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Existence and uniqueness theorem are the tool which makes it possible for us to conclude that there exists only one solution to a given problem which satisfies a constraint condition. How does it work? Why is it the case? We believe it, but it would be interesting to see the main ideas behind this. To this end, in this paper, we investigate existence, uniqueness, and other properties of solutions of a certain nonlinear fuzzy Volterra integrodifferential equation under strongly generalized differentiability. The main tools employed in the analysis are based on the applications of the Banach fixed point theorem and a certain integral inequality with explicit estimate. Also, some results for characterizing solution by an equivalent system of crisp Volterra integrodifferential equations are presented. In this way, a new direction for the methods of analytic and approximate solutions is proposed.

Many important real-world problems of analytical dynamics are described by the nonlinear mathematical models that, as a rule, are presented and modeled by the nonlinear crisp (ordinary) integrodifferential equations (IDEs). Usually, we cannot be sure that this modeling is perfect, because, in many situations, information about the real-world phenomena involved is always pervaded with uncertainty. The uncertainty can arise in experiment part, data collection, and measurement process as well as when determining the constraints conditions. Therefore, it is necessary to have some mathematical apparatus and tools in order to understand this uncertainty. In fact, the aforementioned factors will lead to errors; if the nature of errors is random, then we get a random IDE with random constraints conditions and/or random coefficients. But if the underlying structure is not probabilistic, because of subjective choice, then it may be appropriate to use fuzzy IDE with fuzzy constraints conditions and/or fuzzy coefficients. Anyhow, fuzzy IDEs are utilized to analyze the behavior of phenomena that are subject to imprecise or uncertain factors.

The study of fuzzy IDEs has gained importance in recent times; here, we are focusing our attention on first-order fuzzy Volterra IDEs (VIDEs) subject to given fuzzy initial condition. At the beginning, approaches to fuzzy IDEs and other fuzzy equations can be of three types. The first approach assumes that even if only the initial value is fuzzy, the solution is a fuzzy function, and, consequently, the derivatives in the IDE must be considered as fuzzy derivatives [

Bear in mind that not every fuzzy VIDE is solvable. But that does not mean that a solution does not exist. This is a mathematical subtlety that may not be obvious at first. There is a large divide in math between knowing that something exists and actually constructing it. In fact, we must come to grips with this idea if we are to understand the motivation for the existence and uniqueness theorem. Anyhow, it is worth stating that, in many cases, since fuzzy VIDEs are often derived from problems in physical world, existence and uniqueness are often obvious for physical reasons. Notwithstanding this, a mathematical statement about existence and uniqueness is worthwhile. Uniqueness would be of importance if, for instance, we wished to approximate the solutions. If two solutions passed through a point, then successive approximations could very well jump from one solution to the other with misleading consequences.

The purpose of this paper is to investigate the characterization theorem together with the existence and unicity of two solutions, one solution for each lateral derivative, to first-order fuzzy IDEs of Volterra type under the assumption of strongly generalized differentiability of the general form:

The solvability theory of fuzzy VIDEs has been studied by several researchers by using the strongly generalized differentiability, the Hukuhara derivative, or the Zadeh’s extension principle for the fuzzy-valued mappings of a real variable whose values are normal, convex, upper semicontinuous, and compactly supported fuzzy sets in

The organization of the paper is as follows. In the next section, we present some necessary definitions and preliminary results from the fuzzy calculus theory. The procedure of solving fuzzy VIDEs is presented in Section

The backward of theory of fuzzy VIDEs extremely appears in the references [

Let

A fuzzy number

For each

The question arising here is as follows: if we have an interval-valued function [

Suppose that

In general, we can represent an arbitrary fuzzy number

The metric structure on

For each

For arithmetic operations on fuzzy numbers, the following results are well known and follow from the theory of interval analysis. If

Let

Let

for all

for all

Here, the limit is taken in the metric space (

Let

The subsequent theorems show us a way to translate a fuzzy VIDE into a system of crisp VIDEs without the need to consider the fuzzy setting approach. Anyhow, these two theorems have many uses in the applied mathematics and the numerical analysis fields.

