Numerical solutions for fuzzy Fredholm integral equations of the first kind using Landweber iterative method 1

Abstract

In this paper, we proposed an iterative procedure based on Landweber iterative methods to solve fuzzy Fredholm integral equations of the first kind. In addition, this research be based on a strictly convex fuzzy number space and the Riemann integral of fuzzy-number-valued function which is taken value in the space. The error estimation of the proposed method in terms of uniform and partial modulus of continuity was given. The generalized difference of fuzzy numbers was controlled and estimated reasonably. And it was avoided to compute the generalized difference of fuzzy numbers during the iterations. Finally, two illustrative examples are included in order to demonstrate the accuracy and the convergence of the proposed method.

Keywords

Fuzzy numbers Fuzzy Integral equation Iterative method Uniform modulus of continuity Partial modulus of continuity

1 Introduction

The fuzzy integral equations (FIE) are very useful for solving many problems in several applied fields such as mathematical economics; electrical engineering; medicine; biology and optimal control theory. Since these equations can not usually be solved explicitly, it is required to obtain approximate solutions. Kaleva [26] proposed the existence and uniqueness of the solution of fuzzy differential equations by using the Banach fixed point principle. The Banach fixed point principle is a powerful tool to investigate of the existence and uniqueness of the solution of fuzzy integral equations. The existence and uniqueness of the solution of fuzzy integral equation of the second kind can be found in [5 , 32–34]. The numerical methods for solving fuzzy integral equations involve various techniques. The iterative techniques were applied to fuzzy Fredholm integral equation of the second kind in [8 , 31]. Friedman et al. [20] presented a numerical algorithm to solve fuzzy Fredholm integral equations of the second kind which is based on the successive approximations method. In [10], the successive approximations method is used for nonlinear fuzzy Fredholm integral equations. Bica and Popescu in [11] developed an iterative numerical method to solve nonlinear fuzzy Hammerstein-Volterra integral equations with constant delay. In [12], the same method has been applied to the solutions that take values in the set of right-sided fuzzy numbers for a fuzzy Volterra integral equation with constant delay arising in epidemiology. In recent years, the Picard method and fuzzy expansion method and Chebyshev polynomials method for solving fuzzy linear and nonlinear Volterra-Fredholm integral-differential equations were proposed in [2 –4].

In 2013, the authors presented a numerical solution of nonlinear fuzzy Fredholm integral equations using iterative methods [18]. Afterwards, an iterative method for the numerical solution of two-dimensional nonlinear fuzzy integral equations was proposed in 2015 [35]. In the above mentioned literatures, the proposed methods focus primarily on the fuzzy integral equation of the second kind. However, the fuzzy Fredholm integral equation of the first kind (FFIEs-1) is ill-posed problem. The solution for an ill-posed problem may not exist, and if it exists it may not be unique, or it does not depend continuously on the given Cauchy data and any small perturbation in the given data may cause large change to the solution. So it is more difficult to propose the numerical solution for the fuzzy Fredholm integral equation of the first kind than to propose that of the second kind. In [39] Ill-posedness for fuzzy Fredholm integral equations of the first kind and a regularization method have been proposed.

Here, we propose a new numerical approach for solving fuzzy Fredholm integral equations of the first kind and obtain the error estimation in the approximation of the solution to such fuzzy integral equations. The remainder of this paper is organized as follows. In Section 2, we briefly review some elementary concepts related to fuzzy numbers and a Riemann integral of fuzzy-number-valued function. The iterative method to solve FFIEs-1 is considered in Section 3. In Section 4, the error estimation of the proposed method is presented. In Section 5, we present two examples which are used to highlight the reliability of the proposed method. Our conclusions are given in Section 6.

2 Preliminaries

Let P_k (Rⁿ) denote the family of all nonempty compact convex subset of Rⁿ and define the addition and scalar multiplication in P_k (Rⁿ) as usual. Let A and B be two nonempty bounded subset of Rⁿ. The distance between A and B is defined by the Hausdorff metric

$\begin{matrix} d_{H} (A, B) = \\ max {sup_{a \in A} inf_{b \in B} | | a - b | |, sup_{b \in B} inf_{a \in A} | | b - a | |}, \end{matrix}$ (2.1) where || · || denotes the usual Euclidean norm in Rⁿ [17]. Then (P_k (Rⁿ) ; d_H) is a metric space.

Definition 2.1. Denote $E^{n} = {u : R^{n} \to [0, 1] | u satisfies (1) - (4) below}$ is a fuzzy number space, where

(1) u is normal, i.e. there exists an x₀ ∈ Rⁿ such that u (x₀) =1,

(2) u is fuzzy convex, i.e. u (λx + (1 - λ) y) ≥ min {u (x) , u (y)} for any x, y ∈ Rⁿ and 0 ≤ λ ≤ 1,

(3) u is upper semi-continuous,

(4) [u] ₀ = cl {x ∈ Rⁿ|u (x) >0} is compact.

Here, cl (X) denotes the closure of set X . For 0 < α ≤ 1, the α-level set of u (or simply the α-cut) is defined by [u] _α = {x ∈ Rⁿ|u (x) ≥ α} . The core of u is the set of elements of Rⁿ having membership grade 1, i.e., [u] ₁ = {x|x ∈ Rⁿ, u (x) =1} . Then from above (1)-(4), it follows that the α-level set [u] _α ∈ P_k (Rⁿ) for all 0 < α ≤ 1.

The distance between two fuzzy numbers u and v is defined by

$D (u, v) = sup_{α \in [0, 1]} d_{H} ([u]_{α}, [v]_{α}) .$ (2.2)

Definition 2.2. According to Zadeh’s extension principle, we have addition and scalar multiplication in fuzzy number space E¹ as follows: $[u \oplus v]_{α} = [u]_{α} + [v]_{α} = {x + y | x \in [u]_{α}, y \in [v]_{α}},$ $[k ⊙ u]_{α} = k [u]_{α} = {kx | x \in [u]_{α}}, [0]_{α} = {0} .$ where u, v ∈ E¹ and 0 < α ≤ 1 . [u] _α + [v] _α means the usual addition of two intervals (as subset of R) and k [u] _α means the usual product between a scalar and a subset of R. Also, according to [8, 38], the following algebraic properties for any u, v, w ∈ E¹ hold: (1) u ⊕ (v ⊕ w) = (u ⊕ v) ⊕ w, $(2) u \oplus \tilde{0} = \tilde{0} \oplus u,$ (3) with respect to $\tilde{0}$ , none u ∈ (E¹ - R), $u \neq \tilde{0}$ has opposite in (E¹, ⊕), (4) (a + b) ⊙ u = a ⊙ u + b ⊙ u, ∀ a, b ∈ R with ab ≥ 0, (5) a ⊙ (u ⊕ v) = a ⊙ u ⊕ a ⊙ v, ∀ a ∈ R, (6) a ⊙ (b ⊙ u) = (ab) ⊙ u, ∀ a, b ∈ R and 1⊙ u = u.

Definition 2.3. (See [18, 38]) For arbitrary fuzzy numbers $u, v \in E^{1}, u = [\underline{u} (α), \bar{u} (α)], v = [\underline{v} (α),$ $\bar{v} (α)]$ , the quantity $D (u, v) = sup_{α \in [0, 1]} max {| \underline{u} (α) - \underline{v} (α) |, | \bar{u} (α) - \bar{v} (α) |}$ is the distance between u and v and also the following properties hold: (1) (E¹, D)is a complete metric space, (2) D (u ⊕ w, v ⊕ w) = D (u, v) , ∀ u, v, w ∈ E¹, (3) D (u ⊕ v, w ⊕ e) ≤ D (u, w) + D (v, e) , ∀ u, v, w, e ∈ E¹, $(4) D (u \oplus v, \tilde{0}) \leq D (u, \tilde{0}) + D (v, \tilde{0}), \forall u, v, \in E^{1}$ , (5) D (k ⊙ u, k ⊙ v) = |k|D (u, v) , ∀ u, v, ∈ E¹, k ∈ R, $(6) D (k_{1} ⊙ u, k_{2} ⊙ u) = | k_{1} - k_{2} | D (u, \tilde{0}), \forall u \in E^{1},$ k₁, k₂ ∈ R with k₁ · k₂ ≥ 0. (7) The function of ∥ · ∥ _F : E¹ → R by $∥ u ∥_{F} = D (u, \tilde{0})$ has the usual properties of the norm, that is, ∥u ∥ _F = 0 if and only if $u = \tilde{0}$ , ∥λ ⊙ u ∥ _F = |λ| ∥ u ∥ _F, ∥u ⊕ v ∥ _F ≤ ∥ u ∥ _F + ∥ v ∥ _F, (8) | ∥ u ∥ _F - ∥ v ∥ _F| ≤ D (u, v) , D (u, v) ≤ ∥ u ∥ _F+ ∥ v ∥ _F.

