Fuzzy Generalized Conformable Fractional Derivative

Abstract

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.

1. Introduction

Fuzzy set theory is a powerful tool for modeling uncertainty and for processing vague or subjective information in mathematical models. Their main directions of development have been diversed, and its applications have been varied [1–4].

The 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 the (conformable) fractional derivative of order of f at was defined byand is defined the fractional derivative at 0 as . The aim of this paper is to study and generalize the fuzzy conformable fractional derivative.

2. Preliminaries

Let us denote by the class of fuzzy subsets of the real axis satisfying the following properties:(i)u is normal, i.e., there exists an such that .(ii)u is fuzzy convex, i.e., for and :(iii)u is upper semicontinuous.(iv) is compact.

Then, is called the space of fuzzy numbers. Obviously, . For denotes , then from (i) to (iv) it follows that the α-level sets for all is a closed bounded interval which is denoted by . By we denote the family of all nonempty compact convex subsets of and define the addition and scalar multiplication in as usual.

Theorem 1 (see [8]). If , then(i) for all .(ii) for all .(iii) is a nondecreasing sequence which converges to α, thenConversely, if is a family of closed real intervals verifying (i) and (ii), then defined a fuzzy number such that and

Lemma 1. Let be the fuzzy sets.
Then, if and only if for all see ([9]).

The following arithmetic operations on fuzzy numbers are well known and frequently used below. If , then

Definition 1. Let . If there exists such that , then is called the H-difference of and it is denoted .

Theorem 2 (see [10]).  (i)Let us denoteThen, be a neutral element with respect to , i.e., :(i)With respect to , none of has opposite in .(ii)For any with or and any , we have , for general the above property does not hold.(iii)For any and any , we have .(iv)For any and any , we have .

Define by the following equation:where is the Hausdorff metric:

It is well known that is a complete metric space. We list the following properties of :for all and .

Let be a sequence in converging to A. Then, theorem in [6] gives us an expression for the limit.

Theorem 3 (see [6]). If as , thenLet be an interval. We denote by that the space of all continuous fuzzy functions on I is a complete metric space with respect to the metric:

3. The Fuzzy Conformable Fractional Differentiability

Definition 2. Let be a fuzzy function. order “fuzzy conformable fractional derivative” of F is defined byfor all and . Let stands for . Hence,If F is q-differentiable in some and ,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 is q-differentiable for all andwhere is denoted from the conformable fractional derivative of of order q. The converse result does not hold, since the existence of Hukuhara difference , does not imply the existence of H-difference .
However, for the converse result we have the following.

Theorem 4. Let satisfy the assumptions:(i)For each there exists a such that the H-difference and exists for all .(ii)The set-valued mappings are uniformly Hukuhara conformable fractional derivative of order q with derivative , i.e., for each and there exists a such thatfor all and . Then, F is q-differentiable, and the derivative is given by (14).

Proof. Consider the family By definition is a compact, convex, and nonempty subset of .
If , then by assumption ,and consequentlyLet and be a nondecreasing sequence converging to α. For choose such that equation(15) holds true. Then, the triangle inequality yieldsBy assumption , the rightmost term goes to zero as and henceNow by Theorem 3 and (18) we haveIf , then using equation in [11], we deduce as before thatwhere is a nonincreasing sequence converging to zero and consequentlyThen, from Theorem 1 it follows that there is an element such thatFurthermore, let , , and be as in (ii). Then,for all and similarly for . Hence, has the theorem.

Theorem 5. Let be q-differentiable. Denote , . Then, and are q-differentiable and

Proof. If and , we haveDividing by ε, we haveSimilarly, we obtainand 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 is a function and is q-differentiable for some with , then F is not q-differentiable. To avoid this difficulty, we introduce a more general definition of the conformable fractional derivative for fuzzy-number-valued function.

4. The Generalized Fuzzy Conformable Fractional Differentiability

We consider the following definition.

Definition 3. Let be a fuzzy function and . One says, F is -differentiable at point if there exists an element such that for all sufficiently near to 0 there exist , , and the limits (in the metric d):where F is -differentiable at if for all sufficiently near to 0, then there exist :If F is -differentiable at , we denote its q-derivatives by , for .

Example 1. Let and define by for all , where c is the fuzzy number. If is q-differentiable at , then F is the generalized fuzzy conformable fractional derivative at and we have . For instance, if , F is -differentiable. If , then F is -differentiable.

Remark 2. In the previous definition, -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 with , we have.

Theorem 6. Let be fuzzy function, where , :(i)If F is -differentiable, then and are q-differentiable and(ii)If F is -differentiable, then and are q-differentiable and

Proof.  (i)See demonstration of Theorem 5.(ii)If , , and , then we have and multiplying by , we haveSimilarly, we obtainand passing to the limit we have .

Theorem 7. Let and be two q-differentiable functions at a point (F is the generalized fuzzy conformable fractional derivative Definition 3 and :(i)If and F is -differentiable, then is -differentiable and(ii)If and F is -differentiable, then is -differentiable and

Proof. We present the details only for the case , since the other case is analogous. F is -differentiable at the H-difference exists for sufficiently small, i.e., there exists , such thatAlso,where for sufficiently small.
By Theorem 2, we obtainthat is, the H-difference exists and we haveMultiplying with and passing to limit with , by Proof of Lemma 3.2 [12], we obtainAnalogously, we obtainand the conclusion in the case is obtained.

Theorem 8. Let :(i)If F is -differentiable and F is -differentiable, then(ii)If F is -differentiable and F is -differentiable, then

Note that the definition of -differentiable or for , see [12–14].

Proof. We present the details only for Case (i), since the other case is analogous. Let in Definition 7, and then . Therefore, if and , we haveDividing by ε, we haveand passing to the limit,Similarly, we obtainand passing to the limit and gives .

Example 2. Let and , if is q-differentiable on , then the function defined by Then, F is -differentiable, for , and we have
and. (i)Let , so and , then it is easy to see that , i.e.,(ii)Let , so and , then it is easy to see that , i.e.,

5. Conclusion

We have investigated generalized fuzzy conformable fractional differentiability. The 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 because all fractional derivatives do not satisfy the known formula .

The disadvantage of fuzzy generalized conformable differentiability of a function seems to be that a simple fuzzy differential equation has not got a unique solution, so it may have several solutions. The 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

The authors declare that they have no conflicts of interest.