Existence, Uniqueness and Approximate Solutions of Fuzzy Fractional Differential Equations

In this paper, the Cauchy problem of fuzzy fractional differential equations with fuzzy conformable fractional derivative ( γ -differentiability, where γ ∈ 0, 1 ð (cid:2)Þ are introduced. We study the existence and uniqueness of solutions and approximate solutions for the fuzzy-valued mappings of a real variable, we prove some results by applying the embedding theorem, and the properties of the fuzzy solution are investigated and developed. Also, we show the relation between a solution and its approximate solutions to the fuzzy fractional differential equations of order γ . differential equation.


Introduction
In this paper, we will study Fuzzy solutions to where subject to initial condition u 0 for fuzzy numbers, by the use of the concept of conformable fractional H-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 [1] 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 [7]. 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.

Preliminaries
We now recall some definitions needed in throughout the paper. Let us denote by R F the class of fuzzy subsets of the real axis u : R ! 0, 1 ½ f g satisfying the following properties: ii. u is convex fuzzy set: for all x, t ∈ R and 0 < λ ≤ 1, it holds that iii. u is upper semicontinuous: for any x 0 ∈ R, it holds that Then R F is called the space of fuzzy numbers see [8]. Obviously, R ⊂ R F . If u is a fuzzy set, we define u ∀α ∈ 0, 1 ½ . Additionally if we denote0 ¼ χ 0 f g , then0 ∈ R F is a neutral element with respert to þ: Let d : R F Â R F ! R þ ∪ 0 f g by the following equation: where d H is the Hausdorff metric defined as: The following properties are well-known see [10]: and R F , d ð Þis a complete metric space. Definition 1 The mapping u : 0, a ½ ! R F for some interval 0, a ½ is called a fuzzy process. Therefore, its α-level set can be written as follows: 2 Fuzzy Systems Theorem 1.1 [11] Let u : 0, a ½ ! R F be Seikkala differentiable and denote u t Then, the boundary function u α 1 t ð Þ and u α 2 t ð Þ are differentiable and is defined levelwise by the following equation: for all 0 ≤ α ≤ 1. In [12], if u : 0, a ½ ! R F is continuous, it is fuzzy integrable. Theorem 1.2 [13] If u ∈ R F , then the following properties hold: ii. α k f g⊂ 0, 1 ½ is a nondecreasing sequence which converges to α then , we have the following theorems: Theorem 1. 3 There exists a real Banach space X such that  F can be the embedding as a convex cone C with vertex 0 into X. Furthermore, the following conditions hold: i. the embedding j is isometric, ii. addition in X induces addition in  F , i.e, for any u, v ∈  F , iii. multiplication by a nonnegative real number in X induces the corresponding operation in  F , i.e., for any u ∈  F , iv. C-C is dense in X, v. C is closed.

Fuzzy conformable fractional differentiability and integral
and the limits (in the metric d).
integral for a ≥ 0 and γ ∈ 0, 1 ð : where the integral is the usual Riemann improper integral.
where F is any continuous function in the domain of I a γ . Theorem 1.8 [5] Let γ ∈ 0, 1 ð and F be γ-differentiable in 0, a ð Þ and assume that the conformable derivative F γ ð Þ is integrable over 0, a ð Þ. Then for each s ∈ 0, a ð Þ we have

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  F , d ð Þand we prove the uniqueness theorem of solution for the Cauchy problem of fuzzy fractional differential equations of order γ ∈ 0, 1 ð .

Solution and its approximate solutions
where u 0 ∈  F and γ ∈ 0, 1 ð : From Theorems (1.5), (1.7) and (1.8), it immediately follows: Theorem 1.9 A mapping u : 0, a ð Þ !  F is a solution to the problem (25) if and only if it is continuous and satisfies the integral equation for all t ∈ 0, a ð Þ and γ ∈ 0, 1 ð : In the following we give the relation between a solution and its approximate solutions. We where ε n > 0, ε n ! 0, B n t ð Þ ∈ C t 0 , t 0 þ η ½ , X ð Þ , and j s the isometric embedding from  F , d ð Þonto its range in the Banach space X. For each t ∈ t 0 , t 0 þ η ½ there exists an β > 0 such that the H-differences u n t þ εt 1Àγ ð Þ⊖ u n t ð Þ and u n t ð Þ ⊖ u n t À εt 1Àγ ð Þexist for all 0 ≤ ε < β and n ¼ 1, 2, …: If we have Proof: By (28) we know that u t It is well know that From F ∈ C 1 Δ 0 ,  F ð Þ, is know that for any ε > 0, there exists β 1 > 0 such that Take β > 0 such that β < β 1 and By the definition of G t, n ð Þ and (27), we have ∀γ ∈ 0, 1 ð We choose ψ ∈ X * such that ∥ψ∥ ¼ 1 and for all γ ∈ 0, 1 ð where t 1 ≤t ≤ t: In view of (40), we have has only the solution y t ð Þ 0 on t 0 , t 0 þ θ ½ .

Conclusion
In this work, we introduce the concept of conformable differentiability for fuzzy mappings, enlarging the class of γ-differentiable fuzzy mappings where γ ∈ 0, 1 ð . Subsequently, by using the γ-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 γ-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 [4].
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.