Definition 2.4. (B. Bede and L. Stefanini [9]) The generalized Hukuhara difference between two fuzzy numbers u, v ∈ Eⁿ is defined as follows $u \underset{gH}{⊖} v = w \Leftrightarrow {\begin{matrix} (i) u = v + w; \\ or (ii) v = u + (- w) . \end{matrix}$ (2.3)

In terms of the α- levels, we have $[u \underset{gH}{⊖} v]_{α} = [min {\underline{u} (α) - \underline{v} (α), \bar{u} (α) - \bar{v} (α)}, max {\underline{u} (α) - \underline{v} (α), \bar{u} (α) - \bar{v} (α)}]$ and if the H-difference exists, then $u ⊖ v = u \underset{gH}{⊖} v$ ; the conditions for the existence of $w = u \underset{gH}{⊖} v \in E^{n}$ are $case (i) {\begin{matrix} \underline{w} (α) = \underline{u} (α) - \underline{v} (α) and \bar{w} (α) = \bar{u} (α) - \bar{v} (α), \forall α \in [0, 1], \\ with \underline{w} (α) increasing, \bar{w} (α) decreasing, \underline{w} (α) \leq \bar{w} (α) . \end{matrix}$ (2.4) $case (ii) {\begin{matrix} \underline{w} (α) = \bar{u} (α) - \bar{v} (α) and \bar{w} (α) = \underline{u} (α) - \underline{v} (α), \forall α \in [0, 1], \\ with \underline{w} (α) increasing, \bar{w} (α) decreasing, \underline{w} (α) \leq \bar{w} (α) . \end{matrix}$ (2.5)

It is easy to show that (i) and (ii) are both valid if and only if w is a crisp number. In the fuzzy case, it is possible that the gH-difference of two fuzzy numbers does not exist. To address this shortcoming, a new difference between fuzzy numbers was proposed in [9].

Definition 2.5. (See [9]) The generalized difference (g-difference) of two fuzzy numbers u, v ∈ Eⁿ is given by its level sets as $[u \underset{g}{⊖} v]_{α} = cl ⋃_{β \geq α} ([u]_{β} \underset{gH}{⊖} [v]_{α}), \forall α \in [0, 1],$ (2.6) where the gH-difference $\underset{gH}{⊖}$ is with interval operands [u] _β and [v] _α.

Theorem 2.1. (See[21]) Let u ∈ E¹ be a fuzzy number. Then the functions $\underline{u}$ , $\bar{u}$ : [0, 1] → R, defining the endpoints of the α-level sets of u, satisfy the following conditions:

(i) $\underline{u}$ is a bounded, non-decreasing, left-continuous function in (0, 1] and it is right-continuous at 0.

(ii) $\bar{u}$ is a bounded, non-increasing, left-continuous function in (0, 1] and it is right-continuous at 0.

(iii) $\underline{u} (1) \leq \bar{u} (1)$ . Reciprocally, given two functions that satisfy the above conditions, they uniquely determine a fuzzy number.

Definition 2.6. (See [1])A fuzzy-number-valued function f : [a, b] → E¹ is said to be continuous at t₀ ∈ [a, b] if for each ɛ > 0 there is δ > 0 such that D (f (t) , f (t₀)) < ɛ whenever |t - t₀| < δ. If f is continuous for each t ∈ [a, b] then we say that f is fuzzy continuous on [a, b].

Definition 2.7. (See [1]) A fuzzy-number-valued function f : [a, b] → E¹ is said to bounded iff there is M > 0 such that $D (f (t), \tilde{0}) = ∥ f (u) ∥ \leq M$ for all t ∈ [a, b]. Equivalently we get χ_-M ≤ f (x) ≤ χ_M, ∀ x ∈ [a, b].

Definition 2.8. (See [37]) If f, g : [a, b] → E¹ are fuzzy continuous functions, then the function F : [a, b] → R₊ defined by F (x) = D (f (x) , g (x)) is continuous on [a, b]. Also $D (f (x), \tilde{0}) \leq M, \forall x \in [a, b], M > 0$ , that is f is fuzzy bounded.

Definition 2.9. (See [21]) Let f : [a, b] → E¹. We say that f is Fuzzy-Riemann integrable to I ∈ E¹ if for any ɛ > 0, there exists δ > 0 such that for any division P = {[u, v] ; ξ} of [a, b] with the norms △ (P) < δ, we have $D (\sum_{P}^{*} (v - u) ⊙ f (ξ); I) < ɛ,$ where ∑^∗ denotes the fuzzy summation. We choose to write $I : = (FR) \int_{a}^{b} f (x) dx .$

We also call an f as above (FR)-integrable.

Theorem 2.2. (See [21]) If f, g : [a, b] → E¹ are (FR)-integrable fuzzy functions, and α, β are real numbers, then $\begin{matrix} (FR) \int_{a}^{b} α ⊙ f (x) \oplus β ⊙ g (x) dx \\ = α ⊙ (FR) \int_{a}^{b} f (x) dx \oplus β ⊙ (FR) \int_{a}^{b} g (x) dx . \end{matrix}$

Theorem 2.3. (See [22]) Let f : [a, b] → E¹, then f is (FR)-integrable if and only if ${\underline{f}}_{α}$ and ${\bar{f}}^{α}$ are Riemann integrable for any α ∈ [0, 1]. Furthermore, for any α ∈ [0, 1], $\begin{matrix} [(FR) \int_{a}^{b} f (x) dx]_{α} \\ = [(R) \int_{a}^{b} [\underline{f (x)}]_{α} dx, (R) \int_{a}^{b} [\bar{f (x)}]^{α} dx] . \end{matrix}$

Definition 2.10. (See [16]) Let $F_{b}^{st} = {u | u : R \to I ≜ [0, 1]}$ satisfy the following properties:

(1) u is normal, that is ∃ m ∈ R, such that u (m) =1;

(2) [u] ⁰ = cl {ξ ∈ R|u (ξ) >0} is bounded;

(3) for ∀x, y ∈ [u] ⁰, λ ∈ (0, 1) , $u (λ x + (1 - λ) y) > min {u (x), u (y)};$

(4) u is upper semi-continuous on R. then $u \in F_{b}^{st}$ is called continuous fuzzy number that the center is the only one. Obviously $F_{b}^{st} \subset E^{1}$ .

Theorem 2.4. (See [16]) Let $u \in F_{b}^{st}$ , and [u] ^α ={x ∈ R : u (x) ≥ α} , (0 < α ≤ 1), u₁ (α) = min [u] ^α; u₂ (α) = max [u] ^α (α ∈ I), then the following properties hold:

(1) u₁ (α) , u₂ (α) ∈ C (I, R) = C [0, 1];

(2) u₁ (α) is a monotonically increasing function, u₂ (α) is a monotonically decreasing function;

(3) u₁ (1) = u₂ (1).

Conversely, if i (α) , s (α) : I → R meets the above properties (1)-(3), and let $u (x) = {\begin{matrix} sup {α \in I | i (α) \leq x \leq s (α)}, x \in [i (0), s (0)], \\ 0, x \notin [i (0), s (0)], \end{matrix}$ then there is only one $u \in F_{b}^{st}$ , such that [u] ^α = [i (α) , s (α)] , u₁ (α) = i (α) , u₂ (α) = s (α) , α ∈ I .

