On the Existence and Uniqueness Results for Fuzzy Linear and Semilinear Fractional Evolution Equations Involving Caputo Fractional Derivative

Abstract

In this manuscript, we establish new existence and uniqueness results for fuzzy linear and semilinear fractional evolution equations involving Caputo fractional derivative. The existence theorems are proved by using fuzzy fractional calculus, Picard’s iteration method, and Banach contraction principle. As application, we conclude this paper by giving an illustrative example to demonstrate the applicability of the obtained results.

1. Introduction

Fuzzy fractional calculus and fuzzy fractional differential equations are a natural way to model dynamical systems subject to uncertainties. In the past few years, many works have been done by several authors in the theory of fuzzy fractional differential equations (see [1–3]). This theory has been proposed to handle uncertainty due to incomplete information that appears in many mathematical or computer models of some deterministic real-world phenomena. Recently, fractional differential equations have attracted a considerable interest both in mathematics and in applications such as material theory, transport processes, fluid flow phenomena, earthquakes, solute transport, chemistry, wave propagation, signal theory, biology, electromagnetic theory, thermodynamics, mechanics, geology, astrophysics, economics heat conduction in materials with memory, and control theory (see basic books and interesting papers in [4–8]).

In many cases, when a real physical phenomenon is modelled by a fractional initial value problem, we cannot usually be sure that the model is perfect. For example, the initial value of this problem may not be known precisely. In order to get a perfect model with a precise initial condition, Agarwal et al. in [9] proposed the concept of solutions for fuzzy fractional differential equations. Arshad and Lupulescu in [10] proved some results on the existence and uniqueness of solution for the fuzzy fractional differential equations under Hukuhara fractional Riemann-Liouville differentiability. Later, Alikhani and Bahrami in [11] have proved the existence and uniqueness results for nonlinear fuzzy fractional integral and integrodifferential equations by using the method of upper and lower solutions. The authors in [12, 13] discussed the concepts about generalized Hukuhara fractional Riemann-Liouville and Caputo differentiability of fuzzy-valued functions, and the equivalence between fuzzy fractional differential equation and fuzzy fractional integral equation is discussed in [14]. For many basic works related to the theory of fractional differential equations and fuzzy fractional differential equations, we refer the readers to the articles [15–20] and references therein.

Motivated by the above works, in the present paper, we study the existence result of solution for the following fuzzy linear fractional evolution equation:and for the following fuzzy semilinear fractional evolution equation:where is the fuzzy Caputo derivative of at order , , , is a bounded linear operator, and is a fuzzy continuous function.

The paper is organized as follows. In Section 2, we give some basic properties of fuzzy sets, operations of fuzzy numbers, and some detailed definitions of fuzzy fractional integral and fuzzy fractional derivative which will be used in the rest of this paper. In Section 3, we introduce the existence and uniqueness results of solution for the fuzzy linear fractional evolution equation (1). In Section 4, we discussed the existence and uniqueness results for the fuzzy semilinear fractional evolution equation (2). An illustrative example is presented in Section 5 followed by conclusion and future work in Section 6.

2. Preliminaries

In this section, we will briefly give some of notations, definitions, and results from the literature of fuzzy set theory and fuzzy fractional calculus which will be used in the rest of this paper.

Definition 1 (see [21]). A fuzzy number is mapping such that(1) is upper semicontinuous(2) is normal; that is, there exist such that (3) is fuzzy convex, that is, for all and (4) is compact

The of a fuzzy number is defined as follows:

Moreover, we also can present the of fuzzy number by

Example 1. Let be a fuzzy number defined by the following function:If , then the of the fuzzy number is given by .

Notations. (i)We denote by the collection of all fuzzy numbers(ii)We denote by the space of all fuzzy-valued functions which are continuous on (iii)We denote by the set of all bounded and closed intervals of (iv)We denote also by the fuzzy zero defined by

Definition 2 (see [14]). Let and such that
We define the diameter of set of the fuzzy set as follows:

Definition 3 (see [14]). The generalized Hukuhara difference of two fuzzy numbers is defined as follows:

Property 4 (see [22]). If and , then the following properties hold:(1)If exists then it is unique(2)(3)(4)

Definition 5 (see [21]). According to Zadeh’s extension principle, the addition on is defined byAnd scalar multiplication of a fuzzy number is given by

Remark 6 (see [23]). Let and ; then, we have

Definition 7 (see [24]). Let and ; then, the Hausdorf distance between and is given by

Proposition 8 (see [25]). is a metric on and has the following properties:(1) is a complete metric space(2), (3), and (4),

Remark 9. Let . It is easy to see that the space is a Banach space where

Definition 10 (see [26]). Let and . We say that is Hukuhara differentiable at if there exists such that

Definition 11 (see [25]). (1)A function is strongly measurable if , the set-valued mapping defined by is Lebesgue measurable(2)A function is called integrably bounded, if there exists an integrable function such that

Definition 12. Let . The integral of on denoted by is defined byfor all .

