Abstract and Applied Analysis

Volume 2012 (2012), Article ID 248709, 11 pages

http://dx.doi.org/10.1155/2012/248709

## A Coupled System of Nonlinear Fractional Differential Equations with Multipoint Fractional Boundary Conditions on an Unbounded Domain

^{1}School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China^{2}Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 26 January 2012; Accepted 24 March 2012

Academic Editor: Dumitru Baleanu

Copyright © 2012 Guotao Wang 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

This paper investigates the existence of solutions for a coupled system of nonlinear fractional differential equations with *m*-point fractional boundary conditions on an unbounded domain. Some standard fixed point theorems are applied to obtain the main results. The paper concludes with two illustrative examples.

#### 1. Introduction

In the last few decades, the subject of fractional calculus has gained considerable popularity and importance as it finds its applications in numerous fields of science and engineering. Some of the areas of recent applications of fractional models include fluid mechanics, solute transport or dynamical processes in porous media, material viscoelastic theory, dynamics of earthquakes, control theory of dynamical systems, and biomathematics. In the afore-mentioned areas, there are phenomena with estrange kinetics involving microscopic complex dynamical behaviour that cannot be characterized by classical derivative models. It has been learnt through experimentation that most of the processes associated with complex systems have nonlocal dynamics possessing long-memory in time, and the integral and derivative operators of fractional order do have some of these characteristics. Thus, due to the modeling capabilities of fractional integrals and derivatives for complex phenomena, the fractional modelling has emerged as a powerful tool and has accounted for the rapid development of the theory of fractional differential equations. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes [1]. The presence of memory term in such models not only takes into account the history of the process involved but also carries its impact to present and future development of the process. For more details and applications, we refer the reader to the books [2–6]. For some recent work on the topic, see [7–27] and references therein.

The study of coupled systems involving fractional differential equations is also important as such systems occur in various problems of applied nature. For some recent results on systems of fractional differential equations, see [28–35].

Much of the work on fractional differential equations has been considered on finite domain and there are few papers dealing with infinite domain [36–43]. In this paper, we discuss the existence and uniqueness of the solutions of a coupled system of nonlinear fractional differential equations with -point boundary conditions on an unbounded domain. Precisely, we consider the following problem: where , , , and denote Riemann-Liouville fractional derivatives of order and , respectively, and are such that and .

#### 2. Preliminaries

For the convenience of the readers, in this section we first present some useful definitions and lemmas.

*Definition 2.1 (see [5]). *The Riemann-Liouville fractional derivative of order for a continuous function is defined by

provided that the right-hand side is pointwise defined on .

*Definition 2.2 (see [5]). *The Riemann-Liouville fractional integral of order for a function is defined asprovided that such integral exists.

For the forthcoming analysis, we define the spaces

equipped with the norms

Obviously and are Banach spaces.

Lemma 2.3 (see [38]). *Let . For , the fractional boundary value problem
*

has a unique solution

where

with

Lemma 2.4 (see [38]). *For , , where
*

#### 3. Main Results

This section is devoted to some existence and uniqueness results for problem (1.1).

Define the space

equipped with the norm

Clearly is a Banach space.

Let an operator be defined by

where , with

Observe that the problem (1.1) has a solution if and only if the operator defined by (3.3) has a fixed point.

Lemma 3.1. *For , one has
**
where
*

Theorem 3.2. *Assume that*

()* there exist nonnegative functions such that
*()* there exist nonnegative functions such that
**Then the system (1.1) has a solution.*

*Proof. *Let us take
and define
Obviously, is a bounded closed and convex set of .

As a first step, we show that the operator is .

For any , we have
Similarly, we can get
That is, . Thus, .

Next, we show that is completely continuous. By continuity of , and , it follows that is continuous. On the other hand, by a similar process used in [38], we can easily prove that the operators are equicontinuous. Therefore it follows that is an equicontinuous set. Also, it is uniformly bounded as . Thus, we conclude that is a completely continuous operator. Hence, by Schauder fixed point theorem, there exists a solution of (1.1). This completes the proof.

Theorem 3.3. *Assume that*

()* there exist and nonnegative functions such that
*()* there exist and nonnegative functions such that
**Then the system (1.1) has a solution.*

*Proof. *In this case, we take
The rest of the proof is similar to that of Theorem 3.2. So we omit it.

*Remark 3.4. *By taking (instead of , ) in () and (), one can show that (1.1) has a solution.

Theorem 3.5. *Assume that**() the functions and satisfy Lipschitz condition; that is, there exist nonnegative functions and such that**
Then the problem (1.1) has a unique solution if
*

*Proof. *For any , we have

Similarly, it can be shown that

Thus, we get
Obviously, is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle. This completes the proof.