Theorem 2.5. [16] $F_{b}^{st}$ is a closed convex cone on Banach space X = C [0, 1] × C [0, 1] and the zero element θ is the vertex. Thus $F_{b}^{st}$ is complete metric space by the the Hausdorff metric in fuzzy number space E¹.

Definition 2.11. [16] Let $f : T = [a, b] \to F_{b}^{st}, J \in F_{b}^{st} .$ A fuzzy-number-valued function f (t) is said to be Riemann integrable to $J \in F_{b}^{st}$ on [a, b] if for T = [a, b] any division Δ: $a = t_{0} < t_{1} < \dots < t_{n - 1} < t_{n} = b$ and for any τ_k ∈ [t_k-1, t_k] , we have $lim_{λ (Δ) \to 0} \sum_{k = 1}^{n} f (τ_{k}) Δ t_{k} = J,$ where $λ (Δ) = max_{1 \leq k \leq n} {Δ t_{k}}, Δ t_{k} = t_{k} - t_{k - 1} (k = 1, 2, \dots, n)$ . We write $(FR) \int_{a}^{b} f (t) dt = J$ .

From Definition 2.11 and $F_{b}^{st}$ is a complete metric space by the the Hausdorff metric, we have a theorem 2.6 as follows.

Theorem 2.6. [16] If $f : T \to F_{b}^{st}$ is Riemann integrable on T = [a, b], then $\int_{a}^{b} f (t) dt \in F_{b}^{st}$ .

Theorem 2.7. [16] If $f : T \to F_{b}^{st}$ is Riemann integrable on T = [a, b], then f is (K) integrable on T = [a, b] and the two kinds of integrals are equal.

Theorem 2.8. If $f : T \to F_{b}^{st}$ is Riemann integrable on T = [a, b], then $\underline{f} (t) (α), \bar{f} (t) (α)$ are Lebesgue integrable on T = [a, b], for any α ∈ [0, 1].

Proof. Let $f : T \to F_{b}^{st}$ , f is (K) integrable on T = [a, b] if and only if $\underline{f} (t) (α), \bar{f} (t) (α)$ are Lebesgue integrable on T = [a, b], for any α ∈ [0, 1]. Then from Theorem 2.7 we obtain Theorem 2.8.

3 The iterative method to solve FFIEs-1

Let $X = {f : [a, b] \to F_{b}^{st}}$ , and f is a continuous fuzzy-number-valued function, then X is a continuous fuzzy-number-valued function space. We consider a fuzzy Fredholm integral equation of the first kind as follows. $f (x) = (FR) \int_{a}^{b} k (x, y) ⊙ g (y) dy, x \in [a, b] .$ (3.0)

Where a and b are constants, f (x) is the known fuzzy-number-valued function, and k (x, y) is the kernel of the integral equation and k (x, y) >0, and g (y) is the unknown fuzzy-number-value function that will be determined. Equation(3.0) is called the fuzzy Fredholm integral equations of the first kind characterized by the occurrence of the unknown function g (y) only inside the integral sign. The existence of g (y) inside the integral sign causes special difficulties. So we consider using the iterative method to solve FFIEs-1. Let k (x, y) be a continuous binary function, then it is a uniformly continuous function with respect to the variables x and there exists an M₁ > 0 such that $M_{1} = max_{a \leq x, y \leq b} | k (x, y) |$ .

It will be convenient to employ the customary operator notation for integral transforms, viz. $kg \equiv (FR) \int_{a}^{b} k (x, y) ⊙ g (y) dy, x \in [a, b], g (y) \in X .$ $Rg \equiv (FR) \int_{a}^{b} R (x, y) ⊙ g (y) dy, x \in [a, b], g (y) \in X .$ Where $X = {f : [a, b] \to F_{b}^{st}}$ is a continuous fuzzy-number-valued function space.

Now, we define the operator A : X → X by $\begin{matrix} A (g) = F (x) \oplus g (x) \underset{g}{⊖} Rg = F (x) \oplus G (g (x)), F, g \in X, \end{matrix}$ (3.1) where $G (g (x)) ≜ g (x) \underset{g}{⊖} Rg .$

Then a set of continuous functions g₁ (t) , g₂ (t) , ⋯ is defined by the iteration formula $g_{n} (x) = F (x) \oplus g_{n - 1} (x) \underset{g}{⊖} {Rg}_{n - 1},$ (3.2) where g_n (x) , g_n-1 (x) , F (x) ∈ X .

Theorem 3.1. Let the function k (x, y) be continuous and positive for (x, y) ∈ [a, b] × [a, b]. Assume that there exists L > 0, with $\begin{matrix} D (G (g_{1} (u)), G (g_{2} (v))) \leq L \cdot D ({Rg}_{1}, {Rg}_{2}), \\ (u, v) \in [a, b] \times [a, b] \end{matrix}$ (3.3)

If LM (b - a) = C < 1, then the FFIEs-1 (3.0) has a unique solution g ∈ X, and it can be obtained by the iteration formula (3.2). Moreover, the sequence of successive approximations g_n (x) converges to the solution g. Furthermore, the following error bound holds: $D (g (x), g_{m} (x)) \leq \frac{(C)^{m + 1}}{LM (1 - C)} J, \forall x \in [a, b], m \geq 1,$ (3.4) where $J = sup_{a \leq t \leq b} D (K (t, s) f (s), R (s, t) g_{0} (s))$ , $M = max_{a \leq s, t \leq b} | R (s, t) |$ , f (t) is the known fuzzy-number-valued function, and g₀ (t) is the 0-th approximation.

Proof. We first show that A maps X into X(i.e.A (X) ⊂ X). To the end, we show that the operator A is uniformly continuous. Since the fuzzy-number-valued function $F (x) : [a, b] \to F_{b}^{st}$ is continuous, and [a, b] is a compact convex set, we deduce that it is uniformly continuous and hence for ɛ₁ > 0 exists δ₁ > 0 such that $\begin{matrix} D (F (t_{1}), F (t_{2})) < ɛ_{1}, \\ whenever | t_{1} - t_{2} | < δ_{1}, \forall t_{1}, t_{2} \in [a, b] . \end{matrix}$

In a similar way, for $(s, t) \in [a, b] \times [a, b], R (s, t) = \int_{a}^{b} k (y, s) k (y, t) dy$ is uniformly continuous with respect to the variable t and hence for ɛ₂ > 0 exists δ₂ > 0 such that $\begin{matrix} | R (s, t_{1}), R (s, t_{2}) | < ɛ_{2}, \\ whenever | t_{1} - t_{2} | < δ_{2}, \forall t_{1}, t_{2} \in [a, b] . \end{matrix}$

By choosing δ = min {δ₁, δ₂}, then whenever |t₁ - t₂| < δ, ∀ t₁, t₂ ∈ [a, b], we have $\begin{matrix} D (A (g) (t_{1}), A (g) (t_{2})) \\ = D (F (t_{1}) \oplus g (t_{1}) \underset{g}{⊖} Rg, F (t_{2}) \oplus g (t_{2}) \underset{g}{⊖} Rg) \\ \leq D (F (t_{1}), F (t_{2})) + D (g (t_{1}) \underset{g}{⊖} Rg, g (t_{2}) \underset{g}{⊖} Rg)) \\ \leq D (F (t_{1}), F (t_{2})) + LD (Rg, Rg)) \\ \leq ɛ_{1} + LD ((FR) \int_{a}^{b} R (s, t_{1}) ⊙ g (s) ds, (FR) \\ \int_{a}^{b} R (s, t_{2}) ⊙ g (s) ds) \\ \leq ɛ_{1} + L \int_{a}^{b} D (R (s, t_{1}) ⊙ g (s), R (s, t_{2}) ⊙ g (s)) ds \\ \leq ɛ_{1} + L | R (s, t_{1}) - R (s, t_{2}) | \int_{a}^{b} D (g (s), \tilde{0}) ds \\ \leq ɛ_{1} + L (b - a) T_{0} ɛ_{2}, \end{matrix}$ where $T_{0} = sup_{a \leq s \leq b} ∥ g (s) ∥_{F} .$

By choosing $ɛ_{1} = \frac{ɛ}{2}, ɛ_{2} = \frac{ɛ}{2 L (b - a) T_{0}}$ , we have $D (A (g) (t_{1}), A (g) (t_{2})) \leq ɛ .$

This shows that A (g) is uniformly continuous on the interval [a, b], hence A (X) ⊂ X.

