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Existence and Uniqueness of Solutions for Coupled Systems of Higher-Order Nonlinear Fractional Differential Equations
Fixed Point Theory and Applications volume 2010, Article number: 364560 (2010)
Abstract
We study an initial value problem for a coupled Caputo type nonlinear fractional differential system of higher order. As a first problem, the nonhomogeneous terms in the coupled fractional differential system depend on the fractional derivatives of lower orders only. Then the nonhomogeneous terms in the fractional differential system are allowed to depend on the unknown functions together with the fractional derivative of lower orders. Our method of analysis is based on the reduction of the given system to an equivalent system of integral equations. Applying the nonlinear alternative of Leray-Schauder, we prove the existence of solutions of the fractional differential system. The uniqueness of solutions of the fractional differential system is established by using the Banach contraction principle. An illustrative example is also presented.
1. Introduction
In recent years, the applications of fractional calculus in physics, chemistry, electrochemistry, bioengineering, biophysics, electrodynamics of complex medium, polymer rheology, aerodynamics, continuum mechanics, signal processing, electromagnetics, and so forth are highlighted in the literature. The methods of fractional calculus, when defined as a Laplace, Sumudu, or Fourier convolution product, are suitable for solving many problems in emerging biomedical research. The electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The fractional derivative accurately describes natural phenomena that occur in common engineering problems such as heat transfer, electrode/electrolyte behavior, and subthreshold nerve propagation. Application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law stress-strain relationship for modeling biomaterials. A systematic presentation of the applications of fractional differential equations can be found in the book of Oldham and Spanier [1]. For more details, see the monographs of Miller and Ross [2], Samko et al. [3], Podlubny [4], and Kilbas et al. [5]. In consequence, the subject of fractional differential equations is gaining much importance and attention; see [6–31] and the references therein. There has also been a surge in the study of the theory of fractional differential systems. The study of coupled systems involving fractional differential equations is quite important as such systems occur in various problems of applied nature; for instance, see [32–35] and the references therein. Recently, Su [36] discussed a two-point boundary value problem for a coupled system of fractional differential equations. Ahmad and Nieto [37] studied a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Ahmad and Graef [38] proved the existence of solutions for nonlocal coupled systems of nonlinear fractional differential equations. For applications and examples of fractional order systems, we refer the reader to the papers in [39–47]. Motivated by the recent work on coupled systems of fractional order, we consider an initial value problem for a coupled differential system of fractional order given by
where are given functions, denotes the Caputo fractional derivative, , and are suitable real constants. We also discuss the case when the nonlinearities and in (1.1) are of the form and , that is, and depend on and in addition to and respectively.
2. Preliminaries
First of all, we recall some basic definitions [3–5].
Definition 2.1.
For a function , the Caputo derivative of fractional order is defined as
Definition 2.2.
The Riemann-Liouville fractional integral of order , inversion of is the expression given by
Definition 2.3.
The Riemann-Liouville fractional derivative of order for a function is defined by
Now we state a known result [48] which provides a relationship between (2.1) and (2.2).
Lemma 2.4.
For let and . Then
(i);
(ii);
(iii);
(iv)if there exist such that for each with and , then
Remark 2.5.
In the sequel, will be understood in the sense of the limit, that is, . We also point out that the fractional order derivatives do not satisfy the relation of the form (in general).
For the sequel, we need the following results [26].
Lemma 2.6.
Assume that and . Then, for all and for all , the following relations hold:
Lemma 2.7.
Let be a continuously differentiable function with and on a compact subinterval of . Then, for with and , a function is a solution of the initial value problem
if and only if
where is a solution of the integral equation
Proof.
For the sake of completeness and later use, we outline the proof. Using (2.4) with yields
On the other hand, in view of (2.1), we have
Using (2.2) and Lemma 2.4 (ii) together with the substitution , we obtain
Applying the initial conditions (2.7) and the fact that , (2.12) transforms to (2.9).
Conversely, suppose that is a solution of (2.9). Then
As , it follows by Lemma 2.4 (i) and Lemma 2.6 that
Thus, is a solution of (2.6). Now, differentiating (2.9), we obtain
for each Since , the second term in the above expression becomes zero as . Thus, we have
which implies that . Also, it is easy to infer that . Hence we conclude that is a solution of (2.6) and (2.7).
3. Existence Result
For the forthcoming analysis, we introduce the following assumptions:
(A1)let be a continuously differentiable function with and on a compact subinterval of ;
(A2)let be a continuously differentiable function with and on a compact subinterval of ;
(A3)there exist nonnegative functions such that
Now we state a result which describes the nonlinear alternative of Leray and Schauder [49].
