A Fixed-Point Approach to the Hyers–Ulam Stability of Caputo–Fabrizio Fractional Differential Equations

: In this paper, we study Hyers–Ulam and Hyers–Ulam–Rassias stability of nonlinear Caputo–Fabrizio fractional differential equations on a noncompact interval. We extend the corresponding uniqueness and stability results on a compact interval. Two examples are given to illustrate our main results.


Introduction
In 1940, Ulam posed a question concerning the stability of homomorphisms into metric groups, a question which is regarded as the origin of the problem of stability in the theory of functional equations. In 1941, Hyers [1] answered the problem for a linear functional equation on the Banach space and established a new concept on the stability of functional equation, now called Hyers-Ulam stability. In 1978, Rassias [2] introduced a new definition of generalized Hyers-Ulam stability by the constant ε by a variable, and obtained the stability of Hyers-Ulam-Rassias for functional equation. There is a rich literature on this topic for standard integer-order equations (see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). In addition, the same stability concepts are introduced to find approximate solutions to fractional differential equations, see [18,19] and the references therein.
In 2015, Caputo and Fabrizio [20] gave a new definition of fractional derivative with a smooth kernel. Losada and Nieto [21] introduced Caputo-Fabrizio fractional differential equation the newly developed Caputo-Fabrizio fractional derivative and obtained the existence and uniqueness results under some strong restriction. Baleanu et al. [22] obtained the approximate solution for some infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential equations. Goufo [23] used the fractional derivative of the newly developed Caputo-Fabrizio without singular kernel to establish the Korteweg-de Vries-Burgers equation with two perturbation levels. Atangana and Nieto [24] studied the numerical approximation of this new fractional derivative and established an improved RLC circuit model. Moore et al. [25] developed and analyzed a Caputo-Fabrizio fractional derivative model for the HIV epidemic which includes an antiretroviral treatment compartment. Dokuyucu et al. [26] applied the fractional derivative of Caputo-Fabrizio to model the cancer treatment by radiotherapy.
and applied fixed-point theorems to derive the existence and uniqueness of solution to nonlinear equations as follows and obtained the generalized Hyers-Ulam-Rassias stability via the Gronwall's inequality.
Observing that ( [4], Theorem 3) adopted the generalized Banach fixed-point theorem instead of the standard Banach contraction mapping and weakened the condition a α L + b α TL < 1 in ( [21], Theorem 1) to a α L < 1 where k > 0 denoted by the Lipschtiz constant of g, T denoted by the step of the interval and and M(·) denotes a normalization constant depending on ·. Based on the above observation, we apply a new fixed-point approach to show the existence and uniqueness and stability for (1) on a compact interval to a noncompact interval J = [τ 0 , τ 0 + k), k > 0.

Preliminaries
Definition 1 (see [20]). Let 0 < γ < 1, the Caputo-Fabrizio fractional derivative of order γ for a function f can be written as where M(γ) is a normalization constant depending on γ. Please note that ( CF D γ )( f ) = 0 if and only if f is a constant function.
Definition 2 (see [21] or ( [4], Definition 2)). Let 0 < γ < 1. The Caputo-Fabrizio fractional integral of order γ for a function f is defined as Let Ω be a nonempty set, we present the following definition of generalized metric on Ω.
Definition 3 (see [3]). A function ρ : Ω × Ω → [0, ∞] is called a generalized metric on Ω if and only if ρ satisfies Theorem 1 (see [28]). Let (Ω, ρ) is a generalized complete metric space. Suppose P : Ω → Ω is a strictly contractive operator with the Lipschitz constant K < 1. If there exists a nonnegative integer l such that ρ(P l+1 τ, P l τ) < ∞ for some τ ∈ Ω, then the followings are true: (i) The sequence {P n τ} converges to a fixed point τ * of P; (ii) τ * is the unique fixed point of P in Definition 4 (see [4]). Let g : J × R → R be a continuous function. Equation (7) is Hyers-Ulam stable if there exists a real number N > 0, such that for each > 0 and for any solution f ∈ C(J, R) of there exists a solution h ∈ C(J, R) of (1) with Definition 5 (see [4]). Let φ : J → R + and g : J × R → R be continuous functions. Equation (7) is generalized Hyers-Ulam-Rassias stable with respect to φ ∈ C(J, R + ), if there exists a constant c f ,φ > 0 such that for any solution f ∈ C(J, R) of there exists a solution h ∈ C(J, R) of (1) with

Main Results
Throughout this section, we denote the set Y of all continuous functions on J by We give the following conditions: [A 1 ] The function g : J × R → R is continuous and locally Lipschitz in τ.
[A 2 ] There exists a constant L > 0 such that Now, we prove the Hyers-Ulam stability of (7).
for all τ ∈ J and for some > 0, then there exists a unique solution f (τ) of for all τ ∈ J, where a γ and b γ are defined in (2). (5) with where Next, we consider the operator P : Y → Y as follows: for any f , g ∈ Y, where f 0 = f (τ 0 ). Please note that any fixed point of P solves (7). Indeed, the function u − a γ g(τ, u) = v in (10) is invertible, it is increasing. We denote its inverse u = G(τ, v), and G is globally Lipschitz in v and locally Lipschitz in τ by our assumptions. So, any fixed point of (10) satisfies g(s, f (s))ds + f 0 ). (11) Now clearly the function τ → b γ τ τ 0 g(s, f (s))ds + f 0 is locally Lipschitz in τ, we see that the composition function τ → G(τ, b γ τ τ 0 g(s, f (s))ds + f 0 ) is also locally Lipschitz in τ. So, any fixed point f (τ) of (10) is a locally Lipschitz function, and thus it is locally absolute continuous on J. So really (10) gives solutions of (7). As a matter of fact, we need just that u − a γ g(τ, u) = v is invertible, i.e., u − a γ g(τ, u) is strictly monotonic in u, and we can extend our results for more general case. We shall consider (11) instead of (10).
We prove that P f is continuous. Let τ 1 , τ 2 ∈ J, and τ 1 < τ 2 , we have Then, for all f ∈ Y, as τ 1 → τ 2 , the right-hand side of the above inequality tends to zero (due to [A 1 ] and f ∈ Y). Thus, P f is continuous, i.e., P f ∈ Y for all f ∈ Y.
When k = 1 and Y = Ω * , the operator P satisfies all the conditions of Theorem 1.

Remark 1. From Definition 4, (8) shows
for all τ ∈ J and for some G : J → (0, ∞) is a nondecreasing continuous function satisfying for all τ ∈ J, then there exists a unique solution f (τ) of (7) satisfying for all τ ∈ J.
Then, for each l, n ∈ Y, we have for all τ ∈ J. Thus, for any l, n ∈ Y and all τ ∈ J, we have that is, d 2 (Pl, Pn) ≤ L L+1 M l,n , ∀ τ ∈ J. Hence, we obtain Therefore, P is strictly contractive on Y. When k = 1 and Y = Ω * , the operator P satisfies all the conditions of Theorem 1.
Multiply both sides of (18) by e −K(τ−τ 0 ) , then, Then By Theorem 1, there exists a unique solution f : J → R of (7) satisfying By (17), we have which implies (16) holds. The proof is complete.

Remark 3.
Compared to ([4], Theorems 3 and 5), we extend the existence and uniqueness result and the generalized Hyers-Ulam-Rassias stability result for (1) on the noncompact interval and also remove the condition L|a α | < 1 from the assumptions.