Next, we will prove the operator A is a contraction mapping. Let g₁, g₂ ∈ X and t ∈ [a, b], we can obtain $\begin{matrix} D (A (g_{1}) (t), A (g_{2}) (t)) \\ = D (F (t) \oplus G (g_{1}), F (t) \oplus G (g_{2})) \\ = D (G (g_{1}), G (g_{2})) \\ \leq L \cdot D ({Rg}_{1}, {Rg}_{2}) \\ \leq L \cdot D ((FR) \int_{a}^{b} R (s, t) g_{1} (s) ds, \\ (FR) \int_{a}^{b} R (s, t) g_{2} (s) ds) \\ \leq L \cdot \int_{a}^{b} D (R (s, t) g_{1} (s), R (s, t) g_{2} (s)) ds \\ \leq LM (b - a) \cdot D (g_{1}, g_{2}) \\ \leq LM (b - a) \cdot D^{*} (g_{1}, g_{2}) . \end{matrix}$ (3.5)

Consequently, $D^{*} (A (g_{1}) (t), A (g_{2}) (t)) \leq LM (b - a) \cdot D^{*} (g_{1}, g_{2}) .$ (3.6)

From the condition LM (b - a) = C < 1, we can obtain the operator A : X → X is a contraction mapping on the complete metric space (X, D^∗). Hence, the Banach fixed principle implies that the FFIEs-1 (3.0) has a unique solution g ∈ X.

In addition, $\begin{matrix} D^{*} (g, g_{m}) = D^{*} (A (g), A (g_{m - 1})) \\ \leq LM (b - a) \cdot D^{*} (g, g_{m - 1}) \\ \leq LM (b - a) \cdot [D^{*} (g, g_{m}) + D^{*} (g_{m - 1}, g_{m})] . \end{matrix}$ (3.7)

So, we have $D^{*} (g, g_{m}) \leq \frac{LM (b - a)}{1 - LM (b - a)} D^{*} (g_{m - 1}, g_{m}) .$ (3.8)

From the equation (3.6), for a set number of iterations, we have $\begin{matrix} D (g (t), g_{m} (t)) \leq D^{*} (g, g_{m}) \\ \leq \frac{LM (b - a}{1 - LM (b - a)} D^{*} (g_{m - 1}, g_{m}) \\ = \frac{LM (b - a)}{1 - LM (b - a)} D^{*} (A (g_{m - 2}), A (g_{m - 1})) \\ \leq \frac{(LM (b - a))^{2}}{1 - LM (b - a)} D^{*} (g_{m - 2}, g_{m - 1}) \\ \leq \dots \leq \frac{(LM (b - a))^{m}}{1 - LM (b - a)} D^{*} (g_{0}, g_{1}) . \end{matrix}$ (3.9)

On the other hand, $\begin{matrix} D^{*} (g_{0}, g_{1}) = sup_{a \leq t \leq b} D (g_{0} (t), F (t) \oplus G (g_{0} (t))) \\ = sup_{a \leq t \leq b} D (g_{0} (t), F (t) \oplus g_{0} (t) \underset{g}{⊖} {Rg}_{0}) \\ = sup_{a \leq t \leq b} D (\tilde{0}, F (t) \underset{g}{⊖} {Rg}_{0}) \\ = sup_{a \leq t \leq b} D (F (t), {Rg}_{0}) \\ = sup_{a \leq t \leq b} D ((FR) \int_{a}^{b} K (t, s) f (s) ds, \\ (FR) \int_{a}^{b} R (t, s) g_{0} (s) ds) \\ \leq sup_{a \leq t \leq b} \int_{a}^{b} D (K (t, s) f (s), R (s, t) g_{0} (s)) ds \\ \leq (b - a) sup_{a \leq t \leq b} D (K (t, s) f (s), R (s, t) g_{0} (s)) \\ = J (b - a) \end{matrix}$ (3.10)

From the the equation (3.9) and (3.10), we can obtain

$\begin{matrix} D (g (x), g_{m} (x)) \leq \frac{(LM (b - a))^{m}}{1 - LM (b - a)} J (b - a) \\ = \frac{(C)^{m}}{1 - C} J (b - a) = \frac{(C)^{m + 1}}{LM (1 - C)} J, \\ \forall x \in [a, b], m \geq 1 . \end{matrix}$

4 The error estimation for the iterative method

First, we will introduce the uniform modulus of continuity of a fuzzy-number-valued function f and its some properties, which will be used in this section.

Definition 4.1. ([8, 18]) Let f : [a, b] → E¹ be a bounded fuzzy-number-valued function. Then the function ω_[a,b] (f, ·) : R₊ ∪ {0} → R₊ defined by $\begin{matrix} ω_{[a, b]} (f, δ) = sup {D (f (x), f (y)) : \\ x, y \in [a, b], | x - y | \leq δ}, \end{matrix}$ is called the modulus of oscillation of on [a, b], where R₊ is the set of positive real numbers. If f is a continuous fuzzy-number-valued function, then ω_[a,b] (f, δ) is called uniform modulus of continuity of f.

Some properties of the modulus of continuity of f are given as follows:

(1) D (f (x) , f (y)) ≤ ω_[a,b] (f, |x - y|) , ∀ x, y ∈ [a, b].

(2) ω_[a,b] (f, δ) is increasing function of δ.

(3) ω_[a,b] (f, 0) =0.

(4) ω_[a,b] (f, δ₁ + δ₂) ≤ ω_[a,b] (f, δ₁) + ω_[a,b] (f, δ₂) , ∀ δ₁, δ₂ ≥ 0.

(5) ω_[a,b] (f, nδ) ≤ nω_[a,b] (f, δ) , ∀ δ ≥ 0, n ∈ N.

(6) ω_[a,b] (f, λδ) ≤ (1 + λ) ω_[a,b] (f, δ) , ∀ δ, λ ≥ 0.

(7) If [c, d] ⊆ [a, b], then ω_[c,d] ≤ ω_[a,b].

(8) $D ((FH) \int_{a}^{b} f (t) dt, (b - a) ⊙ f (\frac{a + b}{2})) \leq \frac{b - a}{2} ω_{[a, b]} (f, \frac{b - a}{2})$ .

(9) $D ((FH) \int_{a}^{b} f (t) dt, \frac{b - a}{2} ⊙ [f (a) \oplus (b)] \leq \frac{b - a}{2} ω_{[a, b]} (f, \frac{b - a}{2})$ .

(10) $D ((FH) \int_{a}^{b} f (t) dt, \frac{b - a}{6} ⊙ [f (a) \oplus 4 ⊙$ $f (\frac{b - a}{2}) \oplus f (b)] \leq 3 (b - a) ω_{[a, b]} (f, \frac{b - a}{6})$ .

Corollary 4.1. If a fuzzy-number-valued function be Riemann integrable function, then it will be a Henstock integrable function, too. Hence, the above properties (8)-(10) are true for the Riemann integrable fuzzy-number-valued function.

Now, we introduce the numerical method to find the approximate solution of the FFIEs-1 (3.0), and we obtain an error estimation between the exact and approximate solutions.

