Well-posedness and stability for fuzzy fractional differential equations

. In this article, we consider the existence and uniqueness of solutions for a class of initial value problems of fuzzy Caputo–Katugampola fractional differential equations and the stability of the corresponding fuzzy fractional differential equations. The discussions are based on the hyperbolic function, the Banach ﬁxed point theorem and an inequality property. Two examples are given to illustrate the feasibility of our theoretical results.


Introduction
In this paper, we are concerned with the existence, uniqueness and stability of solutions for a class of initial value problems of fuzzy fractional differential equations (FFDE) of the following form: where C D q,p 0 + is the fuzzy Caputo-Katugampola fractional generalized Hukuhara derivative of order q ∈ (0, 1], p > 0 is a fixed real number, λ ∈ R, f : (0, T ] × R F → R F is a continuous fuzzy nonlinear mapping, and R F is the space of fuzzy numbers.
Fractional-order differential equation can be regarded as a generalization of ordinary integer-order differential equation, and we refer the reader to [13,17,24,26]. However, due to errors caused by observations, experiments and maintenance, the variables and parameters that we get are usually fuzzy, incomplete and inaccurate. These uncertainties are introduced into fractional differential equations called fractional fuzzy differential equations.
In recent years, there has been some research on fractional fuzzy differential equations. Except for various numerical solutions, most of the methods transform fractional fuzzy differential equations into fractional fuzzy integral equations and then use nonlinear analysis methods to discuss the qualitative properties of the solutions. In 2010, Agarwal et al. obtained the solution of the initial value problem by studying the corresponding fuzzy integral equation of the initial value problem in [2]. In 2012, Allahviranloo et al. [5] studied the analytical solution to the initial value problem for a class of Riemann-Liouville-type fractional differential equations under the strong generalized Hukuhara differentiability introduced in [8]. Then Allahviranloo et al. studied the initial value problem of the Volterra-Fredholm-type fuzzy integro-differential equation, and established the existence and uniqueness of the solution by using a compact mapping theorem and an iterative method [3]. Recently, Ngo presented results on the existence and uniqueness of solutions for two kinds of fractional fuzzy functional integral equations and fuzzy functional differential equations using the contraction mapping principle and the successive approximation method [11,12]. For research on solutions of initial boundary value problems for fractional fuzzy differential equations, more information can be found in [1,4,6,13,22,27,32,35] and the references therein.
The study of Ulam stability can provide an important theoretical basis for the existence and even uniqueness of the solution of the differential equation and it can also provide a reliable theoretical basis for the approximate solution of the corresponding equation. In 1993, Obloza studied the stability of the differential equation in [23]. Miura and others established Ulam stability theory of differential equations in different abstract spaces [18,19,31]. In 2013, Rezaei et al. [29] established Hyers-Ulam stability of nth-order linear differential equations with constant coefficients using the Laplace transform method. Mortici et al. [20] studied the general solution of the inhomogeneous Euler equation and the Hyers-Ulam stability on a bounded domain using the integral method. In 2016, Bahyrycz et al. discussed Ulam stability of the generalized Frechet equation in a Banach space using a fixed point theorem in [7]. In 2018, Onitsuka [25] established the Ulam stability of first-order nonhomogeneous linear difference equations.
The purpose of this paper is to introduce fuzzy Caputo-Katugampola fractional differential equations, and discuss the existence, uniqueness and stability of solutions of fuzzy fractional differential equations (1). The structure of the paper is as follows: some preliminaries are given in Section 2. In Section 3, we establish the existence and uniqueness of solutions to problem (1). In Section 4, we discuss the stability of the solution. In Section 5, some examples are given to illustrate the feasibility of the results.
We denote the set of all real numbers by R and the set of all fuzzy numbers on R is indicated by R F . A fuzzy number is a mapping u : R → [0, 1] with the following properties: (a) u is upper semicontinuous; (b) u is fuzzy convex, i.e., u(λx + (1 − λ)y) min{u(x), u(y)} for all x, y ∈ R, λ ∈ [0, 1]; (c) u is normal, i.e., there exists x 0 ∈ R for which u(x 0 ) = 1; is the support of the u, and its closure cl(supp u) is compact.
Then it is well known that the α-level set of u, [u] α = [u α , u α ], is a closed interval for all α ∈ [0, 1], where u and u represent the upper and lower branches of the fuzzy set u ∈ R F , respectively. A fuzzy number function defined on the real set R and valued in R F is called a fuzzy-valued function, that is, f : R → R F . Let the α-level representation of the fuzzy-valued function f : and v, and it is denoted by u v. In this paper, the sign " " always stands for the H-difference.
The Hausdorff distance between fuzzy numbers is given by Then it is easy to see that d is a metric in R F and the following properties of the metric d are valid (see [28]): if the function f is continuous in the metric d and its definite integral exist, and we have Definition 1. (See [20].) The generalized Hukuhara difference of two fuzzy numbers u, v ∈ R F (gH-difference for short) is defined as follows: Definition 2. (See [9].) Let u : (a, b) → R F and t ∈ (a, b). The fuzzy function u is said to be generalized Hukuhara differentiable (gH-differentiable) at t if there exists an element u (t) ∈ E such that Now, we consider the fractional hyperbolic functions and their properties that will be used in the next section. The Mittag-Leffler function frequently used in the solution of fractional-order systems (see [15]), is defined as follows: Lemma 1. (See [10].) Set δ > 0. Now E α (·) and E α,β (·) have the following properties: [21].) For λ > 0 and u is d-increasing, or λ < 0 and u is d-decreasing, (by applying the definition of the Mittag-Leffler function) the solution of problem (1) is expressed by where t ∈ [0, T ] (for the case of λ > 0 and u is d-increasing), whereas if λ < 0 and u is d-decreasing, t ∈ [0, T ], then we obtain the solution of problem (1) is

