Fuzzy Fractional Differential Equation Involving the Fuzzy Conformable Derivative and the α − Semigroups

In this work, we discuss solutions to the abstract Cauchy problem of fuzzy conformal fractional. In addition, the method of fuzzy fractional semigroups is used to obtain analytical solutions to the fractional differential equation. We use the concept of Krasnoselskii's fixed point theorem to determine the existence and uniqueness of the solution. An application is also given to illustrate our main abstract results.


Introduction
Fuzzy fractional diferential equations (FFDEs) have proved to be one of the most efective tools for modelling many phenomena in various felds of physics, mechanics, chemistry, and engineering.Tey have a large number of applications in the nonlinear oscillation of earthquakes, in many physical phenomena such as seepage fow in porous media, and in fuid dynamic circulation modelling.Tey have a large number of applications in the nonlinear oscillation of earthquakes, in many physical phenomena such as seepage fow in porous media, and in fuid dynamic circulation models.For more details on this theory and its applications, we refer to [1-7] [31-33].
Recently, Khalil et al. in [8] introduced a new diferential operator, called the conformable derivative.Tis concept of conformal fractional calculus is a relatively new area of research which generalises the classical fractional calculus of noninteger degrees.Tis derivative of fractional order satisfes many well-known properties of the integer derivative, including linearity, the product rule, and the division rule.In addition, Rolle's theorem and the mean value theorem can also be applied, see [8].One of the topics covered in this area concerns fractional semigroups, a type of operator semigroup whose generator is the conformable derivative.In 2015, Abdeljawad in [9] presented detailed results for the conformable derivative.In [10], Horani et al. describe fractional semigroups of operators and they further studied fractional abstract Cauchy problems with conformable derivatives introduced in [8].Khalil et al. in [11] presented a geometric meaning of the conformable derivative through fractional sequences based on the conformable derivative.Abbeljawad et al. introduced the so-called C 0 -α-semigroup T(τ) { } τ≥0 , which is a generalization of the classical strong continuous semigroup with its infnitesimal generator.Moreover, in [12], the authors introduced fuzzy fractional semigroups of operators associated with the congruent fuzzy fractional derivative.Numerous research papers using the congruent derivative have been published in recent years (see [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]).
Te purpose of this work is to study the existence and uniqueness of fuzzy abstract Cauchy problems involving the fuzzy conformable derivatives using the method of fuzzy fractional semigroups.
We consider the following abstract fuzzy fractional conformal Cauchy problem: where Ψ (α) (τ) denotes the fuzzy conformable derivative, Tis paper is organized as follows: in Section 2, we provide all the necessary tools for fuzzy conformal fractional calculus and fuzzy fractional semigroups.In Section 3, we study the existence and uniqueness of the solution to problems (1) and (2).In Section 4, we give applications illustrating our abstract results.Finally, the article ends with a conclusion.
Let X � C([a, b], R F ) be the space of continuous [a, b] valued functions in R F and L(X, X) be the space of all bounded operators on X.

Preliminaries
Tis section introduces the main properties of fuzzy number space and some other spaces based on it.
Defnition 1 (see [28]).Te space of fuzzy numbers denoted by R F is defned as the class of fuzzy subsets of the real axis R, i.e., of u: R ⟶ [0, 1], having the following four properties: (i) Normalization: sup x∈R u(x) � 1.
(ii) Convexity: for all x, y ∈ R and all λ ∈ [0, 1], u(λx + (1 − λ)y) ≥ min(u(x), u(y)).(iii) Upper semicontinuity: for all x ∈ R, the set For all r ∈ (0, 1], the r-level of an element of R F is defned by We can write in interval form as follows: (i) Te addition of two fuzzy numbers is defned by (ii) And multiplication by a scalar is defned by Under r-level, we get for all r ∈ [0, 1], (iii) Te subtraction in R F is called the H-diference and is noted ⊖ defned as follows: Clearly, ω ⊖ ϑ does not always exist.
We also defne the generalized Hukuhara (gH) diference noted by ⊖ gH of two fuzzy numbers as follows: For all, ω, ϑ ∈ R F .
Defnition 2 (see [28]).Te distance between two elements of R F is given by verifying the following properties, for all ω, ξ, κ, ϑ ∈ R F .
In what follows, we introduce the following Hukuhara's derivative.
We say that Remark 5. Clearly, Hukuhara diferentiability implies generalized diferentiability but not vice versa.

