ULAM STABILITY AND DATA DEPENDENCE FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH CAPUTO DERIVATIVE

In this paper, Ulam stability and data dependence for fractional differential equations with Caputo fractional derivative of orderare studied. We present four types of Ulam stability results for the fractional differential equation in the case of 0 < � < 1 and b = +1 by virtue of the Henry-Gronwall inequality. Meanwhile, we give an interesting data dependence results for the fractional differential equation in the case of 1 < � < 2 and b < +1 by virtue of a generalized Henry-Gronwall inequality with mixed integral term. Finally, examples are given to illustrate our theory results.

On the other hand, numerous monographs have discussed the data dependence in the theory of ordinary differential equations (see for example [4,9,10,14,22,24]). Meanwhile, there are some special data dependence in the theory of functional equations such as Ulam-Hyers, Ulam-Hyers-Rassias and Ulam-Hyers-Bourgin. The stability properties of all kinds of equations have attracted the attention of many mathematicians. Particularly, the Ulam-Hyers-Rassias stability was taken up by a number of mathematicians and the study of this area has the grown to be one of the central subjects in the mathematical analysis area. For more information, we can see the monographs Cadariu [8], Hyers [15] and Jung [16].
The first author acknowledges the support by the Tianyuan Special Funds of the National Natural Science Although, there are some work on the local stability and Mittag-Leffler stability for fractional differential equations (see [11,18,19]), to the best of my knowledge, there are very rare works on the Ulam stability for fractional differential equations. Motivated by [1,25,32], we will study the Ulam stability for the following fractional differential equation where c D α is the Caputo fractional derivative of order α ∈ (0, 1) and the function f satisfies some conditions will be specified later. Meanwhile, we will study the data dependence for the following fractional differential where the Caputo fractional derivative of order α ∈ (1, 2).
In the present paper, we introduce four types of Ulam stability definitions for fractional differential equations: Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability. We present the four types of Ulam stability results for a fractional differential equation in the case 0 < α < 1 and b = +∞ by virtue of a Henry-Gronwall inequality. Meanwhile, we give data dependence results for a fractional differential equation in the case 1 < α < 2 and b < +∞ by virtue of Henry-Gronwall inequality with mixed integral term. Finally, examples are given to illustrate our theory results.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. We denote (B, | · |) be a Banach space. Let a ∈ R, b ∈ R, a < b ≤ +∞, Let We need some basic definitions and properties of the fractional calculus theory which are used further in this paper. For more details, see [17].
Definition 2.1. The fractional integral of order γ with the lower limit zero for a function f is defined as provided the right side is point-wise defined on [0, ∞), where Γ(·) is the gamma function.
Definition 2.2. The Riemann-Liouville derivative of order γ with the lower limit zero for a function f : [0, ∞) → R can be written as Let ǫ be a positive real number, f : [a, b) × B → B be a continuous operator and ϕ : [a, b) → R+ be a continuous function. We consider the following differential equation

One can have similar remarks for the inequations (2.3) and (2.4).
So, the Ulam stabilities of the fractional differential equations are some special types of data dependence of the solutions of fractional differential equations.
Remark 2.10. Let 0 < α < 1, if y ∈ C 1 ([a, b), B) is a solution of the inequality (2.2) then y is a solution of the following integral inequality Indeed, by Remark 2.9 we have that Then This implies that From this it follows that We have similar remarks for the solutions of the inequations (2.3) and (2.4).
In what follows, we collect the Henry-Gronwall inequality (see Lemma 7.1.1, [12]), which can be used in fractional differential equations with initial value conditions.
To end this section, we collect a generalized Henry-Gronwall inequality with mixed integral term, which can be used in boundary value problems for fractional differential equations.
x(a) = y(a). Then we have By differential inequality (2.3), we havę From these relation it follows By Lemma 2.11 and Remark 2.12(i), there exists a constant M * f > 0 independent of λϕϕ(t) such that Thus, the equation (2.1) (b = +∞) is generalized Ulam-Hyers-Rassias stable.
The following result is interesting although the proof is not very difficult. Denote by x the solution of the following fractional boundary value problem Then the following relation holds: Proof. By Lemma 3.17 of [1], it is clear that the solution of the fractional boundary value problem (4.1) given by By differential inequality (2.2), we havę From these relation it follows Applying Lemma 2.13 to the above inequality and yields the aim inequality (4.2).

Example
In this section, some examples are given to illustrate our theory results.
Let 0 < α < 1. We consider in the case B := R the equation

1) and the inequation
Let y ∈ C 1 [a, b) be a solution of the inequation (5.2). Then there exists g ∈ C[a, b) such that: ∈ [a, b).