Fuzzy Fractional Evolution Equations and Fuzzy Solution Operators

In this paper, the fuzzy fractional evolution equations of order q (FFEE) with fuzzy Caputo fractional derivative are introduced. We study the existence and uniqueness of mild solutions for FFEE under some conditions. Also, we generalize the deﬁnition of the fuzzy fractional integral and derivative order q . The fuzzy Laplace transform is presented and proved. The solvability of the problem (FFEE) and the properties of the fuzzy solution operator and its generator are investigated and developed.


Introduction
e fuzzy fractional differential equations (FFDEs) can also offer a more comprehensive account of the process or phenomenon. is has recently captured much attention in FFDEs.As the derivative of a function is defined in the sense of Riemann-Liouville, Grünwald-Letnikov, or Caputo in fractional calculus, the used derivative is to be specified and defined in FFDEs as well.FFDEs are examined in [1][2][3][4][5].We adopted the fuzzy Laplace transform method to solve FFDEs because it has the advantage that it solves problems directly without determining a general solution and obtaining nonhomogeneous differential equations [5].
C. G. Gal and S. G. Gal studied [6], with more details, fuzzy linear and semilinear (additive and positive homogeneous) operator theory on the complete metric space.
Let q ∈ R + , a > 0, and R F be the set of fuzzy real numbers.Our aim in this paper is to investigate the existence and uniqueness of the fuzzy mild solution for the fuzzy fractional evolution equation: D q u(t) � Au(t) + f(t, u(t)), t ∈ T � [0, a], D j u(t)     t�0 � u j (0), j � 0, 1, 2, . . ., k � [q],

⎧ ⎨ ⎩ (1)
where A: D(A) ⊂ R F ⟶ R F is the infinitesimal generator of a q-resolvent family (S q (t)) t≥0 , defined as R F , f ∈ T × R F ⟶ R F satisfies some conditions that will be specified later, and the fuzzy fractional derivative D q is understood here in the caputo sense.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.
Let us denote by R F � u: R ⟶ [0, 1] { } the class of fuzzy subsets of the real axis satisfying the following properties [7]: (i) u is normal, i.e., there exists an x 0 ∈ R, such that u(x 0 ) � 1.
Where P K (R) denotes the family of all nonempty compact convex subsets of R and defines the addition and scalar multiplication in P K (R) as usual.For later purposes, we define Theorem 1 (see [8]).If u ∈ R F , then is a nondecreasing sequence which converges to α and then, Lemma 1 (see [9]).Let u, v : X ⟶ [0, 1] be the fuzzy sets.
e following arithmetic operations on fuzzy numbers are well known and frequently used below.If u, v ∈ R F , then where d H is the Hausdorff metric defined in P K (R).
It is well known that (R F , d) is a complete metric space [10].We list the following properties of d(u, v): for all u, v, w ∈ R F and λ ∈ R.
Let T � (0, a] ⊂ R be a compact interval.We denote by C(T, R F ) the space of all continuous fuzzy functions on T and is a complete metric space with respect to the metric Also, we denote by L 1 (T, R F ) the space of all fuzzy functions f: T ⟶ R F which are Lebesgue integrable on the bounded interval T of R.
Let u: T ⟶ R F be a fuzzy function.We denote e derivative u ′ (t) of a fuzzy function u is defined by [11] u provided this equation defines a fuzzy number u provided that the Lebesgue integrals on the right exist.
From [13], we have the following theorems: ere exists a real Banach space X such that R F can be the embedding as a convex cone C with vertex 0 into X.Furthermore, the following conditions hold: (i) e embedding j is isometric (ii) Addition in X induces addition in R F , i.e., for any u, v ∈ R F (iii) Multiplication by a non-negative real number in X induces the corresponding operation in R F , i.e., for Theorem 3. Let X be a Banach space and j an embedding as in eorem 2, u: T ⟶ R F , and assume j ∘ u is Bochner integrable over T. en,  u ∈ R F , and Remark 2. By the definition of fuzzy integral [14], the above equality yields 2 Advances in Fuzzy Systems

