Aumann Fuzzy Improper Integral and Its Application to Solve Fuzzy Integro-Differential Equations by Laplace Transform Method

We introduce the Aumann fuzzy improper integral to define the convolution product of a fuzzy mapping and a crisp function in this paper. The Laplace convolution formula is proved in this case and used to solve fuzzy integro-differential equations with kernel of convolution type. Then, we report and correct an error in the article by Salahshour et al. dealing with the same topic.


Introduction
Integrals of set-valued functions have been studied in connection with statistical problems and have arisen in connection with economic problems.The basic theory of such integrals was developed by Aumann [1].Ralescu and Adams defined in [2] the fuzzy integral of a positive, measurable function, with respect to a fuzzy measure, and studied some properties of this integral.Dubois and Prade [3] generalized the Riemann integral over a closed interval to fuzzy mappings.Their approach was more directly related to the works by Aumann [1] and Debreu [4] on multifunctions integration.Puri and Ralescu [5] generalized the integral of a setvalued function to define the concepts of fuzzy random variable and its expectation.Wu proposed in [6] two types of the fuzzy Riemann integral; the first one was based on the crisp compact interval and the second one was considered on the fuzzy interval, provided a numerical method to approximate this integral by invoking the Simpson's rule, and transformed its membership function into nonlinear programming problem.
In [7], Allahviranloo et al. proposed an integral method for solving fuzzy linear differential equations, under the assumption of strongly generalized differentiability, but they omitted the proofs of their main results.Extending their method, we developed in [8] a more general integral operator method for solving some first-order fuzzy linear differential equations with variable coefficients, and we gave the general formula's solution with necessary proofs.
The notions of the fuzzy improper Riemann integral, the fuzzy random variable, and its expectation were also investigated and studied by Wu in [9] using a different approach.
This concept of improper fuzzy Riemann integral was later exploited by Allahviranloo and Ahmadi in [10] to introduce the fuzzy Laplace transform, which they used to solve some first-order fuzzy differential equations (FDEs).Salahshour and Allahviranloo gave in [11] some applications of fuzzy Laplace transform and studied sufficient conditions ensuring its existence.Recently in [12], we extended and used the fuzzy Laplace transform method to solve second-order fuzzy linear differential equations under strongly generalized Hukuhara differentiability.Then we established in [13] some important results about continuity and strongly generalized Hukuhara differentiability of functions defined via improper fuzzy Riemann integrals, and we proved some properties of fuzzy Laplace transforms for two variables functions, which we applied to solve fuzzy linear partial differential equations of first order.
In the same context, Salahshour et al. developed in [14] the fuzzy Laplace transform method to solve fuzzy convolution Volterra integral equation (FCVIE) of the second kind.

