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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 615707, 6 pages
Unique Solution of a Coupled Fractional Differential System Involving Integral Boundary Conditions from Economic Model
1Department of Electrical Engineering, North China Electric Power University, Baoding 071003, China
2Department of Management and Economic, North China Electric Power University, Baoding 071003, China
3School of Land Science and Technology, China University of Geosciences, Beijing 100083, China
4Institute of Finance and Banking, Chinese Academy of Social Sciences, Beijing, China
Received 13 May 2013; Accepted 1 July 2013
Academic Editor: Yonghong Wu
Copyright © 2013 Rui Li 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 study the existence and uniqueness of the positive solution for the fractional differential system involving the Riemann-Stieltjes integral boundary conditions , , , , , and , where , , and and are the standard Riemann-Liouville derivatives, and are functions of bounded variation, and and denote the Riemann-Stieltjes integral. Our results are based on a generalized fixed point theorem for weakly contractive mappings in partially ordered sets.
Mathematical model is an important tool designed to describe the operation of the economy of a country or a region. Because fractional operators are non-local, they are more suitable for constructing models possessing memory effect with the long time periods, and then fractional differential equations possess large advantage in describing economic phenomena over the time periods. In this paper, we focus on a fractional model arising from economy. Assuming that are continuous and nondecreasing with respect to the second variable on , we discuss the existence and uniqueness of positive solutions for the following system of fractional differential equation with nonlocal Riemann-Stieltjes integral boundary conditions: where , , and and are the standard Riemann-Liouville derivatives, and are functions of bounded variation, and and denote the Riemann-Stieltjes integral.
Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. The idea of using a Riemann-Stieltjes integral with a signed measure is due to Webb and Infante in [1, 2]. The papers [1–3] contain several new ideas and give a unified approach to many BVPs. This implies that the Riemann-Stieltjes integral boundary value problem is a more generalized case which includes multipoints, integral boundary conditions and many nonlocal boundary conditions, as special cases. For some recent work on boundary value problems of fractional differential equation, we refer the reader to some recent papers (see [1–14]).
Recently, motivated by [1–3], Hao et al.  studied the existence of positive solutions for theth-order singular nonlocal boundary value problem where may be singular at and also may be singular at , but there is no singularity at . The existence of positive solutions of the BVP (2) is obtained by means of the fixed point index theory in cones. In , Zhang and Han considered the existence of positive solutions of the following singular fractional differential equation: where , and can be a signed measure. Some growth conditions were adopted to guarantee that (3) has a unique positive solution. Recently, by using a new fixed point theorem for weakly contractive mappings in partially ordered sets, Tao et al.  considered the existence and uniqueness of positive solution for the following problem: where , , and , and is the standard Riemann-Liouville derivative.
Motivated by the above work, in this paper, we study the existence and uniqueness of positive solution for the system of fractional differential equations with nonlocal Riemann-Stieltjes boundary integral conditions. Our main tool is the fixed point theorem for weakly contractive mappings in partially ordered sets, which is obtained by papers [6, 15].
2. Preliminaries and Lemmas
Definition 2 (see [16–18]). The Riemann-Liouville fractional derivative of order of a function is given by where , denotes the integer part of number , provided that the right-hand side is pointwise defined on .
To ensure the nonnegativity of Green functions, we use the following elementary assumption in this paper. are increasing functions of bounded variation such that , for and , , where are defined by (14).
Lemma 5. Let , , and let hold; then and satisfy
Lemma 6 (see ). Let be a partially ordered set, and suppose that there exists a metric in such that is a complete metric space. Assume that satisfies the following condition: if is a nondecreasing sequence in such that , then for all . Let be a nondecreasing mapping, and there exists a constant such that where is a nondecreasing function. If there exists with , then has a fixed point.
Now, we mean by the Banach space of all continuous functions on with the usual . Note that this space can be equipped with a partial order given by It has been proved in [20, 21] that with the classic metric given by satisfies the following condition.
If is a nondecreasing sequence in such that , then for all . Moreover, for , the function is continuous in , and satisfies condition (19).
Next, define a subcone of as follows: Note that is a closed set of ; is a complete metric space.
Clearly, is a solution of system (1) if and only if is a solution of the following nonlinear integral system of equations: Consequently, system (23) is equivalent to the following integral equation: Thus, for , define the operator by Then from the assumption on , and Lemma 5, we have .
3. Main Results
Now, define the class of functions ,
Remark 8. The standard function ; for example, , , , , and so forth.
Remark 9. Clearly, given that, then.
Theorem 10. Suppose that holds, and there exist two functionsand constants , , , and which satisfy such that forwith, . Then, problem (1) has a unique nonnegative solution.
Proof. To prove that the problem (1) has a unique nonnegative solution, it is sufficient to check that the hypotheses of Lemma 6 are satisfied.
By the monotonicity ofand, we know that the operatoris nondecreasing. Now, denote Then,. For any, from (27) and (28), we have
Since, which implies that , andis nondecreasing. Thus, for, we findandsuch that
Finally, taking into account the zero function,, by Lemma 6, problem (1) has a unique nonnegative solution.
Proof. It follows from Theorem 10 that the problem (1) has a unique nonnegative solution. We prove that it is also a positive solution of the problem (1).
Otherwise, there exists such that ; that is; Then, Consequently, Note that , ; then, we have
In addition, it follows from , that , and by the continuity of, we can find a set such that andfor any, whereandstands for the Lebesgue measure, which contradicts with (35). Consequently, we have, . In the same way, we also have and, Hence, the problem (1) has a unique positive solution.
Example 12. Consider the following boundary value problem with fractional order, : where The BVP (37) becomes the 4-point BVP with coefficients: By Theorem 11, the BVP (37) has a unique positive solution.
Proof. Obviously,, , and
By simple calculation, we know that , , and , are all increasing. So, holds.
Take Then, for any, , where Thus,, and all of the conditions of Theorem 10 are satisfied.
On the other hand,, , and by Theorem 11, the BVP (37) has a unique positive solution.
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