Hyers–Ulam Stability and Existence of Solutions to the Generalized Liouville–Caputo Fractional Differential Equations

: The aim of this paper is to study the stability of generalized Liouville–Caputo fractional differential equations in Hyers–Ulam sense. We show that three types of the generalized linear Liouville–Caputo fractional differential equations are Hyers–Ulam stable by a ρ -Laplace transform method. We establish existence and uniqueness of solutions to the Cauchy problem for the corresponding nonlinear equations with the help of ﬁxed point theorems.


Introduction
Because fractional calculus has a good global correlation performance to reflect the historical dependence process of the development of system functions, and can also describe the attributes of the dynamic system itself, it becomes a powerful mathematical tool to describe some complex movements, irregular phenomena, memory features, and other aspects. Fractional calculus theory was widely used by mathematicians as well as chemists, engineers, economists, biologists, and physicists (see [1][2][3][4][5]). In 1876, Riemann proposed the definition of the Riemann-Liouville derivative. Caputo first proposed another definition of fractional derivative via a modified Riemann-Liouville fractional integral at the beginning of the 20th century, namely a Caputo fractional derivative. Caputo and Fabrizio [6] introduced a new nonlocal derivative without a singular kernel and obtained the new Caputo-Fabrizio fractional derivative of order 0 < α < 1. Theoretical research and application of Caputo-Fabrizio fractional can be referred to [7][8][9][10][11][12][13]. Butzer et al. [14][15][16][17][18] study properties of the Hadamard fractional integral and the derivative. In [19,20], Katugampola introduced a new fractional integral and fractional derivative, which generalizes the Riemann-Liouville and the Hadamard integrals and derivative into a single form, respectively.
Hyers-Ulam stability has been one of the most active research topics in differential equations, and obtained a series of results (see [21][22][23][24][25][26][27][28][29][30]). Recently, Alqifiary et al. [22] obtained generalized Hyers-Ulam stability of linear differential equations. Razaei et al. [31] proved that the Hyers-Ulam stability of linear differential equations. Wang et al. [32] proved that two types of fractional linear differential equations are Hyers-Ulam stable. Shen et al. [33] deal with the Ulam stability of linear fractional differential equations with constant coefficients. Liu et al. [34] proved the Hyers-Ulam stability of linear Caputo-Fabrizio fractional differential equations. Liu et al. [35] studied the Hyers-Ulam stability of linear Caputo-Fabrizio fractional differential equations with the Mittag-Leffler kernel. Laplace transform method is used to deal with linear equations and fixed point approach and Gronwall inequality are used to deal with nonlinear equations.
For some differential equations describing physical models and practical problems, it is very difficult to find their exact solutions and the method of finding its exact solution (if they exist) is also very complicated. In order to construct explicit solutions to differential equations with constant coefficients and in the frame of Riemann-Liouville, Caputo and Riez fractional derivatives, integral transforms including Laplace, Mellin, and Fourier were found to be strong tools. One of the main difficulty is to find some appropriate transformations in order to find analytic solutions to some classes of fractional differential equations. In order to extend the possibility of working in a large class of functions, Jarad et al. [36] present a modified Laplace transform that it call ρ-Laplace transform, study its properties, and prove its own convolution theorem.
Motivated by [36], we apply the ρ-Laplace transform method to study the Hyers-Ulam stability of the following linear differential equations: and and where D α,ρ c f denotes the left generalized α order Liouville-Caputo fractional derivative for f with the parameter ρ (see Definition 2).
Next, we study to Cauchy problem for nonlinear equations as follows: and show the existence and uniqueness of solutions via Banach fixed point theorem and Schaefer's fixed point theorem and obtain the generalized Hyers-Ulam-Rassias stability via an extended Gronwall's inequality.

Preliminaries
Let C(I, R) be the Banach space of all continuous functions from I into R with the norm y C := sup{|y(x)| : x ∈ I}.

Definition 3.
(see [36]) Let 0 < α < 1, ρ > 0. The generalized left fractional integrals of the function f is expressed in the form Theorem 4. (see [36], Corollary 3.3) Let α ∈ (0, 1), ρ > 0. The ρ-Laplace transform of the function of the generalized fractional derivative in the Liouville-Caputo sense is expressed in the following form: The ρ-Laplace transform of the function f is given in the form where L{ f } is the usual Laplace transform of f .

Definition 5.
(see [36], Definition 2.9) Let f and g be two functions which are piecewise continuous at each interval [0, T] and of exponential order. The ρ−convolution of f and g is given by Theorem 6. (see [36], Theorem 2.11) Let f and g be two functions which are piecewise continuous at each interval [0, T] and of exponential order e c t ρ ρ . Then, Proof. It is easy to check the following facts: The proof is complete.
From Lemma 8, we derive the following result.
Proof. One can see The proof is finished.

Hyers-Ulam Stability for Linear Problems
Theorem 13. Let 0 < α < 1, ρ > 0 and g(t) be a given real continuous function on [0, ∞). If a function f : [0, ∞) → R satisfies the following inequality Taking the ρ-Laplace transform of (9) via Theorem 4, we have where L ρ {F(·)} denotes the ρ-Laplace transform of the function F. From (10), one has Taking the ρ-Laplace transform of (11), one has Note that which yields that f a (·) is a solution of Equation (1), since, according to the one-to-one transformation of L in (6), we can get that L ρ is the one-to-one transformation. From (11) and (12), we have This implies that Thus, The proof is complete.
Theorem 15. Let 0 < α < 1, ρ > 0, λ ∈ R, and g(t) be a given real continuous function on [0, ∞). If a function f : [0, ∞) → R satisfies the following inequality: Taking the ρ-Laplace transform of (14) via Theorem 4, we have where L ρ {F 1 (·)} denotes the ρ-Laplace transform of the function F 1 . From (15), one has Set Taking the ρ-Laplace transform of (17), one has By Definition 4 and (18), we obtain which yields that f a is a solution of Equation (2) , since L ρ is one-to-one. From (16) and (18), we have This implies that Thus, The proof is complete.
If a function f : [0, ∞) → R satisfies the following inequality for each t ≥ 0 and some function G(t) > 0, where F 2 is defined in (14) .

Existence and Stability Results for the Nonlinear Equation
We introduce the following conditions: [A2] : There exists a L > 0 such that [A3] : There exists a constant L g > 0 such that for each t ∈ [0, T] and all t ∈ R.
From the condition L( T ρ ρ ) α < 1, Λ is a contraction mapping, and, by applying the Banach contraction mapping principle, we know that the operator Λ has a unique fixed point on [0, T].
Next, we show that the existence of solutions for (4) via Schaefer's fixed point theorem.
Proof. Consider Λ as in (26). We divide our proof into several steps.
Indeed, we prove that for all r > 0, there exists a k > 0 such that for every f ∈ B r = { f ∈ C([0, T], R) : f C ≤ r}, we have Λ f C ≤ k. In fact, for any t ∈ [0, T], from [A3], we have which implies that Step 3. P maps bounded sets into equicontinuous sets in C([0, T], R).
Then, the right-hand side of the above inequality tends to zero as t 1 → t 2 . Thus, Λ is equicontinuous.
We can conclude that Λ is completely continuous from Steps 1-3 with the Arzela-Ascoli theorem.
For the sake of discussion, the following inequality is given In the following, we consider (4) and (27) to discuss the generalized Ulam-Hyers-Rassias stability. We need the following condition.

An Example
In this section, an example is given to illustrate our main results.

Example 25.
We consider the following fractional problem