(ii)If , for all , then .
3. The Main Results
Theorem 3.1. Suppose that conditions, hold. Assume that also satisfies(), ; (), , where and satisfy Then, the boundary value problem (1.1) has a positive solution such that is between and .
Proof of Theorem 3.1. Without loss of generality, we suppose that . For any , we have
We now define two open subset and of
For , by (3.3), we have
For , if holds, we have Therefore, we have On the other hand, as , we have and by, we know Thus Therefore, by (3.7), (3.9), Theorem 2.7 and , we have that has a fixed point . Obviously, is a positive solution of the problem (1.1) and . The proof of Theorem 3.1 is complete.
Proof of Theorem 3.2. Firstly, by condition, we can have . Then, there exists an adequately small positive number such that and there exists a constant such that
Set , for any , by (3.10), we have
For , we have
Therefore, we have
Then by Theorem 2.8, we have
Next, by condition (), we have . Then, there exists an adequately small positive number such that , and there exists a constant such that We choose a constant , obviously, . Set , then for any , by (3.15), we have For , we have Therefore, we have Then by Theorem 2.8, we have
Finally, set , For any , by, Lemma 2.3 and proceeding as for the proof of Theorem 3.1, we have Then by Theorem 2.8, we have Therefore, by (3.14), (3.19), (3.21) and , we have Then have fixed points and . Obviously, and are all positive solutions of the BVP (1.1). The proof of Theorem 3.2 is complete.
In order to illustrate the application of our results, we give an example in this section.
Example 4.1. Consider the following singular boundary value problem (SBVP):
By computing, we know that the Green's function is
It is easy to note that and conditions,,, hold.
Next, by computing, we know that . We choose , as and , because of the monotone increasing of on , then Thus condition holds. Hence by Theorem 3.2, SBVP (4.1) has at least two positive solutions and .
The authors were supported financially by Shandong Province Natural Science Foundation (ZR2009AQ004, ZR2010GL013), National Natural Science Foundation of China (11071141).
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