On the stability of a fractional-order differential equation with

The topic of fractional calculus (integration and differentiation of fractional-order), which concerns singular integral and integro-differential operators, is enjoying interest among mathematicians, physicists and engineers (see [1]-[2] and [5]-[14] and the references therein). In this work, we investigate initial value problem of fractional-order differential equation with nonlocal condition. The stability (and some other properties concerning the existence and uniqueness) of the solution will be proved.

Definition 1.2 The (Caputo) fractional-order derivative D α of order α ∈ (0, 1] of the function g(t) is defined as (see [12] - [14]) Now the following theorem (some properties of the fractional-order integration and the fractional-order differentiation) can be easily proved.
In ( [3]) the nonlocal initial value problem for first-order differential inclusions: was studied, where Our objective in this paper is to investigate, by using the Banach contraction fixed point theorem, the existence of a unique solution of the following fractional-order differential equation: with the nonlocal condition: where x 0 ∈ and 0 < t 1 < t 2 < • • • < t m < 1, and a k = 0 for all k = 1, 2, • • • , m.Then we will prove that this solution is uniformly stable.
To facilitate our discussion, let us first state the following assumptions: Proof: 1) -( 2), then by using the definitions and properties of the fractional-order integration and fractional-order differentiation equation ( 1) can be written as Operating by I α on both sides of the last equation, we obtain by substituting for the value of x(0) from (2), we get If we put t = t k in (3), we obtain Then subtract (3) from (4) to get Substitute from ( 5) in (3), we get  Now, differentiate (6) to obtain Therefore we obtain that x ∈ L 1 [0, 1].
To complete the equivalence of equation ( 6) with the initial value problem (1) -( 2), let x(t) be a solution of (6), differentiate both sides, and get Then operate by I 1−α on both sides to obtain And if t = 0 we find that the nonlocal condition (2) is satisfied.Which proves the equivalence.
Theorem 3.1 The solution of the initial-value problem (1) -( 2) is uniformly stable Proof: Let x(t) be a solution of ) and let x(t) be a solution of equation ( 8) such that x(0) = x 0 − m k=1 a k x(t k ).Then

2 Definition 2 . 1 1 ,
is a function which is absolutely continuous, (iii) b(t) is a function which is absolutely continuous.EJQTDE, 2008 No. 29, p.By a solution of the initial value Problem (1) -(2) we mean a function x ∈ C[0, 1] with dx dt ∈ L 1 [0, 1].Theorem 2.1 If the above assumptions (i) -(iii) are satisfied such that then the initial value Problem (1) -(2) has a unique solution.

EJQTDE, 2008
No. 29, p. 3 Now define the operator T : C → C by which proves that the map T : C → C is contraction.Applying the Banach contraction fixed point theorem we deduce that(7) has a unique fixed point x ∈ C[0, 1].EJQTDE, 2008 No. 29, p. 4