The Ulam Stability of Fractional Differential Equation with the Caputo-Fabrizio Derivative

Ulam [1] proposed to study the approximation degree of the approximate solution and the exact solution of the equation in 1940. Hyers [2] responded to Ulam’s proposal and defined the Hyers-Ulam stability of equation in 1941. Later on, Rassias [3] extended Hyers’s work and defined the Hyers-Ulam-Rassias stability of equation in 1978. The Hyers-Ulam stability and Hyers-Ulam-Rassias stability are collectively referred to as the Ulam stability. Subsequently, researchers initiated a research on the Ulam stability of integer-order differential equations (see [4–10]). Obloza [4], Cemil and Emel [5] proved the Hyers-Ulam stability and Hyers-Ulam-Rassias stability of the first-order differential equation, respectively. Wang et al. [6] studied the Ulam stability of the first-order differential equation with a boundary value condition. Otrocol and Ilea [7] obtained the Ulam stability of the first-order delay differential equation. Huang and Li [8] also obtained the Hyers-Ulam stability of another class of the first-order delay differential equation. Zada et al. [9] studied the Hyers-Ulam-Rassias stability of the higher order delay differential equation. However, the study on the Ulam stability of fractional differential equations is in its infancy. Fractional differential equations are widely applied in physics [11, 12], control systems [13], chemical technology [14], and biosciences [15]. Fractional integral boundary value problems have been explored by many researchers. In particular, the integral boundary value problem provides a feasible method for the modeling of population dynamics and chemical engineering problems (see [16–18]). Although fractional integral boundary value problems are widely used, it is not easy to solve the equation, and the exact solution is often not obtained. Therefore, it is necessary to study the Ulam stability of fractional differential equations and use the approximate solution to replace the exact solution. So far, researchers have studied the Ulam stability and the existence and uniqueness of a solution for fractional differential equations with Hilfer-Hadamard, Caputo, and CaputoFabrizio fractional derivatives (see [19–22]). Abbas et al. [19] proved the existence and the Ulam stability of a fractional differential equation with the Hilfer-Hadamard derivative. In [20], Wang et al. established the Ulam stability and data dependence for the Caputo fractional differential equation


Introduction
Ulam [1] proposed to study the approximation degree of the approximate solution and the exact solution of the equation in 1940. Hyers [2] responded to Ulam's proposal and defined the Hyers-Ulam stability of equation in 1941. Later on, Rassias [3] extended Hyers's work and defined the Hyers-Ulam-Rassias stability of equation in 1978. The Hyers-Ulam stability and Hyers-Ulam-Rassias stability are collectively referred to as the Ulam stability. Subsequently, researchers initiated a research on the Ulam stability of integer-order differential equations (see [4][5][6][7][8][9][10]). Obloza [4], Cemil and Emel [5] proved the Hyers-Ulam stability and Hyers-Ulam-Rassias stability of the first-order differential equation, respectively. Wang et al. [6] studied the Ulam stability of the first-order differential equation with a boundary value condition. Otrocol and Ilea [7] obtained the Ulam stability of the first-order delay differential equation. Huang and Li [8] also obtained the Hyers-Ulam stability of another class of the first-order delay differential equation. Zada et al. [9] studied the Hyers-Ulam-Rassias stability of the higher order delay differential equation. However, the study on the Ulam stability of fractional differential equations is in its infancy.
Fractional differential equations are widely applied in physics [11,12], control systems [13], chemical technology [14], and biosciences [15]. Fractional integral boundary value problems have been explored by many researchers. In particular, the integral boundary value problem provides a feasible method for the modeling of population dynamics and chemical engineering problems (see [16][17][18]). Although fractional integral boundary value problems are widely used, it is not easy to solve the equation, and the exact solution is often not obtained. Therefore, it is necessary to study the Ulam stability of fractional differential equations and use the approximate solution to replace the exact solution. So far, researchers have studied the Ulam stability and the existence and uniqueness of a solution for fractional differential equations with Hilfer-Hadamard, Caputo, and Caputo-Fabrizio fractional derivatives (see [19][20][21][22]). Abbas et al. [19] proved the existence and the Ulam stability of a fractional differential equation with the Hilfer-Hadamard derivative.
In [20], Wang et al. established the Ulam stability and data dependence for the Caputo fractional differential equation In [21], Dai et al. studied the Ulam stability of the Caputo fractional differential equation with an integral boundary condition where I γ 0 + ð·Þ is the Riemann-Liouville fractional integral, γ > 0.
Equation (4) is a new kind of the Korteweg-de Vries-Bergers (KDVB) equation model. In [23], Equation (4) is used to describe unusual irregularities and nonlinearities in wave dynamics and liquids motions.
The main contributions are as follows: Firstly, we give the definitions of the Hyers-Ulam stability and Hyers-Ulam-Rassias stability for Equation (4). Then, we obtain a sufficient condition to derive the uniqueness of the solution for Equation (4) by the Banach contraction principle. Next, we give a sufficient condition to prove the existence of the solution for Equation (4) by Krasnoselskii's fixed point theorem. On this basis, we give the Ulam stability results for Equation (4) by the Laplace transform and inequality results.
The rest of our article is arranged as follows. Some basic definitions and necessary theorems are presented in Section 2. We establish sufficient conditions to show existence and uniqueness of solution for the Caputo-Fabrizio fractional differential equation in Section 3. In Section 4, we prove the Ulam stability of the Caputo-Fabrizio fractional differential equation. Two examples are provided in Section 5 to illustrate our theorems.