In this way, we consider the uniform partition of [a, b] as $▵_{t} : a = t_{0} \leq t_{1} \leq \dots \leq t_{n - 1} \leq t_{n} = b,$ with t_i = a + ih, where $h = \frac{b - a}{n}$ . Then the following procedure gives the approximate solution of (3.0) in point t. $y_{0} (t) = g_{0} (t),$ $\begin{matrix} y_{m} (t) = F (t) \oplus G (y_{m - 1}) \\ = F (t) \oplus y_{m - 1} (t) \underset{g}{⊖} {Ry}_{m - 1} \\ = F (t) \oplus y_{m - 1} (t) \\ \underset{g}{⊖} ({\sum_{j = 0}^{n - 1}}^{*} \frac{h}{2} [R (t_{j}, t) ⊙ y_{m - 1} (t_{j}) \oplus \\ R (t_{j + 1}, t) ⊙ y_{m - 1} (t_{j + 1})] .) \end{matrix}$ (4.1)

Theorem 4.1. Consider the fuzzy Fredholm integral equation of the first kind (3.0) with continuous and positive kernel function k (s, t) for any (s, t) ∈ [a, b] × [a, b]. The operator G is continuous on E₁, g₀ (t) is continuous on [a, b]. In addition there exists L > 0 such that $\begin{matrix} D (G (g_{1} (u)), G (g_{2} (v))) \leq L \cdot D ({Rg}_{1}, {Rg}_{2}), \\ (u, v) \in [a, b] \times [a, b] . \end{matrix}$

If LM (b - a) = C < 1, then the iterative procedure (4.1) converges to the unique solution g of (3.0), and its error estimate is as follows: $\begin{matrix} D^{*} (g, y_{m}) \leq \frac{(C)^{m + 1}}{LM (1 - C)} J \\ + \frac{C}{2 (1 - C)} ω_{[a, b]} (F, h) + (\frac{C^{2} + 2 C}{2 M (1 - C)}) \\ (L_{1} ω_{s} (R, h) + L_{2} ω_{t} (R, h)) . \end{matrix}$ where $\begin{matrix} ω_{s} (R, h) = \\ sup_{a \leq t \leq b} {sup | R (s_{1}, t) - R (s_{2}, t) | : | s_{1} - s_{2} | \leq h}, \\ ω_{t} (R, h) = \\ sup_{a \leq t \leq b} {sup | R (s, t_{1}) - R (s, t_{2}) | : | t_{1} - t_{2} | \leq h}, \\ M = max_{(s, t) \in [a, b] \times [a, b]} | R (s, t) | . \end{matrix}$

Proof. For any t ∈ [a, b], $g_{1} (t) = F (t) \oplus G (g_{0}) = F (t) \oplus g_{0} (t) \underset{g}{⊖} {Rg}_{0} = F (t) \oplus g_{0} (t) \underset{g}{⊖} \int_{a}^{b} R (s, t) g_{0} (s) ds$ , we have

$\begin{matrix} D (g_{1} (t), y_{1} (t)) \\ = D (F (t) \oplus G ({Rg}_{0}), F (t) \oplus GR (y_{0})) \\ = D (G ({Rg}_{0}), G ({Ry}_{0})) \\ \leq LD ({Rg}_{0}, {Ry}_{0}) \\ = LD ((FR) \int_{a}^{b} R (s, t) g_{0} (s) ds, ({\sum_{j = 0}^{n - 1}}^{*} \frac{h}{2} \oplus [R (t_{j}, t) ⊙ y_{0} (t_{j}) \oplus R (t_{j + 1}, t) ⊙ y_{0} (t_{j + 1})])) \\ = LD ({\sum_{j = 0}^{n - 1}}^{*} (FR) \int_{t_{j}}^{t_{j + 1}} R (s, t) g_{0} (s) ds, ({\sum_{j = 0}^{n - 1}}^{*} \frac{h}{2} \oplus [R (t_{j}, t) ⊙ y_{0} (t_{j}) \oplus R (t_{j + 1}, t) ⊙ y_{0} (t_{j + 1})])) \end{matrix}$ $\begin{matrix} \leq L \sum_{j = 0}^{n - 1} D ((FR) \int_{t_{j}}^{t_{j + 1}} R (s, t) g_{0} (s) ds, \frac{h}{2} \oplus [R (t_{j}, t) ⊙ y_{0} (t_{j}) \oplus R (t_{j + 1}, t) ⊙ y_{0} (t_{j + 1})]) \\ \leq L \sum_{j = 0}^{n - 1} D ((FR) \int_{t_{j}}^{t_{j + 1}} R (s, t) g_{0} (s) ds, \frac{h}{2} \oplus [R (s, t) ⊙ y_{0} (t_{j}) \oplus R (s, t) ⊙ y_{0} (t_{j + 1})]) \\ + L \sum_{j = 0}^{n - 1} D (\frac{h}{2} \oplus [R (s, t) ⊙ y_{0} (t_{j}) \oplus R (s, t) ⊙ y_{0} (t_{j + 1})], \\ \frac{h}{2} \oplus [R (t_{j}, t) ⊙ y_{0} (t_{j}) \oplus R (t_{j + 1}, t) ⊙ y_{0} (t_{j + 1})]) . \end{matrix}$

Hence, from the property (9) of Definition 4.1 and the Lemma 4 in [11], we have $\begin{matrix} D (g_{1} (t), y_{1} (t)) \\ \leq L \sum_{j = 0}^{n - 1} | R (s, t) | D ((FR) \int_{t_{j}}^{t_{j + 1}} g_{0} (s) ds, \frac{h}{2} \oplus [g_{0} (t_{j}) \oplus g_{0} (t_{j + 1})]) \\ + L \sum_{j = 0}^{n - 1} D (\frac{h}{2} \oplus [R (s, t) ⊙ g_{0} (t_{j})], \frac{h}{2} \oplus [R (t_{j}, t) ⊙ g_{0} (t_{j})]) \\ + L \sum_{j = 0}^{n - 1} D (\frac{h}{2} \oplus [R (s, t) ⊙ g_{0} (t_{j + 1})], \frac{h}{2} \oplus [R (t_{j + 1}, t) ⊙ g_{0} (t_{j + 1})]) \\ \leq L \frac{h}{2} \sum_{j = 0}^{n - 1} | R (s, t) | ω_{[t_{j}, t_{j + 1}]} (g_{0}, \frac{h}{2}) + L \frac{h}{2} \sum_{j = 0}^{n - 1} | R (s, t) - R (t_{j}, t) | D (g_{0} (t_{j}), \tilde{0}) \\ + L \frac{h}{2} \sum_{j = 0}^{n - 1} | R (s, t) - R (t_{j + 1}, t) | D (g_{0} (t_{j + 1}), \tilde{0}) \\ \leq \frac{LM (b - a)}{2} ω_{[a, b]} (g_{0}, h) + Lh [\sum_{j = 0}^{n - 1} | R (s, t) - R (t_{j + 1}, t) | sup_{a \leq s \leq b} D (g_{0} (s), \tilde{0})] \end{matrix}$ $\begin{matrix} \leq \frac{LM (b - a)}{2} ω_{[a, b]} (g_{0}, h) + L (b - a) N_{0} ω_{s} (R, h) \\ = \frac{C}{2} ω_{[a, b]} (g_{0}, h) + \frac{C}{M} N_{0} ω_{s} (R, h) . \end{matrix}$

Where $N_{0} = sup_{a \leq s \leq b} ∥ g_{0} (s) ∥_{F}$ , ω_s (R, h) is the partial modulus of continuity.

