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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 708192, 21 pages
Positive Solutions for Nonlinear Fractional Differential Equations with Boundary Conditions Involving Riemann-Stieltjes Integrals
1School of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, China
2Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia
Received 10 March 2012; Accepted 23 July 2012
Academic Editor: Dumitru Baleanu
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.
We consider the existence of positive solutions for a class of nonlinear integral boundary value problems for fractional differential equations. By using some fixed point theorems, the existence and multiplicity results of positive solutions are obtained. The results obtained in this paper improve and generalize some well-known results.
This paper is concerned with the existence of positive solutions to the following boundary value problem (BVP) for fractional differential equation: where is a linear functional on given by a Riemann-Stieltjes integral with representing a suitable function of bounded variation, is the Riemann-Liouville fractional derivative of order , , satisfies the Carathéodory type conditions, and .
Fractional differential equations arise in the modeling and control of many real-world systems and processes particularly in the fields of physics, chemistry, aerodynamics, electrodynamics of complex media, and polymer rheology. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. Hence, intensive research has been carried out worldwide to study the existence of solutions of nonlinear fractional differential equations (see [1–25]). For example, by means of a mixed monotone method, Zhang  studied a unique positive solution for the singular boundary value problem where , , is the standard Riemann-Liouville derivative, is nonlinear, and and have different monotone properties.
Recently, nonlocal boundary value problems for fractional differential equations were investigated intensively [13–23]. In , Bai concerned the existence and uniqueness of a positive solution for the following nonlocal problem: where , , , is the standard Riemann-Liouville differentiation. The function is continuous on .
In , El-Shahed and Nieto investigated the existence of nontrivial solutions for the following nonlinear -point boundary value problem of fractional type: where , , , . Also the authors considered the analogous problem using the Caputo fractional derivative: Under certain growth conditions on the nonlinearity, several sufficient conditions for the existence of nontrivial solution are obtained by using the Leray-Schauder nonlinear alternative.
Inspired by the work of the above papers, the aim of this paper is to establish the existence and multiplicity of positive solutions of the BVP (1.1). We discuss the boundary value problem with the Riemann-Stieltjes integral boundary conditions, that is, the BVP (1.1), which includes fractional order two-point, three-point, multipoint, and nonlocal boundary value problems as special cases. Moreover, the in (1.1) is a linear function on denoting the Riemann-Stieltjes integral; the in the Riemann-Stieltjes integral is of bounded variation, namely, can be a signed measure. By using the Krasnosel’skii fixed point theorem, the Leray-Schauder nonlinear alternative and the Leggett-Williams fixed point theorem, some existence and multiplicity results of positive solutions are obtained.
The rest of this paper is organized as follows. In Section 2, we present some lemmas that are used to prove our main results. In Section 3, the existence and multiplicity of positive solutions of the BVP (1.1) are established by using some fixed point theorems. In Section 4, we give four examples to demonstrate the application of our theoretical results.
2. Basic Definitions and Preliminaries
We begin this section with some preliminaries of fractional calculus. Let and , where is the largest integer smaller than or equal to . For a function , we define the fractional integral of order of as provided the integral exists. The fractional derivative of order of a continuous function is defined by provided the right-hand side is pointwise defined on . We recall the following properties [26, 27] which are useful for the sequel. For , , we have As an example, we can choose a function such that .
For , the general solution of the fractional differential equation with is given by where . Hence for , we have Set
Lemma 2.1 (see ). Let . Then the boundary value problem, has a unique solution
Lemma 2.3. Let and for , the Green function defined by (2.12) has the following properties: (i) for all ;(ii), .
Proof. By Lemma 2.2, it is easy to prove this lemma, so we omit it.
Let . It follows that is a Banach space, where is defined by the supernorm . . Clearly is a cone of . Now, in the following, we give the assumptions to be used throughout the rest of this paper.(H1) is a function of bounded variation, for and .(H2) satisfies the following conditions of Carathéodory type:(i) is Lebesgue measurable for each fixed ;(ii) is continuous for a.e. .
In order to overcome the difficulty due to the dependance of on derivatives, we consider the following modified problem: where , .
Proof. If is a positive solution of the fractional order boundary value problem (1.1), let . Then from the boundary value conditions of (1.1) and the definition of the Riemann-Liouville fractional integral and derivative, we have
which imply that , . Thus is a positive solution of the nonlinear fractional integrodifferential equation (2.13).
On the other hand, if is a positive solution of the nonlinear fractional integrodifferential equation (2.13), let , then by (2.3) and the definition of the Riemann-Liouville fractional derivative, we have which imply that , , . Moreover, it follows from the monotonicity and property of that . Consequently, is a positive solution of the fractional order boundary value problem (1.1).
In order to prove our main results, we need the following lemmas.
Lemma 2.5 (see ). Let be a real Banach space, be a bounded open subset of , where , is a completely continuous operator. Then, either there exist , such that , or there exists a fixed point .
Lemma 2.6 (see ). Let be a real Banach space, be a cone in . Assume that and are two bounded open sets of with and . Let be a completely continuous operator such that either(i), and , , or(ii), and , .Then T has a fixed point in .
