Advances in Fuzzy Systems
Volume 2019 (2019), Article ID 6457548, 11 pages
https://doi.org/10.1155/2019/6457548

Research Article

Existence and Continuous Dependence on Initial Data of Solution for Initial Value Problem of Fuzzy Multiterm Fractional Differential Equation

Huichol Choi,¹ Kinam Sin,¹ Sunae Pak,¹ and Sungryol So²

¹Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea
²Institute of Advanced Science, Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea

Correspondence should be addressed to Kinam Sin; 15bf12025@hit.edu.cn

Received 17 March 2019; Revised 13 May 2019; Accepted 15 May 2019; Published 19 June 2019

Academic Editor: Rustom M. Mamlook

Copyright © 2019 Huichol Choi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, the fuzzy multiterm fractional differential equation involving Caputo-type fuzzy fractional derivative of order is considered. The uniqueness of solution is established by using the contraction mapping principle and the existence of solution is obtained by Schauder fixed point theorem.

1. Introduction

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 [1–12].

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 [13–15]. Typical numerical methods are the operational matrix method, fractional Euler method, predictor-corrector method, and so on [16–21].

In [22], the existence and uniqueness of the solutions of fuzzy initial value problems of fractional differential equations with the Caputo-type fuzzy fractional derivative have been proved. Under the conditions which the right sides of equations satisfy Hölder continuity or Lipschitz continuity in its all variables, the existence of a solution to the Cauchy problem for fuzzy fractional differential equations was discussed in [23]. In [24], by employing the contraction mapping principle on the complete metric space, the existence and uniqueness result for fuzzy fractional functional integral equation has been proved. The existence results of solutions for fuzzy fractional initial value problem under generalized differentiability conditions are obtained by Banach fixed point theorem in [25]. In [26], researchers discussed the uniqueness and existence of the solutions for FFDEs with Riemann-Liouville H-differentiability of arbitrary order by using Krasnoselskii-Krein type conditions, Kooi type conditions, and Rogers conditions. But, the considerations of researchers in [22–26] were restricted to the case of FFDEs with single derivative term. Ngo et al. [27] presented that the existence and uniqueness results of the solution for fuzzy Caputo-Katugampola (CK) fractional differential equations with initial value and in [28] proved that the fractional fuzzy differential equation is not equal to the fractional fuzzy integral equation in general.

Based on the above facts, in this paper, we study the existence and uniqueness of solutions for fuzzy multiterm fractional differential equations of order with fuzzy initial value under Caputo-type H-differentiability.

The paper is organized as follows. In Section 2, we introduced some definitions and properties of fuzzy fractional calculus. The existence result of solution for proposed problem is described in Section 3. Section 4 presented the continuous dependence on initial data of solution. Finally, the conclusion is summarized in Section 5.

2. Preliminaries and Basic Results

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

Definition 1 (see [29]). Let us denote by the class of fuzzy subsets satisfying the following properties:

(i) is normal, i.e., for which ,(ii) is fuzzy convex, i.e., for all , ,(iii) is upper semicontinuous on ,(iv) is the support of the , and its closure is compact.

Then is called the space of fuzzy number and any is called fuzzy number.
We denote the -cut form of fuzzy number , , by .
Also let us . The metric on is defined as follows:

Definition 2 (see [19]). Let . If there exists such that , then is called the H-difference of and , and it is denoted by . Note that .

Definition 3 (see [19]). Let and . We say that is H-differentiable at , if for sufficiently near to , there exist the H-differences , , and the limits

Then the limit is denoted by .

Theorem 4 (see [14]). Let be H-differentiable and . Then are all differentiable and

Definition 5 (see [14]). A function is said to be Riemann integrable on , if , , , for any division of , with norm , and for any points , ,

We denote the fuzzy Riemann integral of from to by .

Lemma 6 (see [14]). Suppose that are continuous, then
(i) is the fuzzy Riemann integrable on and is differentiable as in Definition 3, namely, .
(ii) .
We introduce following notations:
is the set of all continuous fuzzy-valued functions on .
is the set of all absolutely continuous fuzzy-valued functions on .
is the space of all Lebesque integrable fuzzy-valued functions on , where and without losing generality, we promise that .

Definition 7 (see [19]). Let . The fuzzy Riemann-Liouville fractional integral of the fuzzy-valued function is defined as follows:

where is the Riemann-Liouville integral operator of and is the Gamma function.

Lemma 8 (see [19]). Let . Then Riemann-Liouville integral of the fuzzy-valued function , based on its -cut form, can be expressed as follows:

where

Lemma 9. Let , . Then the following relations are satisfied:

Proof. Let us denote the -cut form of by .
Then we have

Moreover, since , we get

Definition 10 (see [19]). Let . We say that is fuzzy Riemann-Liouville H-differentiable of order if

is H-differentiable. Then fuzzy Riemann-Liouville H-derivative of order of function is denoted by .

Definition 11 (see [19]). Let . We say that is a fuzzy Caputo-type differentiable function if H-difference exists and satisfies. Then fuzzy Caputo-type derivative of order of function is denoted by

Lemma 12 (see [19]). Let and for . If is a fuzzy Caputo-type fractional differentiable function, then

Lemma 13. Let be H-differentiable. Then the following relations hold:

(i).(ii).(iii), .

