Abstract and Applied Analysis

Volume 2014 (2014), Article ID 896871, 13 pages

http://dx.doi.org/10.1155/2014/896871

Research Article

The Existence of Solution for a -Dimensional System of Multiterm Fractional Integrodifferential Equations with Antiperiodic Boundary Value Problems

Dumitru Baleanu,1,2,3 Sayyedeh Zahra Nazemi,⁴ and Shahram Rezapour⁴

¹Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia

²Department of Mathematics, Cankaya University, Ogretmenler Caddesi 14, Balgat, 06530 Ankara, Turkey

³Institute of Space Sciences, Magurele, Bucharest, Romania

⁴Department of Mathematics, Azarbaijan Shahid Madani University, Azarshahr, Tabriz, Iran

Received 8 January 2014; Revised 18 March 2014; Accepted 20 March 2014; Published 24 April 2014

Academic Editor: Hossein Jafari

Copyright © 2014 Dumitru Baleanu 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.

There are many published papers about fractional integrodifferential equations and system of fractional differential equations. The goal of this paper is to show that we can investigate more complicated ones by using an appropriate basic theory. In this way, we prove the existence and uniqueness of solution for a -dimensional system of multiterm fractional integrodifferential equations with antiperiodic boundary conditions by applying some standard fixed point results. An illustrative example is also presented.

1. Introduction

Fractional differential equations have recently been studied by many researchers for a variety of problems (see, e.g., [1–33] and the references therein). Antiperiodic boundary value problems occur in the mathematical modeling of a variety of physical processes (see, e.g., [3–6, 29, 30] and the references therein). On the other hand, the study of a coupled system of fractional order is also very significant because this kind of system can often occur in applications (see, e.g., [7, 15, 24, 28, 29] and the references therein). We are going to investigate a complicated case in this work. Let and . In this paper, we study the existence and uniqueness of solution for the -dimensional system of multiterm fractional integrodifferential equations

with antiperiodic boundary conditions , , and for , where denotes the Caputo fractional derivative, , , for , , and , are continuous functions for all . Hereafter, we will use vector notations. Define the space endowed with the norm . In fact, and the product space endowed with the norm are Banach spaces. The Riemann-Liouville fractional integral of order is defined by ( and ), provided the integral exists. The Caputo derivative of order for a function is defined by for and [25]. Recently, Wang et al. proved the following result [30].

Lemma 1. For each , the unique solution of the boundary value problem

is given by , where is Green's function defined as

One can find the next result in [34].

Theorem 2. Let be a Banach space and a completely continuous operator. Suppose that the set is bounded. Then has a fixed point in .

We will use the last two results for solving the problem (1).

2. Main Results

Now, we are ready to state and prove our main results. For each , put

and , where for all . Define the operator by

where and

for , where

Thus, for each , we have

Theorem 3. The operator is completely continuous.

Proof. First, we show that the operator is continuous. Let and for and let be a sequence in such that . Then, we have

for . Since for , the sequences , , and converge uniformly on and also , , and converge uniformly on for . Since

by using the above inequalities and the continuity of (), we get

Thus, is continuous in . Let be a bounded subset of . Choose positive constants such that for all and . Thus, for each we have

for all . Hence,

for all and so . This implies that the operator is uniformly bounded. Now, we show that is an equicontinuous set. Let . Then, we have

for all . As , the right-hand side of the above inequalities tends to zero. Thus, by using the Arzela-Ascoli theorem one can conclude that the operator is completely continuous. This completes the proof.

Theorem 4. Assume that there exist positive constants , , , , and () such that

and for all , , and . Then problem (1) has at least one solution.

Proof. First, we show that   for  some   is bounded. Let . Then, for each we have

Hence,

for all . Thus, we get

Hence, for . This implies that

and so . Therefore, the set is bounded. Now by using Theorem 2, the operator has at least one fixed point. This implies that the problem (1) has at least one solution.

Theorem 5. Suppose that there exist nonnegative constants , , , and for such that

for all , , and . Then the problem (1) has a unique solution.

Proof. Let for , , and

We show that . Let . Then

for . Hence,

for . Hence, . Now, for each ,  , and we get

for . This implies that

for and so

Since , is a contraction and so, by using the Banach contraction principle, has a unique fixed point. Now, one can easily get that the problem (1) has a unique solution.

3. Example

Here, we provide an example to illustrate one of our results. It is considerable that there are some examples which provide nonuniqueness of solutions for some fractional differential equations (see, e.g., [32]).

Example 1. Consider the following 3-dimensional system:

where , , , , , , , , , , , , , , , , , , , , , , , , , , , ,   , , ,  , , , , and , with , , ,  , , , , , and . Then, we have

Here, , , ,

Thus, by using Theorem 4, 3-dimensional system (30) has at least one solution.