In addition, for any t ∈ [a, b], we have $\begin{matrix} g_{2} (t) = F (t) \oplus G (g_{1}) \\ = F (t) \oplus g_{1} (t) \underset{g}{⊖} {Rg}_{0} \\ = F (t) \oplus g_{1} (t) \underset{g}{⊖} (FR) \int_{a}^{b} R (s, t) g_{1} (s) ds, \end{matrix}$ so, we obtain $\begin{matrix} D (g_{2} (t), y_{2} (t)) = D (F (t) \oplus G ({Rg}_{1}), F (t) \oplus GR (y_{1})) = D (G ({Rg}_{1}), G ({Ry}_{1})) \\ \leq LD ({Rg}_{1}, {Ry}_{1}) \\ = LD ((FR) \int_{a}^{b} R (s, t) g_{1} (s) ds, ({\sum_{j = 0}^{n - 1}}^{*} \frac{h}{2} \oplus [R (t_{j}, t) ⊙ y_{1} (t_{j}) \oplus R (t_{j + 1}, t) ⊙ y_{1} (t_{j + 1})])) \end{matrix}$ $\begin{matrix} \leq LD ((FR) \int_{a}^{b} R (s, t) g_{1} (s) ds, ({\sum_{j = 0}^{n - 1}}^{*} \frac{h}{2} \oplus [R (s, t) ⊙ g_{1} (t_{j}) \oplus R (s, t) ⊙ g_{1} (t_{j + 1})])) \\ + LD ({\sum_{j = 0}^{n - 1}}^{*} \frac{h}{2} \oplus [R (s, t) ⊙ g_{1} (t_{j}) \oplus R (s, t) ⊙ g_{1} (t_{j + 1})], \\ {\sum_{j = 0}^{n - 1}}^{*} \frac{h}{2} \oplus [R (s, t) ⊙ y_{1} (t_{j}) \oplus R (s, t) ⊙ y_{1} (t_{j + 1})]) \\ + LD ({\sum_{j = 0}^{n - 1}}^{*} \frac{h}{2} \oplus [R (s, t) ⊙ y_{1} (t_{j}) \oplus R (s, t) ⊙ y_{1} (t_{j + 1})], \\ {\sum_{j = 0}^{n - 1}}^{*} \frac{h}{2} \oplus [R (t_{j}, t) ⊙ y_{1} (t_{j}) \oplus R (t_{j + 1}, t) ⊙ y_{1} (t_{j + 1})]) \end{matrix}$ $\begin{matrix} \leq \frac{LM (b - a)}{2} ω_{[a, b]} (g_{1}, h) + \frac{LM (b - a)}{2} [D (g_{1} (t_{j}), y_{1} (t_{j})) + D (g_{1} (t_{j + 1}), y_{1} (t_{j + 1}))] \\ + L (b - a) N_{1} ω_{s} (R, h) \\ = \frac{C}{2} ω_{[a, b]} (g_{1}, h) + \frac{C)}{2} [D (g_{1} (t_{j}), y_{1} (t_{j})) + D (g_{1} (t_{j + 1}), y_{1} (t_{j + 1}))] + \frac{C}{M} N_{1} ω_{s} (R, h), \end{matrix}$ where $N_{1} = sup_{a \leq s \leq b} ∥ y_{1} (s) ∥_{F}$ .

By induction, for m ≥ 3, using (3.1) and (4.1), we obtain $\begin{matrix} D (g_{m} (t), y_{m} (t)) \\ \leq \frac{C}{2} ω_{[a, b]} (g_{m - 1}, h) + \frac{C)}{2} [D (g_{m - 1} (t_{j}), y_{1} (t_{m - 1})) + D (g_{m - 1} (t_{j + 1}), y_{m - 1} (t_{j + 1}))] \\ + \frac{C}{M} N_{m - 1} ω_{s} (R, h), \end{matrix}$ (4.2) where $N_{m - 1} = sup_{a \leq s \leq b} ∥ y_{m - 1} (s) ∥_{F}$ .

For a ≤ t ≤ b, taking the supremum from (4.2), we have $\begin{matrix} D^{*} (g_{m}, y_{m}) \leq \frac{C}{2} ω_{[a, b]} (g_{m - 1}, h) + {CD}^{*} (g_{m - 1}, y_{m - 1}) + + \frac{C}{M} N_{m - 1} ω_{s} (R, h), \end{matrix}$ $\begin{matrix} D^{*} (g_{m - 1}, y_{m - 1}) \leq \frac{C}{2} ω_{[a, b]} (g_{m - 2}, h) + {CD}^{*} (g_{m - 2}, y_{m - 2}) + + \frac{C}{M} N_{m - 2} ω_{s} (R, h), \end{matrix}$ $\dots$ $\begin{matrix} D^{*} (g_{1}, y_{1}) \leq \frac{C}{2} ω_{[a, b]} (g_{0}, h) + {CD}^{*} (g_{0}, y_{0}) + \frac{C}{M} N_{0} ω_{s} (R, h) . \end{matrix}$

Multiplying the above inequalities by 1, C, C², ⋯ , C^m-1, respectively and summing, we have $\begin{matrix} D^{*} (g_{m}, y_{m}) \leq \frac{C}{2} (ω_{[a, b]} (g_{m - 1}, h) + C ω_{[a, b]} (g_{m - 2}, h) + \dots + C^{m - 1} ω_{[a, b]} (g_{0}, h)) \\ + \frac{C}{M} ω_{s} (R, h) (N_{m - 1} + {CN}_{m - 2} + \dots + C^{m - 1} N_{0}) . \end{matrix}$ (4.3)

On the other hand, we also see that for t₁, t₂ ∈ [a, b] with |t₁, t₂| ≤ h, we obtain $\begin{matrix} D (g_{m} (t_{1}), g_{m} (t_{2}) = D (F (t_{1}) \oplus g_{m - 1} (t_{1}) \underset{g}{⊖} {Rg}_{m - 1}, F (t_{2}) \oplus g_{m - 1} (t_{2}) \underset{g}{⊖} {Rg}_{m - 1}) \\ \leq D ((F (t_{1}), F (t_{2})) + D (G (g_{m - 1}), G (g_{m - 1}) \\ \leq D ((F (t_{1}), F (t_{2})) + LD ({Rg}_{m - 1}, {Rg}_{m - 1}) \\ = D ((F (t_{1}), F (t_{2})) + LD ((FR) \int_{a}^{b} (s, t_{1}) g_{m - 1} (s) ds, (FR) \int_{a}^{b} (s, t_{2}) g_{m - 1} (s) ds) \\ \leq D ((F (t_{1}), F (t_{2})) + L \int_{a}^{b} | R (s, t_{1}) - R (s, t_{2}) | D (g_{m - 1} (s), \tilde{0}) ds \\ \leq D ((F (t_{1}), F (t_{2})) + L (b - a) ω_{t} (R, h) N_{m - 1}^{'} \\ \leq D ((F (t_{1}), F (t_{2})) + \frac{C}{M} ω_{t} (R, h) N_{m - 1}^{'}, \end{matrix}$ where ω_t (R, h) is the partial modulus of continuity with respect to t, $N_{m - 1}^{'} = sup_{a \leq \leq bs} ∥ g_{m - 1} (s) ∥_{F}$ .

In the above mentioned inequality, then we obtain the relation between the modulus of continuity of g_m and F as follows: $ω_{[a, b]} (g_{m}, h) \leq ω_{[a, b]} (F, h) + \frac{C}{M} ω_{t} (R, h) N_{m - 1}^{'} .$ (4.4)

By substitution (4.4) into (4.3), we obtain $\begin{matrix} D^{*} (g_{m}, y_{m}) \leq \frac{C}{2} (1 + C + \dots + C^{m - 1}) ω_{[a, b]} (F, h) \\ + \frac{C}{2 M} ω_{t} (R, h) ({CN}_{m - 2}^{'} + C^{2} N_{m - 3}^{'} + \dots + C^{m - 1} N_{0}^{'}) \\ + \frac{C}{M} ω_{s} (R, h) (N_{m - 1} + {CN}_{m - 2} + \dots + C^{m - 1} N_{0}) . \\ \leq \frac{C}{2} (\frac{1 - C^{m}}{1 - C}) ω_{[a, b]} (F, h) \\ + \frac{C}{2 M} ω_{t} (R, h) ({CN}_{m - 2}^{'} + C^{2} N_{m - 3}^{'} + \dots + C^{m - 1} N_{0}^{'}) \\ + \frac{C}{M} ω_{s} (R, h) (N_{m - 1} + {CN}_{m - 2} + \dots + C^{m - 1} N_{0}) . \end{matrix}$

Let $L_{1} = max_{0 \leq i \leq m - 1} {N_{i}}$ , $L_{2} = max_{0 \leq i \leq m - 2} {N_{i}^{'}}$ , we have $\begin{matrix} D^{*} (g_{m}, y_{m}) \leq \frac{C}{2} (\frac{1 - C^{m}}{1 - C}) ω_{[a, b]} (F, h) \\ + \frac{C}{2 M} ω_{t} (R, h) (C + C^{2} + \dots + C^{m - 1}) L_{2} \\ + \frac{C}{M} ω_{s} (R, h) (1 + C + \dots + C^{m - 1}) L_{1} . \end{matrix}$ (4.5)

