Research Article | Open Access
On the Existence and Uniqueness for High Order Fuzzy Fractional Differential Equations with Uncertainty
A class fuzzy fractional differential equation (FFDE) involving Riemann-Liouville -differentiability of arbitrary order is considered. Using Krasnoselskii-Krein type conditions, Kooi type conditions, and Rogers conditions we establish the uniqueness and existence of the solution after determining the equivalent integral form of the solution.
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 [1, 2]. It was also applied for modeling the behavior of viscoelastic and viscoplastic materials under external influences [3, 4] and the continuum and statistical mechanics for viscoelasticity problems . Some of the several research papers were published to consider the uniqueness of the solution for fractional differential equations under Nagumo like conditions (see, e.g., [6–13] and references therein). In [11, 12], Lakshmikantham and Leela established the uniqueness of the solution for the problem , where . Then, in , Yoruk et al. proved the uniqueness of the solution via Krasnoselskii-Krein, Rogers, and Kooi conditions, for .
On the other hand, in order to obtain more realistic modeling of phenomena, one has to take uncertainty; see [14–16] and the references cited therein. Many researchers have worked in the theoretical and numerical aspect of fractional and fuzzy differential equations; the reader is kindly referred to [2, 17–31] and the references therein.
In , Allahviranloo and Ahmadi introduced the fuzzy Laplace transform, which they used under the strongly generalized differentiability. Recently, ElJaoui et al.  developed it further. The newly defined fuzzy Laplace transform  for high order fuzzy derivatives is one of the most useful methods as mentioned by Jafarian et al. in : “…, one of the important and interesting transforms in the problems of fuzzy equations is Laplace transforms. The fuzzy Laplace transform method solves fuzzy fractional differential equations and fuzzy boundary and initial value problems [35–38] ….”
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 : where and is a continuous fuzzy-valued function with where is the Hausdorff distance.
The organization of this paper is as follows. Section 2 contains some basic definitions concerning fuzzy set theory and Riemann-Liouville generalized -differentiability. In Section 3, using the fuzzy Laplace transform we determine the equivalent integral problem. Section 4 is devoted to the main results: a Krasnoselskii-Krein type of uniqueness theorem, a Kooi type uniqueness theorem, and a Rogers type uniqueness theorem; then we prove that the successive approximations converge to the unique solution.
First, let us recall some basic definitions about fuzzy numbers and fuzzy sets. Here and in the rest of the paper, we denote by the Gamma function and the integer part of .
As defined in , is the space of fuzzy numbers:(A1) is normal; that is, there exists such that .(A2) is fuzzy convex; that is, whenever and .(A3) is upper semicontinuous; that is, for any and there exists such that whenever , .(A4)The closure of the set is compact.
The set is called -level set of .
It follows from that the -level sets are convex compact subsets of for all . The fuzzy zero is defined by
Definition 1 (see ). A fuzzy number in the parametric form is a pair of functions , , which satisfy the following conditions:(1) is a bounded nondecreasing left continuous function in and right continuous at ;(2) is a bounded nonincreasing left continuous function in and right continuous at ;(3), .Moreover, we also can present the -cut representation of fuzzy numbers as for all .
According to Zadeh’s extension principle, we have the following properties of fuzzy addition and multiplication by scalar on : Seeking simplicity, we note by the usual . The Hausdorff distance between the fuzzy numbers is denoted by , such that . And is a complete metric space.
Definition 2. Let . If there exists such that , then is called the -difference of and , and it is denoted by .
Remark 3. Note that the sign stands for -difference and .
We denote by the space of all fuzzy-valued functions which are continuous on and the space of all Lebesgue integrable fuzzy-valued functions on , where . We also denote by the space of fuzzy-valued functions which have continuous -derivatives up to order on such that in .
Definition 4 (see ). Let . The fuzzy fractional integral of the fuzzy-valued function is defined by where
Definition 5 (see [41, Definition 6]). Let , , and , where . One says that is fuzzy Riemann-Liouville fractional differentiable of order at , if there exists an element , such that, for all sufficiently small, one has(i) or(ii)
Denote by the space of fuzzy-valued functions on the bounded interval which have continuous -derivative up to order such that . is a complete metric space endowed by the metric such that for every
In the rest of the paper, we say that a fuzzy-valued function is -differentiable if it is differentiable as in Definition 5 case (i) and is -differentiable if it is differentiable as in Definition 5 case (ii).
Definition 6 (see [41, Theorem 7]). Let , , and , where such that ; then one has the following:(i)if is -differentiable fuzzy-valued function, then or(ii)if is -differentiable fuzzy-valued function, then where
The following theorem is an important one about the fuzzy Laplace transform of the Riemann-Liouville -derivative for fuzzy-valued functions.
Theorem 7. Suppose that ; one has the following:(i)if is -differentiable fuzzy-valued function, then or(ii)if is -differentiable fuzzy-valued function, then
The proof runs along similar lines as that of [41, Theorem 16], and we omit it here.
3. Fuzzy Fractional Integral Equation
In this section, we study the relation between problem (1) and the fuzzy integral form using the well-known fuzzy Laplace transform.
