On the Existence and Uniqueness Results for Fuzzy Linear and Semilinear Fractional Evolution Equations Involving Caputo Fractional Derivative

In this manuscript, we establish new existence and uniqueness results for fuzzy linear and semilinear fractional evolution equations involving Caputo fractional derivative. The existence theorems are proved by using fuzzy fractional calculus, Picard’s iteration method, and Banach contraction principle. As application, we conclude this paper by giving an illustrative example to demonstrate the applicability of the obtained results.


Introduction
Fuzzy fractional calculus and fuzzy fractional differential equations are a natural way to model dynamical systems subject to uncertainties. In the past few years, many works have been done by several authors in the theory of fuzzy fractional differential equations (see [1][2][3]). This theory has been proposed to handle uncertainty due to incomplete information that appears in many mathematical or computer models of some deterministic real-world phenomena. Recently, fractional differential equations have attracted a considerable interest both in mathematics and in applications such as material theory, transport processes, fluid flow phenomena, earthquakes, solute transport, chemistry, wave propagation, signal theory, biology, electromagnetic theory, thermodynamics, mechanics, geology, astrophysics, economics heat conduction in materials with memory, and control theory (see basic books and interesting papers in [4][5][6][7][8]).
In many cases, when a real physical phenomenon is modelled by a fractional initial value problem, we cannot usually be sure that the model is perfect. For example, the initial value of this problem may not be known precisely. In order to get a perfect model with a precise initial condition, Agarwal et al. in [9] proposed the concept of solutions for fuzzy fractional differential equations. Arshad and Lupulescu in [10] proved some results on the existence and uniqueness of solution for the fuzzy fractional differential equations under Hukuhara fractional Riemann-Liouville differentiability. Later, Alikhani and Bahrami in [11] have proved the existence and uniqueness results for nonlinear fuzzy fractional integral and integrodifferential equations by using the method of upper and lower solutions. The authors in [12,13] discussed the concepts about generalized Hukuhara fractional Riemann-Liouville and Caputo differentiability of fuzzy-valued functions, and the equivalence between fuzzy fractional differential equation and fuzzy fractional integral equation is discussed in [14]. For many basic works related to the theory of fractional differential equations and fuzzy fractional differential equations, we refer the readers to the articles [15][16][17][18][19][20] and references therein.
Motivated by the above works, in the present paper, we study the existence result of solution for the following fuzzy linear fractional evolution equation: and for the following fuzzy semilinear fractional evolution equation: is the fuzzy Caputo derivative of xðtÞ at order 0 < q < 1, t 0 ≥ 0, T > 0, AðtÞ is a bounded linear operator, and f is a fuzzy continuous function.
The paper is organized as follows. In Section 2, we give some basic properties of fuzzy sets, operations of fuzzy numbers, and some detailed definitions of fuzzy fractional integral and fuzzy fractional derivative which will be used in the rest of this paper. In Section 3, we introduce the existence and uniqueness results of solution for the fuzzy linear fractional evolution equation (1). In Section 4, we discussed the existence and uniqueness results for the fuzzy semilinear fractional evolution equation (2). An illustrative example is presented in Section 5 followed by conclusion and future work in Section 6.

Preliminaries
In this section, we will briefly give some of notations, definitions, and results from the literature of fuzzy set theory and fuzzy fractional calculus which will be used in the rest of this paper.
Definition 1 (see [21]). A fuzzy number is mapping u : R ⟶ ½0, 1 such that (1) u is upper semicontinuous (2) u is normal; that is, there exist x 0 ∈ R such that u ðx 0 Þ = 1 (3) u is fuzzy convex, that is, uðλx + ð1 − λÞyÞ ≥ min fuðxÞ, uðyÞg for all x, y ∈ R and λ ∈ ½0, 1 The α − cut of a fuzzy number u is defined as follows: Moreover, we also can present the α − cut of fuzzy number u by ½u α = ½u l ðαÞ, u r ðαÞ: Example 1. Let u be a fuzzy number defined by the following function: If α = 1, then the α − cut of the fuzzy number u is given by ½u α = ½u 1 = f2g.

