Analysis of global solution of functional fuzzy integral equations using Krasnoselskii-Burton fixed point theorem

Abstract

Fixed point theorems are very important and advantageous concepts in mathematical analysis. In this paper, we establish Krasnoselskii-Burton fixed point theorem for fuzzy Banach space. Using fuzzy Krasnoselskii-Burton and applying some notions and definitions in fuzzy functions and fuzzy normed space such as compactness and convexity of fuzzy sets, contraction fuzzy mappings and equicontinuous fuzzy functions we prove the existence of global solution of functional fuzzy integral equations.

Keywords

Fixed point theorem convex fuzzy set functional fuzzy integral equations global solutions fuzzy operator equations

1 Introduction

One of the most important concepts in the theory of integral equations and differential equations, especially fuzzy differential and integral equations, is the investigate and prove the existence and uniqueness of the solution of these equations. To this end, various theorems and techniques have been used by many researchers. One of the most widely used tools to prove the existence of solution of integral equations and convergence of numerical and analytical methods in literature, is to use fixed point theorems. In mathematics, a fixed point theorem is a result saying that a function F will have at least one fixed point (a point x which F (x) = x), under some conditions of F that can be stated in general terms. Results of this kind are amongst the most generally in mathematics. Some of the best known fixed point theorems in mathematical analysis, algebra and discrete mathematics include: Banach, Brouwer, Lefschetz, Borel, Schauder fixed point theorems. Two main results of fixed point theory are Schauder’s theorem [25] and the contraction mapping principle. Krasnoselskii combined them into a new fixed point theorem [19 , 26]. Indeed, he considered the sum of a compact mapping and a contraction mapping. This combination can occur in practice; in dealing with a perturbed differential operator we may find that the perturbations leads to a contraction mapping while inversion of the differential operator gives a compact mapping. Burton in 1997 introduced an improved type of Krasnoselskii’s fixed point theorem that is well-known and useful in the literature for the applications to the differential and integral equations [10]. As we know, many studies have been conducted so far on fuzzy integral equations, for example see [8 , 23].

In this paper, at first we intend to generalize the Krasnoselskii’s and Burtons’s fixed point theorems in fuzzy normed spaces and then we establish a fuzzy variant of Krasnoselskii-Burton fixed point theorem via employing a special case of fixed point theorem in Banach spaces due to the present in [11]. In addition, using this result the existence of the global solutions of the functional fuzzy integral equations (FFIE)

$\begin{matrix} x (t) & = a (t) \oplus h (t, x (t), x (η (t))) \\ \oplus (FR) \int_{0}^{θ (t)} H (t, s, x (s), x (ϑ (s))) ds \end{matrix}$ (1) for all $t \in ℝ^{+}$ are proved, where $a : ℝ^{+} \to R_{ϝ}, h : ℝ^{+} \times R_{ϝ} \times R_{ϝ} \to R_{ϝ}, H : ℝ^{+} \times ℝ^{+} \times R_{ϝ} \times R_{ϝ}$ →R_ϝ and $η, θ, ϑ : ℝ^{+} \to ℝ^{+}$ . The set of all fuzzy numbers is denoted by R_ϝ which will be introduced later. By a solution of (1) we mean a function $x \in C (ℝ^{+}, R_{ϝ})$ that satisfies the Equation. (1), where $C (ℝ^{+}, R_{ϝ})$ is the space of continuous fuzzy-valued functions defined on $ℝ^{+}$ . The fuzzy integral equations (1) is a general case of fuzzy integral equations, since it is includes several functional and functional integral equations that have been studied in literatures. We organize this paper as follows: in Section 2, are presented some notions about the structure of the set of fuzzy numbers and some mathematical analysis properties for fuzzy-number-valued functions such as fuzzy Henstock and fuzzy Riemann integrability, continuity, convergence, convexity, boundedness and compactness. Furthermore, some definitions of fuzzy geometry such as fuzzy line, fuzzy line segment, convex fuzzy space are discussed in this section. Section 3 which is the main section of paper, firstly, we reconstruct the Krasnoselskii, Burton and Dhage fixed point theorems in fuzzy Banach spaces and then we provide some conditions which under these conditions the fuzzy operator equations (1) has at least one global solution in fuzzy Banach spaces. The existence of this global solution using above reorganized fixed point theorems is proved in this section too. Section 4 is devoted to the numerical examples. Finally we conclude in Section 5.

