ULAM ’ S STABILITIES OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

The study of stability problems for various functional equations originated from a famous talk given by Ulam In 1940 (see [1]). In the talk, Ulam discussed a problem concerning the stability of homomorphisms. A significant breakthrough came in 1941, when Hyers [2] gave a partial solution to Ulam’s problem.. In 1978, Rassias [3] provided a remarkable generalization of the Ulam-Hyers stability of mappings by considering variables. During the last two


Introduction
The study of stability problems for various functional equations originated from a famous talk given by Ulam In 1940 (see [1]).In the talk, Ulam discussed a problem concerning the stability of homomorphisms.A significant breakthrough came in 1941, when Hyers [2] gave a partial solution to Ulam's problem..In 1978, Rassias [3] provided a remarkable generalization of the Ulam-Hyers stability of mappings by considering variables.During the last two decades very important contirbutions to the stability problems of functional equations were given by many mathematicians (see [4][5][6][7][8][9][10][11]).A generalization of Ulam's problem was recently proposed by replacing functional equations with differential equations: The differential equation F (t, y(t), y ′ (t), ..., y (n) (t)) = 0 has the Hyers-Ulam stability if for given ε > 0 and a function y such that F (t, y(t), y ′ (t), ..., y (n) (t)) ≤ ε there exists a solution y 0 of the differential equation such that and lim ε→0 K(ε) = 0. Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [12,13]).Thereafter, Alsina and Ger published their paper [14], which handles the Hyers-Ulam stability of the linear differential equation y ′ (t) = y(t): If a differentiable function y(t) is a solution of the inequality |y ′ (t) − y(t)| ≤ ε for any t ∈ (a, ∞), then there exists a constant c such that |y(t) − ce t | ≤ 3ε for all t ∈ (a, ∞).Recently, the Hyers-Ulam stability problems of linear differential equations of first order and second order with constant coefficients were studied by using the method of integral factors (see [15,16]).The results given in [17][18][19] have been generalized by Popa and Rus [20,21] for the linear differential equations of nth order with constant coefficients.For more details on Hyers-Ulam stability and the generalized Hyers-Ulam stability, we refer the reader to the papers [22][23][24][25][26][27][28].
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary order (non-integer).In recent years, fractional differential equations arise naturally in various fields such as rheology, fractals, chaotic dynamics, modeling and control theory, signal processing, bioengineering and biomedical applications, etc; Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes, [29][30].Some researchers have used the fixed point approach to investigate the Hyers-Ulam stability for fractional differential equations [e.g.31,33].The objective of this article is to investigate the Hyers-Ulam-Rassias Stability for the stability and Hyers-Ulam Criteria for fractional integrodifferential equations with the initial condition where Rdenotes the set of real numbers,J = [t 0 , t+a], a > 0, , and x 0 is a real constant.

Preliminaries
In this section, we give some basic definitions and Lemmas which we used to prove the main results.
provided that this integral exists, where Γ is the Gamma function.
Definition 2. Assume that for continuously differentiable functions g : J → R , G : J × J → R and satisfying fractional differential inequality for all t ∈ J and for each ε > 0,where x (α) (t) denotes the fractional drivative of order α.there exists a solution y 0 : J → Y of the fractional initial value problem (1.1)and (1.2) such that |x(t) − x 0 (t)| ≤ Kε, for all t ∈ J .Then we say that the above fractional initial value problem (1.1)and (1.2) has the Hyers-Ulam stability.Definition 3. We say that equation (1.1) with initial condition (1.2) has the Hyers-Ulam-Rassias stability with respect to ϕ if there exists a positive constant K > 0 with the following property: For each x(t) satisfying then there exists some solution x 0 (t) of the equation (1.1) with (1.2) such that |x(t) − x 0 (t)| ≤ Kϕ(t).
Lemma 2. (Gronwall's lemma).Let u(t) and v(t) be nonnegative continuous functions on some interval 0 < t 0 ≤ t ≤ t 0 + a.Also, let the function f (t)be positive, continuous, and monotonically nondecreasing on [t 0, t 0 + a] and satisfy the inequality then, there holds the inequality Proof.For the proof of Lemma 1.2, see [35].

Main Results On Hyers-Ulam and Hyers-Ulam-Rassias Stability
In this section, we will prove our main results, and establish the HU stability of solution of (1.1) satisfying (1.2).
Theorem 4. let the function g satisfy the inequality and let G satisfy the inequality where β(t) and γ(t) are continuous and nonnegative functions such that Then the problem (1.1),(1.2) is stable in the sense of Hyers and Ulam.
Applying the integral operator (2.1) to the inequality (3.9) we obtain On can easily show that z(t) ∈ C[J, R] defined by In view of (3.3) there exists a positive constant K such that which completes the proof.Now we will prove the Hyers-Ulam-Rassias stability (HUR) of problem (1.1), (1.2).Theorem 5. Let the function g satisfy the inequality and let G satisfy the inequality where γ(t) and β(t) are continuous and nonnegative functions such that then the problem (1.1),(1.2) is stable in the sense of HUR.
x (α) (t) − g(t, x(t)) Applying the integral operator (2.1) to the inequality (3.9) we obtain On can easily show that z(t) ∈ C[J, R] defined by G(τ, s, z(s))dτ ds satisfies the IVP (1.1),(1.2).Now we estimate the difference  Proof.Applying the same approach used in the Theorem 2, we can get the proof of theorem .

Conclusion
In this work, the problem of the Hyers-Ulam and Hyers-Ulam-Rassias Stability of solution of Ulam's Stabilities of Fractional Integro-Differential Equations has been investigated and solved using the direct method.