For any m ∈ N, according to $\frac{1 - C^{m}}{1 - C} \leq \frac{1}{1 - C}$ , we have $D^{*} (g_{m}, y_{m}) \leq \frac{C}{2 (1 - C)} ω_{[a, b]} (F, h) + (\frac{C^{2} + 2 C}{2 M (1 - C)}) (L_{1} ω_{s} (R, h) + L_{2} ω_{t} (R, h)) .$ (4.6)

Considering the inequality (3.8) and (3.9), we obtain $\begin{matrix} D^{*} (g, y_{m}) \leq D^{*} (g, g_{m}) + D^{*} (g_{m}, y_{m}) \\ \leq \frac{(C)^{m + 1}}{LM (1 - C)} J \\ + \frac{C}{2 (1 - C)} ω_{[a, b]} (F, h) + (\frac{C^{2} + 2 C}{2 M (1 - C)}) (L_{1} ω_{s} (R, h) + L_{2} ω_{t} (R, h)) . \end{matrix}$ (4.7)

Since C < 1, it follows that $lim_{m \to \infty} C^{m + 1} = 0$ , and $\begin{matrix} lim_{h \to 0} ω_{[a, b]} (F, h) = 0, lim_{h \to 0} ω_{s} (R, h) = 0, lim_{h \to 0} ω_{t} (R, h) = 0 . \end{matrix}$

Hence, we obtain $lim_{m \to \infty, h \to 0} D^{*} (g, y_{m}) = 0 .$

That shows the convergence of the method.

5 Numerical Examples

In order to illustrate the efficiency of the presented method, we give two examples. Also, we compare the numerical solution obtained by using the proposed method with the exact solution.

Example 5.1. Consider the following fuzzy Fredholm integral equation of the first kind: $\begin{matrix} (FR) \int_{0}^{1} k (x, s) ⊙ g (s) ds = f (x), x \in [0, 1], \end{matrix}$ (5.1) where $\begin{matrix} k (x, s) = e^{xs}, f (x) = \frac{e^{x + 1} - 1}{x + 1} ⊙ \tilde{v} (x) . \end{matrix}$ (5.2)

Let $\tilde{v}$ be given by the following triangular fuzzy number: $\tilde{v} (x) = {\begin{matrix} x + 1, x \in (- 1, 0), \\ - x + 1, x \in (0, 1), \\ 0, x \in (- \infty, - 1] \cup [1, + \infty) . \end{matrix}$ (5.3) $\begin{matrix} \underline{f} (x, α) = (α - 1) \frac{e^{x + 1} - 1}{x + 1}, \end{matrix}$ (5.4) $\begin{matrix} \bar{f} (x, α) = (1 - α) \frac{e^{x + 1} - 1}{x + 1} . \end{matrix}$ (5.5)

The exact solution of (5.1) is given by $\begin{matrix} \tilde{g} (x) = e^{x} ⊙ \tilde{v} (x), \\ \underline{g} (x, α) = (α - 1) e^{x}, \bar{g} (x, α) = (1 - α) e^{x} . \end{matrix}$ (5.6)

Notes 1: In our numerical computations, we always take step size $h = \frac{1}{100}$ and only consider the cases when x = 0.5. To compare the error based on the iteration steps m = 5, n = 20 and the iteration steps m = 10, n = 20, see respectively Tables 1 and 2. To compare the error based on the iteration steps m = 15, n = 100 and the iteration steps m = 20, n = 100, see respectively Table 3 and Table 4. The numerical results for the exact solutions $\underline{g}$ and $\bar{g}$ and the approximation solutions $\underline{y}$ and $\bar{y}$ with α = 0.5 and m = 20, n = 25 are shown in Fig. 1.

Table 1

x = 0.5, m = 5, n = 20

α-level	0.0	0.2	0.4	0.6	0.8	1.0
$\| \bar{g} - {\bar{y}}_{m} \|$	9.1243e-8	8.2102e-8	7.2892e-6	3.3909e-7	9.4692e-6	4.5521e-10
$\| \underline{g} - {\underline{y}}_{m} \|$	1.2394e-9	2.901e-9	9.9136e-8	8.5997e-9	7.4353e-10	6.2013e-10

Table 2

x = 0.5, m = 10, n = 20

α-level	0.0	0.2	0.4	0.6	0.8	1.0
$\| \bar{g} - {\bar{y}}_{m} \|$	2.9013e-10	2.5906e-9	2.2984e-10	2.0131e-10	1.7362e-11	1.4298e-11
$\| \underline{g} - {\underline{y}}_{m} \|$	1.2011e-9	1.0765e-8	9.5816e-10	8.3811e-11	7.1838e-9	5.9865e-11

Table 3

x = 0.5, m = 15, n = 100

α-level	0.0	0.2	0.4	0.6	0.8	1.0
$\| \bar{g} - {\bar{y}}_{m} \|$	1.7762e-10	1.1014e-10	1.4217e-11	1.2436e-12	1.1596e-11	8.8701e-12
$\| \underline{g} - {\underline{y}}_{m} \|$	1.0703e-9	9.6319e-11	8.6102e-10	7.4901e-10	6.4213e-11	5.3511e-12

Table 4

x = 0.5, m = 20, n = 100

α-level	0.0	0.2	0.4	0.6	0.8	1.0
$\| \bar{g} - {\bar{y}}_{m} \|$	9.1105e-12	8.1042e-10	2.6218e-11	2.7846e-11	4.9406e-10	4.6174e-12
$\| \underline{g} - {\underline{y}}_{m} \|$	1.3182e-10	1.2681e-12	2.3983e-11	4.8126e-11	3.8961e-12	5.9018e-12

Fig. 1

α = 0.5, n = 25, m = 20.

Example 5.2. Consider the following fuzzy Fredholm integral equation of the first kind with a real symmetric kernel function: $\begin{matrix} (FR) \int_{0}^{1} k (x, s) ⊙ g (s) ds = f (x), x \in [0, 1], \end{matrix}$ (5.7) where $k (x, s) = {\begin{matrix} (1 - x) s, 0 \leq s \leq x, \\ (1 - s) x, x \leq s \leq 1 . \end{matrix}$ (5.8)

$\begin{matrix} f (x) = sin π x ⊙ \tilde{v} (x) . \end{matrix}$ (5.9)

Let $\tilde{v}$ be given by the equation (5.3). $\begin{matrix} \underline{f} (x, α) = (α - 1) sin π x, \end{matrix}$ (5.10) $\begin{matrix} \bar{f} (x, α) = (1 - α) sin π x . \end{matrix}$ (5.11)

The exact solution of (5.7) is given by $\begin{matrix} \tilde{g} (x) = π^{2} sin π x ⊙ \tilde{v} (x), \underline{g} (x, α) \\ = (α - 1) π^{2} sin π x, \bar{g} (x, α) = (1 - α) π^{2} sin π x . \end{matrix}$ (5.12)

Notes 2: In our numerical computations, we always take step size $h = \frac{1}{100}$ and only consider the cases when x = 0.5. To compare the error based on the iteration steps m = 5, n = 20 and the iteration steps m = 10, n = 20, see respectively Table 5 and Table 6. To compare the error based on the iteration steps m = 15, n = 100 and the iteration steps m = 20, n = 100, see respectively Table 7 and Table 8. The numerical results for the exact solutions $\underline{g}$ and $\bar{g}$ and the approximation solutions $\underline{y}$ and $\bar{y}$ with α = 0.5 and m = 20, n = 25 are shown in Fig.2.