2 Preliminaries

In this section, we briefly recall some definitions, notions and basic facts concerning fuzzy numbers, fuzzy-valued functions and mathematical analysis in fuzzy Banach spaces which will be referred them throughout this paper.

Definition 1. [17] A fuzzy number is a function $u : ℝ \to [0, 1]$ satisfying the following properties: u is upper semicontinuous on R, u (x) =0 outside of some interval [c, d], there are the real numbers a and b with c ≤ a ≤ b ≤ d, such that u is increasing on [c, a], decreasing on [b, d] and u (x) =1 for each x ∈ [a, b] and u is fuzzy convex set (that is u (λx + (1 - λ) y) ≥ min {u (x) , u (y)} , ∀ x, y ∈ R, λ ∈ [0, 1]). The set of all fuzzy numbers is denoted by R_ϝ and any real number $α \in ℝ$ can be interpreted as a fuzzy number $\tilde{α} = χ_{{α}}$ and therefore $ℝ \subset R_{ϝ}$ . Also, the neutral element respect to ⊕ in R_ϝ denoted by $\tilde{0} = χ_{{0}}$ . According to [13, 27] for any 0 < r ≤ 1 an arbitrary fuzzy number is represented, in parametric form, by an ordered pair of functions $(u_{-}^{r}, u_{+}^{r})$ , which satisfies the following properties: $u_{-}^{r}$ is bounded left continuous non-decreasing function over [0, 1], $u_{+}^{r}$ is bounded left continuous non-increasing function over [0, 1] and $u_{-}^{r} < u_{+}^{r}$ .

Definition 2. [15] For any u ∈ R_ϝ the r-level set of u is denoted by [u] ^r and defined by $[u]^{r} = {x \in ℝ ∣ u (x)$ ≥r}, where 0 < r ≤ 1. Also, [u] ⁰ = $\bar{{x \in ℝ : u (x) > 0}}$ is called the support of u and is a compact set, where $\bar{A}$ denotes the closure of A. It follows that the level sets of u are closed and bounded intervals in $ℝ$ . It is well-known that the addition and multiplication operations of real numbers can be extended to R_ϝ. In other words, for any u, v ∈ R_ϝ and $λ \in ℝ$ , we define uniquely the sum u ⊕ v and the product λ ⊙ u by [u ⊕ v] ^r = [u] ^r + [v] ^r, [λ ⊙ u] ^r = λ [u] ^r, for all r ∈ [0, 1], where [u] ^r ⊕ [v] ^r means the usual addition of two intervals (as subset of $ℝ$ ) and λ [u] ^r means the usual product between a scalar and a subset of R. We use the same symbol ∑ both for the sum of real numbers and for the sum ⊕ (when the terms are fuzzy numbers). Also according to [3, 27], the following algebraic properties for any u, v, w ∈ R_ϝ hold:

u ⊕ (v ⊕ w) = (u ⊕ v) ⊕ w,

$u \oplus \tilde{0} = \tilde{0} \oplus u = u,$

with respect to $\tilde{0},$ none $u \in (R_{ϝ} - ℝ), u \neq \tilde{0}$ has opposite in (R_ϝ, ⊕) ,

$(a + b) ⊙ u = a ⊙ u \oplus b ⊙ u, \forall a, b \in ℝ$ with ab ≥ 0,

$a ⊙ (u \oplus v) = a ⊙ u \oplus a ⊙ v, \forall a \in ℝ,$

$a ⊙ (b ⊙ u) = (ab) ⊙ u, \forall a, b \in ℝ$ and 1⊙u = u .

Definition 3. [16, 27] For arbitrary fuzzy numbers $u = (u_{-}^{r}, u_{+}^{r})$ , $v = (v_{-}^{r}, v_{+}^{r})$ the quantity $D (u, v) = sup max {| u_{-}^{r} - v_{-}^{r} |, | u_{+}^{r} - v_{+}^{r} |}$ where r ∈ [0, 1], is the distance between u and v. It is demonstrated that (R_ϝ, D) is a complete metric space and followingproperties hold:

D (u ⊕ w, v ⊕ w) = D (u, v) ∀ u, v, w ∈ R_ϝ,

D (k ⊙ u, k ⊙ v) = |k|D (u, v) ∀ u, v ∈ R_ϝ, ∀ k $\in ℝ,$

D (u ⊕ v, w ⊕ e) ≤ D (u, w) + D (v, e) ∀ u, v, w, e ∈ R_ϝ .