Fuzzy Fractional Integral and Fuzzy Fractional Derivative
Let

], and let
Lemma 2 (see [1]).The family A α ; α ∈ [0, 1]  , given by ( 16), defined a fuzzy number We define where δ n (t) is the nth derivative of the delta function and Γ(•) is the gamma function (for the properties of ϕ q (t), see [15,16]).ese functions satisfy the semigroup property: The Sobolev spaces can be defined in the following way [17]: , and let According to lemma (2) and by the notation of φ α i and

Fuzzy Fractional Integral and Derivative
Define the fuzzy fractional primitive of order q > 0 of u: by For q � 1, we obtain I 1 u(t) �  t a u(s)ds, t ∈ T, that is, the integral operator.Also, the following properties are obvious: for all t ∈ T, and α ∈ [0, 1], the fuzzy fractional differential operator in the Riemann-Liouville sense, is defined for all u satisfying Advances in Fuzzy Systems by In fact, and the fuzzy fractional differential operator in the Caputo sense is defined: by For k � 0, we obtain provided that the equation defines a fuzzy number D q c u(t) ∈ R F .In fact, Some simple but relevant results valid for q, p, t > 0 are If moreover (25) holds, then 0 .So, we can apply [D q u] α to [I q u] α and thanks to the semigroup property (24) If u satisfies (25), then according to (19), where Convolving both sides of (37) with ϕ q and applying the semigroup property (18), we obtain An application of D k+1 to both sides gives and then, which together with (38) implies (34).If and then, 4 Advances in Fuzzy Systems erefore, and the first identity is valid for all

Fuzzy Laplace
. We define the fuzzy Laplace transform [18] by where λ > 0 and real.
Theorem 4. [18].Let u and v are continuous fuzzy-valued functions.Suppose that c 1 and c 2 are constants.en, Lemma 3. Let u is continuous fuzzy-valued function and c ∈ R: As in [5], we can introduce laplace transforms of derivatives by: Proof.We prove (49) because the proof of (48) is similar.
For arbitrary fixed α ∈ [0, 1], we have We properties of the Laplace transform, and since Lϕ q (λ) � λ − q , we obtain en, we conclude that by linearity of L, Using (31) leads to obtain (54) □ 3.3.Fuzzy Solution Operators.We adopt the general definition and theorem of operator theory on R F in [6].Let A: R F ⟶ R F is linear if and is continuous at each x ∈ R F and L + (R F ) is semilinear and continuous at  0. Let us consider the metric Φ: where ‖x‖ F � d(  0, x) and we have Theorem 5. Let A be a bounded linear operator on R F [6].
Here, i: R F ⟶ R F denotes the identity function of R F .
Proof.Recall that x by ( eorem 3.5 and Corollary 3.6 in [6]), it suffices to show that P n x is a Cauchy sequence in the complete metric space (L(R F ), Φ).First A ∈ L(R F ), it follows A n ∈ L(R F ) for n � 0, 1, 2, . ... en for m < n we have However, and, so by induction, one can obtain We obtain and passing to supremum with ||x|| F ≤ 1 we get and □ Motivated by the above definitions in [19,20], we can give the following definition.
Proof.We assume that Φ(R(λ q , A),  O) ≤ (M/|λ q − ω|) on R F .By the properties of j we have, for all u ∈ C, It follows that It follows that sup which completes the proof.□ Next, we need to define the fuzzy solution operator (or fuzzy q-resolvent family), which is similar to that given in [19].
Consider the following particular case of (82) for the Caputo fractional derivative evolution equation of order q(k < q < k + 1) is an integer: Advances in Fuzzy Systems where A : R F ⟶ R F .Applying (44), we obtain that the Cauchy problem ( 26) and then we define the solution operator of (70) in terms of the corresponding integral equation (71).
Definition 3. A family of bounded linear operators S q (t)   t≥0 on R F is called a solution operator for (71) (or qresolvent family), if the following conditions are satisfied: (1) S q (t) is strongly continuous for t ≥ 0 and S q (0) � i, the identity mapping on R F (2) S q (t)D(A) ⊂ D(A) and AS q (t)u 0 � S q (t)Au 0 for all u 0 ∈ D(A), t ≥ 0 (3) S q (t)u 0 is a solution of (71) for all u 0 ∈ D(A), t ≥ 0 Definition 4.
e solution operator S q (t) is called exponentially bounded if there are constants M ≥ 1 and w ≥ 0 such that Proposition 4. If S q (t) is the solution operator of (70) and u 0 ∈ R F , then if for t > 0, the Hukuhara difference S q (t)u 0 − h u 0 exists, we define which this limit exists in the metric space (R F , d).
Taking v(t) � (D q S q )(t)u 0 and using (70) and (44), we obtain (73) A is the generator of a fuzzy q-resolvent family S q (t)   t≥0 on R F if and only if A 1 is the generator of an q-resolvent family S 1 q (t)   t≥0 defining on the convex closed set C by S 1 q � jS q (t)j − 1 .
Proof.Follows from the definition of S q (t)   t≥0 and S 1 q (t)   t≥0 (see [19]), then S q (t)   t≥0 is the fuzzy solution operator on R F if and only if S 1 q (t)   t≥0 is the solution operator on C.
We assume that A is the generator of a fuzzy q-resolvent family S q (t)   t≥0 on R F .By the properties of j we have, for all u 0 ∈ j − 1 (D(A)), Conversely, if A 1 is the generator of a q-resolvent family S 1 q (t)   t≥0 on C, then for all u 0 ∈ D(A) □ Lemma 6.Let A is a operator in (70) and j an embedding as in eorem 2, the solution operators S q (t) of ( 70) is defined by Proof.Taking the fuzzy Laplace transform of (70) on both sides, we obtain and using the j, λ q j(L(u(t))(λ)) − j(AL(u(t))(λ)) � λ q− 1 j u 0 , where λ q > 0, Since (λ q I − A 1 ) − 1 exist, i.e, λ q ∈ ρ(A 1 ) (see [20]), from the above equation, we obtain Now (87) follows easily by taking the inverse of Laplace transform and applying j − 1 is completes the proof.