Conformable Derivative and Fuzzy α− Semigroup of
Operator.Te conformable derivative is a relatively new concept in fractional calculus.It is based on the basic limit defnition of the derivative and difers from other fractional derivatives, such as the Riemann fractional derivative, in some key properties [9].Te conformable derivative has been studied in various contexts, including ordinary differential equations, complex-valued functions, and commutative algebras [9,30].
Te conformable derivative has been extended to arbitrary time scales, leading to the development of a new kind of conformable derivative on arbitrary time scales [5].It has also been defned in fnite-dimensional commutative associative algebras, demonstrating its applicability in various mathematical structures [6].
Here are some key properties and defnitions of the conformable derivative: exists, then we defne the fuzzy conformable derivative of φ of order ϑ from where T ϑ (φ)(τ) can also be represented by φ (ϑ) (τ).If the fuzzy conformable derivative of φ of order ϑ exists, then we simply say φ is ϑ gh -diferentiable , then φ is a fuzzy continuous function at τ 0 .
Proof.Indeed, we have Passing the limit, we get lim Terefore, the function φ is a fuzzy continuous function.

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Remark 9 (i) Te conformable derivative of a constant function is zero, which is not the case for Riemann fractional derivatives [9].(ii) Te fuzzy conformable derivation satisfes all the classical properties of the derivation.
Below we give some basic defnitions of α− semigroups.Let X � C([a, b], R F ) be the space of continuous [a, b] valued functions in R F and L(X, X) is the space of all bounded operators on X.
In the following, we defne T: [a, b] ⟶ L(X, X) by It is clear that T is well a fuzzy α− semigroup.Indeed, what needed to be shown.

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Theorem 17 (Krasnoselskii's fxed point theorem).Let M be a closed convex and nonempty subset of a Banach space X.
Let A, B be two operators such that (i) Ax + By ∈ M whenever x, y ∈ M, (ii) A is compact and continuous, (iii) B is a contraction mapping.
Ten, there exists z ∈ M such that z � Az + Bz.

Main Results
In this section, we show how our theory can be applied to provide solutions to specifc problems.In particular, we would like to use the fractional semigroup concept to solve fuzzy diferential problems.
Indeed, given that we have where Hence, To conclude, we have, for τ ≥ 0, Tus, Tis completes the proof.
In the following, let us show the existence and uniqueness of the solution.
Proof.Consider a closed convex subset that is not empty where with Defne the operators T 1 and T 2 on B k by Tus, Ψ being a fxed point of the operator TΨ � T 1 Ψ + T 2 Ψ is a solution of equations ( 37) and (38).
In the frst step, we prove that T maps B k into B k , i.e., for any Ψ, ϕ ∈ B k .
We have to show that T 1 Ψ + T 2 ϕ ∈ B k : Hence, T 1 is continuous.Now, we show that T 1 (B k ) resides in a relatively compact set.Taking τ 1 ≤ τ 2 ≤ T, we have As τ 1 ⟶ τ 2 , we get T 1 Ψ(τ 1 ) ⟶ T 1 Ψ(τ 2 ).Hence, T 1 (B k ) resides in a relatively compact.Now, we show that T 2 is contraction.Letting Ψ, ϕ ∈ B k , we have For λ ≤ ‖A‖b α− 1 , then T 2 is contraction.Hence, according to Teorem 17, T has a fxed point in B k which is a solution of equations ( 37) and (38).

. Application to Fuzzy Differential Equation
In this section, we apply the main results of the previous sections to solve fuzzy fractional diferential equations.We consider the following fuzzy fractional partial diferential equation with initial conditions.
where ψ (α) means the fuzzy conformable derivative of ψ with respect to τ.
Let us put Ten, problem (68) can be written as where is defned by Clearly, A is a linear operator and continuous at each Ψ.Ten, according to Teorem 19, the solution of (68) is written as where We will calculate exp(1/ατ α ⊙ A).In this case, we get that Tus, (78) As τ ⊙ sin(ψ(τ, x)) is Lipschitz with respect to τ and satisfes the hypothesis of Teorem 20, then ψ(τ, x) and ϕ(τ, x) are unique.

Conclusion
In this study, the initial problem of fuzzy conformal orders is discussed in the context of conformable generalized Hukuhara diferentiability.Fuzzy conformable derivatives based on generalized Hukuhara diferentiability are introduced, and many relevant properties of this topic are shown [27].Terefore, the fuzzy semifractional group method is used to determine the analytical solution of the conformable fractional diferential equation.To ensure the existence and uniqueness of the solution, we use Krasnoselskii's fxed point theorem.Finally, the application of abstract Cauchy problems is highlighted to demonstrate the efectiveness and efciency of these methods.Te results show that the conformable fuzzy fractional-order semigroup method is an efective and practical tool for solving conformable fuzzy fractional-order diferential equations.We obtain interesting results here that can be used in future studies of fuzzy fractional partial diferential equations under conformable gH diferentiability.

□ 3 . 2 .
Existence and Uniqueness of the SolutionTheorem 20.Suppose a bounded continuous function g satisfes the following condition: