Existence, Uniqueness, and E q –Ulam-Type Stability of Fuzzy Fractional Differential Equation

: This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order q ∈ ( 1,2 ] , c 0 D q 0 + u ( t ) = λ u ( t ) ⊕ f ( t , u ( t )) ⊕ B ( t ) C ( t ) , t ∈ [ 0, T ] with initial conditions u ( 0 ) = u 0 , u (cid:48) ( 0 ) = u 1 . Moreover, by using direct analytic methods, the E q –Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations.


Introduction
In real-life phenomena, numerous physical processes are used to present fractionalorder sets that may change with space and time. The operations of differentiation and integration of fractional order are authorized by fractional calculus. The fractional order may be taken on imaginary and real values [1][2][3]. The theory of fuzzy sets is continuously drawing the attention of researchers. This is mainly due to its extended adaptability in various fields including mechanics, engineering, electrical, processing signals, thermal system, robotics, control, signal processing, and in several other areas [4][5][6][7][8][9][10]. Therefore, it has been a topic of increasing concern for researchers during the past few years.
Fuzzy fractional differential equations appeared for the first time in 2010 when an idea of the solution was initially proposed by Agarwal et al. [11]. However, the Riemann-Liouville H derivative based on the strongly generalizing Hukuhara differentiability [12,13] was defined by Allahviranloo and Salahshour [14,15]. They worked on solutions to Cauchy problems under this kind of derivative.
In the above equation, α ∈ (0, 1] and λ > 0. Shen studied the Ulam stability under the generalization of Hukuhara differentiability of a first-order linear fuzzy differential equation in 2015 [24]. Later, Shen et al. investigated the Ulam stability of a nonlinear fuzzy fractional equation with the help of fixed-point techniques in 2016 [25], by focusing on the initial condition where RL 0 D q 0 + denoted Riemann-Liouville H derivative with respect to order q ∈ (0, 1], f : (0, T] × E 1 → E 1 , T ∈ R + , and λ ∈ R.
More results can be observed that are related to Ulam-type stability in [26][27][28]. Motivated by the above-cited papers, we aim to deal with fuzzy fractional differential equations of the form, with initial conditions u(0) = u 0 , u (0) = u 1 , Here, c 0 D q 0 + denotes the Caputo derivative of order q ∈ (1, 2], f : (0, T] × E 1 → E 1 , T ∈ R + and λ ∈ R. This paper focuses on facilitating, with as few conditions as possible, to assure the uniqueness and existence of a solution to Cauchy problems (1) and (2). It establishes a link between fuzzy fractional differential equations and the Ulam-type stability, which enhances and generalizes some familiar outputs in the existing literature.

Basic Concepts
Assume that P k (R) denotes the collection of all nonempty convex and compact subsets of R and define sums and scalar products in P k (R) in the usual manner. Let A and B be two nonempty bounded subsets in R. The distance between A and B is defined through the Hausdorff metric, In the above equality, ||x|| stands for the usual Euclidean norm in R. Now it is well known that the metric D turns the space (P k (R), D) into a complete and separable metric space [26]. Denote where (1)-(4) stands for the following properties of the function u: (1) u is normal in the sense that there exists an s 0 ∈ R such that u(s 0 ) = 2; (2) u is fuzzy convex, that is u(qs + (1 − q)y) min{u(s), u(y)} for any s, y ∈ R and q ∈ (1, 2]; (3) u is an upper semicontinuous function on R; (4) The set [u] 1 defined by [u] 1 = {t ∈ R|u(t) > 1} is compact.
Define u as the lower branch andū as the upper branch of the fuzzy number u ∈ E 1 . The set [u] q = {t ∈ R|u(t) ≥ q} := [u q , u q ] is known as the q-level set of fuzzy number u, where q ∈ (1, 2]. The length of q-level set is calculated as diam[u] q = u q − u q . Lemma 1 ([29,30]). If u, v, s, y ∈ E 1 , then Through a generalization of the Hausdorff-Pompeiu metric on convex and compact sets, the metric D on E 1 can be defined by Definition 1 ([13]). Assume that F ∈ C E (a, b] L E (a, b]. The fuzzy Riemann-Liouville integral for a fuzzy-valued function F is defined by 1,2]. For q = 1 we obtain T 1 F(t) = t a F(x)ds, which is the classical fuzzy integral operator.
It is said that F is Caputo H-differentiable of order 1 < q 2 at t 1 , if there exists an element c 0 D q t F(t 1 ) ∈ E 1 such that the following fuzzy equalities are valid: Here, we use only the first two cases [23]. These derivatives are trivial because they reduce to crisp elements. Regarding other fuzzy cases, the reader is referred to [23]. Furthermore, regarding this simplicity, a fuzzy-valued function F is called c [(i)-GH]-differentiable or c [(ii)-GH]differentiable if it is differentiable according to concept (i) or to (ii) of Definition 2, respectively.
The Mittag-Leffler and fractional hyperbolic functions frequently occur in solutions to fractional systems; see, e.g., [16,23]. The Mittag-Leffler functions in the form of a single and a double parameter are defined by, respectively, Some properties of these functions can be found in [31][32][33].