Lemma 2.7 (see [30, 31]). Let be a cone in a real Banach space , , be a nonnegative continuous concave functional on such that for all , and . Suppose that is completely continuous and there exist positive constants such that(C1) and for ,(C2) for ,(C3) for with .Then T has at least three fixed points , and satisfying
Remark 2.8. If , then condition (C1) of Lemma 2.7 implies condition (C3) of Lemma 2.7.
For notational convenience, we introduce the following constants: and a nonnegative continuous concave functional on the cone defined by
3. Main Results
In this section, we present and prove our main results.
Theorem 3.1. Assume that (H1) and (H2) hold and there exist nonnegative functions such that
for almost every and all , .
If then BVP (1.1) has a unique positive solution.
Proof. We will show that is a contraction mapping. For any and , by the definition of fractional integral, we obtain So, for any , by (3.3) and Lemma 2.3, we have This implies that where . By the Banach contraction mapping principle, we deduce that has a unique fixed point . Thus, by Lemma 2.4, is a unique positive solution of BVP (1.1).
Lemma 3.2. Assume that (H1) and (H2) hold and the following conditions are satisfied.(H3) There exist nonnegative real-valued functions such that
for almost every and all .
Then is a completely continuous operator.
Proof. For any , as for all , we have , so . Let be any bounded set. Then there exists a constant such that for any . Moreover for any , , . Proceeding as for (3.3), we obtain
Therefore, is uniformly bounded.
Now we show that is equicontinuous on . Since is continuous on , is uniformly continuous on . Hence, for any , there exists a constant such that for any , , when , it holds Consequently, for any and , we have This implies that is equicontinuous. Thus according to the Ascoli-Arzela Theorem, is a relatively compact set.
In the end, we show that is continuous. Assume that , , then and (), where is a positive constant. Keeping in mind that satisfies Carathéodory conditions on , we have Proceeding as for (3.3), for we obtain This together with (3.6), The Lebesgue dominated convergence theorem gives Now we deduce from (3.15), Lemma 2.3 that , as . So is continuous. Therefore is completely continuous.
Remark 3.3. If is continuous, by similar argument as above, we can show that is completely continuous.
Theorem 3.4. Assume that (H1)–(H3) hold. If then BVP (1.1) has at least one positive solution.
Proof. Let we have . From Lemma 3.2, we know that is completely continuous. If there exists such that then by (H3) and (3.19), we have which implies that This means that . By Lemma 2.5, has a fixed point . By Lemma 2.4, BVP (1.1) has at least one positive solution .
Proof. Let . For any , we have and for every . Similar to (3.7), for , we have
It follows from condition (i) that
Thus, for any , by (3.23) and Lemma 2.3, we have
which means that
On the other hand, let . For any , we have and for every . Similar to (3.23), from condition (ii), we can get Hence for any , , by (3.26) and Lemma 2.3 we have Thus we get By (3.25), (3.28), and Lemma 2.6, has a fixed point such that . By Lemma 2.4, BVP (1.1) has at least one positive solution .
Theorem 3.6. Assume that (H1)–(H3) hold. If there exist constants such that (I), for ,(II), for ,(III), for ,where , are defined by (2.19), then BVP (1.1) has at least three positive solutions , , and satisfying
Proof. We will show that all conditions of Lemma 2.7 are satisfied.
First, if , then . So we have , . Similar to (3.23), it follows from condition (II) that Thus, for any , by (3.30), we have which means that , . Therefore, . By Lemma 3.2, we know that is completely continuous.
Next, similar to (3.30) and (3.31), it follows from condition (I) that if then . So the condition of Lemma 2.7 holds.
Now, we take , . It is easy to see that , and so where is defined by (2.20). This proves that .
On the other hand, if , then , . By the definition of fractional integral, for any , , we obtain It follows from (3.33) and condition (III) that Hence we have which implies that , for . This shows that condition (C1) of Lemma 2.7 is also satisfied.
By Lemma 2.7 and Remark 2.8, BVP (2.13) has at least three positive solutions , and such that , , and , . By Lemma 2.4, BVP (1.1) has at least three positive solutions , . By (2.3), we have , . So are three positive solutions of BVP (1.1) satisfying The proof of Theorem 3.6 is completed.
Example 4.1. Consider the following problem: Let . Then Let Then is a nonnegative continuous function on and, for any and , satisfies So we have where denotes a Beta function. So all conditions of Theorem 3.1 are satisfied. Thus, by Theorem 3.1, BVP (4.1) has at least one positive solution.
Example 4.2. Consider the following problem: where with Set Then is continuous and As in , , , where denotes a Beta function and is defined by (4.2). So all conditions of Theorem 3.4 are satisfied. Thus, by Theorem 3.4, BVP (4.7) has at least one positive solution.
Example 4.3. Consider the following problem: where with as given by (4.8). Set Then is continuous and By Example 4.2, , , where is defined by (4.2). As in [1, 3], we also take , then Choosing , we have So all conditions of Theorem 3.5 are satisfied. Thus, by Theorem 3.5, BVP (4.12) has at least one positive solution.
Example 4.4. Consider the following problem: where with as given by (4.8). Set