Fuzzy Fractional Evolution Equations and Fuzzy Solution Operators

Abstract

In this paper, the fuzzy fractional evolution equations of order q (FFEE) with fuzzy Caputo fractional derivative are introduced. We study the existence and uniqueness of mild solutions for FFEE under some conditions. Also, we generalize the definition of the fuzzy fractional integral and derivative order q. The fuzzy Laplace transform is presented and proved. The solvability of the problem (FFEE) and the properties of the fuzzy solution operator and its generator are investigated and developed.

1. Introduction

The fuzzy fractional differential equations (FFDEs) can also offer a more comprehensive account of the process or phenomenon. This has recently captured much attention in FFDEs. As the derivative of a function is defined in the sense of Riemann–Liouville, Grünwald–Letnikov, or Caputo in fractional calculus, the used derivative is to be specified and defined in FFDEs as well. FFDEs are examined in [1–5]. We adopted the fuzzy Laplace transform method to solve FFDEs because it has the advantage that it solves problems directly without determining a general solution and obtaining nonhomogeneous differential equations [5].

C. G. Gal and S. G. Gal studied [6], with more details, fuzzy linear and semilinear (additive and positive homogeneous) operator theory on the complete metric space.

Let , and be the set of fuzzy real numbers. Our aim in this paper is to investigate the existence and uniqueness of the fuzzy mild solution for the fuzzy fractional evolution equation:where is the infinitesimal generator of a q-resolvent family , defined as , satisfies some conditions that will be specified later, and the fuzzy fractional derivative is understood here in the caputo sense.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.

Let us denote by the class of fuzzy subsets of the real axis satisfying the following properties [7]:(i)u is normal, i.e., there exists an , such that .(ii)u is fuzzy convex, i.e., for and ,(iii)u is upper semicontinuous.(iv) is compact.

Then, is called the space of fuzzy numbers. Obviously, . For , denote , and then from (i) to (iv), it follows that the α-level set for all is a closed bounded interval which we denote by .

Where denotes the family of all nonempty compact convex subsets of and defines the addition and scalar multiplication in as usual. For later purposes, we define as

Theorem 1. (see [8]). If , then(i) for all .(ii) for all .(iii) is a nondecreasing sequence which converges to α and then,

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

Lemma 1. (see [9]). Let be the fuzzy sets.

Then, if and only if for all

The following arithmetic operations on fuzzy numbers are well known and frequently used below. If , then

For , if there exists such that , then is the Hukuhara difference of u and denoted by .

Define by the following equation:where is the Hausdorff metric defined in .

It is well known that is a complete metric space [10]. We list the following properties of :for all and .

Let be a compact interval. We denote by the space of all continuous fuzzy functions on T and is a complete metric space with respect to the metric

Also, we denote by the space of all fuzzy functions which are Lebesgue integrable on the bounded interval T of .

Let be a fuzzy function. We denote

The derivative of a fuzzy function u is defined by [11]provided this equation defines a fuzzy number . The fuzzy integral , is defined by [12]provided that the Lebesgue integrals on the right exist.

From [13], we have the following theorems:

Theorem 2. There exists a real Banach space X such that can be the embedding as a convex cone C with vertex 0 into X. Furthermore, the following conditions hold:(i)The embedding j is isometric(ii)Addition in X induces addition in , i.e., for any (iii)Multiplication by a non-negative real number in X induces the corresponding operation in , i.e., for any (iv)C-C is dense in X(v)C is closed

Remark 1. Let as . It verifies the following properties:

Theorem 3. Let X be a Banach space and j an embedding as in Theorem 2, , and assume is Bochner integrable over T. Then, , and

Remark 2. By the definition of fuzzy integral [14], the above equality yields

3. Fuzzy Fractional Integral and Fuzzy Fractional Derivative

Let be such that for all and . Suppose that for all , and let

Lemma 2. (see [1]). The family , given by (16), defined a fuzzy number such that .
We definewhere is the nth derivative of the delta function and is the gamma function (for the properties of , see [15, 16]). These functions satisfy the semigroup property:The Sobolev spaces can be defined in the following way [17]:Note that . LetSo, , if for some .

Remark 3. Let be such that for all and . Suppose that for all , and letNote that .
According to lemma (2) and by the notation of and the family , given by (21), defined a fuzzy number such that .

3.1. Fuzzy Fractional Integral and Derivative

Let . Define the fuzzy fractional primitive of order of :by

For , we obtain , that is, the integral operator. Also, the following properties are obvious:(i) for each constant (ii)

Proposition 1. [1]. If , then we have

Definition 1. Let be a given function such that for all , and , the fuzzy fractional differential operator in the Riemann–Liouville sense, is defined for all u satisfyingbyIn fact,and the fuzzy fractional differential operator in the Caputo sense is defined:byFor , we obtain , provided that the equation defines a fuzzy number . In fact,Some simple but relevant results valid for are

Proposition 2. Let and . Then, for any ,

If moreover (25) holds, then

Proof. Let and be such that for all and . If for all , then satisfies (25): (i.e., ) andThat is, . So, we can apply to and thanks to the semigroup property (24)If u satisfies (25), then according to (19),where and . Therefore,Convolving both sides of (37) with and applying the semigroup property (18), we obtainAn application of to both sides givesand then,which together with (38) implies (34). If , that is, , , we have .

Remark 4. and then,Therefore,and the first identity is valid for all , the second for .

