On the Existence and Uniqueness for High Order Fuzzy Fractional Differential Equations with Uncertainty

In the last several years, fractional differential equations attracted more and more researchers and have proven to be very useful tools formodeling phenomena in physics, finance, and many other areas. The Riemann-Liouville formulation arises in a natural way for problems such as transport problems from the continuum randomwalk schemeor generalizes Chapman-Kolmogorov models [1, 2]. It was also applied for modeling the behavior of viscoelastic and viscoplastic materials under external influences [3, 4] and the continuum and statistical mechanics for viscoelasticity problems [5]. Some of the several research papers were published to consider the uniqueness of the solution for fractional differential equations underNagumo like conditions (see, e.g., [6–13] and references therein). In [11, 12], Lakshmikantham and Leela established the uniqueness of the solution for the problem


Introduction
In the last several years, fractional differential equations attracted more and more researchers and have proven to be very useful tools for modeling phenomena in physics, finance, and many other areas.The Riemann-Liouville formulation arises in a natural way for problems such as transport problems from the continuum random walk scheme or generalizes Chapman-Kolmogorov models [1,2].It was also applied for modeling the behavior of viscoelastic and viscoplastic materials under external influences [3,4] and the continuum and statistical mechanics for viscoelasticity problems [5].Some of the several research papers were published to consider the uniqueness of the solution for fractional differential equations under Nagumo like conditions (see, e.g., [6][7][8][9][10][11][12][13] and references therein).In [11,12], Lakshmikantham and Leela established the uniqueness of the solution for the problem   () = (, ()), where 0 <  < 1.Then, in [13], Yoruk et al. proved the uniqueness of the solution via Krasnoselskii-Krein, Rogers, and Kooi conditions, for 1 <  < 2.
In [32], Allahviranloo and Ahmadi introduced the fuzzy Laplace transform, which they used under the strongly generalized differentiability.Recently, ElJaoui et al. [33] developed it further.The newly defined fuzzy Laplace transform [32] for high order fuzzy derivatives is one of the most useful methods as mentioned by Jafarian et al. in [34]: ". .., one of the important and interesting transforms in the problems of fuzzy equations is Laplace transforms.The fuzzy Laplace transform method solves fuzzy fractional differential equations and fuzzy boundary and initial value problems [35][36][37][38] . ..." Motivated by the above works, we adopted the fuzzy Laplace transform to prove the uniqueness and existence for the following initial value problems (FFDE) for arbitrary order  > 1: where  0 ∈ E and  : E 0 → E is a continuous fuzzy-valued function with where  is the Hausdorff distance.

Preliminaries
First, let us recall some basic definitions about fuzzy numbers and fuzzy sets.Here and in the rest of the paper, we denote by Γ the Gamma function and [] the integer part of .
According to Zadeh's extension principle, we have the following properties of fuzzy addition and multiplication by scalar on E: Seeking simplicity, we note ⊕, ⊙ by the usual +, ⋅.The Hausdorff distance between the fuzzy numbers is denoted by  : We denote by  F [0, ] the space of all fuzzy-valued functions which are continuous on [0, ] and  F [0, ] the space of all Lebesgue integrable fuzzy-valued functions on [0, ], where  > 0. We also denote by  (−1)F [0, ] the space of fuzzy-valued functions  which have continuous derivatives up to order  − 1 on [0, ] such that  (−1) in  F [0, ].
The proof runs along similar lines as that of [41,Theorem 16], and we omit it here.

Fuzzy Fractional Integral Equation
In this section, we study the relation between problem (1) and the fuzzy integral form using the well-known fuzzy Laplace transform.
In fact, by taking Laplace transform on both sides of we get Based on the type of Riemann-Liouville -differentiability, we obtain two cases.
and based on the lower and upper functions of    the above equation becomes where In order to solve system (18), and for the sake of simplicity, we assume that where  1 (; ) and  1 (; ) are solutions of the previous system (18); it yields and based on the lower and upper functions of    the above equation becomes where In order to solve system (23), and for the sake of simplicity, we assume that L [ (; )] =  2 (; ) , where  2 (; ) and  2 (; ) are solutions of the previous system (23).Then we obtain Taking into account the initial conditions of problem (1) and using the linearity of the inverse Laplace transform on systems (20) and (26), we obtain the following for both cases.
is a solution for problem (1) if and only if  is a solution for the following integral equation: in the sense of  [() − ]-differentiability, and in the sense of  [() − ]-differentiability.
The proof of this theorem is essentially based on Lemma 11.
Also, if  > 0, then from the condition (K1) for small , we have ( (, ) ,  (, )) Proof.Without loss of generality, we prove Theorem 13 for the sequence {  } in the sense of  [() − ]-differentiability using Ascoli-Arzela Theorem.The convergence of the sequence {x  } in the sense of  [() − ]-differentiability is completely similar so we omit it.
Step Step 2. We prove that the functions  and  are continuous in Let us note The right-hand side in the above inequalities is at most for every  ≤  − 1.And since  is arbitrary and  1 ,  2 can be interchangeable, we get The same goes for (), and we obtain These imply that  and  are continuous on [0, ].
)|}.And (, E) is a complete metric space.Definition 2. Let ,  ∈ E. If there exists  ∈ E such that  =  + , then  is called the -difference of  and , and it is denoted by  ⊖ .