Research Article | Open Access
Huichol Choi, Kinam Sin, Sunae Pak, Sungryol So, "Existence and Continuous Dependence on Initial Data of Solution for Initial Value Problem of Fuzzy Multiterm Fractional Differential Equation", Advances in Fuzzy Systems, vol. 2019, Article ID 6457548, 11 pages, 2019. https://doi.org/10.1155/2019/6457548
Existence and Continuous Dependence on Initial Data of Solution for Initial Value Problem of Fuzzy Multiterm Fractional Differential Equation
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.
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 , 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 . In , 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 . In , 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.  presented that the existence and uniqueness results of the solution for fuzzy Caputo-Katugampola (CK) fractional differential equations with initial value and in  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 ). 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 ). Let . If there exists such that , then is called the H-difference of and , and it is denoted by . Note that .
Definition 3 (see ). Let and . We say that is H-differentiable at , if for sufficiently near to , there exist the H-differences , , and the limitsThen the limit is denoted by .
Theorem 4 (see ). Let be H-differentiable and . Then are all differentiable and
Definition 5 (see ). 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 ). Suppose that are continuous, then
(i) is the fuzzy Riemann integrable on and is differentiable as in Definition 3, namely, .
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 ). 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 ). 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 haveMoreover, since , we get
Definition 10 (see ). 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 ). 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 ). 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), .
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 haveLet 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.
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 obtainBy (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 getOn the other hand, when , by Lemma 8holds . 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 asConversely 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 getMoreover since we obtain
Now we employ the following metric structure in :
Obviously we can see that is a complete metric space (see ).
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 haveBy (ii) of Lemma 14, we haveConsequently the following inequality is obtained:Multiplying the both sides of the above equation by , we obtain thatandThus 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.
Let andThen 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 haveTherefore 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