□
Advances in Fuzzy Systems

Fuzzy Fractional Differential Equations
Consider the following fuzzy fractional differential equation: where A generator of q-resolvent family (S q (t)) t≥0 on R F , D q is the fuzzy caputo fractional differential operator define by (29) and f ∈ T × R F ⟶ R F is continuous.Firstly, we consider the following Cauchy problem e roblem (70) is particular case of (83), and if (70) has a solution operator S q (t), then the corresponding problem (83) is uniquely solvable with the solution provided u j ∈ D(A), j � 0, 1, 2, . . ., k.For this reason, we restrict ourselves to problem (71) (in crisp, see [19]).Next, we consider the particular case of (82).
where A is a operator and f is an abstract function defined on T × R F and with values in R F .Theorem 6.Let A is an operator, f ≔ T × R F ⟶ R F be continuous on T and if f satisfies a Hölder condition with an exponent of β ∈ (0, 1] The function u(t) ∈ C(T, R F ) is a solution of (85) if and only if where Proof.Now applying the fuzzy Riemann-Liouville fractional integral operator (23) in (85) on both sides, we get and taking the fuzzy Laplace transform of (89) on both sides, we obtain (90) By using j and if (λ q I − A 1 ) − 1 exists, i.e., λ q ∈ ρ(A 1 ), from the above equation, we obtain Now, (87) follows easily by taking the inverse of Laplace transform: and applying j − 1 and Lemma 4, is completes the proof.(101)
It follows that ψ p is a contraction, and then ∃!u ∈ C(T, R F ) such that ψ p u � u.
ψ p (u) � u implies that ψ p+1 (u) � ψu, one can write ψ p (ψu) � ψu it follows that ψu is a fixed point of ψ p , and since the fixed point is unique, we get ψ(u) � u.
Hence, (85) has a unique solution. □ Theorem 7 leads to the following appropriate definition of a mild solution to (82).5.A function u(t) ∈ C(T, R F ) is called a mild solution of (82), if it satisfies the operator equation: Let A is a operator and f: T × R F ⟶ R F be continuous on T. Assume that for every v(t), w(t) ∈ R F , t ∈ T. en, problem (82) has a unique mild solution u(t) ∈ C(T, R F ).Proof.Let ψ : C(T, R F ) ⟶ C(T, R F ) be the operator defined byWe have to show that u is mild solution of (82) if and only if ψu � u and v(t), w(t) ∈ C(T, R F ), we find that d(ψv(t), ψw(t)) � d  Lte ωa h(v, w).