Abstract and Applied Analysis

Volume 2014 (2014), Article ID 457965, 8 pages

http://dx.doi.org/10.1155/2014/457965

Research Article

Positive Solutions of a Singular Nonlocal Fractional Order Differential System via Schauder’s Fixed Point Theorem

Xinguang Zhang,1,2 Cuiling Mao,¹ Yonghong Wu,² and Hua Su³

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.

Abstract.

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.

1. Introduction

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 [17] 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 [18], 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 [5] 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. [11].

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 .

Lemma 1. Let , , , and . Then system (3) is turned into the equivalent one:

and if is a solution of the problem (7), then is a solution of the system (3).

Proof. By using semigroup property of the fractional integration operator (see [1] page 73, Lemma 2.3 or [4] 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 [18], which are important to the proof of our main results.

Lemma 2 (see [18]). 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 [18]). 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.

Acknowledgments

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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