4. Conclusions

Fractional integrodifferential equations, system of fractional differential equations, and their applications represent a topic of high interest in the area of fractional calculus and its applications in various fields of science and engineering. Antiperiodic boundary value problems occur in the mathematical modeling of a variety of physical processes. The goal of this paper is to investigate a complicated case by using an appropriate basic theory. In this way, we prove the existence and uniqueness of solution for a new -dimensional system of multiterm fractional integrodifferential equations with antiperiodic boundary conditions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Research of the second and third authors was supported by Azarbaijan Shahid Madani University. Also, the authors express their gratitude to the referee for the helpful suggestions which improved the final version of this paper.

References

R. P. Agarwal, M. Belmekki, and M. Benchohra, “A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative,” Advances in Difference Equations, vol. 2009, Article ID 981728, 47 pages, 2009.

R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010.

R. P. Agarwal and B. Ahmad, “Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1200–1214, 2011.

B. Ahmad and J. J. Nieto, “Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory,” Topological Methods in Nonlinear Analysis, vol. 35, no. 2, pp. 295–304, 2010.

B. Ahmad, “Existence of solutions for fractional differential equations of order q ∈ (2,3] with anti-periodic boundary conditions,” Journal of Applied Mathematics and Computing, vol. 34, no. 1-2, pp. 385–391, 2010.

B. Ahmad and J. J. Nieto, “Anti-periodic fractional boundary value problems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1150–1156, 2011.

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.

B. Ahmad and S. Sivasundaram, “On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 480–487, 2010.

Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.

D. Baleanu, R. P. Agarwal, H. Mohammadi, and Sh. Rezapour, “Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces,” Boundary Value Problems, vol. 2013, article 112, 2013.

D. Baleanu, H. Mohammadi, and Sh. Rezapour, “Positive solutions of an initial value problem for nonlinear fractional differential equations,” Abstract and Applied Analysis, vol. 2012, Article ID 837437, 7 pages, 2012.

D. Baleanu, Sh. Rezapour, and H. Mohammadi, “Some existence results on nonlinear fractional differential equations,” Philosophical Transactions of the Royal Society A. Mathematical, Physical and Engineering Sciences, vol. 371, no. 1990, Article ID 20120144, 7 pages, 2013.

D. Baleanu, H. Mohammadi, and Sh. Rezapour, “On a nonlinear fractional differential equation on partially ordered metric spaces,” Advances in Difference Equations, vol. 2013, article 83, 2013.

D. Baleanu, H. Mohammadi, and Sh. Rezapour, “The existence of solutions for a nonlinear mixed problem of singular fractional differential equations,” Advances in Difference Equations, vol. 2013, article 359, 2013.

D. Baleanu, S. Z. Nazemi, and Sh. Rezapour, “The existence of positive solutions for a new coupled system of multiterm singular fractional integrodifferential boundary value problems,” Abstract and Applied Analysis, vol. 2013, Article ID 368659, 15 pages, 2013.

D. Baleanu, S. Z. Nazemi, and Sh. Rezapour, “Existence and uniqueness of solutions for multi-term nonlinear fractional integro-differential equations,” Advances in Difference Equations, vol. 2013, article 368, 2013.

D. Baleanu, S. Z. Nazemi, and Sh. Rezapour, “Attractivity for a k-dimensional system of fractional functional differential equations and global attractivity for a k-dimensional system of nonlinear fractional differential equations,” Journal of Inequalities and Applications, vol. 4014, article 31, 2014.

M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations with fractional order and nonlocal conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2391–2396, 2009.

M. A. Darwish and S. K. Ntouyas, “On initial and boundary value problems for fractional order mixed type functional differential inclusions,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1253–1265, 2010.

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, Amsterdam, The Netherlands, 2006.

V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008.

S. M. Momani, “Some existence theorems on fractional integro-differential equations,” Abhath Al-Yarmouk Journal, vol. 10, pp. 435–444, 2001.

S. M. Momani and R. El-Khazali, “On the existence of extremal solutions of fractional integro-differential equations,” Journal of Fractional Calculus, vol. 18, pp. 87–92, 2000.

S. K. Ntouyas and M. Obaid, “A coupled system of fractional differential equations with nonlocal integral boundary conditions,” Advances in Difference Equations, vol. 2012, article 130, 2012.

I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.

J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007.

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993.

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.

J. Sun, Y. Liu, and G. Liu, “Existence of solutions for fractional differential systems with antiperiodic boundary conditions,” Computers & Mathematics with Applications, vol. 64, no. 6, pp. 1557–1566, 2012.

X. Wang, X. Guo, and G. Tang, “Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order,” Journal of Applied Mathematics and Computing, vol. 41, no. 1-2, pp. 367–375, 2013.

X. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher, Hong Kong, 2011.

C. Yuan, “Two positive solutions for -type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 930–942, 2012.

W. Zhong, X. Yang, and F. Gao, “A Cauchy problem for some local fractional abstract differential equation with fractal conditions,” Journal of Applied Functional Analysis, vol. 8, no. 1, pp. 92–99, 2013.

D. R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, Mass, USA, 1980.