Research Article | Open Access
Existence, Uniqueness, and Characterization Theorems for Nonlinear Fuzzy Integrodifferential Equations of Volterra Type
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 [1, 2]. These can be done by the use of the Hukuhara derivative for fuzzy-valued functions. Generally, this approach has a drawback; the solution becomes fuzzier as time goes; hence, the fuzzy solution behaves quite differently from the crisp solution. In the second approach, the fuzzy IDE is transformed to a crisp one by interpreting it as a family of differential inclusions [3, 4]. The main shortcoming of using differential inclusions is that we do not have a derivative of a fuzzy-valued function. The third approach is based on the Zadeh’s extension principle, where the associated crisp problem is solved and in the solution the initial fuzzy values are substituted instead of the real constants, and, in the final solution, arithmetic operations are considered to be operations on fuzzy numbers [5, 6]. The weakness of this approach is the need to rewrite the solution in the fuzzy setting which in turn makes the methods of solution are not user-friendly and more restricted with more computation steps. As a conclusion, to overcome the above-mentioned shortcoming, the concept of a strongly generalized differentiability was developed and investigated in [7–14]. Anyhow, using the strongly generalized differentiability, the fuzzy IDE has locally two solutions. Indeed, with this approach, we can find solutions for a larger class of fuzzy IDEs compared to using other types of differentiability.
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: subject to the fuzzy initial condition where and are continuous fuzzy-valued functions that satisfy a Lipchitz condition, , and are real finite constants with .
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 reader is asked to refer to [15–22] in order to know more details about these analyses, including their kinds and history, their modifications and conditions for use, their scientific applications, their importance and characteristics, and their relationship including the differences. But, on the other aspect as well, more details about the solvability theory of crisp VIDEs can be found in [23, 24] and more details about the characterization theorem can be found in [25, 26].
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 3. In Section 4, existence and uniqueness of two solutions are introduced. In Section 5, we utilize two characterization theorems and some properties for the solution of fuzzy VIDEs. This paper ends at Section 6 with some concluding remarks.
2. Background Material of Fuzzy Calculus Theory
The backward of theory of fuzzy VIDEs extremely appears in the references [15–22] side by side with their applications in the field of engineering problems, applied mathematics, theoretical physics, and mathematical finance. Anyhow, for the reader’s convenience, we present some necessary definitions from fuzzy calculus theory and preliminary results. For the concept of fuzzy derivative, we will adopt strongly generalized differentiability, which is a modification of the Hukuhara differentiability and has the advantage of dealing properly with fuzzy VIDEs.
Let be a nonempty set. A fuzzy set in is characterized by its membership function . Thus, is interpreted as the degree of membership of an element in the fuzzy set for each . A fuzzy set on is called convex if for each and , , is called upper semicontinuous if is closed for each , and is called normal if there is such that . The support of a fuzzy set is defined as .
Definition 1 (see ). A fuzzy number is a fuzzy subset of the real line with a normal, convex, and upper semicontinuous membership function of bounded support.
For each , set and , where denotes the closure of . Then, it easy to establish that is a fuzzy number if and only if is compact convex subset of for each and . Thus, if is a fuzzy number, then , where and for each . The symbol is called the -cut representation or parametric form of a fuzzy number . We will let denote the set of fuzzy numbers on .
The question arising here is as follows: if we have an interval-valued function  defined on , then is there a fuzzy number such that ? The next theorem characterizes fuzzy numbers through their -cut representations.
Theorem 2 (see ). Suppose that and satisfy the following conditions: first, is a bounded increasing function and is a bounded decreasing function with ; second, for each , and are left-hand continuous functions at ; third, and are right-hand continuous functions at . Then, , defined by is a fuzzy number with parameterization given by . Furthermore, if is a fuzzy number with parameterization , then the functions and satisfy the aforementioned conditions.
In general, we can represent an arbitrary fuzzy number by an order pair of functions () which satisfy the requirements of Theorem 2. Frequently, we will write simply and instead of and , respectively.
The metric structure on is given by such that for arbitrary fuzzy numbers and , where is the Hausdorff metric between and . This metric is defined as , where the two sets and are the -neighborhoods of and , respectively. It is shown in  that is a complete metric space.
