In this paper, the Cauchy problem of fuzzy fractional differential equationsTγut=Ftut, ut0=u0,
- fuzzy conformable fractional derivative
- fuzzy fractional differential equations
- existence and uniqueness of solution
- approximate solutions
- Cauchy problem of fuzzy fractional differential equations
In this paper, we will study Fuzzy solutions to
where subject to initial condition for fuzzy numbers, by the use of the concept of conformable fractional -differentiability, we study the Cauchy problem of fuzzy fractional differential equations for the fuzzy valued mappings of a real variable. Several import-extant results are obtained by applying the embedding theorem in  which is a generalization of the classical embedding results [2, 3].
In Section 2 we recall some basic results on fuzzy number. In Section 3 we introduce some basic results on the conformable fractional differentiability [4, 5] and conformable integrability [5, 6] for the fuzzy set-valued mapping in . In Section 4 we show the relation between a solution and its approximate solution to the Cauchy problem of the fuzzy fractional differential equation, and furthermore, and we prove the existence and uniqueness theorem for a solution to the Cauchy problem of the fuzzy fractional differential equation.
We now recall some definitions needed in throughout the paper. Let us denote by the class of fuzzy subsets of the real axis satisfying the following properties:
is normal: there exists with
is convex fuzzy set: for all and , it holds thatE3
is upper semicontinuous: for any , it holds that
Then is called the space of fuzzy numbers see . Obviously, . If is a fuzzy set, we define the -level (cut) sets of , with . Also, if then -cut of denoted by
Lemma 1 see  Letbe the fuzzy sets. Thenif and only iffor all
For and the sum and the product are defined by
. Additionally if we denote , then is a neutral element with respert to
Let by the following equation:
where is the Hausdorff metric defined as:
The following properties are well-known see :
and is a complete metric space.
Definition 1 The mapping for some interval is called a fuzzy process. Therefore, its -level set can be written as follows:
Theorem 1.1  Let be Seikkala differentiable and denote. Then, the boundary functionandare differentiable and
Definition 2  Let . The fuzzy integral, denoted by , is defined levelwise by the following equation:
for all . In , if is continuous, it is fuzzy integrable.
Theorem 1.2  If , then the following properties hold:
is a nondecreasing sequence which converges tothen
Conversely if is a family of closed real intervals verifyingand, thendefined a fuzzy numbersuch that
From , we have the following theorems:
Theorem 1.3 There exists a real Banach space such thatcan be the embedding as a convex conewith vertexinto X. Furthermore, the following conditions hold:
the embedding is isometric,
addition in induces addition in , i.e, for any
multiplication by a nonnegative real number in induces the corresponding operation in , i.e., for any
C-C is dense in
3. Fuzzy conformable fractional differentiability and integral
Definition 3  Letbe a fuzzy function.order “fuzzy conformable fractional derivative” ofis defined by
for all. Letstands for. Hence
Ifis- differentiable in some, andexists, then
and the limits (in the metric d).
Remark 1 From the definition, it directly follows that ifis-differentiable then the multivalued mappingis-differentiable for alland
where is denoted from the conformable fractional derivative of of order . The converse result does not hold, since the existence of Hukuhara difference does not imply the existence of H-difference
Theorem 1.4  Let .
If is differentiable and is -differentiable then
Remark 2 If is -differentiable and for all is continuous, then we denote .
Definition 4  LetDefine the fuzzy fractional.
where the integral is the usual Riemann improper integral.
Theorem 1.7  , for , where is any continuous function in the domain of .
Theorem 1.8  Let and be -differentiable in and assume that the conformable derivative is integrable over . Then for each we have
4. Existence and uniqueness solution to fuzzy fractional differential equations
In this section we state the main results of the paper, i.e. we will concern ourselves with the question of the existence theorem of approximate solutions by using the embedding results on fuzzy number space and we prove the uniqueness theorem of solution for the Cauchy problem of fuzzy fractional differential equations of order .
4.1 Solution and its approximate solutions
Assume that is continuous . Consider the fractional initial value problem
From Theorems (1.5), (1.7) and (1.8), it immediately follows:
Theorem 1.9 A mapping is a solution to the problem (25) if and only if it is continuous and satisfies the integral equation
for all and
In the following we give the relation between a solution and its approximate solutions.
We denote where be two positive real numbers
Theorem 1.10 Let and , such that
where , , and s the isometric embedding from onto its range in the Banach space . For each there exists an such that the H-differences and exist for all and If we have
uniform convergence (u.c) for all , then and
Proof: By (28) we know that . For fixed and any denote and
It is well know that
From is know that for any , there exists such that
whenever and with Take natural number such hat
Take such that and
By the definition of and (27), we have
We choose such that and for all
where In view of (40), we have
Let and applying (35), we have
On the other hand, from the assumption of Theorem (1.9), there exists an such that the H-differences exist for all and
Now let we verify that the fuzzy number-valued sequence uniformly converges on . In fact, from the assumption u.c. for all , we know
Since is complete, there exists a fuzzy number-valued mapping such that u.c to on as
In addition, we have
Let It follows that
Hence the H-difference exist for all
Thus from (60) we have for all
So, Similarty, we have
Hence exists and
from is arbitrary, we know that Eq. (29) holds true and The proof is concluded.
Lemma 2 For all , and
If we replace Eq. (27) by
retain other assumptions, then the conclusions also hold true.
Proof: This is completely similar to the proof of Theorem (1.10), hence itis omitted here.
4.2 Uniqueness solution
In this section, by using existence theorom of approximate solutions, and the embedding results on fuzzy number space , we give the existence and uniqueness theorem for the Cauchy problem of the fuzzy fractional differential equations of order
Let and for all
and for all such that is noncreasing on the fractional initial value problemE74
has only the solution on .
for all and
Then the Cauchy problem (29) has unique solution on where and the successive iterations
uniformly converge to on
Example 1 Let is a constant, taking in the proof of Theorem (4.2), then obtain the proof of Corollary 4.1 in  where , hence . Then the Cauchy problem (29) has unique solution , and the successive iterations (75) uniformly converge to on
In this work, we introduce the concept of conformable differentiability for fuzzy mappings, enlarging the class of -differentiable fuzzy mappings where . Subsequently, by using the -differentiable and embedding theorem, we study the Cauchy problem of fuzzy fractional differential equations for the fuzzy valued mappings of a real variable. The advantage of the -differentiability being also practically applicable, and we can calculate by this derivative the product of two functions because all fractional derivatives do not satisfy see .
On the other hand, we show and prove the relation between a solution and its approximate solutions to the Cauchy problem of the fuzzy fractional differential equation, and the existence and uniqueness theorem for a solution to the problem (2) are proved.
For further research, we propose to extend the results of the present paper and to combine them the results in citeref for fuzzy conformable fractional differential equations.
Conflict of interest
The authors declare no conflict of interest.