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Stability of the solutions for nonlinear fractional differential equations with delays and integral boundary conditions
Advances in Difference Equations volume 2013, Article number: 43 (2013)
Abstract
In this article, we establish sufficient conditions for the existence, uniqueness and stability of solutions for nonlinear fractional differential equations with delays and integral boundary conditions.
MSC:34A08, 34A30, 34D20.
1 Introduction
Fractional differential equations is a generalization of ordinary differential equations and integration to arbitrary non-integer orders. The origin of fractional calculus goes back to Newton and Leibniz in the seventeenth century. Fractional differential equations appear naturally in a number of fields such as physics, engineering, biophysics, blood flow phenomena, aerodynamics, electron-analytical chemistry, biology, control theory, etc. An excellent account of the study of fractional differential equations can be found in [1–11] and the references therein. Boundary value problems for fractional differential equations have been discussed in [12–22]. By contrast, the development of stability for solutions of fractional differential equations is a bit slow. El-Sayed, Gaafar and Hamadalla [23] discuss the existence, uniqueness and stability of solutions for the non-local non-autonomous system of fractional order differential equations with delays
where denotes the Riemann-Liouville derivative of order α.
We consider nonlinear fractional differential equations with delay and integral boundary conditions of the form
where , are continuous functions, , are given continuous functions, , are constants.
In this article our aim is to show the existence of a unique solution for (1.1)-(1.3) and its uniform stability.
2 Preliminaries
In this section, we introduce notation, definitions and preliminary facts which are used throughout this paper.
Definition 2.1 The fractional integral of order of a function of order is defined by
provided the right-hand side exists pointwise on . Γ is the gamma function.
For instance, exists for all when ; note also that when , then and moreover .
Definition 2.2 The Riemann-Liouville fractional derivative of order of a function is given by
Definition 2.3 Let be continuous functions and satisfy the Lipschitz conditions
for all .
3 Existence of a unique solution for nonlinear fractional differential equations (1.1)-(1.3)
Let X be the class of all continuous functions defined on with the norm
Theorem 3.1 Let be continuous and satisfy the Lipschitz condition: if
where , then nonlinear fractional differential equations (1.1)-(1.3) have a unique positive solution.
Proof For , equation (1.1) can be written as
Integrating both sides of the above equation, we obtain
then
Applying the operator by on both sides,
differentiating both sides, we obtain
Now, let be defined by
Then
By conditions (1.2), we have
and
Now, choose N large enough such that . So, the map is a contraction and it has a fixed point , and hence there exists a unique which is a solution of integral equation (3.1).
We now prove the equivalence between integral equation (3.1) and nonlinear fractional differential equations (1.1)-(1.3). Indeed, since and , applying the operator on both sides of (3.1), we obtain
Differentiating both sides,
we get
which proves the equivalence of (3.1) and (1.1). We want to prove that . Since are continuous on , there exist constants m, M such that . We have
which implies
which in turn implies
and
Then from (3.1) and from (1.2), we have . □
Now, for , , the solution of nonlinear fractional differential equations (1.1)-(1.3) takes the form
4 Stability of a unique solution for nonlinear fractional differential equations (1.1)-(1.3)
In this section, we study the stability of the solution of nonlinear fractional differential equations (1.1)-(1.3).
The is a solution of the nonlinear fractional differential equations
Definition 4.1 The solution of nonlinear fractional differential equation (1.1) is stable if for any , there exists such that for any two solutions and of nonlinear fractional differential equations (1.1)-(1.3) and respectively, one has , then for all .
Theorem 4.2 The solution of nonlinear fractional differential equations (1.1)-(1.3) is uniformly stable.
Proof Let and be the solutions of nonlinear fractional differential equations (1.1)-(1.3) and respectively, then for , from (3.1), we have
and
Then
and
therefore, for , we can find such that . Then , which proves that the solution is uniformly stable. □
References
Machado JT, Kiryakova V, Mainardi F: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(3):1140-1153. 10.1016/j.cnsns.2010.05.027
Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. Theory and Applications of Fractional Differential Equations 2006.
Diethelm K: The Analysis of Fractional Differential Equations. Springer, Berlin; 2010.
Agarwal RP, O’Regan D, Stanek S: Positive solutions for mixed problems of singular fractional differential equations. Math. Nachr. 2012, 285: 27-41. 10.1002/mana.201000043
Agarwal RP, O’Regan D, Stanek S: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 2010, 371: 57-68. 10.1016/j.jmaa.2010.04.034
Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.
Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.
Samko SG, Kilbas AA, Marichev OI: Fractional Integral and Derivatives. Gordon & Breach, New York; 1993.
Podlubny I Mathematics in Science and Engineering 198. In Fractional Differential Equations. Academic Press, New York; 1999.
Sabatier J, Agrawal OP, Tenreiro Machado JA: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Berlin; 2007.
Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge; 2009.
Yang L, Chen H, Luo L, Luo Z: Successive iteration and positive solutions for boundary value problem of nonlinear fractional q -difference equation. J. Appl. Math. Comput. 2012. doi:10.1007/s12190-012-0622-4
Yang L, Chen H: Nonlocal boundary value problem for impulsive differential equations of fractional order. Adv. Differ. Equ. 2011. doi:10.1155/2011/404917
Su X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 2009, 22: 64-69. 10.1016/j.aml.2008.03.001
Wang G, Liu W: Existence results for a coupled system of nonlinear fractional 2 m -point boundary value problems at resonance. Adv. Differ. Equ. 2011. doi:10.1186/1687-1847-2011-44
Caballero J, Harjani J, Sadarangani K: Positive solutions for a class of singular fractional boundary-value problems. Comput. Math. Appl. 2011, 62: 1325-1332. 10.1016/j.camwa.2011.04.013
Staněk S: The existence of positive solutions of singular fractional boundary-value problems. Comput. Math. Appl. 2011, 62: 1379-1388. 10.1016/j.camwa.2011.04.048
Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 2006., 2006: Article ID 36
Ahmad B, Nieto JJ: Existence of solution for non-local boundary value problems of higher-order nonlinear fractional differential equations. Abstr. Appl. Anal. 2009., 2009: Article ID 494720
Liu S, Jia M, Tian Y: Existence of positive solutions for boundary-value problems with integral boundary conditions and sign changing nonlinearities. Electron. J. Differ. Equ. 2010., 2010: Article ID 163
Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Bound. Value Probl. 2009., 2009: Article ID 708576
Zhang X: Some results of linear fractional order time-delay system. Appl. Math. Comput. 2008, 197(1):407-411. 10.1016/j.amc.2007.07.069
El-Sayed AMA, Gaafar FM, Hamadalla EMA: Stability for a non-local non-autonomous system of fractional order differential equations with delays. Electron. J. Differ. Equ. 2010., 2010: Article ID 31
Acknowledgements
This work was supported by the Natural Science Foundation of Hunan Province (13JJ6068, 12JJ9001), Hunan Provincial Science and Technology Department of Science and Technology Project (2012SK3117) and Construct program of the key discipline in Hunan Province.
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Authors’ contributions
ZG carried out the stability of the solutions for nonlinear fractional differential equations studies and drafted the manuscript. LY and ZL carried out the stability of the solutions for nonlinear fractional differential equations studies. All authors read and approved the final manuscript.
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Gao, Z., Yang, L. & Luo, Z. Stability of the solutions for nonlinear fractional differential equations with delays and integral boundary conditions. Adv Differ Equ 2013, 43 (2013). https://doi.org/10.1186/1687-1847-2013-43
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DOI: https://doi.org/10.1186/1687-1847-2013-43