Lemma 3 (see ). For each with , the metric function satisfies the following properties:(i);(ii);(iii);(iv).
For arithmetic operations on fuzzy numbers, the following results are well known and follow from the theory of interval analysis. If and are two fuzzy numbers, then, for each , we have, firstly, ; secondly, ; thirdly, ; fourthly, if and only if if and only if and . In fact, the collection of all fuzzy numbers with aforementioned addition and scalar multiplication is a convex cone .
Let . If there exists a such that , then is called the -difference of and , denoted by . Here, the sign “” stands always for -difference and let us remark that . Usually, we denote by , while stands for the -difference. It follows that Hukuhara differentiable function has increasing length of support . To avoid this difficulty, we consider the following definition.
Definition 4 (see ). Let and . One says that is strongly generalized differentiable at , if there exists an element such that, either(i)for all being sufficiently close to , the -differences , exist and ;(ii)for all being sufficiently close to , the -differences , exist and .
Here, the limit is taken in the metric space () and at the endpoints of , we consider only one-sided derivatives. For customizing, in Definition 4, the first case corresponds to the H-derivative introduced in , so this differentiability concept is a generalization of the Hukuhara derivative.
Definition 5 (see ). Let . One says that is (1)-differentiable on  if is differentiable in the sense (i) of Definition 4 and its derivative is denoted by . Similarly, one says that is (2)-differentiable on  if is differentiable in the sense (ii) of Definition 4 and its derivative is denoted by .
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.
Theorem 6 (see ). Let and put for each .(i)If is -differentiable, then and are differentiable functions on and ;(ii)If is -differentiable, then and are differentiable functions on and .
A fuzzy-valued function is called continuous at a point provided for arbitrary fixed ; there exists an such that whenever for each . We say that is continuous on if is continuous at each such that the continuity is one-sided at endpoints and .
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 , we define the integral of a fuzzy-valued function using the Riemann integral concept.
Definition 7 (see ). Suppose that , for each partition of  and for arbitrary points , ; let and . Then, the definite integral of over  is defined by provided that the limit exists in the metric space ().
Theorem 8 (see ). Let be continuous fuzzy-valued function and put for each . Then, exist, belonging to , and are integrable functions on , and .
Lemma 9 (see ). Let be integrable fuzzy-valued function and . Then, the following hold:(i) is integrable;(ii), ;(iii);(iv).
It should be noted that the fuzzy integral can be also defined using the Lebesgue-type approach  or the Henstock-type approach . However, if is continuous function, then all approaches yield the same value and results. Moreover, the representation of the fuzzy integral using Definition 7 is more convenient for numerical calculations and computational mathematics. The reader is kindly requested to go through [27, 28, 31–33] in order to know more details about the fuzzy integrals, including their history and kinds, their properties and modification for use, their applications and characteristics, their justification and conditions for use, and their mathematical and geometric properties.
3. Solving Fuzzy Volterra Integrodifferential Equation
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: subject to the crisp initial condition, where and are continuous real-valued function and with .
Assume that the initial condition in (5) is uncertain and modeled by a fuzzy number. Also, assume that the functions and in (4) contain uncertain parameters modeled by a fuzzy number. Then, we obtain the fuzzy VIDEs (1) and (2). In order to solve this problem, we write the fuzzy function in terms of its -cut representation form to get and .
On the other aspect as well, the Zadeh extension principle will lead to the following definition of when is a fuzzy number : . Indeed, according to Nguyen theorem  it follows that where the two term endpoints functions and are defined, respectively, as and . Similarly, the parametric form of is given as
The reader is asked to refer to [34–37] in order to know more details about Zadeh’s extension principle and Nguyen theorem, including their justification and conditions for use, their mathematical and geometric properties, their types and kinds, and their applications and method of calculations.
Definition 10. Let such that or exists. If and satisfy fuzzy VIDEs (1) and (2), one says that is a (1)-solution of fuzzy VIDEs (1) and (2). Similarly, if and satisfy fuzzy VIDEs (1) and (2), one says that is a (2)-solution of fuzzy VIDEs (1) and (2).
The object of the next algorithm is to implement a procedure to solve fuzzy VIDE in parametric form in terms of its -cut representation, where the new obtained system consists of two crisp VIDEs for each type of differentiability.
