Abstract
In this study, fuzzy conformable fractional differential equations are investigated. We study conformable fractional differentiability, and we define fractional integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy fractional differential equation by using the concept of conformable differentiability. This concept is based on the enlargement of the class of differentiable fuzzy mappings; for this, we consider the lateral Hukuhara derivatives of order .
1. Introduction
Fractional calculus is generalization of differentiation and integration to an arbitrary order. The derivative for fuzzy-valued mappings was developed by [1] that generalized and extended the concept of Hukuhara differentiability (H-derivative) for set-valued mappings to the class of fuzzy mappings. Subsequently, using the H-derivative [2, 3] started to develop a theory for FDE. The concept of the fuzzy fractional derivative was introduced by [4] and developed by [5–11], 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 [12–14]. All definitions above satisfy the property that the fuzzy fractional derivative is linear. This is the only property inherited from the first fuzzy derivative by all of the definitions. However, the following are some of the setbacks of the other definitions [15]. The fuzzy conformable derivative may facilitate some computations:(i)It satisfies all concepts and rules of an ordinary derivative such as quotient, product, and chain rules while the other fractional definitions fail to meet these rules(ii)It can be extended to solve exactly and numerically fractional differential equations and systems easily and efficiently
And it was introduced and developed in [16, 17]. The objective of this study is to present some results for fuzzy conformable differentiability and fuzzy fractional integrability of such functions; we study the fuzzy fractional differential equations (FFDEs) by using this derivative and give an existence and uniqueness theorem for a solution of FFDEs.
2. Preliminaries
Let us denote by the class of fuzzy subsets of the real axis satisfying the following properties:(i) is normal, i.e, there exists such that (ii)u is fuzzy convex, i.e, for and ,(iii) is upper semicontinuous(iv) is compact
Then, is called the space of fuzzy numbers. Obviously, . For , denote ; then, from (i) to (iv), it follows that the -level set 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 [7]). If , then(i) for all (ii) for all (iii) is a nondecreasing sequence which converges to , and then,Conversely, if is a family of closed real intervals verifying (i) and (ii), then defined a fuzzy number such that for and .
Lemma 1 (see [18]). Let be the fuzzy sets. Then, if and only if for all .
The following arithmetic operations on fuzzy numbers are well known and frequently used below. If , then
Definition 1 (see [19, 20]). Let . If there exists such as , then is called the -difference of , and it is denoted as .
Definition 2 (see [21]). Let we denoteDefine by the equationwhere 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 . Then, theorem in [2] gives us an expression for the limit.
Theorem 2 (see[2]). If as , then
3. Fuzzy Conformable Fractional Differentiability and Fuzzy Fractional Integral
3.1. Fuzzy Conformable Fractional Differentiability
Now, we present our new definition, which is the simplest and most natural and efficient definition of fractional derivative of order .
Definition 3 (see[17]). Let be a fuzzy function, and order fuzzy conformable fractional derivative of is defined byfor all . Let stands for . Hence,If is -differentiable in some and exists, thenand the limits (in the metric d).
Remark 1. From the definition, it directly follows that if is -differentiable, then the multivalued mapping is -differentiable for all andwhere is denoted from the conformable fractional derivative of of order .
Theorem 3 (see[17]). Let be -differentiable. Denote . Then, and are -differentiable and
Theorem 4. Let is -differentiable on . If with , then there exists such that .
Proof. For each , there exists such that the -differences and exist for all . Then, we can find a finite sequence such that the family covers and . Pick , such that . Then,for some . Hence,
Theorem 5. If is -differentiable, then it is continuous.
Proof. Let with . Then, by properties of equation (7) and the triangle inequality, we havewhere is so small that the -difference exists. By the differentiability, the right-hand side goes to zero as , and hence, is right continuous. The left continuity is proved similarly.
Theorem 6. Let . If is differentiable and is -differentiable, then
The proof is similar to the proof of Theorem 8 case (i) in [17] and is omitted.
Theorem 7. Let , and if are -differentiable and , then and
Proof. Since is -differentiable, it follows that exists, i.e., there exists such thatAnalogously, since is -differentiable, there exists such thatand we getthat is, the -differenceBy similar reasoning, we get that there exist and such thatand sothat is, the -differenceWe observe thatFinally, by multiplying (21) and (24) with and passing to limit with , we get that is -differentiable and . The case (ii) is similar to the previous one.
3.2. Fuzzy Fractional Integral
Let and be such that for all and . Suppose that for all and let
Lemma 2. The family , given by equation (26), defined a fuzzy number such that .
Proof. For , we have and . It follows . Since , we havefor and . Obviously, is integrable on . Therefore, if , then by Lebesque’s dominated convergence theorem, we haveFrom Theorem 1, the proof is complete.
Definition 4. Let define the fuzzy fractional integral for ,by`where the integral for is the usual Riemann improper integral. Also, the following properties are obvious.
Lemma 3. Let and be fractional integrable and . Then,(i)(ii)
Proof. The proof is similar to the proof of Theorem 4.3 cases (i) and (ii) in [2] and is omitted.
Theorem 8. , for , where is any continuous function in the domain of
Proof. Since is continuous, then is clearly -differentiable becauseand is continuous for all ; then, by Theorem 5.6 in [2] and Theorem 6, the fractional integral is -differentiable. Hence,
Theorem 9. Let and be -differentiable in , and assume that the conformable derivative is integrable over . Then, for each , we have
Proof. Let and be fixed. We shall prove thatwhere is the Hukuhara conformable fractional derivative of ; then, using Theorems 3 and 6 gives us the following equation.By equation (29), we haveSo,where is the Hukuhara derivative of ; equation (37) is also true for a fuzzy mapping . The equality (34) now follows Theorem 5.7 in [2].
4. Fuzzy Comformable Fractional Differential Equations
We study the fuzzy initial value problemwhere is the continuous fuzzy mapping, and is the fuzzy number. From Theorems 5, 8, and 9, it immediately follows.
Theorem 10. mapping is a solution to problem (38) if and only if it is continuous and satisfies the integral equation:for all and .
Theorem 11. Let be continuous, and assume that there exists such thatfor all . Then, problem (38) has a unique solution on .
Proof. If in problem (38) we consider the conformable derivative for all Theorem 3, then from Theorem 6.1 in [2] and using Definition 4 and Lemma 1, we can prove that there exists an unique solution on , and the proof is now complete.
Remark 2. In [15], it is observed that if we fuzzify the equivalent ordinary differential equation , then we will get fuzzy differential equations (the equation was fuzzified by adding a forcing term in the right-hand side). That is, if we consider fuzzy differential equation with the same initial condition , we get the result.
Consider the following linear fractional equation:where . Denote , and .
Theorem 12. Equation (41) has a unique solution in , and for given initial , it is given by
Proof. Equation (41) can be written, levelwise, asfor every , so thatThus, for ,and, therefore, it can be deduced thatThis proves that, for ,So,
5. Conclusion
In this study, for developing and proving some results for fuzzy conformable differentiability and fuzzy fractional integrability of such functions, we provided existence and uniqueness solutions to fuzzy fractional problems for order FFDEs, which is interpreted by using the generalized conformable fractional derivatives concept.
For future research, we will solve the fractional fuzzy conformable partial differential equations [22, 23] and a class of linear differential dynamical systems [24] by using the proposed method.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.