Advances in Fuzzy Systems
But the proof proposed for their main result, Theorem 4.1 was invalid and the arguments presented in this demonstration were incorrect.One can remark that it was literally identical to the corresponding proof in the classical case, without taking into consideration the fuzzy nature of the data.
First let us recall and enounce Theorem 4.1 in [14]; then we will show the invalid arguments presented by the authors, to prove the fuzzy convolution formula.
Theorem 1 (convolution theorem: see Theorem 4.1 in [14]).If  and  are piecewise continuous fuzzy-valued functions on [0, ∞[ and of exponential order , then First notice that  and  are fuzzy-valued functions, so both of the improper integrals ∫ ∞ 0  − () and ∫ ∞ 0  − () are fuzzy numbers.Then, we cannot justify the following passage by a simple integral linearity argument: without proving that for each fuzzy number  ∈  : ∫ It was the most important key of their proof as in the crisp case, since it allows us to reverse the order of the double integrals, but unfortunately it is also incorrect, because the notion of the absolute value of a fuzzy number is not defined at least in [14].Furthermore, the concept of the absolute convergence of a fuzzy improper integral does not make sense in the fuzzy literature.
To overcome all of these obstacles, we propose in the actual paper the convolution product of a crisp mapping and a fuzzy function in Section 4, and we intend to investigate rigorously the case of two fuzzy functions in a future work.
The theory of fuzzy integro-differential equations has many applications and have been studied extensively in the fuzzy literature; for the reader, we refer to [15][16][17] and the references therein.Concerning the classical integro-differential equations, one can consult [18][19][20].
The aim of this work is to define the convolution product and to prove a fuzzy Laplace convolution formula, in the purpose of solving the following fuzzy integro-differential equations (FIDEs) with kernel of convolution type: provided that  : [0, ∞[→ ,  : [0, ∞[× →  are continuous fuzzy-valued functions and  : [0, ∞[→ R is a crisp continuous function verifying some assumptions to be mentioned later.
Then we give some examples to illustrate the efficiency of our method for solving FIDEs.
To achieve this goal, we first introduce the Aumann fuzzy improper integral concept, which we utilize instead of the Riemann fuzzy improper integral used in [10,[12][13][14].
This new definition of fuzzy generalized (improper) integral is essentially based on the notion of fuzzy integral and the expectation of a fuzzy random variable, introduced by Puri and Ralescu in [5].
The remainder of this paper is organized as follows.
Section 2 is reserved for some preliminaries.And Section 3 is devoted to the definition of the Aumann fuzzy improper integral.In Section 4, fuzzy Laplace transform is introduced, its basic properties are studied, and a particular case of Laplace convolution is investigated.Then in Section 5, the main result about Laplace convolution is enounced and proved.The procedure for solving fuzzy integro-differential equations by fuzzy Laplace transform is proposed and some numerical examples are given in Section 6.In the last section, we present conclusion and a further research topic.

Preliminaries
Denote by   (R) the family of all nonempty compact convex subsets of R and define the addition and scalar multiplication in   (R) as usual.The distance between two nonempty bounded subsets  and  of R is defined by the Hausdorff metric Define where is the support of , and its closure cl (supp ) is compact.
For 0 <  ≤ 1, the -cut (or level) of  is denoted Then, from (i) to (iv), it follows that the -level set []  ∈   (R) for all 0 ≤  ≤ 1.It is well known that Let  :  ×  → [0, ∞) be a function which is defined by the equation where  is the Hausdorff metric defined in   (R).Then, the following properties hold true (see [5,21]): (1) (, ) is a complete metric space.
The following general definition and properties were developed by Puri and Ralescu in [5], for the fuzzy Aumann integral theory in   .Here, we restrict their theory to  =  1 instead of   .
Let (Ω, A, ) be a probability space where the probability measure  is assumed to be nonatomic.
Definition 4 (Puri and Ralescu [5]).Let (Ω, A, ) be a probability space where the probability measure  is assumed to be nonatomic.A set-valued function is a function  : Ω → P(R) such that () ̸ = 0 for every  ∈ Ω.By  1 () we denote the space of -integrable functions  : Ω → R. We denote by () the set of all  1 () selections of ; that is, The Aumann integral of , denoted by ∫ Ω   or ∫  for short, is defined by Definition 5. A strongly measurable and integrably bounded mapping  : Ω →  is said to be integrable over Ω if ∫ Ω  ∈ .
Remark 8 (Puri and Ralescu [5]).It is important to observe that Theorem 7 can be stated in a different form by replacing convergence in the sense of Kuratowski by convergence in the Hausdorff metric.The statement of the theorem remains unchanged provided that we assume that all functions take values in (R), the set of all nonempty, compact subsets of R.
Now, we define the Hukuhara difference and the strongly generalized differentiability.
The following theorem (see [22]) allows us to consider case (i) or (ii) of the previous definition almost everywhere in the domain of the functions under discussion.

