- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2012 (2012), Article ID 594802, 14 pages

http://dx.doi.org/10.1155/2012/594802

## Existence and Uniqueness of Solutions for the System of Nonlinear Fractional Differential Equations with Nonlocal and Integral Boundary Conditions

^{1}Department of Mathematics, Fatih University, 34500 Buyucekmece, Turkey^{2}ITTU, Ashgabat, Turkmenistan^{3}Institute of Cybernetics, ANAS, and Baku State University, 1141 Baku, Azerbaijan

Received 20 March 2012; Accepted 6 May 2012

Academic Editor: Ravshan Ashurov

Copyright © 2012 Allaberen Ashyralyev and Yagub A. Sharifov. 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

In the present study, the nonlocal and integral boundary value problems for the system of nonlinear fractional differential equations involving the Caputo fractional derivative are investigated. Theorems on existence and uniqueness of a solution are established under some sufficient conditions on nonlinear terms. A simple example of application of the main result of this paper is presented.

#### 1. Introduction

Differential equations of fractional order have been proved to be valuable tools in the modeling of many phenomena of various fields of science and engineering. Indeed, we can obtain numerous applications in viscoelasticity [1–3], dynamical processes in self-similar structures [4], biosciences [5], signal processing [6], system control theory [7], electrochemistry [8], diffusion processes [9], and linear time-invariant systems of any order with internal point delays [10]. Furthermore, fractional calculus has been found many applications in classical mechanics [11], and the calculus of variations [12], and is a very useful means for obtaining solutions of nonhomogenous linear ordinary and partial differential equations. For more details, we refer the reader to [13].

There are several approaches to fractional derivatives such as Riemann-Liouville, Caputo, Weyl, Hadamar and Grunwald-Letnikov, and so forth. Applied problems require those definitions of a fractional derivative that allow the utilization of physically interpretable initial and boundary conditions. The Caputo fractional derivative satisfies these demands, while the Riemann-Liouville derivative is not suitable for mixed boundary conditions. The details can be found in [14–17].

The study of existence and uniqueness, periodicity, asymptotic behavior, stability, and methods of analytic and numerical solutions of fractional differential equations have been studied extensively in a large cycle works (see, e.g., [10, 18–37] and the references therein). However, many of the physical systems can better be described by integral boundary conditions. Integral boundary conditions are encountered in various applications such as population dynamics, blood flow models, chemical engineering, and cellular systems. Moreover, boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two-point, three-point, multipoint, and nonlocal boundary value problems as special cases, see [38–41].

In the present paper, we study existence and uniqueness of the problem for the system of nonlinear fractional differential equations of form with the nonlocal and integral boundary condition where is an identity matrix, is the given matrix, and Here, and are smooth vector functions, is the Caputo fractional derivative of order , .

The organization of this paper is as follows. In Section 2, we provide necessary background. In Section 3, theorems on existence and uniqueness of a solution are established under some sufficient conditions on nonlinear terms. Finally, in Section 4, a simple example of application of the main result of this paper is presented.

#### 2. Preliminaries

In this section, we present some basic definitions and preliminary facts which are used throughout the paper. By , we denote the Banach space of all vector continuous functions from into with the norm

*Definition 2.1. *If and , then the operator , defined by
for , is called the Riemann-Liouville fractional integral operator of order . Here is the Gamma function defined for any complex number as

*Definition 2.2. *The Caputo fractional derivative of order of a continuous function is defined by
where , (the notation stands for the largest integer not greater than ).

*Remark 2.3. *Under natural conditions on , the Caputo fractional derivative becomes the conventional integer order derivative of the function as .

*Remark 2.4. *Let and , then the following relations hold:

Lemma 2.5 (see, [42]). *For , , the homogenous fractional differential equation
**
has a solution
**
where, , , and .*

Lemma 2.6 (see, [42]). *Assume that , with derivative of order that belongs to , then
**
where , and .*

Lemma 2.7 (see, [42]). *Let . Then
**
is satisfied almost everywhere on . Moreover, if , then identity (2.9) is true for all .*

Lemma 2.8 (see, [42]). *If , then for all .*

#### 3. Main Results

Lemma 3.1. *Let and . Then, nonlocal boundary value problem
**
has a unique solution given by
**
where
*

*Proof. *Assume that is a solution of nonlocal boundary value problem (3.1) and (3.2), then using Lemma 2.6, we get
Applying condition (3.2) and identity (3.6), we get
From condition (1.3) it follows that the inverse of the matrix exists. Therefore, we can write
Using formulas (3.6) and (3.8), we obtain
which can be written as (3.3). Lemma 3.1 is proved.