$D (k_{1} ⊙ u, k_{2} ⊙ u) = | k_{1} - k_{2} | D (u, \tilde{0}) \forall k_{1},$ $k_{2} \in ℝ$ with k₁k₂ ≥ 0 and ∀u ∈ R_ϝ,

Throughout this paper we denote that ∥ . ∥ _ϝ = $D (., \tilde{0})$ . Also, according to [4, 7], the pair (R_ϝ, D) is a commutative semigroup with $\tilde{0} = χ_{0}$ zero element, but cannot be a group for pure fuzzy numbers. ∥ . ∥ _ϝ has the properties of a usual norm on R_ϝ, i.e. ∥ . ∥ _ϝ = 0 iff u = 0, ∥λ ⊙ u ∥ _ϝ = |λ| ∥ u ∥ _ϝ and ∥u ⊕ v ∥ _ϝ ≤ ∥ u ∥ _ϝ + ∥ v ∥ _ϝ. Moreover, | ∥ u ∥ _ϝ - ∥ v ∥ _ϝ| ≤ D (u, v) and D (u, v) ≤ ∥ u ∥ _ϝ + ∥ v ∥ _ϝ for any u, v ∈ R_ϝ.

According to [9] and with our notation, if $u = (u_{-}^{r},$ $u_{+}^{r})$ , $v = (v_{-}^{r}, v_{+}^{r})$ are fuzzy numbers, then $u \oplus v = ((u_{-}^{r}, u_{+}^{r}) \oplus (v_{-}^{r}, v_{+}^{r})) = (u_{-}^{r} + v_{-}^{r}, u_{+}^{r} + v_{+}^{r}) = ((u +$ $v)_{-}^{r}, (u + v)_{+}^{r})$ . Also, $u ⊙ v = ((u_{-}^{r}, u_{+}^{r}) ⊙ (v_{-}^{r}, v_{+}^{r}))$ $= (u_{-}^{r} . v_{-}^{r}, u_{+}^{r} . v_{+}^{r}) = ((u . v)_{-}^{r}, (u . v)_{+}^{r})$ , for all u, v with positive supports. In addition, for all strictly increasing and positive real function ψ, we have $ψ (u) = ψ (u_{-}^{r}, u_{+}^{r}) = (ψ (u_{-}^{r}), ψ (u_{+}^{r}))$ .

Definition 4. [3] Let f : R_ϝ → R_ϝ be a fuzzy-valued function, that is f maps any u ∈ R_ϝ to f (u) ∈ R_ϝ, then f is called continuous on fixed u ∈ R_ϝ if for each ɛ > 0 exists δ > 0 such that D (f (u) , f (v)) < ɛ whenever D (u, v) < δ , ∀v ∈ R_ϝ, and is said uniformly continuous on R_ϝ if for ɛ > 0 exists δ > 0 such that D (f (u) , f (v)) < ɛ whenever D (u, v) < δ ,∀u, v ∈ R_ϝ. In addition, if $f, g : [a, b] \subseteq ℝ \to R_{ϝ}$ are fuzzy continuous functions, then the function $F : [a, b] \to ℝ^{+}$ defined by F (x) = D (f (x) , g (x)) is continuous on [a, b]. Also $D (f (x), \tilde{0}) = ∥ f (x) ∥_{ϝ} \leq M, \forall x \in [a, b], M > 0$ , that is f is fuzzy bounded. Equivalently we get χ_-M ≤ f (x) ≤ χ_M , ∀ x ∈ [a, b].

On the set $C (ℝ, R_{ϝ}) = {f : ℝ \to R_{ϝ} : f is$ continuous}, we define $D^{*} (f, g) = sup D (f (t), g (t)), \forall f, g \in C (ℝ, R_{ϝ}) .$

It is obvious that $(C (ℝ, R_{ϝ}), D^{*})$ is a complete metric space.

Lemma 5. [24] Iff, g : R_ϝ → R_ϝare fuzzy continuous functions onR_ϝ, then the functionΨ : R_ϝ → R_ϝdefined byΨ (u) = (f ∘ g) (u) = f (g (u)) is continuous onR_ϝ.