Aumann Fuzzy Improper Integral
Since the Aumann integral over [, ] is linear (see [24]), then from Theorem 13, we deduce the linearity of the Aumann improper fuzzy integral over [, ∞[.Then, we said that a fuzzy mapping  : R →  is integrable over R, if it is integrable over ] − ∞, ] and over [, ∞[, for each real .In this case, we define For more details concerning Aumann fuzzy improper integral, one can see [5].
Remark 16.The concepts of the fuzzy improper integral, the fuzzy random variable, and its expectation were defined and studied in a different way by Wu in [9].His proposal of the improper fuzzy Riemann integral was an appropriate attempt for finding the expectations of fuzzy random variables numerically.He stated that the developments in [5] were in measuretheoretic sense; thus, it was difficult to provide a numerical method in applications.
However, this statement seems to be false because of the approach developed in our present article and precisely by the identities ( 16) and ( 17); the Aumann fuzzy improper integral (and the integral over a compact subset of R) has the same properties and qualities as well as the improper fuzzy Riemann integral.
Proof.To prove Theorem 18, one can adopt the proof in [10] using Aumann fuzzy improper integral instead of Riemann fuzzy improper integral.
If  1 and  2 are two crisp functions defined from [0, ∞[ into R, then, we recall the well-known classical convolution Laplace formula: Then from (30)-( 33) and since L[()] ≤ 0, we deduce In both cases, we have Remark 22. Now let us recall the error in [25] Example 1.
The authors studied the following fuzzy integro-differential equation using fuzzy differential transform method (DTM): But  = ( + 1,  − 2) is not a fuzzy number in the parametric form, since the function  =  − 2 is not decreasing.Note that the second initial data   (0) can be obviously deduced by taking  = 0 in the equation.(39) In this case, since () is (i)-differentiable, the solution is valid. , Then by the inverse Laplace transform the lower and upper functions of solution of ( 37 (41) In this case, () is (ii)-differentiable only for  ∈ [7/4, 2] and the solution is acceptable only over this interval.
By the inverse Laplace transform we get the lower and upper functions of solution of ( 42  (46) In this case, () is (ii)-differentiable only for  ∈ [3/2, 2], so the solution is valid only over this interval.

Main Result
To overcome all the obstacles and to avoid the error in [14], we propose in this paper the convolution product of crisp and fuzzy functions, and we intend to investigate rigorously the case of two fuzzy functions in a future work.Now, we enounce our main result giving the convolution Laplace formula generalizing the result in Section 4. (47) Proof.Let  ≥ 0 and  > 0. It is obvious that [( * )()]  = ( *   )().

Fuzzy Laplace Transform Algorithm for Solving Fuzzy Integro-Differential Equations
Our aim now is to solve the following fuzzy integro-differential equation using fuzzy Laplace transform method under strongly generalized differentiability: Please notice that Theorem 10 allows us to use only (i) or (ii) type of strongly generalized differentiability.
Assume in a first time that L[()] ≥ 0. By using the fuzzy Laplace transform and Theorem 25, we have Then, we have the following alternatives for solving (56).(66) By using the inverse Laplace transform, we get Remark 26.Similarly, if we assume that L[()] < 0, we obtain the following results.
By the inverse Laplace transform we get the lower and upper functions of solution of (72) for  ≥ 0  (, ) In this case, the solution is invalid over [0, ∞[, since () is not (i)-differentiable.
By solving the linear system (75) and using the inverse Laplace transform, we get One can verify that in this case the solution is acceptable over a closed interval [, ] such that  (81) In this case, the solution is valid over [0, ∞[, since () is (i)-differentiable.

Conclusion
In this paper, we have introduced the Aumann fuzzy improper integral, and also we have applied Laplace transform method for solving FIDEs, with kernel of convolution type, under the assumption of strongly generalized differentiability.Clearly, the suggested formula allows us to solve more difficult FIDEs by Laplace method compared to the previously reported works.
Indeed, in the most fuzzy examples studied before, the considered kernels () were real and nonnegative constants.
But in this paper, we treated various cases for this kernel (): positive or negative in the first and second examples, respectively; () = sin() and () = exp(−) were nonconstant functions of  in the third and fourth ones.