Case 1. If is (1)-differentiable, then and solving fuzzy VIDEs (1) and (2) translates into the following subroutine.
Step 1. Solve the following system of crisp VIDEs for and : subject to the crisp initial conditions,
Step 2. Ensure that the solution  and its derivative  are valid level sets for each .
Step 3. Use (3) to construct a (1)-solution such that for each .
Case 2. If is (2)-differentiable, then and solving fuzzy VIDEs (1) and (2) translates into the following subroutine.
Step 1. Solve the following system of crisp VIDEs for and subject to the crisp initial conditions,
Step 2. Ensure that the solution  and its derivative  are valid level sets for each .
Step 3. Use (3) to construct a (2)-solution such that for each .
Sometimes, we cannot decompose the membership function of the fuzzy solution as a function defined on for each . Then, using identity (3), we can leave the solution in terms of its -cut representation form.
Next, we construct a procedure based on Algorithm 11 in order to obtain the analytic or the approximate solutions of fuzzy VIDEs (1) and (2). Anyhow, by considering the (1)-differentiability concept and without loss of generality, we assume that the function takes the form . So, based on this, fuzzy VIDE (1) can be written in a new discretized form as , in which the -cut representation form of should be of the form
In order to design a scheme for solving fuzzy VIDEs (1) and (2), we first replace it by the following equivalent crisp system of VIDEs: subject to the crisp initial conditions, where the new functions are given, respectively, as
Prior to applying the analytic or the numerical methods for solving system of crisp VIDEs (13) and (14), we suppose that the kernel function is nonnegative for and nonpositive for . Therefore, system of crisp VIDEs (13) can be translated again into the following equivalent form:
4. Existence and Uniqueness of Two Fuzzy Solutions
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 (1) and (2) exist? Secondly, under what conditions can we be sure that there are two unique solutions, one solution for each lateral derivative, to fuzzy VIDEs (1) and (2)?
Remark 12. 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 is called continuous at a point () in provided for arbitrary ; there exists an such that whenever and for each and . Similarly, for with the need for attention to change the metric use on .
Denote by the set of all continuous mapping from  to . The supremum metric on is defined by such that for each , where is fixed. It is shown in  that () is a complete metric space.
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.
Lemma 13. The fuzzy VIDEs (1) and (2), where and are supposed to be continuous, are equivalent to one of the following fuzzy integral equations:(i);(ii), depending on the strong differentiability considered, -differentiability or -differentiability, respectively.
Proof. Since and are continuous functions, so they are integrable. Now, we determine the equivalent integral forms of fuzzy VIDEs (1) and (2) under each type of strongly generalized differentiability as follows. Firstly, let us consider that is (1)-differentiable; then the equivalent integral form of fuzzy VIDEs (1) and (2) can be written by implementation of fuzzy integration on both sides of the original equation one time as follows: Secondly, let us consider that is (2)-differentiable; then the equivalent integral form of fuzzy VIDEs (1) and (2) can be written as which is equivalent to the form of part (ii) of Lemma 13.
Lemma 13 can be reformulated as follows: a mapping is a solution to the fuzzy VIDEs (1) and (2) if and only if is continuous and satisfy Case (i) or Case (ii) in the sense of -differentiability or in the sense of -differentiability, respectively.
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 14 and Theorem 15) were collected from .
Definition 14. Let be a metric space. A mapping is said to be a contraction mapping, if there exists a positive real number with such that for each .
We observe that applying to each of the two points of the space contracts the distance between them; obviously, is continuous. Anyhow, a point is called a fixed point of the mapping if . Next, we present the Banach fixed-point theorem.
Theorem 15. Any contraction mapping of a nonempty complete metric space into itself has a unique fixed point.
Lemma 16. The real-valued functions with , represented by are continuous nondecreasing functions on . Furthermore, , , and .
Proof. Clearly are continuous functions on for each . Since and for each and , thus, are nondecreasing functions. As a result, one can conclude that and . On the other aspect as well, using the limit functions techniques, yields that
It should be mentioned here that Lemma 16 guarantees the existence of a unique fixed point for the next theorem. In other words, an existence of a unique solution for fuzzy VIDEs (1) and (2) for each type of differentiability.