Lemma 3.2. *Assume that and . Then, the vector function is a solution of the boundary value problem (1.1) and (1.2) if and only if it is a solution of the integral equation
*

*Proof. *If solves boundary value problem (1.1) and (1.2). Then, by the same manner as in Lemma 3.1, we can prove that is solution of integral equation (3.10). Conversely, let is solution of integral equation (3.10). We denote that
Then, by Lemmas 2.7 and 2.8, we obtain
The application of the fractional differential operator to both sides of (3.12) yields
Hence, solves fractional differential equation (1.1). Also, it is easy to see that satisfies nonlocal boundary condition (1.2).

The first main statement of this paper is an existence and uniqueness of boundary value problem (1.1) and (1.2) result that it is based on a Banach fixed point theorem.

Theorem 3.3. *Assume that: *(H1)* There exists a constant such that
for each and all . *(H2)* There exists a constant such that
for each and all .**If
**
then boundary value problem (1.1) and (1.2) has unique solution on .*

*Proof . *Transform problem (1.1) and (1.2) into a fixed point problem. Consider the operator
defined by
Clearly, the fixed points of the operator are solution of problem (1.1) and (1.2). We will use the Banach contraction principle to prove that under assumption (3.16) operator has a fixed point. It is clear that the operator maps into itself and that
for any and . From condition (1.3) it follows that
Then, using estimates (3.19) and (3.20), we get
Consequently, by assumption (3.16) operator is a contraction. As a consequence of Banach's fixed point theorem, we deduce that operator has a fixed point which is a solution of problem (1.1) and (1.2). Theorem 3.3 is proved.

The second main statement of this paper is an existence of boundary value problem (1.1) and (1.2) result that it is based on Schaefer's fixed point theorem.

Theorem 3.4. *Assume that:*(H3)* The function is continuous.*(H4)* There exists a constant such that for each and all .*(H5) *The function is continuous.*(H6) *There exists a constant such that for each and all .** Then, boundary value problem (1.1) and (1.2) has at least one solution on .*

*Proof. *We will divide the proof into four main steps in which we will show that under the assumptions of theorem operator has a fixed point.

*Step 1. *Operator under the assumptions of theorem is continuous. Let be a sequence such that in . Then, for each
Since and are continuous functions, we have
as .

*Step 2. *Operator maps bounded sets in bounded sets in . Indeed, it is enough to show that for any , there exists a positive constant such that for each , we have . By assumptions (H4) and (H6), we have for each ,
Hence,
Thus,

*Step 3. *Operator maps bounded sets into equicontinuous sets of .

Let be a bounded set of as in Step 2, and let . Then,
As , the right-hand side of the above inequality tends to zero. As a consequence of Steps 1 to 3 together with the Arzela-Ascoli theorem, we can conclude that the operator is completely continuous.

*Step 4. *A priori bounds. Now, it remains to show that the set
is bounded.

Let for some . Then, for each we have
This implies by assumptions (H4) and (H6) (as in Step 2) that for each we have
Thus, for every , we have
Therefore,
This shows that the set is bounded. As a consequence of Schaefer's fixed point theorem, we deduce that has a fixed point which is a solution of problem (1.1) and (1.2). Theorem 3.4 is proved.

Moreover, we will give an existence result for problem (1.1) and (1.2) by means of an application of a Leray-Schauder type nonlinear alternative, where conditions (H4) and (H6) are weakened.

Theorem 3.5. *Assume that (H3), (H5) and the following conditions hold.*(H7)* There exist and continuous and nondecreasing
such that for each and all .*(H8)* There exist and continuous nondecreasing
such that for each and all .*(H9)* There exists a number such that
**Then, boundary value problem (1.1) and (1.2) has at least one solution on .*

*Proof. * Consider the operator defined in Theorems 3.3 and 3.4. It can be easily shown that operator is continuous and completely continuous. Let for each . Then, from assumptions (H7) and (H8) if follows that for each
Thus,
Then, by condition (H9), there exists such that .

Let
The operator is continuous and completely continuous. By the choice of , there exists such that for some . As a consequence of the nonlinear alternative of Leray-Schauder type [43], we deduce that has a fixed point in , which is a solution of problem (1.1) and (1.2). This completes of proof of Theorem 3.5.