Proposition 13 (see [25]). Let be a fuzzy function. If is strongly measurable and integrably bounded, then it is integrable.

2.1. Fractional Integral and Fractional Derivative of Fuzzy Function

Proposition 14. If , then the following properties hold:(1) if (2)If is a nondecreasing sequence which converges to , then

Conversely, if is a family of closed real intervals verifying (1) and (2), then defined a fuzzy number such that

Let ; the fractional integral of order of a real function is given by

Let such that . Suppose that for all and letwhere is the Euler gamma function.

We have the following lemma.

Lemma 15 (see [10]). The family given by (17) defined a fuzzy number such that .

Definition 16 (see [10]). Let .
The fuzzy fractional integral at order of denoted byis defined by

Proposition 17 (see [10]). Let and ; then, we have(1)(2)(3), where

Example 2. Let be a constant fuzzy function such that .
If , then

Definition 18 (see [13]). Let . The function is called fuzzy Caputo fractional differentiable of order at if there exists an element such that

Remark 19 [13]. Since for each , thenwhere

Example 3. Let . If , then

3. Fuzzy Linear Fractional Evolution Equation

Definition 20 (see [14]). A fuzzy function is called-increasing(-decreasing) on if for every , the real function is nondecreasing (nonincreasing), respectively.

Remark 21. If is -increasing or -decreasing on , then we say that is -monotone on .

Definition 22. A fuzzy function is a solution of problem (1) if and only if(1) is continuous and for all (2) exists and continuous on , where (3) satisfies (1)

Lemma 23. A -monotone fuzzy function is a solution of problem (1) if and only if(1) is continuous and for all (2) satisfies the following integral equation(3)The function is -increasing on

Proof. See the proof of Theorem 3 in [14].

Theorem 24. Assume that
is a bounded linear operator on for each .
The function is continuous.
If the assumptions hold, then problem (1) has a unique solution on .

Proof. To show that problem (1) has a unique solution defined on , we use Picard’s iteration method (see [27]).
Letand let be the operator defined as follows:Let : then, we haveand by induction, we can writeIt follows thatFinally, since for large , then by the well-known generalization of the Banach contraction principle, the operator has a unique fixed point which is the solution of problem (1).

4. Fuzzy Semilinear Fractional Evolution Equation

Definition 25. A fuzzy function is a solution of problem (2) if and only if(1) is continuous and for all (2) satisfies (2)

Lemma 26. A -monotone fuzzy function is a solution of problem (2) if and only if(1) is continuous and for all (2) satisfies the following integral equation:(3)The function is -increasing on

Proof. See the proof of Theorem 3 in [14].

Theorem 27. Assume that
The function f is continuous and there exists a positive constant K such that
for all .
If the assumptions H₁–H₃ hold andthen problem (2) has a unique solution on .

Proof. Let be the operator defined as follows:Let be a positive real number such that , where .
We can prove that the operator transforms the ball into itself, whereFor this purpose, let ; then, we haveIt follows that .
For , we haveIt follows thatwhich implies thatThus,Finally, is a contraction mapping and therefore, there exists a unique fixed point such that which is the solution of problem (2).

5. Illustrative Example

In this section, we give an example to illustrate the practical usefulness of the results that we establish in the paper. By using Theorem 27, we can solve the following fuzzy semilinear fractional evolution equation:

We define the operator by.where is the identity mapping defined on .

And the function is given by .

It is clear that is a bounded linear operator and the function is continuous. Hence, the assumptions and are verified.

On the other hand, we have

Hence, the assumption in Theorem 27 holds with .

It remains to be checked that condition (32) in Theorem 27 is also satisfied with , , , , and .

For this purpose, we have

Finally, all the conditions of Theorem 27 are satisfied for problem (40); then, it has a unique solution on [0,1].

Remark 28. We can show that the fuzzy semilinear fractional evolution equation (40) has a unique solution on [0,1] for some fractional-order . Indeed, we shall check that condition (32) in Theorem 27 is satisfied for some . We haveThen, by using Theorem 27, the fuzzy semilinear fractional evolution equation (40) has a unique solution on [0,1] for the values of satisfying (44).

6. Conclusion and Future Work

In this manuscript, we studied the existence and uniqueness results of fuzzy linear and semilinear fractional evolution equations involving Caputo fractional derivative. The existence theorems are proved by using fuzzy fractional calculus, Picard’s iteration method, and Banach fixed-point theorem. As application, we present an illustrative example to show the applicability of our main results.

Our future work is to extend the obtained results in this paper to the infinite dimensional case by using the theory of fuzzy operator semigroups.

Data Availability

The data used to support the findings of this study are included in the references within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.