Fuzzy Conformable Fractional Semigroups of Operators

In this paper, we introduce a fuzzy fractional semigroup of operators whose generator will be the fuzzy fractional derivative of the fuzzy semigroup at 
 
 t
 =
 0
 
 . We establish some of their proprieties and some results about the solution of fuzzy fractional Cauchy problem.


Introduction
Fractional semigroups are related to the problem of fractional powers of operators initiated first by Bochner [1]. Balakrishnan [2] studied the problem of fractional powers of closed operators and the semigroups generated by them. e fractional Cauchy problem associated with a Feller semigroup was studied by Popescu [3]. Abdeljawad et al. [4] studied the fractional semigroup of operators. e semigroup generated by linear operators of a fuzzy-valued function was introduced by Gal and Gal [5]. Kaleva [6] introduced a nonlinear semigroup generated by a nonlinear function.In the last few decades, fractional differentiation has been used by applied scientists for solving several fractional differential equations and they proved that the fractional calculus is very useful in several fields of applications and real-life problems such as, but certainly not limited, in physics (quantum mechanics, thermodynamics, and solid-state physics), chemistry, theoretical biology and ecology, economics, engineering, signal and image processing, electric control theory, viscoelasticity, fiber optics, stochastic-based, finance, tortoise walk, Baggs and Freedman model, normal distribution kernel, time-fractional nonlinear dispersive PDEs, fractional multipantograph system, time-fractional generalized Fisher equation and time-fractional k(m, n) equation, and nonlinear time-fractional Schrodinger equations [7][8][9][10][11][12][13][14]. e concept of fuzzy fractional derivative was introduced by [15] and developed by [16][17][18][19], but these researchers tried to put a definition of a fuzzy fractional derivative. Most of them used an integral from the fuzzy fractional derivative. Two of which are the most popular ones, Riemann-Liouville definition and Caputo definition. All definitions mentioned above satisfy the property that the fuzzy fractional derivative is linear. is is the only property inherited from the first fuzzy derivative by all of the definitions. e obtained fractional derivatives in this calculus seemed complicated and lost some of the basic properties that usual derivatives have such as the product rule and the chain rule. However, the semigroup properties of these fractional operators behave well in some cases. Recently, Harir et al. [20] defined a new well-behaved simple fractional derivative called "the fuzzy conformable fractional derivative" depending just on the basic limit definition of the derivative. ey proved the product rule and the fractional mean value theorem and solved some (conformable) fractional differential equations [18].
Here, we introduce the fuzzy fractional semigroups of operators associated with the fuzzy conformable fractional derivative, for proving to be a very fruitful tool to solve differential equations. en, we show that this semigroup is a solution to the fuzzy fractional Cauchy problem x (q) (t) � f(x(t)), x(0) � x 0 , and q ∈ (0, 1] according to the fuzzy conformable fractional derivative which was introduced in [20].

Preliminaries
Let us denote by { } the class of fuzzy subsets of the real axis satisfying the following properties [21]: (i) u is normal, i.e., there exists an x 0 ∈ R such that u(x 0 ) � 1, (ii) u is the fuzzy convex, i.e., for x, y ∈ R and 0 < λ ≤ 1, (1) Here, P K (R) denotes the family of all nonempty compact convex subsets of R and defines the addition and scalar multiplication in P K (R) as usual.
Lemma 1 (see [22]). Let u, v: e following arithmetic operations on fuzzy numbers are well known and frequently used below. If u, v ∈ R F , then Let us define d: where d H is the Hausdorff metric defined in P K (R).
Theorem 1 (see [23]). (R F , d) is a complete metric space. We list the following properties of d(u, v): for all u, v, w ∈ R F and λ ∈ R.
Theorem 2 (see [24]). ere exists a real Banach space X such that R F can be the embedded as a convex cone C with vertex 0 in X. Furthermore, the following conditions hold true:

Fuzzy q-Semigroup of Operators
Definition 1 (see [20]). Let F: (0, a) ⟶ R F be a fuzzy function. q th order "fuzzy conformable fractional derivative" of F is defined by (where the limit is taken in the metric space (R F , d)).
where the integral is the usual Riemann improper integral.

x⊖x(T(t)x⊖x) exists and we have
en, Consequently, We will write A for such generator.

Lemma 2. Let
A is the operator of the fuzzy q-semigroup T(t) { } t≥0 on R F if and only if A 1 is the operator of the q-semigroup T 1 (t) t≥0 defining on the convex closed set C and T 1 � jT(t)j − 1 .
By using Definition 5, the proof is similar to the proof of Lemma 5 in [18] and is omitted.
Proof. Let q ∈ (0, 1] and x ∈ R F , for t ≥ 0, and we have
en, T(t) is obviously a c 0 -q-semigroup of contraction on R F .

Remark 2.
If M � 1 and w � 0 in Definition 4, we say that For q ∈ (0, 1], T(0) � I and T(t)f ∈ R F whenever f ∈ R F and that

Fuzzy Fractional Cauchy Problems
Let F: R F ⟶ R F be continuous and consider the fractional initial value problem where q ∈ (0, 1). It is well known that instead of the differential equation (24), it is possible to study an equivalent fractional integral equation.
A solution x(t) of equation (24) is independent of the initial time t 0 . In fact, let k 0 < a and denote y(t) � x(k 0 + (1/q)t q ). en, and y(t 0 ) � x(k 0 + (1/q)t q 0 ) � y 0 . Hence, y(t) and x(t) are solutions of the same fractional differential equation with a different initial value. q ∈ (0, 1).

Theorem 4. Let
If x(t) is a solution to the fuzzy fractional initial value problem, Proof. Let q ∈ (0, 1) and k > 0. As obtained above, y(t) � x(k + (1/q)t q ) is a solution of the fractional initial value problem y (q) (t) � F(y(t)), y(0) � x(k). Hence, We set k � (1/q)s, then and Being a solution to a differential equation, Suppose that a fuzzy semigroup for all x ∈ R F . en, T(t)(x 0 ) is a solution to the fractional initial value problem where F(x(t)) � T (q) (0)(x 0 ).
Proof. By the q-semigroup property and using proof of eorem 3, we obtain and Finally, we show that the fuzzy exponential function is a generalization of the fuzzy semigroup introduced in [5]. □ Theorem 6. If A: R F ⟶ R F is a bounded linear operator, then the fuzzy exponential function has a power series representation e t q /q ( )A (x) � ∞ k�0 t kq q k k! A k x, t ≥ 0.
Proof. Let A: R F ⟶ R F be a bounded linear operator as defined by Gal and Gal in [5]. en, and hence by [6] satisfies the condition. Consequently, is a solution to the Cauchy problem x (q) (t) � Ax(t), x(0) � x 0 . Define S(t) by a power series as Now, by eorem 3.9 in [5], (pose (s q /q) � t with e (s q /q)A and S(s)) in [5], so S(t) is a International Journal of Differential Equations 5 fuzzy semigroup, and hence by eorem 5, S(t)(x 0 ) is a solution to the problem Since a bounded linear operator is Lipschitzian, it follows by eorem 6.1 in [25] that the problem x (q) (t) � Ax(t), x(0) � x 0 , has a unique solution. Hence, e (t q /q)A (x 0 ) � S(t)(x 0 ), for all x 0 ∈ R F .

Data Availability
No data were used to support this study.

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