#### 4. Example

*Example 4.1. *Consider the following multipoint boundary value problem on an unbounded domain:

Here , , , , , , , , and . One has

For , , , , by direct calculation we find that

Thus all conditions of Theorem 3.2 are satisfied. Therefore, by Theorem 3.2, the couple system of nonlinear fractional differential (4.1) has at least one solution.

*Example 4.2. *Consider the following problem on an unbounded domain:
Here , , , , , , , , and , .

With
we have
where , . So, the condition () holds. Let us assume that
For example, condition (4.7) holds if we take
Thus all the conditions of Theorem 3.5 are satisfied. Therefore, by the conclusion of Theorem 3.5, the coupled system (4.4) has a unique solution.

#### 5. Conclusion

We have shown the existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations with multipoint fractional boundary conditions on a semi-infinite domain. Our existence results are based on Schauder’s fixed point theorem, while the uniqueness result is obtained by applying Banach’s contraction mapping principle. The existence of solutions for (1.1) has been addressed for different kinds of growth conditions. Our approach is simple and can easily be applied to a variety of problems. This has been demonstrated by solving two examples.

#### Acknowledgment

The research of B. Ahmad was supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

#### References

- M. P. Lazarević and A. M. Spasić, “Finite-time stability analysis of fractional order time-delay systems: Gronwall's approach,”
*Mathematical and Computer Modelling*, vol. 49, no. 3-4, pp. 475–481, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - I. Podlubny,
*Fractional Differential Equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, San Diego, Calif, USA, 1999. - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods (Series on Complexity, Nonlinearity and Chaos)*, World Scientific, 2012. - R. L. Magin,
*Fractional Calculus in Bioengineering*, Begell House, Connecticut, Conn, USA, 2006. - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204 of*North-Holland Mathematics Studies*, Elsevier Science, Amsterdam, The Netherlands, 2006. - J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds.,
*Advances in Fractional Calculus*, Springer, Dordrecht, The Netherlands, 2007. View at Publisher · View at Google Scholar - S. Liang and J. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equation,”
*Nonlinear Analysis*, vol. 71, no. 11, pp. 5545–5550, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Zhang, “Existence results of positive solutions to boundary value problem for fractional differential equation,”
*Positivity*, vol. 13, no. 3, pp. 583–599, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Ahmad and J. J. Nieto, “Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations,”
*Abstract and Applied Analysis*, vol. 2009, Article ID 494720, 9 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Caballero Mena, J. Harjani, and K. Sadarangani, “Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems,”
*Boundary Value Problems*, vol. 2009, Article ID 421310, 10 pages, 2009. View at Google Scholar · View at Zentralblatt MATH - S. Zhang, “Positive solutions to singular boundary value problem for nonlinear fractional differential equation,”
*Computers & Mathematics with Applications*, vol. 59, no. 3, pp. 1300–1309, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Ahmad, “Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations,”
*Applied Mathematics Letters*, vol. 23, no. 4, pp. 390–394, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,”
*Nonlinear Analysis*, vol. 72, no. 2, pp. 916–924, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Băleanu, O. G. Mustafa, and R. P. Agarwal, “An existence result for a superlinear fractional differential equation,”
*Applied Mathematics Letters*, vol. 23, no. 9, pp. 1129–1132, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. Magin, “Fractional Bloch equation with delay,”
*Computers & Mathematics with Applications*, vol. 61, no. 5, pp. 1355–1365, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Ahmad and J. J. Nieto, “Anti-periodic fractional boundary value problems,”
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1150–1156, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Ahmad and S. K. Ntouyas, “A four-point nonlocal integral boundary value problem for fractional differential equations of arbitrary order,”
*Electronic Journal of Qualitative Theory of Differential Equations*, no. 22, pp. 1–15, 2011. View at Google Scholar - A. Cabada and G. Wang, “Positive solutions of nonlinear fractional differential equations with integral boundary value conditions,”
*Journal of Mathematical Analysis and Applications*, vol. 389, no. 1, pp. 403–411, 2012. View at Google Scholar - B. Ahmad and R. P. Agarwal, “On nonlocal fractional boundary value problems,”
*Dynamics of Continuous, Discrete & Impulsive Systems. Series A*, vol. 18, no. 4, pp. 535–544, 2011. View at Google Scholar · View at Zentralblatt MATH - J. D. Ramírez and A. S. Vatsala, “Monotone method for nonlinear Caputo fractional boundary value problems,”
*Dynamic Systems and Applications*, vol. 20, no. 1, pp. 73–88, 2011. View at Google Scholar - Y. Zhao, S. Sun, Z. Han, and M. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equations,”
*Applied Mathematics and Computation*, vol. 217, no. 16, pp. 6950–6958, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Wang, S. K. Ntouyas, and L. Zhang, “Positive solutions of the three-point boundary value problem for fractional-order differential equations with an advanced argument,”
*Advances in Difference Equations*, vol. 2011, article 2, 2011. View at Google Scholar - B. Ahmad and J. J. Nieto, “Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions,”
*Boundary Value Problems*, vol. 2011, article 36, 2011. View at Google Scholar - B. Ahmad, J. J. Nieto, A. Alsaedi, and M. El-Shahed, “A study of nonlinear Langevin equation involving two fractional orders in different intervals,”
*Nonlinear Analysis*, vol. 13, no. 2, pp. 599–606, 2012. View at Publisher · View at Google Scholar - B. Ahmad, “On nonlocal boundary value problems for nonlinear integro-differential equations of arbitrary fractional order,”
*Results in Mathematics*. In press. View at Publisher · View at Google Scholar - E. Hernández, D. O'Regan, and K. Balachandran, “On recent developments in the theory of abstract differential equations with fractional derivatives,”
*Nonlinear Analysis*, vol. 73, no. 10, pp. 3462–3471, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Wang, “Monotone iterative technique for boundary value problems of nonlinear fractional differential equation with deviating arguments,”
*Journal of Computational and Applied Mathematics*, vol. 236, pp. 2425–2430, 2012. View at Google Scholar - G. Wang, R. P. Agarwal, and A. Cabada, “Existence results and monotone iterative technique for systems of nonlinear fractional differential equations,”
*Applied Mathematics Letters*, vol. 25, pp. 1019–1024, 2012. View at Google Scholar - C.-Z. Bai and J.-X. Fang, “The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations,”
*Applied Mathematics and Computation*, vol. 150, no. 3, pp. 611–621, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. Daftardar-Gejji, “Positive solutions of a system of non-autonomous fractional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 302, no. 1, pp. 56–64, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Su, “Boundary value problem for a coupled system of nonlinear fractional differential equations,”
*Applied Mathematics Letters*, vol. 22, no. 1, pp. 64–69, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Ahmad and J. 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 - J. Wang, H. Xiang, and Z. Liu, “Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations,”
*International Journal of Differential Equations*, vol. 2010, Article ID 186928, 12 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Babakhani, “Positive solutions for system of nonlinear fractional differential equations in two dimensions with delay,”
*Abstract and Applied Analysis*, vol. 2010, Article ID 536317, 16 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. Gafiychuk, B. Datsko, and V. Meleshko, “Mathematical modeling of time fractional reaction-diffusion systems,”
*Journal of Computational and Applied Mathematics*, vol. 220, no. 1-2, pp. 215–225, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Arara, M. Benchohra, N. Hamidi, and J. J. Nieto, “Fractional order differential equations on an unbounded domain,”
*Nonlinear Analysis*, vol. 72, no. 2, pp. 580–586, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Zhao and W. Ge, “Unbounded solutions for a fractional boundary value problems on the infinite interval,”
*Acta Applicandae Mathematicae*, vol. 109, no. 2, pp. 495–505, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Liang and J. Zhang, “Existence of three positive solutions of $m$-point boundary value problems for some nonlinear fractional differential equations on an infinite interval,”
*Computers & Mathematics with Applications*, vol. 61, no. 11, pp. 3343–3354, 2011. View at Publisher · View at Google Scholar - X. Su, “Solutions to boundary value problem of fractional order on unbounded domains in a Banach space,”
*Nonlinear Analysis*, vol. 74, no. 8, pp. 2844–2852, 2011. View at Publisher · View at Google Scholar - R. P. Agarwal, M. Benchohra, S. Hamani, and S. Pinelas, “Boundary value problems for differential equations involving Riemann-Liouville fractional derivative on the half-line,”
*Dynamics of Continuous, Discrete & Impulsive Systems. Series A*, vol. 18, no. 2, pp. 235–244, 2011. View at Google Scholar · View at Zentralblatt MATH - S. Liang and J. Zhang, “Existence of multiple positive solutions for $m$-point fractional boundary value problems on an infinite interval,”
*Mathematical and Computer Modelling*, vol. 54, no. 5-6, pp. 1334–1346, 2011. View at Publisher · View at Google Scholar - F. Chen and Y. Zhou, “Attractivity of fractional functional differential equations,”
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1359–1369, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Su and S. Zhang, “Unbounded solutions to a boundary value problem of fractional order on the half-line,”
*Computers & Mathematics with Applications*, vol. 61, no. 4, pp. 1079–1087, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH