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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 696283, 21 pages
Positive Solutions for Second-Order Singular Semipositone Differential Equations Involving Stieltjes Integral Conditions
1School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China
2Department 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.
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.
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 , 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 , Lan [5–7], Liu , Ma et al. [9, 10], and Yao . In , 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  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 ,  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  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:
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 , so we omit it.
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
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):
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
On the other hand, from Lemma 2.5, we also obtain
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.
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 Taking