Definition 6. In [27], the fuzzy Riemann integral for a fuzzy-number-valued function f on [a, b] was introduced. For Δ_n : a = x₀ < x₁ < . . . < x_n-1 < x_n = b, a partition of the interval [a, b], we define the modulus of the partition Δ_n by $∥ Δ_{n} ∥ = sup_{1 \leq i \leq n} | x_{i} - x_{i - 1} |$ . The function f is called fuzzy Riemann integrable on [a, b] to I ∈ R_ϝ if for every ɛ > 0 there is a number δ > 0 such that for any Δ_n satisfying ∥Δ_n ∥ < δ and any intermediate points ξ_i ∈ [x_i-1, x_i] , i = 1, . . . , n, we have $D (\sum_{i = 1}^{n} (x_{i} - x_{i - 1}) ⊙ f (ξ_{i}), I) < ɛ .$

Then I is called the fuzzy Riemann integral of f on [a, b] and denoted by $(FR) \int_{a}^{b} f (t) dt$ . If f is fuzzy Riemann integrable to I ∈ R_ϝ, then for each r ∈ [0, 1], the crisp functions $f_{-}^{r} (x), f_{+}^{r} (x)$ are Riemann integrable on [a, b] to $I_{-}^{r}, I_{+}^{r}$ , respectively. If f is continuous on [a, b], then for each r ∈ [0, 1], $f_{-}^{r} (x), f_{+}^{r} (x)$ are continuous on [a, b], and hence, f is fuzzy Riemann integrable on [a, b].

Corollary 7. [3 , 27]

Iff ∈ C_ϝ [a, b], its definite integral exists, and also, $((FR) \int_{a}^{b} f (t) dt)_{-}^{r} = \int_{a}^{b} f_{-}^{r} (t) dt$ and $((FR) \int_{a}^{b} f (t) dt)_{+}^{r} = \int_{a}^{b} f_{+}^{r} (t) dt$ , for allr ∈ [0, 1].

Iff : [a, b] → R_ϝ, be an integrable bounded function, then for any fixedu ∈ [a, b], the functionα_u : [a, b] → R₊, defined byα_u (x) = D (f (u) , f (x)) is Lebesgue integrable on [a, b].

Iffandgare (FR)-integrable functions on [a, b] andD (f (x) , g (x)) is Lebesgue integrable, then

$\begin{matrix} D ((FR) \int_{a}^{b} f (x) dx, (FR) \int_{a}^{b} g (x) dx) \\ \leq (L) \int_{a}^{b} D (f (x), g (x)) dx . \end{matrix}$ (2)

In order to establish and proof the fuzzy Krasnoselskii fixed point theorem and other topics in the rest of this paper we need to apply some notions and definitions of mathematical analysis and geometry in fuzzy space. In the following two remarks we recall some concepts concerning fuzzy analysis, fuzzy geometry and fuzzy Banach spaces.

Remark 8. The concept of convex fuzzy sets was introduced by Zadeh [28] and by Lowen [22]. For example, in the following convex fuzzy set, $β (x) = {\begin{matrix} 1 / (1 + j)^{2} & j < | x | < 1 + j, j = 1, 2, \\ 0 & else, \end{matrix}$ we get β (x) =0.25 for each x ∈ (0, 1), i.e., there is no increasing point of the membership functions in this interval. Almost all definitions and notions of the real and complex numbers space can be extended for fuzzy number space. Furthermore, it is possible to extend most definitions of basic mathematical analysis and geometry for fuzzy metric and normed spaces which for instance we review three cases of them as follows.

Fuzzy line: [2] Let $β : ℝ \to [0, 1]$ be a fuzzy set and $x_{1}, x_{2} \in ℝ$ , $I = {x = λ x_{1} + (1 - λ) x_{2}, λ \in ℝ}$ ,

$β (x) = {\begin{matrix} β (x_{1}) & λ > 1, x \in I \\ λ β (x_{1}) + (1 - λ) β (x_{2}) & λ \in [0, 1], x \in I, \\ β (x_{2}) & λ < 0, x \in I, \\ 0 & x \notin I . \end{matrix}$

Then $β_{I} : ℝ \to [0, 1]$ is a fuzzy line joining x₁ and x₂.

Fuzzy line segment: [2] Let $β : ℝ \to [0, 1]$ be a fuzzy set and x₁ ≠ x₂ ∈ supp (β). Now, we define the fuzzy line segment joining x₁, x₂ as follows:

(i) Closed fuzzy line segment, denoted by β_[x₁,x₂] is $β_{[x_{1}, x_{2}]} = {\begin{matrix} λ β (x_{1}) + (1 - λ) β (x_{2}) & x = λ x_{1} + (1 - λ) x_{2}, λ \in [0, 1], \\ 0 & else . \end{matrix}$

(ii) A fuzzy line segment is open, closed-open and open-closed if above β_[x₁,x₂] holds with λ, belonging to (0, 1), [0, 1) and (0, 1] instead of [0, 1], respectively.

