1 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 is the fuzzy Caputo–Katugampola fractional generalized Hukuhara derivative of order , is a fixed real number, is a continuous fuzzy nonlinear mapping, and 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 .th-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.

2 Preliminaries

In this section, we briefly introduce some definitions, notations and results related to fuzzy functions, which will be referred to throughout this paper.

We denote the set of all real numbers by and the set of all fuzzy numbers on is indicated by . A fuzzy number is a mapping with the following properties:

• is upper semicontinuous;

• is fuzzy convex, i.e., for all , ;

• is normal, i.e., there exists for which ;

• supp is the support of the u, and its closure cl(supp ) is compact.

For denote and . Then it is well known that the -level set of , is a closed interval for all , where and represent the upper and lower branches of the fuzzy set , respectively. A fuzzy number function defined on the real set and valued in is called a fuzzy-valued function, that is, . Let the -level representation of the fuzzy-valued function . be expressed by

For , we define the diameter of the -level set of as . Let . If there exists such that , then is called the -difference of and , and it is denoted by . In this paper, the sign always stands for the -difference.

The Hausdorff distance between fuzzy numbers is given by ,

Then it is easy to see that is a metric in and the following properties of the metric are valid (see [28]):

• for all ;

• 

• for all

• is a complete metric space

For the fuzzy-valued function defined on , we introduce measure . We say that the fuzzy-valued function is integrable on if the function is continuous in the metric and its definite integral exist, and we have

Definition 1. (See [20].) The generalized Hukuhara difference of two fuzzy numbers (-difference for short) is defined as follows:

A function is called -increasing (-decreasing) on if for every , the function is nondecreasing (nonincreasing) on . If is -increasing or -decreasing on , then we say that is -monotone on .

Definition 2. (See [9].) Let and . The fuzzy function is said to be generalized Hukuhara differentiable (-differentiable) at if there exists an element such that

Denote by the set of all continuous fuzzy functions, the set of all absolutely continuous fuzzy functions on the interval with values in .

Theorem 1. (See [21].) If ) is a d-monotone fuzzy function and , then

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 . Now and have the following properties:

• Let . Then and . Moreover, and ;

• Let and . Then and are nonnegative. Additionally, . Then and ;

• 

Theorem 2. (See [21].) For and is -increasing, or and is -decreasing, (by applying the definition of the Mittag-Leffler function) the solution of problem (1)is expressed by

where (for the case of and is -increasing), whereas if and is -decreasing, , then we obtain the solution of problem (1)is

3 Existence and uniqueness results

Let be the space of all continuous fuzzy-valued functions on . Consider the following assumptions:

1. (H1) is continuous;

2. (H2) there exists such that and ;

3. (H3)

Theorem 3. Assume that , is -increasing, and conditions (H1), (H2) and (H3) are satisfied. Then the initial value problem (1)has a unique solution in .

Proof. Consider the operator given by

where , and it is easy to see that is a solution to the initial value problem (1) if and only if . From Lemma 1 we have

for and for each , which implies that

Therefore, the Banach contraction mapping principle guarantees that . has a unique fixed point , so there is a unique solution to problem (1). The proof is completed.

Theorem 4. Assume that , u is -decreasing, conditions (H1).(H3) and the following condition are satisfied:

1. (H4) for any

is nonincreasing in ,

is nonincreasing in , for any and

Then the initial value problem (1)has a unique solution in .

Proof. Consider the operator given by

where It is easy to see that is a solution to the initial value problem (1) if and only if . From Lemma 1 we have

for and for each , which implies that

Therefore, the Banach contraction mapping principle guarantees that has a unique fixed point , so there is a unique solution to problem (1).The proof is completed.

4 Stability results

Motivated by -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 -Ulam-type stability concepts of fuzzy fractional differential equations.