#### 4. An Example

In this section, we give an example to illustrate the usefulness of our main results. Let us consider the following nonlocal boundary value problem for the system of fractional differential equation Evidently, Hence, conditions (H1) and (H2) hold with . We will check that condition (3.16) is satisfied for appropriate values of with . Indeed, Then, by Theorem 3.3 boundary value problem (4.1) has a unique solution on for values of satisfying condition (4.3). For example, if then

#### 5. Conclusion

In this work, some existence and uniqueness of a solution results have been established for the system of nonlinear fractional differential equations under the some sufficient conditions on nonlinear terms. Of course, such type existence and uniqueness results hold under the same sufficient conditions on nonlinear terms for the system of nonlinear fractional differential equations (1.1), subject to multipoint nonlocal and integral boundary conditions where are given matrices and Here, .

Moreover, applying the result of the paper [44] the first order of accuracy difference scheme for the numerical solution of nonlocal boundary value problem (1.1) and (5.1) can be presented. Of course, such type existence and uniqueness results hold under the some sufficient conditions on nonlinear terms for the solution system of this difference scheme.

#### References

- R. I. Bagley, “A theoretical basis for the application of fractional calculus to viscoelasticity,”
*Journal of Rheology*, vol. 27, no. 3, pp. 201–210, 1983. - G. Sorrentinos, “Fractional derivative linear models for describing the viscoelastic dynamic behavior of polymeric beams,” in
*Proceedings of the IMAS Conference and Exposition on Structural Dynamics*, St. Louis, Mo, USA, 2006. - G. Sorrentinos, “Analytic modeling and experimental identification of viscoelastic mechanical systems,” in
*Advances in Fractional Calculus*, Springer, 2007. - F. Mainardi,
*Fractals and Fractional Calculus in Continuum Mechanics*, Springer, New York, NY, USA, 1997. View at Zentralblatt MATH - R. Magin, “Fractional calculus in bioengineering,”
*Critical Reviews in Biomedical Engineering*, vol. 32, no. 1, pp. 1–104, 2004. - M. Ortigueira, “Special issue on fractional signal processing and applications,”
*Signal Processing*, vol. 83, no. 11, pp. 2285–2480, 2003. - B. M. Vinagre, I. Podlubny, A. Hernández, and V. Feliu, “Some approximations of fractional order operators used in control theory and applications,”
*Fractional Calculus & Applied Analysis*, vol. 3, no. 3, pp. 231–248, 2000. View at Zentralblatt MATH - K. B. Oldham, “Fractional differential equations in electrochemistry,”
*Advances in Engineering Software*, vol. 41, no. 1, pp. 9–12, 2010. - R. Metzler and J. Klafter, “Boundary value problems for fractional diffusion equations,”
*Physica A*, vol. 278, no. 1-2, pp. 107–125, 2000. View at Publisher · View at Google Scholar - M. De la Sen, “Positivity and stability of the solutions of Caputo fractional linear time-invariant systems of any order with internal point delays,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 161246, 25 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. M. Rabei, T. S. Alhalholy, and A. Rousan, “Potentials of arbitrary forces with fractional derivatives,”
*International Journal of Modern Physics A*, vol. 19, no. 17-18, pp. 3083–3092, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - O. P. Agrawal, “Formulation of Euler-Lagrange equations for fractional variational problems,”
*Journal of Mathematical Analysis and Applications*, vol. 272, no. 1, pp. 368–379, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Tu, K. Nishimoto, S. Jaw, and S. D. Lin, “Applications of fractional calculus to ordinary and partial differential equations of the second order,”
*Hiroshima Mathematical Journal*, vol. 23, no. 1, pp. 63–77, 1993. View at Zentralblatt MATH - K. Dichelm,
*The Analysis of Fractional Differential Equations*, Springer, Heidelberg, Germany, 2004. - I. Podlubny,
*Fractional Differential Equations*, Academic Press, New York, NY, USA, 1999. - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives*, Gordon and Breach Science Publishers, London, UK, 1993. - J. L. Lavoie, T. J. Osler, and R. Tremblay, “Fractional derivatives and special functions,”
*SIAM Review*, vol. 18, no. 2, pp. 240–268, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations with fractional order,”
*Surveys in Mathematics and its Applications*, vol. 3, pp. 1–12, 2008. View at Zentralblatt MATH - R. W. Ibrahim and S. Momani, “On the existence and uniqueness of solutions of a class of fractional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 334, no. 1, pp. 1–10, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,”
