Research Article Fuzzy Generalized Conformable Fractional Derivative

We give a new deﬁnition of fuzzy fractional derivative called fuzzy conformable fractional derivative. Using this deﬁnition, we prove some results and we introduce new deﬁnition of generalized fuzzy conformable fractional derivative


Introduction
Fuzzy set theory is a powerful tool for modeling uncertainty and for processing vague or subjective information in mathematical models. eir main directions of development have been diversed, and its applications have been varied [1][2][3][4]. e derivative for fuzzy valued mappings was developed by Puri and Ralescu [5], which generalized and extended the concept of Hukuhara differentiability for set-valued mappings to the class of fuzzy mappings. Subsequently, using the H-derivative, Kaleva [6] started to develop a theory for FDE. In [7], a new well-behaved simple fractional derivative called "the conformable fractional derivative" depending just on the basic limit definition of the derivative, namely, for a function f(0, ∞) ⟶ R the (conformable) fractional derivative of order 0 < q ≤ 1 of f at t > 0 was defined by and is defined the fractional derivative at 0 as (T q f)(0) � lim t⟶0 + (T q f)(t). e aim of this paper is to study and generalize the fuzzy conformable fractional derivative.

Preliminaries
Let us denote by R F � u : R ⟶ [0, 1] { } the class of fuzzy subsets of the real axis satisfying the following properties: (i) u is normal, i.e., there exists an x 0 ∈ R such that u(x 0 ) � 1. (ii) u is fuzzy convex, i.e., for x, y ∈ R and 0 < λ ≤ 1: (2) en, R F is called the space of fuzzy numbers. Obviously, By P K (R) we denote the family of all nonempty compact convex subsets of R and define the addition and scalar multiplication in P K (R) as usual.
Theorem 1 (see [8]). If u ∈ R F , then converges to α, then  [9]). e following arithmetic operations on fuzzy numbers are well known and frequently used below. If u, v ∈ R F , then Theorem 2 (see [10]) en, 0 ∈ R F be a neutral element with respect to +, i.e., where d H is the Hausdorff metric: It is well known that (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.
Let (A k ) be a sequence in P K (R) converging to A. en, theorem in [6] gives us an expression for the limit.
Let I � (0, a) ⊂ R be an interval. We denote by C(I, R F ) that the space of all continuous fuzzy functions on I is a complete metric space with respect to the metric:

The Fuzzy Conformable Fractional Differentiability
Definition 2. Let F : I ⟶ R F be a fuzzy function. qth order "fuzzy conformable fractional derivative" of F is defined by for all t > 0 and q Hence, If F is q-differentiable in some (0, a) and lim and the limits exist (in the metric d).
Remark 1. From the definition, it directly follows that if F is q-differentiable then the multivalued mapping F α is q-differentiable for all α ∈ [0, 1] and where for all 0 ≤ ε < δ and α ∈ [0, 1]. en, F is q-differentiable, and the derivative is given by (14).
Proof. Consider the family T q F α | α ∈ [0, 1] By definition T q F α (t) is a compact, convex, and nonempty subset of R.
If α 1 ≤ α 2 , then by assumption (i), and consequently Let α > 0 and α k be a nondecreasing sequence converging to α. For h > 0 choose ε > 0 such that equation (15) holds true. en, the triangle inequality yields By assumption (i), the rightmost term goes to zero as k ⟶ ∞ and hence

Advances in Fuzzy Systems
Dividing by ε, we have Similarly, we obtain and passing to the limit gives the theorem. Note that this definition and theorem of conformable fractional derivative are very restrictive; for instance, if where c is a fuzzy number and g : [a, b] ⟶ R + is a function and is q-differentiable for some q ∈ (0, 1] with g (q) (t) < 0, then F is not q-differentiable. To avoid this difficulty, we introduce a more general definition of the conformable fractional derivative for fuzzynumber-valued function.

The Generalized Fuzzy Conformable Fractional Differentiability
We consider the following definition.
Definition 3. Let F : I ⟶ R F be a fuzzy function and q ∈ (0, 1]. One says, F is q (1) -differentiable at point t > 0 if there exists an element F (q) (t) ∈ R F such that for all ε > 0 sufficiently near to 0 there exist F(t + εt 1− q ) ⊖ F(t), F(t) ⊖ F(t − εt 1− q ), and the limits (in the metric d): where F is q (2) -differentiable at t > 0 if for all ε < 0 sufficiently near to 0, then there exist If F is q (n) -differentiable at t > 0, we denote its q-derivatives (q ∈ (0, 1]) by F (q) n (t), for n � 1, 2.
Example 1. Let g : I ⟶ R + and define F : for all t ∈ I, where c is the fuzzy number. If g is q-differentiable at t 0 ∈ I, then F is the generalized fuzzy conformable fractional derivative at t 0 ∈ I and we have

Remark 2.
In the previous definition, q (1) -differentiable corresponds to Definition 3, so this differentiability concept is a generalization of Definition 2 and obviously more general. For instance, in the previous example, for (ii) If F is q (2) -differentiable, then f α 1 (t) and f α 2 (t) are q-differentiable and F q (2) ( Proof (i) See demonstration of eorem 5.

Conclusion
We have investigated generalized fuzzy conformable fractional differentiability. e conformable q-differentiability introduced here is a very general differentiability concept, being also practically applicable, and we can calculate by the fuzzy conformable derivative of the product of two functions (T q (f · g)) because all fractional derivatives do not satisfy the known formula T q (f · g) � T q (f)g + fT q (g). e disadvantage of fuzzy generalized conformable differentiability of a function seems to be that a simple fuzzy differential equation (y (q) + y � 0, 0 < q ≤ 1, y(0) � y 0 ∈ R F ) has not got a unique solution, so it may have several solutions. e advantage of the existence of these solutions is that we can choose the solution that reflects better the behaviour of the modelled real-world system.
For further research we propose the study for fuzzy fractional differential equations, by using the generalized conformable differentiability concept. In addition, we propose to extend the results of the present paper and to combine them with the results in [15] for fuzzy fractional differential equations.

Data Availability
No data were used to support this study.

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