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In this paper, the fuzzy multiterm fractional differential equation involving Caputo-type fuzzy fractional derivative of order

Nowadays the fractional differential equations (FDEs) are powerful tools representing many problems in various areas such as control engineering, diffusion processes, signal processing, and electromagnetism.

Recently the fuzzy fractional differential equations (FFDEs) have been studied by many researchers in order to analyze some systems with fuzzy initial conditions. But, it is too difficult to find the exact solutions of most FFDEs representing real-world phenomena. Therefore, research about FFDEs can be classified into two classes, namely, existence of solution and numerical methods. Many theoretical researches have been advanced on the existence, uniqueness, and stability of solution of FFDEs [

Also the analytical method and the numerical method are typical methods for solving FFDEs. The analytical method includes the Laplace transform method, monotone iterative method, variation of constant formula, and so on [

In [

Based on the above facts, in this paper, we study the existence and uniqueness of solutions for fuzzy multiterm fractional differential equations of order

The paper is organized as follows. In Section

We introduce some definitions and notations which will be used throughout our paper.

Let us denote by

Then

We denote the

Also let us

Let

Let

Then the limit is denoted by

Let

A function

We denote the fuzzy Riemann integral of

Suppose that

(i)

(ii)

We introduce following notations:

Let

Let

Let

Let us denote the

Then we have

Moreover, since

Let

Let

Let

Let

First we prove (i). From the assumption of Lemma

By Lemma

From the result (i) of lemma, it is obvious that (ii) holds.

Next let prove (iii).

By Lemma

The following facts are true:

(i) Let

(ii) For any positive numbers

(iii) Let

Let us consider the assertion (i). For any

We use the notation

Since

It is well known that

Now we prove (ii).

From the definition of fractional integral, the following is true:

By the definition of Gamma function, we have

Let us consider assertion (iii).

We use the notations

Since

The proof is completed.

Let us consider the existence of solution for initial value problem of following fuzzy multiterm fractional differential equation:

Let

Now let us consider the following:

The solution of initial value problem of fuzzy fractional differential equation (

Let

Also since

Therefore the following relations hold:

From the Caputo-type differentiability of fuzzy-valued function

Therefore we obtain

By (

From (

Conversely, we prove that

On the other hand, when

Since the interval family

Consequently the H-differentiability of

From (

Also it is obvious that

Let

Let

Then if

Applying the operator

Therefore substituting the above results to

Next let

Namely

Consequently from (

Moreover since

Now we employ the following metric structure in

Obviously we can see that

For any positive number

Then the metric

Assume that the function

Since

Therefore for any

Also we define the operator

For any

Thus the fuzzy integral equation (

The existence of solution for the fuzzy integral equation (

For any

Consequently the following inequality is obtained:

Multiplying the both sides of the above equation by

Thus the operator

Next let us consider the existence of solution in case which have not satisfied the Lipschitz condition.

Assume that

Suppose that the followings conditions are satisfied:

(i)

(ii)

Let

Firstly, let us prove that

We have that, for any

Thus we can get that

Also by (iii) of Lemma

Thus we can see that

Therefore we can get that for any

Therefore the operator

Next let us prove (ii). The uniformly boundness of

Let us consider the equicontinuity of

For

Without losing generality, let

Then we have

Now we estimate the second term of the above expression.

Estimating the third term of above equation similarly to above, we have

So we obtain as follows:

From right side of the above expression, we can see that

By Arzelà-Ascoli theorem,

By Theorem

We consider the continuous dependence on initial condition of solution for proposed problem (

The following theorem shows the continuous dependence on initial condition of solution.

Suppose that the following conditions are satisfied:

Then following relation holds:

Since

Now we estimate

Consequently, we get

Next we estimate

By (

We obtained the uniqueness results by employing contraction mapping principle and the existence results by using Schauder fixed point theorem for the solution of the fuzzy multiterm fractional differential equation involving Caputo-type fuzzy fractional derivative of order

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.