In fact, by taking Laplace transform on both sides of we get Based on the type of Riemann-Liouville -differentiability, we obtain two cases.
Case 1. If is -differentiable fuzzy-valued function, then and based on the lower and upper functions of the above equation becomes where In order to solve system (18), and for the sake of simplicity, we assume that where and are solutions of the previous system (18); it yields
Case 2. If is -differentiable fuzzy-valued function, then and based on the lower and upper functions of the above equation becomes where In order to solve system (23), and for the sake of simplicity, we assume that where and are solutions of the previous system (23). Then we obtainTaking into account the initial conditions of problem (1) and using the linearity of the inverse Laplace transform on systems (20) and (26), we obtain the following for both cases.
is a solution for problem (1) if and only if is a solution for the following integral equation:in the sense of -differentiability, and in the sense of -differentiability.
4. Main Results
Now, we state the Krasnoselskii-Krein type conditions for FFDE (1).
Theorem 8. Let satisfy the following Krein type conditions: (H1), and ,(H2), where and are positive constants and ; then in the sense of -differentiability, the solution is unique and in the sense of -differentiability, the solution is unique on , where and is the bound for on : that is, .
Proof. First we establish the uniqueness; suppose and are any two solutions of (1) in -differentiability and let and . Note that .
We define ; clearly .
Using (27) and condition (H2), we get For the sake of simplicity we use the same symbol to denote all different constants arising in the rest of the proof.
We have Since for , multiplying both sides of (30) by and then integrating the resulting inequality, we get Using the fact that for every , (31) becomes This leads to the following estimates on and , for : Define the function for . When either or is the maximum, we get or Since (by assumption), we have So all of the exponents of in the above inequalities are positive. Hence, . Therefore, if we define , the function is continuous in .
We want to prove that . In fact, since the function is continuous, if does not vanish at some points , that is, on , then there exists a maximum reached when is equal to some : such that , for . But, from condition (H1) we get for either cases or which is a contradiction. Thus, the uniqueness of the solution is established in the sense of -differentiability. The second part of the proof is almost completely similar to the -differentiability; thus, we omit it.
Theorem 10 (Kooi’s type uniqueness theorem). Let satisfy the following conditions: (J1), and ;(J2),where and are positive constants and , for ; then in the sense of -differentiability, the solution is unique and in the sense of -differentiability, the solution is unique.
Proof. It is similar to that of Theorem 8; thus, we omit it.
Lemma 11. Let and be two nonnegative continuous functions in the interval for a real number . Let . Assume the following: (i),(ii),(iii),(iv). Then .
Proof. Let . After differentiating and using (ii), we obtain, for , , so that is decreasing. Now, from (iii) and (iv), if then, for a small , we have Hence, which implies that . Finally, is nonnegative due to (i), and thus .
Theorem 12 (Rogers’ type uniqueness theorem). Let the function verify the following conditions: (), uniformly for positive and bounded and on ,(). Then the problem has at most one solution.
The proof of this theorem is essentially based on Lemma 11.
Proof. Suppose and are any two solutions of (1) in -differentiability, and let and ; we get for where is defined as in Lemma 11.
Also, if , then from the condition (K1) for small , we have By applying Lemma 11, we obtain for every , and this proves the uniqueness of the solution of the FFDE (1) in -differentiability. The second part of the proof is almost completely similar; thus, we omit it.
Theorem 13. Let satisfy the conditions of Theorem 8. Then the successive approximations in the sense of -differentiability or in the sense of -differentiability converge to the unique solution of the FFDE (1).
Proof. Without loss of generality, we prove Theorem 13 for the sequence in the sense of -differentiability using Ascoli-Arzela Theorem. The convergence of the sequence in the sense of -differentiability is completely similar so we omit it.
Step 1. The sequences and are well defined and continuous and uniformly bounded on ; in fact For and , we have Moreover, for every we have By induction, the sequences and are well defined and uniformly bounded on .
Step 2. We prove that the functions and are continuous in , where and are defined by such that Let us note where For and for every , we obtain The right-hand side in the above inequalities is at most for large if provided that for every . And since is arbitrary and can be interchangeable, we get The same goes for , and we obtain These imply that and are continuous on .
Step 3. We verify that the family is equicontinuous in and that the family is equicontinuous in .
We may prove that by using condition (H2) and the definition of successive approximations (43) we obtain As a consequence, we obtain the following estimation: By Arzela-Ascoli Theorem, there exists a subsequence of integers , such that Let us note Further, if and as , then the limit of any successive approximation of is the solution of (1), which was proved to be unique in Theorem 8. It follows that a selection of subsequences is unnecessary and that the entire sequence converges uniformly to . For that, it is sufficient to show that and which will lead to and being null.
Setting and by defining , we show that .
Now we shall prove that . Suppose that at any point in ; then there exists such that . Hence, from condition (H1), we obtain or In both cases, we end up with a contradiction. So . Therefore, iteration (43) converges uniformly to the unique solution of (1) on .
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.
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