Convex fuzzy sets: [2] A fuzzy set β is a convex (strictly convex) fuzzy set if and only if the closed (the open) fuzzy line segment joining two points of β is included (strictly included) in β. Equivalently, a fuzzy set β is convex (strictly convex) fuzzy set if and only if

β (λx₁ + (1 - λ) x₂) ≥ (>) λβ (x₁) + (1 - λ) β (x₂) for all x₁, x₂ ∈ supp (β) , λ ∈ [0, 1] (x₁ ≠ x₂, λ ∈ (0, 1)).

Similarly, one can define many of the implications such as boundededness, compactness and connectivity on fuzzy spaces [1 , 21].

Remark 9. [5, 6] Let (R_ϝ, D) be a fuzzy normed space. Let u_n be a sequence in R_ϝ. Then u_n is said to be convergent if there exists u ∈ R_ϝ such that $lim_{n \to \infty} D (u_{n}, u) = 0$ . In this case, u is called the limit of sequence u_n and we denote it by $lim_{n \to \infty} u_{n} = u$ . A sequence u_n in R_ϝ is said to be Cauchy sequence if $lim_{n \to \infty} D (u_{n + m}, u) = 0, m = 1, 2, . . .$ . It is Known that every convergent sequence in a fuzzy normed space is a Cauchy sequence. If every Cauchy sequence is convergent, then the fuzzy normed space is called a fuzzy Banach space.

3 Main results

The main topics of the paper will be presented in this section.

3.1 Krasnoselskii-Burton fixed point theorem in fuzzy space

Here, we give generalized fuzzy type of three theorems which were introduced by Krasnoselskii [26], Burton [10] and Dhage [11] in crisp number space, respectively. Regarding to Remarks 8 and 9, it is known that the argument of these theorems in fuzzy normed space is very straightforward. It is enough to apply the same steps that are done in mentioned references.

Theorem 10.(Fuzzy Krasnoselskii fixed point) LetE_ϝbe a closed convex non-empty subset of a fuzzy Banach spaceR_ϝ. Assume thatTandSmapE_ϝintoR_ϝand that

Tu ⊕ Sv ∈ E_ϝ, ∀u, v ∈ E_ϝ,

T is compact and continuous,

S is a contraction mapping,

then there exists w ∈ E_ϝ such that Tw ⊕ Sw = w.

Theorem 11.(Fuzzy Burton fixed point) LetE_ϝbe a closed, convex and bounded subset of the fuzzy Banach spaceR_ϝand letT : R_ϝ → R_ϝandS : E_ϝ → R_ϝbe two operators such that

T is a contraction mapping,

S is a completely continuous,

∀v ∈ E_ϝ, u = Tu ⊕ Sv ⇒ u ∈ E_ϝ,

then the operator equation Tu ⊕ Sv = u has a solution in E_ϝ.

Theorem 12.(Fuzzy Dhage fixed point) LetE_ϝbe a closed, convex and bounded subset of the fuzzy Banach spaceR_ϝand letT : R_ϝ → R_ϝandS : E_ϝ → R_ϝbe two operators such that

there is a continuous nondecreasing function $ψ : ℝ^{+} \to ℝ^{+}, ψ (q) < q, q > 0$ such that D (Tu, Sv) ≤ ψ (D (u, v)).

S is a completely continuous,

∀v ∈ E_ϝ, u = Tu ⊕ Sv ⇒ u ∈ E_ϝ,

then the operator equation Tu ⊕ Sv = u has a solution in E_ϝ.

In the space $C (ℝ^{+}, R_{ϝ})_{B}$ of continuous and bounded fuzzy-valued functions defined on $ℝ^{+}$ , we use a standard fuzzy supremum norm ∥ . ∥ _ϝs in $C (ℝ^{+}, R_{ϝ})_{B}$ by $∥ . ∥_{ϝ s} = sup_{t \in ℝ^{+}} D (x (t), \tilde{0}) = D^{*} (x, \tilde{0}) .$

It is clear that $C (ℝ^{+}, R_{ϝ})_{B}$ is a fuzzy Banach space with respect to this norm in it.