Existence and uniqueness results
Let C[a, b] be the space of all continuous fuzzy-valued functions on [a, b]. Consider the following assumptions: where t ∈ [0, T ], and it is easy to see that u is a solution to the initial value problem (1) if and only if u = A 1 u. From Lemma 1 we have for u, v ∈ R F and for each t ∈ [0, T ], which implies that Therefore, the Banach contraction mapping principle guarantees that u = A 1 u has a unique fixed point u * ∈ C[0, T ], so there is a unique solution to problem (1). The proof is completed.
Proof. Consider the operator A 2 : C[0, T ] → C[0, T ] given by for u, v ∈ R F and for each t ∈ [0, T ], which implies that Therefore, the Banach contraction mapping principle guarantees that u = A 2 u has a unique fixed point u * ∈ C[0, T ], so there is a unique solution to problem (1). The proof is completed.

Stability results
Motivated by E α -Ulam-type stability concepts of fractional differential equations (see [33]) and Ulam-type stability concepts of fuzzy differential equations (see [34]), we introduce some new E q -Ulam-type stability concepts of fuzzy fractional differential equations. Let ε > 0 and φ : [0, T ] → R + be a continuous function. We consider the equation and the associated three inequalities Definition 5. Equation (2) is E q -Ulam-Hyers-Rassias stable with respect to φ if there exists c φ > 0 such that for each ε > 0 and for each solution y ∈ C[0, T ] to inequality (5) there exists a solution x ∈ C[0, T ] to Eq. (2) with Definition 6. Equation (2) is generalized E q -Ulam-Hyers-Rassias stable with respect to φ if there exists c φ > 0 such that for each solution y ∈ C[0, T ] to inequality (4) there exists a solution x ∈ C[0, T ] to Eq. (2) with T ] if and only if there exists a function g ∈ C[0, T ] (which depends on y) such that: (3) and (4) and we omit them here.

Note one can have similar results for inequations
Proof. The sufficiency is obvious and we only prove the necessity. Let Then we get (ii). Additionally, due to d C D q,p 0 + y(t), λy(t) ⊕ f t, y(t) = d C D q,p 0 + y(t) λy(t) ⊕ f t, y(t) ,0 = d g(t),0 and inequality (5), we can see (i) holds. The proof is completed.
Lemma 3. Let y be a solution of inequality (5) with y(0) = y 0 . Assume that condition (H5) is satisfied. Then y satisfies the integral inequality Proof. From Lemma 2 we see y satisfies if λ > 0 and y is d-increasing, noticing y is a solution to problem (6), we have Then From this it follows that Lemma 4. Let y be a solution of inequality (5) with y(0) = y 0 . Assume that condition (H5) is satisfied. Then y satisfies integral inequality Proof. The proof of Lemma 4 is similar to Lemma 3, so we omit it here.
Proof. Let x be a solution to problem (1), and denote y as a solution to inequality (5) with y(0) = u 0 . According to Lemma 3, we have where t ∈ [0, T ], From condition (H6) it follows that by the generalized Gröwall inequality (see [30,34]), and we obtain Therefore Eq. (2) is E q -Ulam-Hyers-Rassias stable according to Definition 5. The proof is completed.
Proof. The proof of is similar to Theorem 5, so we omit it here.