Let and . be a continuous function. We consider the equation

and the associated three inequalities

Definition 3. Equation (2) is -Ulam–Hyers stable if there exists such that for each and for each solution to inequality (3), there exists a solution to Eq. (2) with

Definition 4. Equation (2) is generalized -Ulam–Hyers stable if there exists a continuous function with such that for each solution to inequality (3), there exists a solution to Eq. (2) with

Definition 5. Equation (2) is -Ulam–Hyers–Rassias stable with respect to if there exists such that for each and for each solution to inequality (5) there exists a solution to Eq. (2) with

Definition 6. Equation (2) is generalized -Ulam–Hyers–Rassias stable with respect to if there exists such that for each solution to inequality (4) there exists a solution to Eq. (2) with

Lemma 2. A function is a solution of inequality (5)with

1. (H5) exists in for all if and only if there exists a function (which depends on) such that:

   • 

   • 

Note one can have similar results for inequations (3)and (4)and we omit them here.

Proof. The sufficiency is obvious and we only prove the necessity. Let

Then we get (ii). Additionally, due to

and inequality (5), we can see (i) holds. The proof is completed.

Lemma 3. Let y be a solution of inequality (5)with . Assume that condition (H5) is satisfied. Then y satisfies the integral inequality

if and y is -increasing, where and

Proof. From Lemma 2 we see satisfies

if and is -increasing, noticing is a solution to problem (6), we have

Let

Then

From this it follows that

Lemma 4. Let y be a solution of inequality (5)with . Assume that condition (H5) is satisfied. Then y satisfies integral inequality

if and y is d-decreasing, where and

Proof. The proof of Lemma 4 is similar to Lemma 3, so we omit it here.

Remark 1. One can have similar results to Lemmas 3, 4 inequalities (3) and (4)

Theorem 5. Assume that and is -increasing, conditions (H1)-(H3) and

1. (H6) there exists a nonnegative, nondecreasing and continuous function such that holds.

holds.

Suppose also that a function satisfies inequality (5)and condition (H5) holds. Then Eq. (2)is -Ulam–Hyers–Rassias stable.

Proof. Let be a solution to problem (1), and denote as a solution to inequality (5) with . According to Lemma 3, we have

where , From condition (H6) it follows that

by the generalized Gröwall inequality (see [30, 34]), and we obtain

Therefore Eq. (2) is -Ulam–Hyers–Rassias stable according to Definition 5. The proof is completed.

Remark 2. Under the assumptions of Theorem 5, we consider Eq. (2) and inequality (4). One can verify that Eq. (2) is generalized -Ulam–Hyers–Rassias stable according to Definition 6. Under the assumptions except (H6) of Theorem 5, we consider Eq. (2) and inequality (3). One can show that Eq. (2) is -Ulam–Hyers stable and generalized -Ulam–Hyers stable according to Definitions 3 and 4, respectively.

Theorem 6. Assume that and u is d-decreasing, conditions (H1)-(H3) are satisfied. Suppose also that a function satisfies inequality (5)and (H5), (H6) hold. Then Eq. (2)is -Ulam–Hyers–Rassias stable.

Proof. The proof of is similar to Theorem 5, so we omit it here.

Remark 3. Under the assumptions of Theorem 6, we consider Eq. (2) and inequality (4). One can verify that Eq. (2) is generalized -Ulam–Hyers–Rassias stable according to Definition 6. Under the assumptions except (H6) of Theorem 5, we consider Eq. (2) and inequality (3). One can show that Eq. (2) is -Ulam–Hyers stable and generalized -Ulam–Hyers stable according to Definitions 3 and 4, respectively.

5 Examples

Example 1. Consider the following initial value problem for the fuzzy fractional differential equation

where , and is a symmetric triangular fuzzy number. Take , clearly, conditions (H1)–(H3) hold, then according to Theorem 3, problem (7) has a unique solution.

Assume that a fuzzy-valued function satisfies

Take and . Then we have

which means condition (H6) holds. Thus Eq. (7) is -Ulam–Hyers–Rassias stable according to Theorem 5.

Example 2. Consider the following initial value problem for fuzzy fractional differential equation:

where , and is a symmetric triangular fuzzy number. Take , obviously, conditions (H1)–(H3) holds, then according to Theorem 4, problem (8) has a unique solution.

Assume that a fuzzy-valued function satisfies

Take and . Then we have

which means condition (H5) holds, thus Eq. (8) is -Ulam–Hyers–Rassias stable according to Theorem 6.

Acknowledgments

The authors thank the referees for valuable comments and sugges- tions, which improved the presentation of this manuscript.

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Notes

Notas de autor