Theorem 3.1.
Let be a normed linear space, be a convex set, and be open in with Let be a continuous and compact mapping. Then either the mapping has a fixed point in or there exist and with .
Lemma 3.2.
Suppose that the assumption holds and . Then, a function is a solution of the initial value problem (1.1) if and only if
where with is a solution of the integral equation
and a function is a solution of the initial value problem (1.2) if and only if
where with is a solution of the integral equation
We do not provide the proof as it is similar to that of Lemma 2.7. Consider the coupled system of integral equations
where and are given by (3.3) and (3.5), respectively.
Let denote the space of all continuous functions defined on Let and be normed spaces with the sup-norm and respectively. Then, is a normed space endowed with the sup-norm defined by
Lemma 3.3.
Assume that are continuous functions. Then is a solution of (1.1)-(1.2) if and only if is a solution of (3.6).
Proof.
For in (2.4), we have
Using the fact
and making the substitutions , we obtain
Using the initial conditions of (1.1) together with and (3.9) becomes (3.3), and an application of Cauchy function yields the first equation of (3.6). The converse of the theorem follows by applying the arguments used to prove the converse of Lemma 2.7. Similarly, it can be shown that satisfying the second equation of (3.6) together with (3.5) is a solution of (1.2) and vice versa. Thus, satisfying (3.6) is a solution of the system (1.1)-(1.2) and vice versa.
Theorem 3.4.
Let the assumptions ()–() hold. Then there exists a solution for the coupled integral equations (3.3) and (3.5) if
where
Proof.
Let us define an operator by
where
and are given by (3.11). In view of ()-(), it follows that is well defined and continuous.
Define a ball in the normed space as
where , and let be such that .
Let . Then and
Similarly, it can be shown that
Hence we conclude that This implies that Now we show that is a completely continuous operator (continuous and compact). To do this, we first set
For and with , we have
Similarly,
Since the functions are uniformly continuous on , it follows from the above estimates that is an equicontinuous set. Also, it is uniformly bounded as Thus, we conclude that is a completely continuous operator.
Now, let us consider the eigenvalue problem
Assuming that is a solution of (3.20) for , we find that
and, in a similar manner,
which imply that Hence, by Theorem 3.1, has a fixed point in such that This completes the proof.
Thus, by Lemma 3.2 and Theorem 3.4, the solution of (1.1)-(1.2) is given by
where
Now we allow the nonlinear in (1.1) to depend on in addition to and in (1.2) to depend on together with . Precisely, for we consider the following fractional differential system:
subject to the initial conditions given by (1.1)-(1.2), where are given functions.
In order to prove the existence of solution for the system (3.25), we need the following assumptions:
(let be a continuously differentiable function with and on a compact subinterval of ;
(let be a continuously differentiable function with and on a compact subinterval of ;
(there exist nonnegative functions such that
In this case, and involved in the coupled system of integral equations (3.6) modify to the following form:
The following corollary presents the analogue form of Theorem 3.4 for the fractional differential system (3.25).
Corollary 3.5.
Suppose that the assumptions ()–() hold. Then there exists a solution for the coupled integral equation (3.27) if
where and are given by (3.11).
The method of proof is similar to that of Theorem .
4. Uniqueness Result
To prove the uniqueness of solutions of (1.1)-(1.2), we need the following assumptions.
For each , there exist nonnegative functions and such that
Theorem 4.1.
Assume that , , and hold. Furthermore,
Then there exists a unique solution for the coupled integral equations (3.3) and (3.5).
Proof.
For , we define
where
As before, we define the operator by which is well defined and continuous. For , using (4.2) and (4.4), we have
Similarly, by using (4.3) and (4.5), it can be shown that . Thus, .
Now, for , we obtain
In a similar manner, we find that
Since , , therefore is a contraction. Hence, by Banach contraction principle, has a unique fixed point in such that , which is a solution of the coupled integral equations (3.3) and (3.5). This completes the proof.
The following Corollary ensures the uniqueness of the solutions of (3.25). We do not provide the proof as it is similar to that of Theorem 4.1.
Corollary 4.2.
Assume that , and the following condition hold:
For each , there exist nonnegative functions , , and such that
Furthermore,
Then there exists a unique solution for the coupled integral equation (3.27).
5. Example
For and , we consider the following coupled system of fractional differential equations:
Here , and Clearly, the assumptions are satisfied with In this case
where . Thus, all the conditions of Theorem 3.4 are satisfied, and hence there exists a solution of (5.1).
To prove the uniqueness of solutions of (5.1), we just need to verify the assumption . With and , we find that
As all the conditions of Theorem 4.1 hold, therefore the conclusion of Theorem 4.1 applies, and hence the coupled system of fractional differential equation (5.1) has a unique solution.
6. Conclusions
We have presented some existence and uniqueness results for an initial value problem of coupled fractional differential systems involving the Caputo type fractional derivative. The nonlinearities in the coupled fractional differential system depend on (i) the fractional derivatives of lower orders, (ii) the unknown functions together with the fractional derivative of lower orders. The proof of the existence results is based on the nonlinear alternative of Leray-Schauder, while the uniqueness of the solutions is proved by applying the Banach contraction principle. The present work can be extended to nonlocal coupled systems of nonlinear fractional differential equations.
References
Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York, NY, USA; 1974:xiii+234.
Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, New York, NY, USA; 1993:xvi+366.
Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Science, Yverdon, Switzerland; 1993:xxxvi+976.
Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.
Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier Science B.V., Amsterdam, The Netherlands; 2006:xvi+523.
Podlubny I, Petras I, Vinagre BM, O'Leary P, Dorck L: Analogue realizations of fractional-order controllers. Nonlinear Dynamics 2002,29(1–4):281–296.
Diethelm K, Ford NJ: Multi-order fractional differential equations and their numerical solution. Applied Mathematics and Computation 2004,154(3):621–640. 10.1016/S0096-3003(03)00739-2
Yu C, Gao G: Existence of fractional differential equations. Journal of Mathematical Analysis and Applications 2005,310(1):26–29. 10.1016/j.jmaa.2004.12.015
Bai Z, Lu H: Positive solutions for boundary value problem of nonlinear fractional differential equation. Journal of Mathematical Analysis and Applications 2005,311(2):495–505. 10.1016/j.jmaa.2005.02.052
Deng W: Numerical algorithm for the time fractional Fokker-Planck equation. Journal of Computational Physics 2007,227(2):1510–1522. 10.1016/j.jcp.2007.09.015
Rida SZ, El-Sherbiny HM, Arafa AAM: On the solution of the fractional nonlinear Schrödinger equation. Physics Letters A 2008,372(5):553–558. 10.1016/j.physleta.2007.06.071
Ladaci S, Loiseau JJ, Charef A: Fractional order adaptive high-gain controllers for a class of linear systems. Communications in Nonlinear Science and Numerical Simulation 2008,13(4):707–714. 10.1016/j.cnsns.2006.06.009
Ibrahim RW, Darus M: Subordination and superordination for univalent solutions for fractional differential equations. Journal of Mathematical Analysis and Applications 2008,345(2):871–879. 10.1016/j.jmaa.2008.05.017
Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(10):3337–3343. 10.1016/j.na.2007.09.025
Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(8):2677–2682. 10.1016/j.na.2007.08.042
Benchohra M, Henderson J, Ntouyas SK, Ouahab A: Existence results for fractional order functional differential equations with infinite delay. Journal of Mathematical Analysis and Applications 2008,338(2):1340–1350. 10.1016/j.jmaa.2007.06.021
Ahmad B, Sivasundaram S: Some basic results for fractional functional integro-differential equations. Communications in Applied Analysis 2008,12(4):467–477.
Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions. Mathematical and Computer Modelling 2009,49(3–4):605–609. 10.1016/j.mcm.2008.03.014
N'Guerekata GM: A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Analysis: Theory, Methods & Applications 2009,70(5):1873–1876. 10.1016/j.na.2008.02.087
Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Boundary Value Problems 2009, Article ID 708576 2009:-11.
Ahmad B, Nieto JJ: Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations. Abstract and Applied Analysis 2009, Article ID 494720 2009:-9.
Ahmad B, Otero-Espinar V: Existence of solutions for fractional differential inclusions with antiperiodic boundary conditions. Boundary Value Problems 2009, Article ID 625347 2009:-11.
Benchohra M, Hamani S: The method of upper and lower solutions and impulsive fractional differential inclusions. Nonlinear Analysis: Hybrid Systems 2009,3(4):433–440. 10.1016/j.nahs.2009.02.009
Ahmad B, Sivasundaram S: Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlinear Analysis: Hybrid Systems 2009,3(3):251–258. 10.1016/j.nahs.2009.01.008
Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge, UK; 2009.
Kosmatov N: Integral equations and initial value problems for nonlinear differential equations of fractional order. Nonlinear Analysis: Theory, Methods & Applications 2009,70(7):2521–2529. 10.1016/j.na.2008.03.037
El-Shahed M, Nieto JJ: Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order. Computers and Mathematics with Applications 2010,59(11):3438–3443. 10.1016/j.camwa.2010.03.031
Agarwal RP, Lakshmikantham V, Nieto JJ: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Analysis: Theory, Methods & Applications 2010,72(6):2859–2862. 10.1016/j.na.2009.11.029
Ahmad B, Nieto, JJ: Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Nonlinear Analysis: Hybrid Systems 2010,35(4):295–304.
Nieto JJ: Maximum principles for fractional differential equations derived from Mittag-Leffler functions. Applied Mathematics Letters 2010, 23: 1248–1251. 10.1016/j.aml.2010.06.007
Ahmad B, Sivasundaram S: On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order. Applied Mathematics and Computation 2010,217(2):480–487. 10.1016/j.amc.2010.05.080
Bai C-Z, Fang J-X: The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations. Applied Mathematics and Computation 2004,150(3):611–621. 10.1016/S0096-3003(03)00294-7
Daftardar-Gejji V: Positive solutions of a system of non-autonomous fractional differential equations. Journal of Mathematical Analysis and Applications 2005,302(1):56–64. 10.1016/j.jmaa.2004.08.007
Chen Y, An H-L: Numerical solutions of coupled Burgers equations with time- and space-fractional derivatives. Applied Mathematics and Computation 2008,200(1):87–95. 10.1016/j.amc.2007.10.050
Gafiychuk V, Datsko B, Meleshko V: Mathematical modeling of time fractional reaction-diffusion systems. Journal of Computational and Applied Mathematics 2008,220(1–2):215–225. 10.1016/j.cam.2007.08.011
Su X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Applied Mathematics Letters 2009,22(1):64–69. 10.1016/j.aml.2008.03.001
Ahmad B, Nieto JJ: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Computers & Mathematics with Applications 2009,58(9):1838–1843. 10.1016/j.camwa.2009.07.091
Ahmad B, Graef JR: Coupled systems of nonlinear fractional differential equations with nonlocal boundary conditions. PanAmerican Mathematical Journal 2009,19(3):29–39.
Podlubny I: Fractional-order systems and -controllers. IEEE Transactions on Automatic Control 1999,44(1):208–214.
Vinagre BM, Monje C, Calderon A: Fractional order systems and fractional order control actions. IEEE Conference on Decision and Control, 2002, Las Vegas, Nev, USA 2550–2554.
Poinot T, Trigeassou J-C: Identification of fractional systems using an output-error technique. Nonlinear Dynamics 2004,38(1–4):133–154.
Espndola JJ, Silva Neto JM, Lopes EMO: A new approach to viscoelastic material properties identification based on the fractional derivative model. Proceedings of the 1st IFAC Workshop on Fractional Differentiation and Its Application, July 2004, Bordeaux, France
Lazarević MP: Finite time stability analysis of fractional control of robotic time-delay systems. Mechanics Research Communications 2006,33(2):269–279. 10.1016/j.mechrescom.2005.08.010
Sira-Ramirez H, Batlle VF: On the GPI-PWM control of a class of switched fractional order systems. In Proceedings of the 2nd IFACWorkshop on Fractional Differentiation and Its Applications, July 2006, Porto, Portugal. The Institute of Engineering of Porto (ISEP);
Ahmad W, Abdel-Jabbar N: Modeling and simulation of a fractional order bioreactor system. In Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, July 2006, Porto, Portugal. The Institute of Engineering of Porto (ISEP);
Li Y, Chen Y-Q, Podlubny I: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Computers & Mathematics with Applications 2010,59(5):1810–1821. 10.1016/j.camwa.2009.08.019
Qian D, Li C, Agarwal RP, Wong PJY: Stability analysis of fractional differential system with Riemann-Liouville derivative. Mathematical and Computer Modelling 2010,52(5–6):862–874. 10.1016/j.mcm.2010.05.016
Li C, Deng W: Remarks on fractional derivatives. Applied Mathematics and Computation 2007,187(2):777–784. 10.1016/j.amc.2006.08.163
Fucik S, Kufner A: Nonlinear Differential Equations, Studies in Applied Mechanics. Volume 2. Elsevier Scientific, Amsterdam, The Netherlands; 1980:359.
Acknowledgments
The authors are grateful to the reviewers and Professor Juan J. Nieto for their suggestions. This paper was supported by Deanship of Scientific Research, King Abdulaziz University through Project no. 429/47-3.
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Ahmad, B., Alsaedi, A. Existence and Uniqueness of Solutions for Coupled Systems of Higher-Order Nonlinear Fractional Differential Equations. Fixed Point Theory Appl 2010, 364560 (2010). https://doi.org/10.1155/2010/364560
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DOI: https://doi.org/10.1155/2010/364560