Existence and Uniqueness Results
In this part, existence and uniqueness of solutions to the Cauchy problem in (1) and ( 2) are discussed. We can start with the lemma given below.

Lemma 4 ([16]
). When λ > 1, the c [(i)-GH]-differentiable solution to problem (1) is given by when λ < 1, the c [(ii)-GH]-differentiable solution to problem (1) is given by (1) is given by (1) is given by (1) is given by By applying Lemma 4 and Remark 4 with f (t, u(t)) ⊕ B(t)C(t) instead of f (t), it follows that the Cauchy problem in (1) and (2) possesses an integral version. In case λ 1 and the function t → u(t), t ∈ [0, T] is assumed to be c [(i)-GH]-differentiable, then the function u satisfies In case λ 1 and the function t → u(t) is supposed to be c [(ii)-GH]-differentiable, then the function u satisfies In case λ < 1 and the function t → u(t) is c [(i)-GH]-differentiable, then the function u satisfies In case λ > 1 and the function t → u(t) is c [(ii)-GH]-differentiable, then the function u satisfies We should formulate the basic assumptions before initiating our main work: is valid and such that λ ∈ R is satisfied; Proof. Let the operator P 1 : (1) and (2) if and only if u = P 1 u. Let u and v belong to E 1 . From the above Lemmas 1 and 2 we infer for u, v ∈ E 1 , and for all t ∈ [0, T], which means that Thus, the Banach contraction mapping (BCM) principle shows the operator P 1 has a unique fixed point u * ∈ C E [0, T]. It represents the unique c [(i)-GH]-differentiable solution to the Cauchy problem (1) and (2).

Theorem 2.
Let λ 1 and suppose the conditions (H 1 )-(H 3 ) are satisfied. Assume that (H 4 ) for any t ∈ (0, T], is non-increasing in α, and for any q ∈ [1, 2] and t ∈ (0, T] Then, the Cauchy problem (1) and (2) has a unique c [(ii) Proof. Let the operator P 2 : For condition (H 4 ) and [35], we know that (1) and (2) if and only if u = P 2 u. Let u and v belong to C E [0, T]. From the above Lemmas 1 and 2 and Remark 2 we infer for u, v ∈ E 1 and for all t ∈ [0, T], which means that Thus, the Banach contraction mapping (BCM) principle shows the operator P 2 has a unique fixed point u * ∈ C E [0, T]. It represents the unique c [(ii)-GH]-differentiable solution to the Cauchy problem (1) and (2). Now, the proof is completed.  (1) and (2) Proof. Let the operator P 3 : . It is not difficult to see that u is a c [(i)-GH]-differentiable solution for Cauchy problem (1) and (2) if and only if u = P 3 u. Let u and v belong to C E [0, T]. From the above Lemmas 1 and 2 and Remarks 2 and 3 we deduce For u, v ∈ E 1 and for all t ∈ [0, T], which signifies as Thus, the Banach contraction mapping (BCM) principle shows the operator P 3 has a unique fixed point u * ∈ C E [0, T]. It represents the unique c [(i)-GH]-differentiable solution to the Cauchy problem (1) and (2). Now the proof is done.

Stability Results
In various studies, E α -Ulam-type stability approaches regarding fractional differential equations [23] and Ulam-type stability approaches regarding fuzzy differential equations [24,25] were established. Afterward, Yupin Wang and Shurong Sun worked on E q -Ulam-type stability concepts regarding fuzzy fractional differential equation where q ∈ (0, 1]. We offer some new E q -Ulam-type stability concepts regarding fuzzy fractional differential equation where q ∈ (1, 2]. Assume that γ > 0 is a constant and that t → ζ(t), t ∈ [0, T] is a positive continuous function. In addition, suppose that t → u(t), t ∈ [0, T] is a continuous function that solves the equation in (1) and consider the following related inequalities: where t ∈ [0, T]. (1) is called E q -Ulam-Hyers stable in case there exist a finite constant c > 1 and a function v ∈ C E [0, T] that satisfies the equation in (1) such that for all γ > 1 and for all solutions u ∈ C E [0, T] of Equation (1) that satisfy the inequality in (3), the following inequality is valid:  (1) is called E q -Ulam-Hyers-Rassias stable in case with respect to ζ, when there exist c ζ > 1 and a function v ∈ C E [0, T] that satisfies the equation in (1) such that for all γ > 1 and for all solutions u ∈ C E [0, T] of the equation in (1) that satisfy the inequality in (5), the following inequality is valid: (1) is called E q -Ulam-Hyers-Rassias stable in case with respect to ζ if there exist c ζ > 1 and a function v ∈ C E [0, T] that satisfies the equation in (1) such that for all γ > 1 and for all solutions u ∈ C E [0, T] of the equation in (1) that satisfy the inequality in (4), the following inequality is valid:

Definition 6. Equation
Proof. The sufficiency begins obviously, and we will only prove the necessity. From condition (H 8 ), we observe that the function t → h(t), t ∈ [0, T], defined by belongs to C E [0, T] and that h(t) belongs to E 1 for all t ∈ (0, T]. Therefore, it follows that the equation in (ii) is satisfied. Additionally, we have From the inequality (3), it then follows that D h(t),0 γ, and therefore, (i) is satisfied. This completes the proof of Lemma 5.