Preliminaries
We will denote by C 1 ½0, 1 the space of continuous differentiable functions on ½0, 1 with norm Definition 1 [24]. The Caputo-Fabrizio fractional derivative of order β of a continuous differentiable function x is given by the normalization function MðβÞ depends on β.
Definition 2 [25]. The Riemann-Liouville fractional integral of order γ of a function x is given by Based on Definition 2 in [5] and Definition 2.1 in [9], we give the definitions of the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability for Equation (4).

Definition 3. Equation (4) has the Hyers-Ulam stability if and only if for any solution xðtÞ of
where ε > 0, there is a constant C > 0 and a solution yðtÞ of Equation (4) satisfying where δðtÞ ∈ Cð½0, 1, R + Þ, there is a constant K k,δ > 0 and a solution yðtÞ of Equation (4) satisfying Theorem 5 [26]. If x is a piecewise continuous function and there exist K > 0 and μ such that then the Laplace transform L½xðtÞðsÞ exists.

Journal of Function Spaces
Theorem 7. The solution of the following fractional problem is given by where Proof. Since xðtÞ is continuous differentiable function on ½0, 1, x ′ ðtÞ is bounded function on ½0, 1. By Definition 1, CF D β xðtÞ is also a bounded function. Then, there exist constants k 1 , k 2 > 0 and μ 1 , μ 2 such that From Theorem 5, the Laplace transform of x′ðtÞ and CF D β xðtÞ exists.
Taking the Laplace transform for the first formula of Equation (14), we conclude Taking the Laplace inverse transform for the above equation, we conclude Þds: ð21Þ Then Since xð1Þ = I γ 0 + xðξÞ, thus Þds: ð23Þ By the definition of Gðt, sÞ, we conclude Remark 8.

Journal of Function Spaces
Thus, there exists a constant E > 0 such that Theorem 9 (Krasnoselskii's fixed point theorem). Let S be a bounded convex closed subset of a Banach space W, and P, Q : S ⟶ W satisfy the following: (i) Px + Qy ∈ S, for all x, y ∈ S (ii) P is completely continuous (iii) Q is a contraction mapping Then, P + Q has at least one fixed point.
Proof. Since k ∈ Cð½0, 1 × ℝ, ℝÞ, there exists T > 0 such that Similar to the proof of Theorem 3 in [22]. Let operator F be given by Firstly, we prove that F maps a closed set into a closed set.
Proof. Since k ∈ Cð½0, 1 × ℝ, ℝÞ, there exists T > 0 such that Let operators P and Q be given by

Journal of Function Spaces
Firstly, for all x 1 , x 2 ∈ U c , using Remark 8, it follows that Hence, we have Px 1 + Qx 2 ∈ U c . Then, for all x 1 , x 2 ∈ C 1 ½0, 1, As ξ γ /ðΓðγ + 1ÞÞ + Ec k < 1, Q is a contraction mapping. Finally, we prove operator P is completely continuous.
Step 1. Operator P is continuous. Let x n be a convergent sequence, x n ⟶ x ∈ C 1 ð½0, 1, ℝÞ, by Remark 8 and ðS 2 Þ; it follows that Since x n ⟶ x, we have Px n ⟶ Px; then operator P is continuous.
Step 2. Operator P is bounded on U c .
Step 3. Operator P is equicontinuous in C 1 ð½0, 1, ℝÞ. Let t 1 , t 2 ∈ ½0, 1 and t 2 < t 1 , x ∈ U c ; it follows that Then, operator P is equicontinuous. From Step 1-Step 3 and the Arzela-Ascoli theorem, P is completely continuous. By Theorem 9, P + Q has at least one fixed point, since From Theorem 7, Equation (4) has at least one solution.

Journal of Function Spaces
Proof. Since ðS 1 Þ and ðS 2 Þ hold, by Theorems 10 and 11, Equation (4) has a unique solution. From Theorem 7, Equation (4) has the unique solution Let yðtÞ satisfy yð0Þ = xð0Þ and be a solution of the inequality Then From the proof of Theorem 7, we conclude Then Thus Þds From the Gronwall-Bellman inequality, we conclude From Definition 3, Equation (4) has the Hyers-Ulam stability.
Proof. Since (S 1 ) and (S 2 ) hold, by Theorems 10 and 11, Equation (4) has a unique solution. From Theorem 7, Equation (4) has the unique solution Let yðtÞ satisfy yð0Þ = xð0Þ and be a solution of the inequality Then by ðS 3 Þ, it follows that Thus From the Gronwall-Bellman inequality, we conclude From Definition 4, Equation (4) has the Hyers-Ulam-Rassias stability on ½0, 1.