Existence and uniqueness of solution for fuzzy integral equations of product type

In this paper, we study the fuzzy Fredholm integral equations of product type. As a special case, integral equations of product type arise, e.g., in the study of the spread of an infectious disease that does not induce permanent immunity. Because of the importance and practicality of such integrals, we prove the existence and uniqueness of the solution of these integral equations with the help of the Banach fixed point theorem. Finally, we obtain an error estimation between the exact solution and the solution of the iterative method. Examples show the applicability of our results.


Introduction
Integral equations have a variety of applications in many domains such as mechanics, potential theory, geophysics, refreshing theory, and medical sciences. Most of the boundary value problems, such as differential equations, can be formulated only in terms of integral equations. Hence, the solution of integral equations is very important. As a special case, integral equations of product type arise, e.g., in the study of the spread of an infectious disease that does not induce permanent immunity [see, e.g., Bailey (1975), Diekmann (1979), Gripenberg (1980), Gripenberg (1981), Waltman (1974), Olaru (2014) and references therein]. Gripenberg (1981) studied the existence and the uniqueness of a bounded, continuous, and nonnegative solution to the integral equation of product type as follows: for τ > 0 under appropriate assumptions on functions P and Q. Olaru (2010) showed the existence and uniqueness of a continuous solution to the integral equation as follows: (τ, s, z(s))ds and χ i is continuous Lipschitzian for i = 1, . . . , m.
Recently, Boulfoul et al. (2018) consider the more general nonlinear integral equation z(τ ) = h(τ, z(τ )) + h 1 t, τ 0 w 1 (τ, s, z(s))ds h 2 τ, τ 0 w 2 (τ, s, z(s))ds , for τ > 0. Pachpatte (1995) provided a new integral inequality that he used to study the boundedness, asymptotic behavior, and growth of solutions of above equation. Abdeldaim (2012) and Li et al. (2011) generalized Pachpatte's inequality to some integral inequalities in order to study the boundedness and the asymptotic behavior of continuous solutions to integral equation of product type.
Contrary to classical logic, the fuzzy logic takes into account ambiguity as a part of system modeling. So, numerical solution of fuzzy integral equations is more important than solving crisp integral equations. Many researchers have studied the solution of the fuzzy integral equations of the first and second kinds (Bica 2008;Bica and Popescu 2013;Bede and Gal 2004;Behzadi et al. 2014), but according to the investigations, numerical solution of fuzzy integral equations of product type has been studied by no researchers. Hence, in the present paper, we examine the existence and uniqueness of the solution of the fuzzy integral equations of product type as follows: where u is an unknown fuzzy function and f (x) is the source fuzzy function. The structure of the paper is as follows: In Sect. 2, we introduce some fuzzy concepts. In Sect. 3, the existence and uniqueness of the solution of the fuzzy integral equations of product type is proved using the Banach fixed point theorem. The error estimation between the exact solution and the solution of the iterative method is obtained in Sect. 4. Section 5 includes an example to check the accuracy of the proposed method . Finally, in Sect. 6 we present our concluding remarks.

Some fuzzy concepts
Definition 1 (See Gal 2000) A fuzzy number is a function η : R → [0, 1] having the properties: The set of all fuzzy numbers is denoted by R .An alternative definition which yields the same R is given by Kaleva (1987).
The addition and scalar multiplication of fuzzy numbers in R are defined as follows: (λu(r ), λu(r )) λ < 0. (6) the product η μ of fuzzy numbers η and μ, based on Zadeh's extension principle, is defined by We say that u and v have the some sign if they are both positive or both negative.
(2) For fuzzy numbers which are not crisp, there is no opposite element (that is, (R , ⊕) cannot be a group).
For arbitrary a, b ∈ R, this property is not fulfilled. (4) For any λ, μ ∈ R and u ∈ R , we have b], and denote the space of all such functions by C [a, b].
is complete metric space. In Ezzati and Ziari (2013), the authors proved that if f ∈ C [a, b], its definite integral exists, and also,

Existence and uniqueness of solution
In this section, we establish the uniqueness and existence of solution of the integral equation (1). We introduce the following conditions: ( Before the establishing the existence and uniqueness of Solution (1), we present Lemmas 2-3.

Lemma 2 For given
Proof By Definition 5, we have using Definition 2(6), it follows that
According to the concept of fuzzy distance, for arbitrary
For u 1 , u 2 ∈ X and x ∈ [a, b], we can write: Deriving the supremum of both sides of the above inequality, we have Since C < 1, the operator H is a contraction. According to the Banach fixed point theorem, Eq. (1) has a unique solution u * ∈ X .

Error estimation
In this section, we derive the error estimation between the exact solution, u * , and the approximate solution of (1.1) in terms of the sequence of successive approximations, (y m ) m∈N .

Theorem 3 Let assumptions of Theorem (2) hold. Then the error estimation between the exact and approximate solu-
where M = max{M 1 , M 2 }.
Proof Using the concepts of fuzzy distance and triangular inequality, we have: Using Lemmas 2 and 3, the result is: Thus By using Defination 5, we have Combining (10) and (11) yields On the other hand, Thus D * y m , y m−1 ≤ C D * y m−1 , y m−2 .

By induction
The same procedure for y 1 and y 0 leads as follows: From (12), (13) and (14), we conclude that

Examples
Example 1 Consider the following integral equation of product type To compare the exact and iterative solutions with m = 5, see Table 1.
Comparison of the results of the iterative method and the exact solution of the integral equation of product type in Table 1 shows that the use of the iterative method in finding the solution of such integral equations has high accuracy.
Example 2 Consider the following integral equation of product type x, r ∈ [0, 1], and kernels Since the solution of the integral equation ,u, is a continuous function, therefore it is uniform continuous and bounded function in [0,1]. Hence, functions k 1 , k 2 and f satisfy all the conditions of Theorem 2. So the aforementioned integral equation has a unique solution. The exact solution of this example is u(x) = u(x, r ), u(x, r ) = (r 2 + r ) sin π x, (4 − r 3 − r ) sin π x .
To compare the exact and iterative solutions with m = 5, see Table 2.

Conclusions
This paper aims the fuzzy Fredholm integral equation of product type. In this research, Banach fixed point theorem was used to prove the existence and uniqueness of the solution of integral equations of product type. Finally, we got the error estimation between the exact solution and the solution of the iterative method. Given the widespread use of this kind of integral equations in physics, mechanics, and medi-