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A class fuzzy fractional differential equation (FFDE) involving Riemann-Liouville

In the last several years, fractional differential equations attracted more and more researchers and have proven to be very useful tools for modeling phenomena in physics, finance, and many other areas. The Riemann-Liouville formulation arises in a natural way for problems such as transport problems from the continuum random walk scheme or generalizes Chapman-Kolmogorov models [

On the other hand, in order to obtain more realistic modeling of phenomena, one has to take uncertainty; see [

In [

Motivated by the above works, we adopted the fuzzy Laplace transform to prove the uniqueness and existence for the following initial value problems (FFDE) for arbitrary order

Our aim is to both generalize and extend the previous uniqueness results of [

The organization of this paper is as follows. Section

First, let us recall some basic definitions about fuzzy numbers and fuzzy sets. Here and in the rest of the paper, we denote by

As defined in [

The closure of the set

The set

It follows from

A fuzzy number

Moreover, we also can present the

According to Zadeh’s extension principle, we have the following properties of fuzzy addition and multiplication by scalar on

Let

Note that the sign

We denote by

Let

Let

or

Denote by

In the rest of the paper, we say that a fuzzy-valued function

Let

if

or

if

where

The following theorem is an important one about the fuzzy Laplace transform

Suppose that

if

or

if

The proof runs along similar lines as that of [

In this section, we study the relation between problem (

In fact, by taking Laplace transform on both sides of

If

If

Taking into account the initial conditions of problem (

Now, we state the Krasnoselskii-Krein type conditions for FFDE (

Let

First we establish the uniqueness; suppose

We define

Using (

We have

We want to prove that

For the case

Let

It is similar to that of Theorem

Let

Let

Let the function

The proof of this theorem is essentially based on Lemma

Suppose

Also, if

Let

Without loss of generality, we prove Theorem

We may prove that by using condition (H2) and the definition of successive approximations (

Setting

Now we shall prove that

In this paper we established the uniqueness and existence of the solution under a fuzzy version of the Krasnoselskii-Krein conditions. Our work generalizes and extends the work of Yoruk et al. to arbitrary order in the fuzzy version. We finally hope to study other classes of fuzzy fractional differential problems in future works.

The authors declare that there are no competing interests regarding the publication of this paper.