About this Journal Submit a Manuscript Table of Contents
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

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 [116]. 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 [14] 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).

References

  1. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” in North-Holland Mathematics Studies, vol. 204, Elservier Science, Amsterdam, The Netherlands, 2006. View at Zentralblatt MATH
  2. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
  3. J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering,, Springer, Dordrecht, The Netherlands, 2007.
  4. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integral and Derivatives, Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993.
  5. A. S. Perelson, “Modeling the interaction of the immune system with HIV,” in Mathematical and Statistical Approaches To AIDS Epidemiology, C. Castillo-Chavez, Ed., vol. 83 of Lecture Notes in Biomathematics, Springer, New York, NY, USA, 1989. View at Zentralblatt MATH
  6. C. S. Goodrich, “Existence of a positive solution to systems of differential equations of fractional order,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1251–1268, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1263–1274, 2012. View at Zentralblatt MATH
  8. B. Ahmad and J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 58, no. 9, pp. 1838–1843, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. C. S. Goodrich, “Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions,” Computers and Mathematics with Applications, vol. 61, no. 2, pp. 191–202, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  10. C. S. Goodrich, “Positive solutions to boundary value problems with nonlinear boundary conditions,” Nonlinear Analysis, Theory, Methods and Applications, vol. 75, no. 1, pp. 417–432, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. A. A. M. Arafa, S. Z. Rida, and M. Khalil, “Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection,” Nonlinear Biomedical Physics, vol. 6, no. 1, article 1, 2012. View at Publisher · View at Google Scholar · View at Scopus
  12. X. Zhang, L. Liu, and Y. Wu, “The uniqueness of positive solution for a singular fractional differential system involving derivatives,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 6, pp. 1400–1409, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  13. X. Zhang, L. Liu, and Y. Wu, “The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8526–8536, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. X. Zhang, L. Liu, and Y. Wu, “Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1420–1433, 2012. View at Publisher · View at Google Scholar · View at Scopus
  15. I. S. Jesus, J. A. Tenreiro MacHado, and J. Boaventure Cunha, “Fractional electrical impedances in botanical elements,” JVC/Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1389–1402, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. X. Zhang, L. Liu, B. Wiwatanapataphee, and Y. Wu, “Positive solutions of eigenvalue problems for a class of fractional differential equations with derivatives,” Abstract and Applied Analysis, vol. 2012, Article ID 512127, 16 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. M. Rehman and R. Khan, “Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations,” Applied Mathematics Letters, vol. 23, pp. 1038–1044, 2010. View at Zentralblatt MATH
  18. X. Zhang, L. Liu, Y. Wu, and Y. Lu, “The iterative solutions of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4680–4691, 2013. View at Publisher · View at Google Scholar