Remark 5.
Similar results as in Lemma 5 can be obtained by using the inequalities in (4) and (5). (1) and (2) and satisfies the inequality in (3) and is such that c 0 D q t u(0) = u 0 . Let the condition in (H 8 ) be satisfied. Then, for every t ∈ [0, T], the function u(t) satisfies the inequality

Lemma 6. Let u(t) be a c [(i)-GH]-differentiable function that solves the Cauchy equation in
when λ < 1, and t ∈ [0, T]. Here, the functions G 1 ( f , t) and G 2 ( f , t) are defined by Proof. Since the function u ∈ C E [0, T] is a solution to the Cauchy problem (1) and (2), we infer Now, regarding clarity, the proof can be divided into two cases.
Suppose λ > 1. Then, we write Observing that u is a c [(i)-GH]-differentiable solution of Equation (6), then Lemma 4 with f (t, u(t)) + B(t)C(t) instead of f (t) shows the equality Now, it follows that When λ < 1, we denote It should be observed that u(t) is a c [(ii)-GH]-differentiable solution of Equation (6) that obeys the inequality in (3). An application of Lemma 5 then yields

Now, it follows that
Now, the proof is completed. (1) and (2) and satisfies the inequality (3) and is such that c 0 D q t u(0) = u 0 . Let the condition in (H 8 ) be satisfied. Then, for every t ∈ [0, T] the function u(t) satisfies the integral inequality

Lemma 7. Let u(t) be a c [(ii)-GH]-differentiable function that solves the Cauchy equation in
Proof. Now, regarding clarity, the proof can be divided into two cases. Case 1.
When λ < 1, observe that u is a c [(ii)-GH]-differentiable solution of Equation (5), then the Lemma 5 with f (x, u(x)) + B(x)C(x) instead of f (t) shows the equality

Remark 6.
We can obtain similar results to those in Lemmas 6 and 7 for inequalities (3) and (4).

Remark 7.
In view of Definition 6 can be verified as according to the assumption in Theorems 5-8, we assume Equation (1) and inequality (4). It can be verified that Equation (1) is generalized E q -Ulam-Hyers-Rassias stable with respect to Definition 6.
T] when we assume q = 1. This means that certain theorems in [25] are special cases of Theorem 5 and 6 in the present paper.

Remark 9.
According to the assumptions excluding (H9) in Theorems 5-8, we consider the equation in (1) and inequality in (3). It can be proved that in terms of Definitions 3 and 4, Equation (1) is E q -Ulam-Hyers.

Examples
In this part, we will show four examples to explain our main results. Example 1. Consider the following Cauchy problem in terms of a Fuzzy fractional differential equation Compared to Equation (1), in the above equations, q = 1.5, λ = 2, T = 2π, f (t, u(t)) = 1.2u(t)cos(t) ⊕ t 2 e t F, and F = (0, 1, 2) ∈ E 1 is a symmetric triangular fuzzy number. Hence, with L = 1.3, the condition (H 1 ) and (H 2 ) are satisfied. It is not difficult to prove that condition (H 3 ) is satisfied. Hence, as a consequence of Theorem 1, the Cauchy problem (7) and (8) has a c [(i)-GH]-differentiable solution. The numerical solutions with respect to the q = 1.5 level are provided by utilizing the Adams-Moultan predictor-corrector method.

Graphical Presentation
We used the Adams-Bashforth-Moulton technique to acquire the numerical solution for this fractional differential equation for graphical representation of the solution of the problem presented in Equations (7), (9), (11) and (13). For simulation, the modified predictor-corrector scheme is used to examine the effect and contribution of the timedelayed factor. A graphical representation of the solution with different variations of the time delay factor, as well as other parameters, is made to check and demonstrate the stability of the model under consideration. We are able to see the Ulam-Hyers stability of varied accuracies and delays from the numerical data. The system will attain Ulam-Hyers stability more quickly with greater accuracy. This is also true when the number of delays increases. Figures 1-4 show the stability of the system (7), (9), (11) and (13)

Conclusions
This paper aims to define the uniqueness and existence of a group of nonlinear fuzzy fractional differential equation of solutions to the Cauchy problem. Moreover, E q -Ulam-type stability of Equation (1) is observed by applying the inequality technique. We obtain uniqueness and existence results with the help of nonlocal conditions of the Caputo derivative. Moreover, future work may include broadening the idea indicated in this task and familiarizing observability, and generalize other tasks. Ulam-type stability of fuzzy fractional differential equations, similar to crisp situations for approximate solutions, provides a reliable theoretical basis. This a fruitful area with wide research projects, and it can bring about countless applications and theories. We have decided to devote much attention to this area. Furthermore, it is fruitful to investigate stability problems in a classical sense for the fuzzy fractional differential equation.