^{1}

^{1}

^{1}

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.

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 [

Puri and Ralescu [

In [

The notions of the fuzzy improper Riemann integral, the fuzzy random variable, and its expectation were also investigated and studied by Wu in [

This concept of improper fuzzy Riemann integral was later exploited by Allahviranloo and Ahmadi in [

In the same context, Salahshour et al. developed in [

But the proof proposed for their main result, Theorem

First let us recall and enounce Theorem 4.1 in [

If

First notice that

Moreover, the authors claimed that due to the hypothesis on

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 [

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

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 [

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:

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 [

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 [

The remainder of this paper is organized as follows.

Section

Denote by

For

A fuzzy number

A crisp number

The following general definition and properties were developed by Puri and Ralescu in [

Let

A mapping

A mapping

Let

A strongly measurable and integrably bounded mapping

If

If

It is important to observe that Theorem

Now, we define the Hukuhara difference and the strongly generalized differentiability.

For

We say that a fuzzy mapping

for all

or

for all

or

for all

or

for all

The following theorem (see [

Let

Let

If

If

Considering the positive measure related to the exponential law on the positive real line

We define the Aumann fuzzy improper integral

A strongly measurable and integrably bounded mapping

Using Lemma

If

Since the Aumann integral over

If

Analogously, we define the integrability and the Aumann fuzzy improper integral

Then, we said that a fuzzy mapping

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 [

He stated that the developments in [

However, this statement seems to be false because of the approach developed in our present article and precisely by the identities (

Let

If

Let

If

If

To prove Theorem

Let

Theorem

Let

Suppose that

If the function

Therefore,

If

Then using (

If the function

Therefore,

Then from (

Now let us recall the error in [

Note that the second initial data

We correct the previous fuzzy Volterra integro-differential equation as follows:

By the inverse Laplace transform, we get the lower and upper functions of solution of (

In this case, since

Then by the inverse Laplace transform the lower and upper functions of solution of (

In this case,

We consider the following fuzzy Volterra integro-differential equation:

By the inverse Laplace transform we get the lower and upper functions of solution of (

In this case, the solution is acceptable since

Using the inverse Laplace transform, we obtain the solution of (

In this case,

To overcome all the obstacles and to avoid the error in [

Let

Let

Let

Hence,

Let

Therefore, (

If we denote

Since

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

Assume in a first time that

By using the fuzzy Laplace transform and Theorem

Then from (

Using

Therefore,

By using the inverse Laplace transform, we get

Then from (

Using

That is,

Then by solving the linear system (

By using the inverse Laplace transform, we get

Similarly, if we assume that

If

where

By using the inverse Laplace transform, we get

If

By using the inverse Laplace transform, we obtain

We consider the following fuzzy integro-differential equation:

By the inverse Laplace transform we get the lower and upper functions of solution of (

In this case, the solution is invalid over

By solving the linear system (

One can verify that in this case the solution is acceptable over a closed interval

Analogously, we can solve the following generalized fuzzy integro-differential equation, with kernel of convolution type via Laplace transform method:

We consider the following known fuzzy integro-differential equation:

By the inverse Laplace transform we get the lower and upper functions of solution of (

In this case, the solution is valid over

By solving the linear system (

Notice that the length of

Taking

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

But in this paper, we treated various cases for this kernel

For future research, we will apply Laplace transform method to solve FIDEs with a fuzzy kernel.

The authors declare that there are no conflicts of interest regarding the publication of this paper.