Let

If

If

A fuzzy-valued function

In order to complete the expert results about the fuzzy calculus theory, we finalize the present section by some preliminary information about the fuzzy integral. Following [

Suppose that

Let

Let

It should be noted that the fuzzy integral can be also defined using the Lebesgue-type approach [

The topic of fuzzy VIDEs is one of the most important modern mathematical fields that result from modeling of uncertain physical, engineering, and economical problems. In this section, we study fuzzy VIDEs using the concept of strongly generalized differentiability in which fuzzy equation is converted into equivalent system of crisp equations for each type of differentiability. Furthermore, we present an algorithm to solve the new system which consists of two crisp VIDEs.

Let us consider the following first-order equation describing the crisp VIDE:

Assume that the initial condition

On the other aspect as well, the Zadeh extension principle will lead to the following definition of

The reader is asked to refer to [

Let

The object of the next algorithm is to implement a procedure to solve fuzzy VIDE in parametric form in terms of its

To find solutions of fuzzy VIDEs (

If

If

Sometimes, we cannot decompose the membership function of the fuzzy solution

Next, we construct a procedure based on Algorithm

In order to design a scheme for solving fuzzy VIDEs (

Prior to applying the analytic or the numerical methods for solving system of crisp VIDEs (

Similarly, if

The topics of fuzzy VIDEs which is growing interest for some time, in particular in relation to fuzzy control, fuzzy population growth model, and fuzzy oscillating magnetic fields, have been rapidly developed in recent years. Anyhow, in this work, we are interested in the following main questions; firstly, under what conditions can we be sure that solutions of fuzzy VIDEs (

Throughout this paper, we will try to give the results of the all theorems and lemmas; however, in some cases, we will switch between the results obtained for the two types of differentiability in order not to increase the length of the paper without loss of generality for the remaining results. Actually, in the same manner, we can employ the same technique to construct the proof for the omitted case.

A fuzzy-valued function

Denote by

The following lemma transforms a fuzzy VIDE into two fuzzy integral equations. Here, the equivalence between two equations means that any solution of an equation is a solution too for the other one with respect to the differentiability used.

The fuzzy VIDEs (

Since

Lemma

In mathematics, the Banach fixed-point theorem, also known as the contraction mapping theorem, is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces and provides a constructive method to find those fixed points. The following results (Definition

Let

We observe that applying

Any contraction mapping

The real-valued functions

Clearly

It should be mentioned here that Lemma

Let

Without loss of generality, we consider the

Anyhow,

On the other aspect as well, differentiate both sides of (

The continuous nonlinear terms

The characterization theorem shows us the following general hint on how to deal with the analytical or the numerical solutions of fuzzy VIDEs. We can translate the original fuzzy VIDE equivalently into a system of crisp VIDEs. The solutions techniques of the system of crisp VIDEs are extremely well studied in the literature, so any method we can consider for the system of crisp VIDEs, since the solution will be as well solution of the fuzzy VIDE under study. As a conclusion, one does not need to rewrite the methods for system of crisp VIDEs in fuzzy setting, but instead, we can use the methods directly on the obtained crisp system.

A function

Consider the fuzzy VIDEs (

there exists real-finite constants

Since the proof procedure is similar for the two types of differentiability, assume that

Conversely, suppose that we have a solution (

The purpose of the next corollary is not to make an essential improvement of Theorem

Suppose that

Here, we consider the (1)-differentiability only; actually, in the same manner, we can employ the same technique in the sense of (2)-differentiability. To this end, assume the hypothesis of Corollary

The readers should pay attention to the fact that the following requirement conditions on

A popular fuzzy real number is the triangular one, which is represented with 3 tuple points

Let

Let

if

if

Let

By using the main properties of Theorems

Similarly, in the sense of (2)-differentiability, the fuzzy VIDE (

Given two triangular fuzzy numbers

Consider the fuzzy VIDEs (

there exists real-finite constants

It is easy to see that the Lipchitz property of

The continuous nonlinear terms

We know that solving fuzzy IDEs requires appropriate and applicable definitions and theorems to accomplish the mathematical construction. In this paper, we presented and proved the existence and uniqueness of two solutions of fuzzy IDEs of Volterra type based on the Hausdorff distance under the assumption of strongly generalized differentiability for the fuzzy-valued mappings of a real variable whose values are normal, convex, upper semicontinuous, and compactly supported fuzzy sets in

The authors declare no conflict of interests.

This project was funded by the deanship of scientific research of KAU under Grant no. 28-130-35-HiCi. The authors, therefore, acknowledge the technical and financial support of KAU.