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.
KeywordsRiemann-Liouvile derivatives nonlinear fractional differential equation delay integral boundary conditions stability
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.
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