Theorem 17. Let and be continuous fuzzy-valued functions. If there exists such that for each , , and , then, the fuzzy VIDEs (1) and (2) have two unique solutions on . One is -differentiable solution and the other one is -differentiable solution.
Proof. Without loss of generality, we consider the -differentiability only; actually, in the same manner, we can employ the same technique for the -differentiability. For each and , define the operator as
Thus, is continuous and . Now, we are going to show that the operator satisfies the hypothesis of the Banach-fixed point theorem. For each , we have
But since from Lemma 16, so, we can choose such that
Anyhow, G is a contractive mapping, whilst the unique fixed point of is in the space . Using the fact that is the integral of a continuous function, we conclude that it is actually in the space . Hence, by the Banach fixed-point theorem, fuzzy VIDEs (1) and (2) have a unique fixed point , that is, a continuous function on  satisfying . As a result, writing ()() out, we have by (23)
On the other aspect as well, differentiate both sides of (26) and substitute to obtain fuzzy VIDEs (1) and (2). Hence, every solution of fuzzy VIDEs (1) and (2) must satisfy (26), and conversely. So, the proof of the theorem is complete.
Remark 18. The continuous nonlinear terms and are said to satisfy a Lipchitz condition relative to their last argument in fuzzy sense with respect to the metric space () if the conditions of (22) of Theorem 17 hold.
5. Generalized Characterization Theorem
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 is said to be equicontinuous if, for any and any , we have , whenever , and uniformly bounded on any bounded set. Similarly, for a function defined on with the need for attention to change the metric used on .
Theorem 19. Consider the fuzzy VIDEs (1) and (2) where and are such that (i), , and , ;(ii) and are equicontinuous functions and uniformly bounded on any bounded set;(iii)there exists real-finite constants such that for each , , and . Then, for -differentiability, the fuzzy VIDEs (1) and (2) and the system of crisp VIDEs (8) and (9) are equivalent; and, in -differentiability, the fuzzy VIDEs (1) and (2) and the system of crisp VIDEs (10) and (11) are equivalent.
Proof. Since the proof procedure is similar for the two types of differentiability, assume that is (1)-differentiable without loss of generality. The equicontinuity of and implies the continuity of and , respectively. Furthermore, the Lipchitz property of condition (iii) ensures that and satisfy a Lipchitz property in the metric space () as follows:
while, on the other hand, by similar fashion, it is easy to conclude that
By the continuity of and , from the last Lipchitz conditions of (28) and (29), and the boundedness property of condition (ii), it follows that fuzzy VIDEs (1) and (2) have a unique solution on . Whilst the solution of fuzzy VIDEs (1) and (2) is (1)-differentiable and so, by Theorem 6 Case (i), the functions and are differentiable on , as a conclusion, one can obtained that () is a solution of crisp VIDEs (8) and (9).
Conversely, suppose that we have a solution () with being fixed of fuzzy VIDEs (1) and (2) (note that this solution exists by property of condition (iii)), whilst the Lipchitz conditions of (28) and (29) imply the existence and uniqueness of fuzzy solution . Indeed, since is (1)-differentiable, then and the endpoints of are a solution of crisp VIDEs (8) and (9) (note that is obviously valid level sets of a fuzzy-valued function). But since the solution of crisp VIDEs (8) and (9) is unique, we have . That is, the fuzzy VIDEs (1) and (2) and the system of crisp VIDEs (8) and (9) are equivalent. This completes the proof of the theorem.
The purpose of the next corollary is not to make an essential improvement of Theorem 19 but rather to give alternate conditions under which fuzzy VIDEs (1) and (2) and the corresponding system of crisp VIDEs are equivalent.
Corollary 20. Suppose that and are such that the condition (i) of Theorem 19 holds. If there exists real-finite constants such that for each , , and , then, for -differentiability, the fuzzy VIDEs (1) and (2) and the system of crisp VIDEs (8) and (9) are equivalent; and, in -differentiability, the fuzzy VIDEs (1) and (2) and the system of crisp VIDEs (10) and (11) are equivalent.
Proof. 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 20; then the conditions (i) and (iii) of Theorem 19 clearly hold. To establish condition (ii), apply the following: fix , choose , and suppose . Then, for each , one can write Next, we want to show that are uniformly bounded on any bounded set. To do so, let be any bounded subset of