Existence and Continuous Dependence on Initial Data of Solution for Initial Value Problem of Fuzzy Multiterm Fractional Differential Equation

In this paper, the fuzzy multiterm fractional differential equation involving Caputo-type fuzzy fractional derivative of order 0<α<1 is considered. The uniqueness of solution is established by using the contraction mapping principle and the existence of solution is obtained by Schauder fixed point theorem.


Introduction
Nowadays the fractional differential equations (FDEs) are powerful tools representing many problems in various areas such as control engineering, diffusion processes, signal processing, and electromagnetism.
Recently the fuzzy fractional differential equations (FFDEs) have been studied by many researchers in order to analyze some systems with fuzzy initial conditions.But, it is too difficult to find the exact solutions of most FFDEs representing real-world phenomena.Therefore, research about FFDEs can be classified into two classes, namely, existence of solution and numerical methods.Many theoretical researches have been advanced on the existence, uniqueness, and stability of solution of FFDEs [1][2][3][4][5][6][7][8][9][10][11][12].
In [22], the existence and uniqueness of the solutions of fuzzy initial value problems of fractional differential equations with the Caputo-type fuzzy fractional derivative have been proved.Under the conditions which the right sides of equations satisfy Hölder continuity or Lipschitz continuity in its all variables, the existence of a solution to the Cauchy problem for fuzzy fractional differential equations was discussed in [23].In [24], by employing the contraction mapping principle on the complete metric space, the existence and uniqueness result for fuzzy fractional functional integral equation has been proved.The existence results of solutions for fuzzy fractional initial value problem under generalized differentiability conditions are obtained by Banach fixed point theorem in [25].In [26], researchers discussed the uniqueness and existence of the solutions for FFDEs with Riemann-Liouville H-differentiability of arbitrary order by using Krasnoselskii-Krein type conditions, Kooi type conditions, and Rogers conditions.But, the considerations of researchers in [22][23][24][25][26] were restricted to the case of FFDEs with single derivative term.Ngo et al. [27] presented that the existence and uniqueness results of the solution for fuzzy Caputo-Katugampola (CK) fractional differential equations with initial value and in [28] proved that the fractional 2 Advances in Fuzzy Systems fuzzy differential equation is not equal to the fractional fuzzy integral equation in general.
Based on the above facts, in this paper, we study the existence and uniqueness of solutions for fuzzy multiterm fractional differential equations of order 0 <  < 1 with fuzzy initial value under Caputo-type H-differentiability.
The paper is organized as follows.In Section 2, we introduced some definitions and properties of fuzzy fractional calculus.The existence result of solution for proposed problem is described in Section 3. Section 4 presented the continuous dependence on initial data of solution.Finally, the conclusion is summarized in Section 5.

Preliminaries and Basic Results
We introduce some definitions and notations which will be used throughout our paper.
Definition 1 (see [29]).Let us denote by R F the class of fuzzy subsets  : R → [0, 1] satisfying the following properties: is the support of the , and its closure cl(supp ) is compact.
Then R F is called the space of fuzzy number and any  ∈ R F is called fuzzy number.
We denote the fuzzy Riemann integral of  from  to  by Lemma 6 (see [14]).Suppose that ,  : [, ] → R F are continuous, then (i)  is the fuzzy Riemann integrable on [, ] and () = ∫   () is differentiable as in Definition 3, namely,  (1) We introduce following notations:   () is the set of all continuous fuzzy-valued functions on .
() is the set of all absolutely continuous fuzzy-valued functions on .
() is the space of all Lebesque integrable fuzzy-valued functions on , where  = [0, ] and without losing generality, we promise that  = 1.
By Lemma 6 (i), we obtain Lemma 14.The following facts are true: (i) Let  ∈   ().For any positive number , the fractional integral   0+ () is continuous in .(ii) For any positive numbers  and , it holds that   0+   ≤   /  .
From the definition of fractional integral, the following is true: By the definition of Gamma function, we have Let us consider assertion (iii).We use the notations Therefore the following evaluations are true: The proof is completed.