*Nonlinear Analysis*, vol. 69, no. 8, pp. 2677–2682, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Xinwei and L. Landong, “Existence of solution for boundary value problem of nonlinear fractional differential equation,”
*Applied Mathematics A*, vol. 22, no. 3, pp. 291–298, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Zhang, “Existence of solution for a boundary value problem of fractional order,”
*Acta Mathematica Scientia B*, vol. 26, no. 2, pp. 220–228, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. P. Agarwal, M. Benchohra, and S. Hamani, “Boundary value problems for fractional differential equations,”
*Georgian Mathematical Journal*, vol. 16, no. 3, pp. 401–411, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Ashyralyev and B. Hicdurmaz, “A note on the fractional Schrödinger differential equations,”
*Kybernetes*, vol. 40, no. 5-6, pp. 736–750, 2011. View at Publisher · View at Google Scholar - A. Ashyralyev, F. Dal, and Z. Pınar, “A note on the fractional hyperbolic differential and difference equations,”
*Applied Mathematics and Computation*, vol. 217, no. 9, pp. 4654–4664, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Ashyralyev and Z. Cakir, “On the numerical solution of fractional parabolic partial differential equations,”
*AIP Conference Proceeding*, vol. 1389, pp. 617–620, 2011. - A. Ashyralyev, “Well-posedness of the Basset problem in spaces of smooth functions,”
*Applied Mathematics Letters*, vol. 24, no. 7, pp. 1176–1180, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Yuan, “Two positive solutions for $(n-1,1)$-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. View at Publisher · View at Google Scholar - M. De la Sen, R. P. Agarwal, A. Ibeas, and S. Alonso-Quesada, “On the existence of equilibrium points, boundedness, oscillating behavior and positivity of a SVEIRS epidemic model under constant and impulsive vaccination,”
*Advances in Difference Equations*, vol. 2011, Article ID 748608, 32 pages, 2011. View at Zentralblatt MATH - M. De la Sen, “About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory,”
*Fixed Point Theory and Applications*, vol. 2011, Article ID 867932, 19 pages, 2011. View at Zentralblatt MATH - C. Yuan, “Multiple positive solutions for semipositone $(n,p)$-type boundary value problems of nonlinear fractional differential equations,”
*Analysis and Applications*, vol. 9, no. 1, pp. 97–112, 2011. View at Publisher · View at Google Scholar - C. Yuan, “Multiple positive solutions for $(n-1,1)$-type semipositone conjugate boundary value problems for coupled systems of nonlinear fractional differential equations,”
*Electronic Journal of Qualitative Theory of Differential Equations*, vol. 12, pp. 1–12, 2011. - 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. View at Zentralblatt MATH - R. P. Agarwal, B. de Andrade, and C. Cuevas, “On type of periodicity and ergodicity to a class of fractional order differential equations,”
*Advances in Difference Equations*, vol. 2010, Article ID 179750, 25 pages, 2010. View at Zentralblatt MATH - R. P. Agarwal, B. de Andrade, and C. Cuevas, “Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations,”
*Nonlinear Analysis*, vol. 11, no. 5, pp. 3532–3554, 2010. View at Publisher · View at Google Scholar - A. S. Berdyshev, A. Cabada, and E. T. Karimov, “On a non-local boundary problem for a parabolic-hyperbolic equation involving a Riemann-Liouville fractional differential operator,”
*Nonlinear Analysis*, vol. 75, no. 6, pp. 3268–3273, 2011. - B. Ahmad and J. J. Nieto, “Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions,”
*Boundary Value Problems*, vol. 2011, Article ID 708576, 11 pages, 2009. View at Zentralblatt MATH - A. Bouncherif, “Second order boundary value problems with integral boundary conditions,”
*Nonlinear Analysis*, vol. 70, no. 1, pp. 368–379, 2009. - R. A. Khan, “Existence and approximation of solutions of nonlinear problems with integral boundary conditions,”
*Dynamic Systems and Applications*, vol. 14, no. 2, pp. 281–296, 2005. - R. A. Khan, M. U. Rehman, and J. Henderson, “Existence and uniqueness of solutions for nonlinear fractional differential equations with integral boundary conditions,”
*Fractional Differential Equations*, vol. 1, no. 1, pp. 29–43, 2011. - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, Elsevier, Amsterdam, The Netherlands, 2006. - A. Granas and J. Dugundji,
*Fixed Point Theory*, Springer, New York, NY, USA, 2003. - A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,”
*Journal of Mathematical Analysis and Applications*, vol. 357, no. 1, pp. 232–236, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH