Stability of the solutions for nonlinear fractional differential equations with delays and integral boundary conditions
© Gao et al.; licensee Springer 2013
Received: 7 July 2012
Accepted: 31 January 2013
Published: 25 February 2013
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.
where denotes the Riemann-Liouville derivative of order α.
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.
In this section, we introduce notation, definitions and preliminary facts which are used throughout this paper.
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 .
for all .
3 Existence of a unique solution for nonlinear fractional differential equations (1.1)-(1.3)
where , then nonlinear fractional differential equations (1.1)-(1.3) have a unique positive solution.
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).
Then from (3.1) and from (1.2), we have . □
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).
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.
therefore, for , we can find such that . Then , which proves that the solution is uniformly stable. □
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.
- 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.2010.05.027MathSciNetView ArticleGoogle Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. Theory and Applications of Fractional Differential Equations 2006.Google Scholar
- Diethelm K: The Analysis of Fractional Differential Equations. Springer, Berlin; 2010.View ArticleGoogle Scholar
- 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.201000043MathSciNetView ArticleGoogle Scholar
- 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.2010.04.034MathSciNetView ArticleGoogle Scholar
- Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.Google Scholar
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
- Samko SG, Kilbas AA, Marichev OI: Fractional Integral and Derivatives. Gordon & Breach, New York; 1993.Google Scholar
- Podlubny I Mathematics in Science and Engineering 198. In Fractional Differential Equations. Academic Press, New York; 1999.Google Scholar
- Sabatier J, Agrawal OP, Tenreiro Machado JA: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Berlin; 2007.View ArticleGoogle Scholar
- Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge; 2009.Google Scholar
- 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/s12190-012-0622-4Google Scholar
- Yang L, Chen H: Nonlocal boundary value problem for impulsive differential equations of fractional order. Adv. Differ. Equ. 2011. doi:10.1155/2011/404917Google Scholar
- 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.2008.03.001MathSciNetView ArticleGoogle Scholar
- 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:10.1186/1687-1847-2011-44Google Scholar
- 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.2011.04.013MathSciNetView ArticleGoogle Scholar
- Staněk S: The existence of positive solutions of singular fractional boundary-value problems. Comput. Math. Appl. 2011, 62: 1379-1388. 10.1016/j.camwa.2011.04.048MathSciNetView ArticleGoogle Scholar
- Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 2006., 2006: Article ID 36Google Scholar
- 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 494720Google Scholar
- 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 163Google Scholar
- 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 708576Google Scholar
- Zhang X: Some results of linear fractional order time-delay system. Appl. Math. Comput. 2008, 197(1):407-411. 10.1016/j.amc.2007.07.069MathSciNetView ArticleGoogle Scholar
- 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 31Google Scholar
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