Abstract and Applied Analysis

Volume 2012 (2012), Article ID 572172, 11 pages

doi:10.1155/2012/572172

Research Article

Positive Solutions for Sturm-Liouville Boundary Value Problems in a Banach Space

Hua Su,¹ Lishan Liu,² and Yonghong Wu³

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.

Abstract.

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.

1. Introduction

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 .

Definition 2.1. A function is said to be a positive solution of the boundary value problem (1.1) if satisfies , and the BVP (1.1).

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 [13]).

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 [1] 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 [13]). 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 [13]). 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

Denote

We now present our main results by Theorems 3.1 and 3.2.

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.

Theorem 3.2. Suppose that conditions,, and in Theorem 3.1 hold. Assume that also satisfies

();().

Then, the boundary value problem (1.1) has at least two solutions.

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.

4. Application

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):

where

Then obviously,

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 .

Acknowledgment

The authors were supported financially by Shandong Province Natural Science Foundation (ZR2009AQ004, ZR2010GL013), National Natural Science Foundation of China (11071141).

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