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Abstract and Applied Analysis

Volume 2012 (2012), Article ID 696283, 21 pages

http://dx.doi.org/10.1155/2012/696283

## Positive Solutions for Second-Order Singular Semipositone Differential Equations Involving Stieltjes Integral Conditions

^{1}School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China^{2}Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

Received 18 March 2012; Accepted 3 May 2012

Academic Editor: Shaoyong Lai

Copyright © 2012 Jiqiang Jiang 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

By means of the fixed point theory in cones, we investigate the existence of positive solutions for the following second-order singular differential equations with a negatively perturbed term: , , , where is a parameter; is continuous; may be singular at , and , and the perturbed term is Lebesgue integrable and may have finitely many singularities in , which implies that the nonlinear term may change sign.

#### 1. Introduction

In this paper, we are concerned with positive solutions of the following second-order singular semipositone boundary value problem (BVP): where is a parameter, , are constants such that , and the integrals in (1.1) are given by Stieltjes integral with a signed measure, that is, , are suitable functions of bounded variation, is a Lebesgue integral and may have finitely many singularities in , is continuous, may be singular at , , and .

Semipositone BVPs occur in models for steady-state diffusion with reactions [1], and interest in obtaining conditions for the existence of positive solutions of such problems has been ongoing for many years. For a small sample of such work, we refer the reader to the papers of Agarwal et al. [2, 3], Kosmatov [4], Lan [5–7], Liu [8], Ma et al. [9, 10], and Yao [11]. In [12], the second-order -point BVP, is studied, where (), , is a positive parameter. By using the Krasnosel’skii fixed point theorem in cones, the authors established the conditions for the existence of at least one positive solution to (1.2), assuming that , , is continuous, and there exists such that for . If the constant is replaced by any continuous function on , also has a lower bound and the existence results are still true.

Recently, Webb and Infante [13] studied arbitrary-order semipositone boundary value problems. The existence of multiple positive solutions is established via a Hammerstein integral equation of the form: where is the corresponding Green function, is nonnegative and may have pointwise singularities, satisfies the Carathéodory conditions and for some . Although is a constant, because of the term , [13] includes nonlinearities that are bounded below by an integral function. It is worth mentioning that the boundary conditions cover both local and nonlocal types. Nonlocal boundary conditions are quite general, involving positive linear functionals on the space , given by Stieltjes integrals.

For the cases where the nonlinear term takes only nonnegative values, the existence of positive solutions of nonlinear boundary value problems with nonlocal boundary conditions, including multipoint and integral boundary conditions, has been extensively studied by many researchers in recent years [14–25]. Kong [17] studied the second-order singular BVP: where is a positive parameter, is continuous, and are nondecreasing, and the integrals in (1.4) are Riemann-Stieltjes integrals. Sufficient conditions are obtained for the existence and uniqueness of a positive solution by using the mixed monotone operator theory.

Inspired by the above work, the purpose of this paper is to establish the existence of positive solutions to BVP (1.1). By using the fixed point theorem on a cone, some new existence results are obtained for the case where the nonlinearity is allowed to be sign changing. We will address here that the problem tackled has several new features. Firstly, as , the perturbed effect of on may be so large that the nonlinearity may tend to negative infinity at some singular points. Secondly, the BVP (1.1) possesses singularity, that is, the perturbed term may has finitely many singularities in , and is allowed to be singular at , , and . Obviously, the problem in question is different from those in [2–13]. Thirdly, and denote the Stieltjes integrals where are of bounded variation, that is, and can change sign. This includes the multipoint problems and integral problems as special cases.

The rest of this paper is organized as follows. In Section 2, we present some lemmas and preliminaries, and we transform the singularly perturbed problem (1.1) to an equivalent approximate problem by constructing a modified function. Section 3 gives the main results and their proofs. In Section 4, two examples are given to demonstrate the validity of our main results.

Let be a cone in a Banach space . For , let , , and . The proof of the main theorem of this paper is based on the fixed point theory in cone. We list here one lemma [26, 27] which is needed in our following argument.

Lemma 1.1. *Let be a positive cone in real Banach space , is a completely continuous operator. If the following conditions hold: *(i)* for ,*(ii)*there exists such that for any and ,**then, has a fixed point in .*

*Remark 1.2. *If (i) and (ii) are satisfied for and , respectively, then Lemma 1.1 is still true.

#### 2. Preliminaries and Lemmas

Denote where Obviously,

Throughout this paper, we adopt the following assumptions.(), , , , and () is a Lebesgue integral and .() For any , where with on and , is continuous and nonincreasing on , is continuous on , and for any constant ,

*Remark 2.1. *If and are two positive measures, then the assumption () can be replaced by a weaker assumption:(), , .

*Remark 2.2. *It follows from (2.4) and () that
For convenience, in the rest of this paper, we define several constants as follows:

*Remark 2.3. *If satisfies (1.1), and for any , then we say that is a positive solution of BVP (1.1).

Lemma 2.4. *Assume that holds. Then, for any , the problem,
**
has a unique solution
**
where
*

*Proof. *The proof is similar to Lemma 2.2 of [28], so we omit it.

Lemma 2.5. *Suppose that holds, then Green’s function defined by (2.12) possesses the following properties: *(i)*, ;*(ii)*, ,**where , , and are defined by (2.9).*

*Proof. *(i) It follows from (2.4) that

(ii) By the monotonicity of , and the definition of , we have
By (2.4) and the left-hand side of inequalities (2.14), we have
Similarly, by (2.4) and the right-hand side of inequalities (2.14), we have
The proof of Lemma 2.5 is completed.