Table 5

x = 0.5, m = 5, n = 20

α-level	0.0	0.3	0.5	0.7	0.9	1.0
$\| \bar{g} - {\bar{y}}_{m} \|$	9.6724e-9	9.0352e-9	8.5426e-8	8.4250e-8	9.0146e-9	8.3965e-10
$\| \underline{g} - {\underline{y}}_{m} \|$	8.5218e-10	9.6390e-8	8.2791e-9	9.0193e-10	8.6726e-9	7.9737e-8

Table 6

x = 0.5, m = 10, n = 20

α-level	0.0	0.3	0.5	0.7	0.9	1.0
$\| \bar{g} - {\bar{y}}_{m} \|$	4.3627e-9	3.8532e-10	3.9745e-11	2.9783e-11	3.0358e-10	2.9319e-10
$\| \underline{g} - {\underline{y}}_{m} \|$	3.7497e-10	2.8954e-11	5.7461e-10	4.8547e-12	5.9735e-10	6.4259e-9

Table 7

x = 0.5, m = 15, n = 100

α-level	0.0	0.3	0.5	0.7	0.9	1.0
$\| \bar{g} - {\bar{y}}_{m} \|$	2.8831e-11	3.4584e-11	3.0153e-12	2.5714e-10	2.2843e-10	4.9673e-11
$\| \underline{g} - {\underline{y}}_{m} \|$	3.9071e-10	8.5742e-12	6.3761e-11	5.9357e-11	5.9876e-12	4.6592e-11

Table 8

x = 0.5, m = 20, n = 100

α-level	0.0	0.3	0.5	0.7	0.9	1.0
$\| \bar{g} - {\bar{y}}_{m} \|$	8.8736e-11	9.0052e-12	3.9746e-12	6.8726e-10	5.8319e-11	6.9836e-12
$\| \underline{g} - {\underline{y}}_{m} \|$	7.9758e-11	4.9702e-11	6.0937e-12	8.1653e-10	7.7715e-11	6.6905e-13

Fig. 2

α = 0.5, n = 25, m = 20.

6 Conclusions

In this paper, based on the strictly convex fuzzy number space and a new Riemann integral of fuzzy-number-valued function, we proposed a numerical method to solve fuzzy Fredholm integral equations for the first kind by construction of an iterative procedure ${\tilde{g}}_{n} (x) = \tilde{F} (x) \oplus {\tilde{g}}_{n - 1} (x) \underset{g}{⊖} R {\tilde{g}}_{n - 1}$ . Also, we have obtained the error estimation for approximating the solution of fuzzy Fredholm integral equation for the first kind using the trapezoidal quadrature formula, in terms of uniform and partial modulus of continuity, and proved the convergence of the method. In the iterative procedure, the generalized difference of fuzzy numbers ${\tilde{g}}_{n - 1} (x) \underset{g}{⊖} R {\tilde{g}}_{n - 1}$ was controlled and estimated reasonably and it was avoided to compute the generalized difference of fuzzy numbers during the iterations. Finally, two examples show that the proposed method is effective.

Comparing and outlooks. The study of Fuzzy integral equations (FIEs) began with the investigations of Kaleva [26] and Siekkala [36] for the fuzzy Volterra integral equations which were equivalent to the initial value problem for the first order fuzzy differential equations in 1987. After which these concepts were developed by many researchers and the focus of current research focuses on fuzzy Fredholm integral equations of the second kind (FFIEs-2) or fuzzy Volterra integral equations, such as the mentioned literatures [5–8 , 31–35] in Introduction of this paper and [13 , 30] and Solving the Volterra-Fredholm integral-differential equations [2 –4] and so on. Solving the second kind Fuzzy Integral Equations are well-posed according to the Hadamard’s definition of well-posedness that have been proposed in 1923 [15]. The equations is called the integral equations of the second kind characterized by the occurrence of the unknown function u (x) that will be determined inside and outside the integral sign, such as the equation [2 –4] $\begin{matrix} {\tilde{u}}^{″} = \tilde{f} (x) + μ_{1} \int_{a}^{x} k_{1} (x, t) G_{1} \\ (t, \tilde{u} (t)) dt + μ_{2} \int_{a}^{b} k_{2} (x, t) G_{2} (t, \tilde{u} (t)) dt, \end{matrix}$ and $\tilde{f} (s) + \int_{a}^{b} k_{1} (s, t) \tilde{u} (t) dt + \int_{a}^{b} k_{2} (s, t) \tilde{u} (t) dt = \tilde{u} (t),$ and $\tilde{u} (s) = \tilde{f} (s) + \frac{λ}{π} \int_{a}^{b} \frac{\tilde{u} (t)}{t - s} dt .$

However, in our paper, we consider the fuzzy Fredholm integral equations of the first kind (FFIEs-1) $f (x) = (FR) \int_{a}^{b} k (x, y) ⊙ g (y) dy, x \in [a, b],$ where integral lower limit a and integral upper limit b are constants, f (x) is the known fuzzy-number-valued function, k (s, t) is the kernel of the integral equation, and g (y) is the unknown fuzzy-number-valued function that will be determined. The equation is called the fuzzy Fredholm integral equations of the first kind characterized by the occurrence of the unknown function g (y) only inside the integral sign. The existence of g (y) inside the integral sign causes special difficulties. And, Fuzzy Fredholm integral equations of the first kind are often ill-posed problems [39], the numerical solution researching for FFIEs-1 is more difficult than fuzzy Fredholm integral equations of the second kind (FFIEs-2). For now, we only retrieved two literatures [28, 39] about the numerical solution researching for the ill-posed problems of FIEs-1.

The paper [28] “Solving the First Kind Fuzzy Integral Equations Using a Hybrid Regularization Method and Bernstein Polynomials” that is closely related to our work is published on May 9, 2016. In the paper [28] the authors pay much attention on the Abel integral equation $f (x) = (FR) \int_{0}^{x} \frac{u (t)}{(x - t)^{α}} dt, x \in [a, b],$ where integral upper limit x is variable, u (x) is the unknown function that will be determined. The Abel integral equation is a special kind of Volterra type integral equation of the first kind. In general, solving the Volterra type integral equation of the first kind is well-posed. But the kernel function of the Abel integral operator is unbounded, this characteristic makes that it is an ill-posed problem to solving the Abel integral equation. In the paper [28] the authors neither discussed ill-posedness of the fuzzy Abel integral equation nor used the difference of the fuzzy number. The paper [28] focuses on solving the obtained second kind fuzzy integral equation by approximating Bernstein polynomials.

The paper [39] focuses on studying a fuzzy Fredholm integral equation of the first kind $F (t) = λ ⊙ (FR) \int_{a}^{b} k (s, t) ⊙ Y (s) ds, t \in [c, d] .$

Its contribution is as follows.

(1) Authors systematically define ill-posedness of the fuzzy Fredholm integral equation of the first kind based on endpoint functions of fuzzy-number-valued function.

(2) Authors explicitly make use of the generalized Hukuhara difference of the fuzzy numbers and its properties in the process of theorem proving and the calculation of examples.

(3) A regularization method have been presented to transform the fuzzy Fredholm integral equation of the first kind into the fuzzy integral equation of the second kind based on the technique elaborated by Philips (1962) and Tikhonov (1963). And by converting the first kind to a second kind, then the existing techniques of the second kind to the transformed equation can be apply. But the method involves the selection of regularization parameters, which is a difficult point.

In this current paper, our contribution as follows in contrast with the paper [39].

(1) The iterative method be proposed to solve fuzzy Fredholm integral equations of the first kind. The advantage of this method is that it does not involve the selection of regularization parameters.

(2) We make use of the generalized Hukuhara difference of the fuzzy numbers and its properties in the process of theorem proving, such as the iterative procedure ${\tilde{g}}_{n} (x) = \tilde{F} (x) \oplus {\tilde{g}}_{n - 1} (x) \underset{g}{⊖} R {\tilde{g}}_{n - 1}$ , and so on.

(3) This research be based on a strictly convex fuzzy number space that is different from the paper [39].

For the future works, we can apply the proposed method in this paper for solving two-dimensional fuzzy Fredholm integral equation of the first kind and the Cauchy problem with fuzzy parameters. Also, as future works, by trying on use the conjugate gradient method to solve fuzzy Fredholm integral equation of the first kind.

Footnotes

Acknowledgment

This work was supported by Natural Science Foundation of China (No. 61763044) and Natural Science Foundation of Gansu Province (No. 18JR3RA096) and the improvement Plan for the Scientific Research Ability of Young Teachers in Northwest Normal University (No. 385). The authors would like to thank the Prof. Editors and Editor-in-Chief and reviewers for their constructive comments and valuable suggestions, which improved the quality of this paper.

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