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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 457965, 8 pages
Positive Solutions of a Singular Nonlocal Fractional Order Differential System via Schauder’s Fixed Point Theorem
1School of Mathematical and Informational Sciences, Yantai University, Yantai, Shandong 264005, China
2Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia
3School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, Shandong 250014, China
Received 24 March 2014; Revised 12 May 2014; Accepted 12 May 2014; Published 26 May 2014
Academic Editor: Xinan Hao
Copyright © 2014 Xinguang Zhang 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 establish the existence of positive solutions to a class of singular nonlocal fractional order differential system depending on two parameters. Our methods are based on Schauder’s fixed point theorem.
Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Particularly, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of many materials and processes. With this advantage, fractional order models are more realistic and practical than the classical integer-order models in physics, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, fitting of experimental data, and so forth [1–16]. Recently, Rehman and Khan  studied the fractional order multipoint boundary value problem: where , , , , with . By using the contraction mapping principle, the existence and uniqueness of positive solutions were established. In , Zhang et al. discussed the existence and uniqueness of positive solutions for the following fractional differential equation with derivatives: where , , , , with , and is the standard Riemann-Liouville derivative. is continuous, and may be singular at . By means of monotone iterative technique, the existence and uniqueness of the positive solution for a fractional differential equation with derivatives are established, and the iterative sequence of the solution, an error estimation, and the convergence rate of the positive solution are also given.
However, the research on the systems of fractional differential equations has not received much attention. So motivated by the results mentioned above, in this paper, we study the existence of positive solutions for the following singular nonlocal fractional order differential system depending on two parameters: where , , , , , , , , with , , , are the standard Riemann-Liouville derivatives, and and are positive parameters. and are continuous and may be singular at and . The system (3) is an abstract model arising from biological dynamic system, which was introduced by Perelson  to describe the primary infection with HIV in integer-order version, and was extended to a fractional order version of HIV-1 infection of T-cells by Arafa et al. .
The present paper has several interesting features. Firstly, the system depends on two parameters and the nonlinear terms and are allowed to have different nonlinear character; that is, is decreasing on and is increasing on ; secondly, may be singular at and ; so far fewer work was done when can be singular at ; thirdly, the boundary conditions of the system are nonlocal and involve fractional derivatives of the unknown functions.
2. Preliminaries and Lemmas
In this section, we firstly define an appropriate invariant set and then make a change of variables for the system (3) so that Schauder’s fixed point theorem can be applied. Our work is based on fractional framework; for further background knowledge of fractional calculus, we refer readers to the monographs [1–4] or the papers [6, 8, 17, 18] and the references therein.
Throughout this paper, we mean by the Banach space of all continuous functions on with the usual norm . Let then is a normal cone in the Banach space . Thus the space can be equipped with a partial order given by Now define a subcone of as follows: Obviously, is nonempty since .
Proof. By using semigroup property of the fractional integration operator (see  page 73, Lemma 2.3 or  Sections 2.3 and 2.5), one has
And then, it follows from (8) that
In the same way, we also have
It follows from and that and . So substituting the above formulas into (3), we obtain (7).
On the other hand, if is a solution of (7), (8) yields So from (11), we have Moreover Consequently, is a positive solution of (3).
Now we recall some useful lemmas by , which are important to the proof of our main results.
Lemma 2 (see ). Let , if ; then the unique solution of the linear problem is given by where Moreover, for any ,
By Lemma 2, similar results are valid for the problem For convenience, we adopt the following corresponding notations:
Lemma 3 (see ). The Green functions and have the following properties:(1) and for ,(2)there exist functions , , , and such that where
Clearly, the following maximum principle is direct conclusion of Lemma 2.
Lemma 4. If and satisfies and for any , then for .
It is well known that is a solution of the system (7) if and only if is a solution of the nonlinear integral system of equations and the system (23) is equivalent to the following nonlinear integral equation
Let us define a nonlinear operator by Then the existence of solutions to the system of (7) is equivalent to the existence of fixed point of the nonlinear operator ; that is, if is a fixed point of in , then system (7) has at least one solution which can be written by and then system (3) has at least one solution:
In order to find the fixed point of , we need the definitions of the upper solution and lower solution for the following integrodifferential equation:
Definition 5. A continuous function is called a lower solution of the problem (28), if it satisfies
Definition 6. A continuous function is called an upper solution of the problem (28), if it satisfies
3. Main Result
For the convenience in presentation, we now present some assumptions to be used in the rest of the paper.(A1) is continuous and decreasing on in and is continuous and increasing on in .(A2)For any real numbers , ,
Theorem 7. Suppose (A1) and (A2) hold; then for any , the system (3) has at least one positive solution , and there exist positive constants , , , and such that
Proof. We start by showing that (3) has at least one positive solution . For this purpose, we firstly prove that the operator is well defined and .
For any , there exist two positive numbers such that , so it follows from (25), (20), and (31) that
Take then So is well defined and . It follows from (A1) that the operator is decreasing on . Moreover, by Lemma 2, we have
Let if , then is positive solution of (7). Thus the system (3) has at least one solution which satisfies Theorem 7. If , we have
Let From (A1), we know that is nonincreasing on ; thus by (37) and (40), one gets and and . Thus from (36)–(42), we have So it follows from (42)-(43) that and are, respectively, upper solution and lower solution of the problem (28) and , .
Define a function as follows: Obviously .
Next we define an operator in by and consider the following boundary value problem: It is easy to see that the fixed point of is the solution of (46).
Note that then is uniformly bounded. From the uniform continuity of and Lebesgue dominated convergence theorem, we get is equicontinuous. So is completely continuous. It follows from Schauder’s fixed point theorem that has at least one fixed point such that .
In what follows, we prove In fact, as is a fixed point of , we have We firstly prove . Otherwise, suppose . According to the definition of , we have On the other hand, as is an upper solution of (28), we have Letting , (50)-(51) imply that It follows from Lemma 4 that that is, on , which contradicts . Thus , . In the same way, . Consequently, and then is a positive solution of the problem (28).
It follows from , that there exist four numbers , , , such that and (54)-(55) yield Furthermore, where By Lemma 1, (56), and (57), we obtain that the problem (3) has a positive solution , which satisfies The proof is completed.
Example 8. Consider the following singular fractional order differential system
where and are parameters, , , , and , . Obviously,
Letting , , then (60) is turned into
where and ; it is easy to see that is decreasing with respect to and is increasing with respect to , and then (A1) is satisfied.
It follows from (20) and (21) that there exists a positive number such that and for any constants , , we have where . So and thus (A2) is satisfied. So from Theorem 7, for any , the system (60) has at least one positive solution , and there exist positive constants , , , and such that
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors were supported financially by the National Natural Science Foundation of China (no. 11371221) and the Project of Shandong Province Higher Educational Science and Technology Program (no. J13LI12).
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