Lemma 2.6. *Suppose that and hold. Then, the boundary value problem,
**
has unique solution
**
which satisfies
*

*Proof. *It follows from (2.11), Lemma 2.5, and that (2.18) and (2.19) hold.

Let be a real Banach space with the norm for . We let where . Clearly, is a cone of .

For any , let us define a function :

Next, we consider the following approximate problem of (1.1):

Lemma 2.7. *If is a positive solution of problem (2.22) with for any , then is a positive solution of the singular semipositone differential equation (1.1).*

*Proof. *If is a positive solution of (2.22) such that for any , then from (2.22) and the definition of , we have
Let , then , which implies that
Thus, (2.23) becomes
that is, is a positive solution of (1.1). The proof is complete.

To overcome singularity, we consider the following approximate problem of (2.22): where is a positive integer. For any , let us define a nonlinear integral operator as follows: It is obvious that solving (2.26) in is equivalent to solving the fixed point equation in the Banach space .

Lemma 2.8. *Assume that – hold, then for each , , , is a completely continuous operator.*

*Proof. *Let be fixed. For any , by (2.27) we have
which implies that is nonnegative and concave on . For any and , it follows from Lemma 2.5 that
Thus,
On the other hand, from Lemma 2.5, we also obtain
So,
This yields that .

Next, we prove that is completely continuous. Suppose and with . Notice that
This, together with the continuity of , implies
Using the Lebesgue dominated convergence theorem, we have
So, is continuous.

Let be any bounded set, then for any , we have , . Therefore, we have
By (), we have
It is easy to show that is uniformly bounded. In order to show that is a compact operator, we only need to show that is equicontinuous. By the continuity of on , for any , there exists such that for any and , we have
By (2.36)–(2.37), and (2.27), we have
where
This means that is equicontinuous. By the Arzela-Ascoli theorem, is a relatively compact set. Now since and are given arbitrarily, the conclusion of this lemma is valid.

#### 3. Main Results

Theorem 3.1. *Assume that conditions – are satisfied. Further assume that the following condition holds.** There exists an interval such that**
Then, there exists such that the BVP (1.1) has at least one positive solution provided . Furthermore, the solution also satisfies for some positive constant .*

*Proof. *Take . Let
where is defined by (2.40). For any , , noticing that , we have
For any , by (3.3), we have
which means that

On the other hand, choose a real number such that , where , is defined by (2.9). By , there exists such that for any , we have
Take . Next, we take , and for any , , , we will show
Otherwise, there exist and such that
From , we know that . Then, for , we have
So, by (3.6), (3.9), we have
This implies that , which is a contradiction. This yields that (3.7) holds. By (3.5), (3.7), and Lemma 1.1, for any and , we obtain that has a fixed point in .

Let be the sequence of solutions of the boundary value problems (2.26). It is easy to see that they are uniformly bounded. Next, we show that are equicontinuous on . From , we know that
For any , by the continuity of in , there exists such that for any and , we have
This, combined with (2.11) and (2.37), implies that for any and , we have
By the Ascoli-Arzela theorem, the sequence has a subsequence being uniformly convergent on . Without loss of generality, we still assume that itself uniformly converges to on . Since , we have . By (2.26), we have
From (3.14), we know that is bounded sets. Without loss of generality, we may assume as . Then, by (3.14) and the Lebesgue dominated convergence theorem, we have
By (3.15), direct computation shows that
On the other hand, letting in the following boundary conditions:
we deduce that is a positive solution of BVP (2.22).

Let and . By (3.11) and the convergence of sequence , we have , . It then follows from Lemma 2.7 that BVP (1.1) has at least one positive solution satisfying for any . The proof is completed.

Theorem 3.2. *Assume that conditions – are satisfied. In addition, assume that the following condition holds.** There exists an interval such that
where and
Then there exists such that the BVP (1.1) has at least one positive solution provided . Furthermore, the solution also satisfies for some positive constant .*

*Proof. *By (3.18), there exists such that, for any , , we have
Choose . Let as . Next, we take , and for any , , , we will show that
Otherwise, there exist and such that
From , we know that , and
So, we have on , . Then, by (3.20) we have
This implies that , which is a contradiction. This yields that (3.21) holds.

On the other hand, by (3.19) and the continuity of on , we have
where is defined by (2.40). For
there exists such that when , for any , we have . Take
Then, for any , we have
It follows from (3.19) and (3.28) that
which means that
By (3.21), (3.30), and Lemma 1.1, for any and , we obtain that has a fixed point in satisfying . The rest of proof is similar to Theorem 3.1. The proof is complete.

*Remark 3.3. *From the proof of Theorem 3.2, we can see that if is replaced by the following condition.() There exists an interval such that
then, the conclusion of Theorem 3.2 is still true.

#### 4. Applications

In this section, we construct two examples to demonstrate the application of our main results.

*Example 4.1. *Consider the following 4-point boundary value problem:
where is a parameter and
The BVP (4.1) can be regarded as a boundary value problem of the form of (1.1). In this situation, and
Let
and let , , . By direct calculation, we have , , and
Clearly, the conditions – hold. Taking , we have
Thus also holds. Consequently, by Theorem 3.1, we infer that the singular BVP (4.1) has at least one positive solution provided is small enough.

*Example 4.2. *Consider the following problem:
where is a parameter. Let
Then, , . Here, , so the measure changes sign on . By direct calculation, we have
where