Proof. First we prove (i). From the assumption of Lemma 13, we have

By Lemma 6 (i), we get

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

By Lemma 6 (i), we obtain

Lemma 14. The following facts are true:
(i) Let . For any positive number , the fractional integral is continuous in .
(ii) For any positive numbers and , it holds that .
(iii) Let . For any positive number , it holds that

Proof. Let us consider the assertion (i). For any , it is enough to prove that

We use the notation for -cut representation of .
Since , , and are continuous in . And by Lemma 8, the following expression holds:

It is well known that , , are continuous with respective to . Thus we can see 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 for -cut representations of , respectively.
Since , are continuous in . Therefore the following evaluations are true:

The proof is completed.

3. Existence of Solution for Fuzzy Multiterm Fractional Differential Equation

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

where , , and is fuzzy Caputo-type derivative.

Definition 15. Let . We say that is the solution of initial value problem (24) if holds and satisfies (24).
Now let us consider the following:

where .

Lemma 16. The solution of initial value problem of fuzzy fractional differential equation (25) is represented as

Proof. Let be the solution of initial value problem (25). Then we have

Also since is the fuzzy continuous, it is the fuzzy integrable and exists for .
Therefore the following relations hold:

From the Caputo-type differentiability of fuzzy-valued function , we get

and since the space of the absolutely continuous functions coincides with the space of primitive functions of Lebesque integrable functions, the following relation is satisfied:

Therefore we obtain

By (30) and Lemma 6 (i), the left side of (28) exchanges as

From (28) and (31), is satisfied. Namely, we have

Conversely, we prove that denoted by (26) is the solution of fuzzy initial value problem (25). Since

holds. Also as , we get

On the other hand, when , by Lemma 8

holds . Therefore

Since the interval family generates obviously a fuzzy number, we can see that there exist the H-differences as

Consequently the H-differentiability of is leaded.
From (35), we obtain

Also it is obvious that satisfies the initial condition.

Theorem 17. Let of (24) be a fuzzy continuous with respect to every variable. If is the solution of initial value problem (24), the fuzzy-valued function which is constructed by is the solution in of fuzzy integral equation as

Conversely if is the solution in of fuzzy integral equation (40), which is constructed by (26) is the solution of initial value problem (24).

Proof. Let be the solution of initial value problem (24). Namely, let assume that satisfies

Then if is denoted by , by Lemma 16, the following relation is leaded:

Applying the operator to the both side of above equation, by Lemma 13 (iii), we get

Therefore substituting the above results to , of (41), we obtain (40).
Next let be the solution of fuzzy integral equation (40). Then it is obvious that

satisfies the initial condition of problem (24) from the continuousness of .
Namely

Consequently from (44)

is leaded and regarding the above equation, we get

Moreover since

we obtain

Now we employ the following metric structure in :

Obviously we can see that is a complete metric space (see [24]).

For any positive number , we can consider the metric structure as

Then the metric is equivalent to the metric . Namely,

Theorem 18. Assume that the function in (40) is continuous in its all variables and especially, for any , satisfies the following condition:

Then the fuzzy integral equation (40) has a unique solution.

Proof. Since , there exists that inequality is true.
Therefore for any satisfying this inequality, we put as follows:

Also we define the operator by

For any , by (i) of Lemma 14, are continuous and is continuous by assumptions of theorem. Thus the operator is a map from to .
Thus the fuzzy integral equation (40) is represented as

The existence of solution for the fuzzy integral equation (40) is equivalent to the existence of the fixed point of the operator in .
For any , let us evaluate .

By (iii) of Lemma 14, we have

By (ii) of Lemma 14, we have

Consequently the following inequality is obtained:

Multiplying the both sides of the above equation by , we obtain that

and

Thus the operator is contractive on with respect to the distance and we obtain the unique fixed point of the operator by contraction mapping principle. By the way, since the distance is equivalent to in , is also the unique fixed point in sense of the distance . This completes the proof of theorem.

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

Lemma 19 (Schauder fixed point theorem). Assume that is the complete metric space, is a nonempty convex closed subset of , and is a continuous mapping of into itself such that is contained in a compact subset of , then has a fixed point in .

Theorem 20. Suppose that the followings conditions are satisfied:
(i) ,

where and is zero fuzzy number.
(ii) .
Let

and

Then the following results hold:

(i),(ii) is relatively compact.

Proof. Firstly, let us prove that .
We have that, for any ,

Thus we can get that .
Also by (iii) of Lemma 14, we have that, for any ,

Thus we can see that .
Therefore we can get that for any ,

By (iii) of Lemma 14, we have

Therefore the operator is a continuous mapping from convex closed subset into itself; namely,

Next let us prove (ii). The uniformly boundness of is obvious from (70).
Let us consider the equicontinuity of . For any , there exists which satisfies .
For , we estimate .
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 is equicontinuous.
By Arzelà-Ascoli theorem, is relatively compact.

By Theorem 20 and Schauder fixed point theorem, we can see the operator has a fixed point in .

4. Continuous Dependence on Initial Condition of Solution

We consider the continuous dependence on initial condition of solution for proposed problem (24). Let , are solutions with initial values and , are solutions of corresponding integral equation.

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

Theorem 21. Suppose that the following conditions are satisfied:

(i),

(ii).

Then following relation holds:

Proof. Since are solutions of the integral equation (40), we have

Now we estimate .

By (iii) of Lemma 14, we have

Consequently, we get

Next we estimate .

By (iii) of Lemma 14, we have

By (81), we obtain

5. Conclusions

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 on complete metric space of continuous fuzzy number value functions. We also established the continuous dependence of solution on its initial condition. Our results can be expanded to the case in which the right side of equation involves more than two derivative terms.

Data Availability

No data were used to support this study.

Conflicts of Interest

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

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