Notations.
(i) We denote by E 1 the collection of all fuzzy numbers (ii) We denote by CðJ, E 1 Þ the space of all fuzzy-valued functions which are continuous on J (iii) We denote by P c ðRÞ the set of all bounded and closed intervals of R (iv) We denote also by 0 E 1 the fuzzy zero defined by Definition 2 (see [14]). Let α ∈ ½0, 1 and u ∈ E 1 such that ½u α = ½u l ðαÞ, u r ðαÞ: We define the diameter of α − level set ½u α of the fuzzy set u as follows: Definition 3 (see [14]). The generalized Hukuhara difference of two fuzzy numbers u, v ∈ E 1 is defined as follows: Property 4 (see [22]). If u ∈ E 1 and v ∈ E 1 , then the following properties hold: (1) If u ⊖ gH v exists then it is unique Definition 5 (see [21]). According to Zadeh's extension principle, the addition on E 1 is defined by And scalar multiplication of a fuzzy number is given by Remark 6 (see [23]). Let u, v ∈ E 1 and α ∈ ½0, 1; then, we have 2 Journal of Function Spaces Definition 7 (see [24]). Let u, v ∈ E 1 and α ∈ ½0, 1; then, the Hausdorf distance between u and v is given by Proposition 8 (see [25]). D is a metric on E 1 and has the following properties: (1) ðE 1 ; DÞ is a complete metric space Definition 10 (see [26]). Let f : J ⟶ E 1 and t 0 ∈ J. We say that f is Hukuhara differentiable at t 0 if there exists f ′ðt 0 Þ ∈ E 1 such that Definition 11 (see [25]).
Proposition 13 (see [25]). Let F : J ⟶ E 1 be a fuzzy function. If F is strongly measurable and integrably bounded, then it is integrable.

Fuzzy Linear Fractional Evolution Equation
Definition 20 (see [14]). A fuzzy function x : J ⟶ E 1 is calledd-increasing(d-decreasing) on J if for every α ∈ ½0, 1, the real function t ⟶ dð½xðtÞ α Þ is nondecreasing (nonincreasing), respectively. (2) xðtÞ satisfies the following integral equation Proof. To show that problem (1) has a unique solution defined on J, we use Picard's iteration method (see [27]). Let and let T : CðJ, E 1 Þ ⟶ CðJ, E 1 Þ be the operator defined as follows: Let x, y ∈ CðJ, E 1 Þ: then, we have and by induction, we can write

Illustrative Example
In this section, we give an example to illustrate the practical usefulness of the results that we establish in the paper. By using Theorem 27, we can solve the following fuzzy semilinear fractional evolution equation: We define the operator A by.
where I is the identity mapping defined on E 1 .
And the function f is given by f ðt, uÞ = ðe −t /ð9 + e t ÞÞuðtÞ .
It is clear that AðtÞ is a bounded linear operator and the function t ⟶ AðtÞ is continuous. Hence, the assumptions H 1 and H 2 are verified.
On the other hand, we have Hence, the assumption H 3 in Theorem 27 holds with K = 1/10.
For this purpose, we have Finally, all the conditions of Theorem 27 are satisfied for problem (40); then, it has a unique solution on [0,1].
Remark 28. We can show that the fuzzy semilinear fractional evolution equation (40) has a unique solution on [0,1] for some fractional-order 0 < β < 1. Indeed, we shall check that condition (32) in Theorem 27 is satisfied for some β ∈ 0, 1½. We have Then, by using Theorem 27, the fuzzy semilinear fractional evolution equation (40) has a unique solution on [0,1] for the values of β satisfying (44).

Conclusion and Future Work
In this manuscript, we studied the existence and uniqueness results of fuzzy linear and semilinear fractional evolution equations involving Caputo fractional derivative. The existence theorems are proved by using fuzzy fractional calculus, Picard's iteration method, and Banach fixed-point theorem. As application, we present an illustrative example to show the applicability of our main results.
Our future work is to extend the obtained results in this paper to the infinite dimensional case by using the theory of fuzzy operator semigroups.

Data Availability
The data used to support the findings of this study are included in the references within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.