Existence of Solution for Fuzzy Multiterm Fractional Differential Equation
Let us consider the existence of solution for initial value problem of following fuzzy multiterm fractional differential equation: where  (24).Now let us consider the following: where () ∈   ().
Lemma 16.The solution of initial value problem of fuzzy fractional differential equation ( 25) is represented as Proof.Let  be the solution of initial value problem (25).Then we have Also since  is the fuzzy continuous, it is the fuzzy integrable and   0+ () exists for  ∈ .Therefore the following relations hold: From the Caputo-type differentiability of fuzzy-valued function , we get and since the space of the absolutely continuous functions coincides with the space of primitive functions of Lebesque integrable functions, the following relation is satisfied: Therefore we obtain By (30) and Lemma 6 (i), the left side of (28) exchanges as From ( 28) and (31), ()⊝   0 =   0+ () is satisfied.Namely, we have Conversely, we prove that  denoted by (26) is the solution of fuzzy initial value problem (25).Since ()⊝   0 =   0+ () holds.Also as  ∈   (), we get On the other hand, when [()]  fl [ 1 (, ),  2 (, )], by Lemma 8 Since the interval family {[∫ +ℎ   1 (, ), ∫ +ℎ   2 (, )]}  generates obviously a fuzzy number, we can see that there exist the H-differences as Consequently the H-differentiability of  is leaded.From (35), we obtain Also it is obvious that  satisfies the initial condition.
Theorem 17.Let  of ( 24) be a fuzzy continuous with respect to every variable.If () is the solution of initial value problem (24), the fuzzy-valued function () which is constructed by () fl    0+ () is the solution in   () of fuzzy integral equation as Conversely if () is the solution in   () of fuzzy integral equation (40), () which is constructed by (26) is the solution of initial value problem (24).
Proof.Let () be the solution of initial value problem (24).Namely, let assume that () satisfies Then if () is denoted by () fl    1 (), by Lemma 16, the following relation is leaded: Applying the operator    0+ to the both side of above equation, by Lemma 13 (iii), we get Therefore substituting the above results to    0+ (),    0+ (), () of (41), we obtain (40).Next let () be the solution of fuzzy integral equation (40).Then it is obvious that satisfies the initial condition of problem ( 24) from the continuousness of ().Namely Consequently from ( 44) is leaded and regarding the above equation, we get Moreover since we obtain Now we employ the following metric structure in   (): Obviously we can see that (  (),  * ) is a complete metric space (see [24]).
For any positive number , we can consider the metric structure as Then the metric  *  is equivalent to the metric  * .Namely, Theorem 18. Assume that the function  in ( 40) is continuous in its all variables and especially, for any  1 ,  2 ,  1 ,  2 ∈ R F ,  satisfies the following condition: Then the fuzzy integral equation ( 40) has a unique solution.
Thus the fuzzy integral equation ( 40) is represented as The existence of solution for the fuzzy integral equation ( 40) is equivalent to the existence of the fixed point of the operator  in   ().

𝑑 (𝑇𝑧
By (iii) of Lemma 14, we have By (ii) of Lemma 14, we have Consequently the following inequality is obtained: Multiplying the both sides of the above equation by  − *  , we obtain that Advances in Fuzzy Systems 7 and Thus the operator  is contractive on   () with respect to the distance  *  * and we obtain the unique fixed point   () of the operator  by contraction mapping principle.By the way, since the distance  *  * is equivalent to  * in   (),  * is also the unique fixed point in sense of the distance  * .This completes the proof of theorem.
Next let us consider the existence of solution in case which have not satisfied the Lipschitz condition.
Lemma 19 (Schauder fixed point theorem).Assume that (, ) is the complete metric space,  is a nonempty convex closed subset of , and  is a continuous mapping of  into itself such that () is contained in a compact subset of , then  has a fixed point in .
We have that, for any  ∈   ( 0), Thus we can get that Also by (iii) of Lemma 14, we have that, for any  ∈   ( 0),
Then we have Now we estimate the second term of the above expression.

𝑑 ( 1 Γ (𝛼)
∫ Estimating the third term of above equation similarly to above, we have So we obtain as follows: From right side of the above expression, we can see that { |  ∈   ( 0)} is equicontinuous.
By Theorem 20 and Schauder fixed point theorem, we can see the operator  has a fixed point in   ( 0).
The following theorem shows the continuous dependence on initial condition of solution.

Conclusions
We obtained the uniqueness results by employing contraction mapping principle and the existence results by using Schauder fixed point theorem for the solution of the fuzzy multiterm fractional differential equation involving Caputotype fuzzy fractional derivative of order 0 <  < 1 on complete metric space of continuous fuzzy number value functions.We also established the continuous dependence of solution on its initial condition.Our results can be expanded to the case in which the right side of equation involves more than two derivative terms.