3.2. Fuzzy Laplace Transform

Let be such that for all . Suppose that for all . We define the fuzzy Laplace transform [18] bywhere and real.

Theorem 4. [18]. Let u and are continuous fuzzy-valued functions. Suppose that and are constants. Then,

Lemma 3. Let u is continuous fuzzy-valued function and :

As in [5], we can introduce laplace transforms of derivatives by:

Proposition 3. Let , and then

Proof. We prove (49) because the proof of (48) is similar.
For arbitrary fixed , we haveWe apply the properties of the Laplace transform, and since , we obtainThen, we conclude thatby linearity of L,Using (31) leads to obtain

3.3. Fuzzy Solution Operators

We adopt the general definition and theorem of operator theory on in [6]. Let is linear iffor all , , where is and is continuous at each and is semilinear and continuous at .

Let us consider the metric where and we have , where is given by .

Theorem 5. Let A be a bounded linear operator on [6]. If , then is invertible; moreover,Here, denotes the identity function of .

Proof. Recall that Setting by (Theorem 3.5 and Corollary 3.6 in [6]), it suffices to show that is a Cauchy sequence in the complete metric space . First , it follows for . Then for we haveHowever,and, so by induction, one can obtainWe obtainand passing to supremum with we getTherefore, is a uniform Cauchy sequence on .and . It now follows that ; therefore, its limit exists in . So, holds uniformly on each compact interval .

Motivated by the above definitions in [19, 20], we can give the following definition.

Definition 2. The resolvent set of A, denoted by , is the set of all real numbers such that is bijective, i.e.,

Remark 5. Let . Then, . It follows from Theorem 5 that exists, and

Lemma 4. Let and tow operator.
A is the operator of the resolvent on if and only if is the operator of the resolvent defining on the convex closed set C and and .

Proof. We assume that on . By the properties of j we have, for all ,It follows thatConversely, if on C, then for all It follows thatwhich completes the proof.□
Next, we need to define the fuzzy solution operator (or fuzzy q-resolvent family), which is similar to that given in [19].
Consider the following particular case of (82) for the Caputo fractional derivative evolution equation of order is an integer:where .
Applying (44), we obtain that the Cauchy problem (26)and then we define the solution operator of (70) in terms of the corresponding integral equation (71).

Definition 3. A family of bounded linear operators on is called a solution operator for (71) (or q-resolvent family), if the following conditions are satisfied:(1) is strongly continuous for and , the identity mapping on (2) and for all , (3) is a solution of (71) for all ,

Definition 4. The solution operator is called exponentially bounded if there are constants and such that

Proposition 4. If is the solution operator of (70) and , then if for , the Hukuhara difference exists, we definewhich this limit exists in the metric space .

Proof. Let , we haveTaking and using (70) and (44), we obtain (73)

Lemma 5. Let and tow operator.
A is the generator of a fuzzy q-resolvent family on if and only if is the generator of an q-resolvent family defining on the convex closed set C by .

Proof. Follows from the definition of and (see [19]), then is the fuzzy solution operator on if and only if is the solution operator on C.
We assume that A is the generator of a fuzzy q-resolvent family on . By the properties of j we have, for all ,Conversely, if is the generator of a q-resolvent family on C, then for all

Lemma 6. Let A is a operator in (70) and j an embedding as in Theorem 2, the solution operators of (70) is defined by

Proof. Taking the fuzzy Laplace transform of (70) on both sides, we obtainand using the j,Since exist, i.e, (see [20]), from the above equation, we obtainNow (87) follows easily by taking the inverse of Laplace transform and applying This completes the proof.

4. Fuzzy Fractional Differential Equations

Consider the following fuzzy fractional differential equation:where A generator of q-resolvent family on , is the fuzzy caputo fractional differential operator define by (29) and is continuous.

Firstly, we consider the following Cauchy problem

The roblem (70) is particular case of (83), and if (70) has a solution operator , then the corresponding problem (83) is uniquely solvable with the solutionprovided , . For this reason, we restrict ourselves to problem (71) (in crisp, see [19]).

Next, we consider the particular case of (82).where A is a operator and f is an abstract function defined on and with values in .

Theorem 6. Let A is an operator, be continuous on T and if f satisfies a Hölder condition with an exponent of The function is a solution of (85) if and only ifwhere

Proof. Now applying the fuzzy Riemann–Liouville fractional integral operator (23) in (85) on both sides, we getand taking the fuzzy Laplace transform of (89) on both sides, we obtainBy using j and if exists, i.e., , from the above equation, we obtainNow, (87) follows easily by taking the inverse of Laplace transform:and applying and Lemma 4,This completes the proof.

Theorem 7. Let A be a operator. If satisfies (86), then the solutions of the Cauchy problem (82) are fixed points of the operator equation:Theorem 7 leads to the following appropriate definition of a mild solution to (82).

Definition 5. A function is called a mild solution of (82), if it satisfies the operator equation:

Theorem 8. Let A is a operator and be continuous on T. Assume that there exists and such that  there exists a constant , such thatfor every , . Then, problem (82) has a unique mild solution .

Proof. Let be the operator defined byWe have to show that u is mild solution of (82) if and only if and , we find thatWe can deduce thatand it follows thatSince , such that .
It follows that is a contraction, and then such that .
We prove .
implies that , one can write it follows that is a fixed point of , and since the fixed point is unique, we get .
Hence, (85) has a unique solution.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.