3.2 Global solution of FFIE

Now, we provide some conditions which under these conditions the FFIE (1) will have at least one solution. We consider the operator equations $Tx (t) = a (t) \oplus h (t, x (t), x (η (t)),$ (3) and $Sx (t) = (FR) \int_{0}^{θ (t)} H (t, s, x (s), x (ϑ (s))) ds .$ (4) with the following four conditions:

(C1) The functions $η, θ, ϑ : ℝ^{+} \to ℝ^{+}$ are continuous and $lim_{t \to \infty} η (t) = \infty$ .

(C2) The function $a : ℝ^{+} \to R_{ϝ}$ is fuzzy continuous and fuzzy bounded with $p = sup_{t \in ℝ^{+}} D (a (t), \tilde{0}) = D^{*} (a, \tilde{0}) = ∥ a ∥_{ϝ s}$ .

(C3) The function $h : ℝ^{+} \times R_{ϝ} \times R_{ϝ} \to R_{ϝ}$ is fuzzy continuous and there exists a bounded function $b : ℝ^{+} \to ℝ^{+}$ with bound M and positive constant K ≥ M such that for all $t \in ℝ^{+}$ $\begin{matrix} D (h (t, x_{1} (t), x_{2} (t)), h (t, y_{1} (t), y_{2} (t))) \\ \leq \frac{b (t) max (D (x_{1} (t), y_{1} (t)), D (x_{2} (t), y_{2} (t)))}{K + max (D (x_{1} (t), y_{1} (t)), D (x_{2} (t), y_{2} (t)))} . \end{matrix}$ Also, let $ρ = sup_{t \in ℝ^{+}} D (h (t, \tilde{0}, \tilde{0}), \tilde{0}) \leq D^{*} (h, \tilde{0})$ .

(C4) The function $H : ℝ^{+} \times ℝ^{+} \times R_{ϝ} \times R_{ϝ} \to R_{ϝ}$ is fuzzy continuous and there exists a continuous function $τ : ℝ^{+} \times ℝ^{+} \to ℝ^{+}$ such that $D (H (t, s, x, y), \tilde{0}) \leq τ (s, t)$ . Also, we assume that $lim_{t \to \infty} (L) \int_{0}^{θ (t)} τ (s, t) ds = 0$ .

Corollary 13.SinceK ≥ Mand max(D (x₁ (t) , y₁ (t)) , D (x₂ (t) , y₂ (t))) ≤ D (x₁ (t) , y₁ (t)) + D (x₂ (t) , y₂ (t)), we infer that Condition (C3) satisfied if in particular the functionhsatisfies the following condition

$\begin{matrix} D (h (t, x_{1} (t), x_{2} (t)), h (t, y_{1} (t), y_{2} (t))) \\ \leq \frac{b (t) (D (x_{1} (t), y_{1} (t)) + D (x_{2} (t), y_{2} (t)))}{2 K + (D (x_{1} (t), y_{1} (t)) + D (x_{2} (t), y_{2} (t)))} \\ \leq \frac{b (t)}{2 K} (D (x_{1} (t), y_{1} (t)) + D (x_{2} (t), y_{2} (t))) \\ \leq \frac{M}{2 K} (D (x_{1} (t), y_{1} (t)) + D (x_{2} (t), y_{2} (t))), \end{matrix}$ (5) where $M = sup_{t \in ℝ^{+}} | b (t) |$ . Therefore, the function b defined in Condition (C3) can be yields to the usual Lipschitz condition on the function h.

Corollary 14.If Condition (C4) is satisfied, then the function $δ : ℝ^{+} \to ℝ$ defined byδ (t) = $(L) \int_{0}^{θ (t)} τ (t, s) ds$ is continuous and therefore, there exists a constantΔ > 0 such that $Δ = sup δ (t) = sup ((L) \int_{0}^{θ (t)} τ (t, s) ds), \forall t \in ℝ^{+} \cup {0} .$ Now, we present the main theorem of this paper which is a generalized fuzzy type of theorems that already have been introduced in literature for real and complex metric and Banach spaces.

Theorem 15.Suppose that all Conditions (C1)-(C4) hold. Then the FFIE (1) has at least one solution in the fuzzy space $C (ℝ^{+}, R_{ϝ})_{B}$ .

Proof. Let $J = C (ℝ^{+}, R_{ϝ})_{B}$ , we define the closed ball $\bar{N_{ρ} (0)}$ in J centered at origin 0 of radius ρ = p + ρ + Δ + M > 0 and for all t > 0 consider mappings T and S with two operator equations (3), (4) on J and on $\bar{N_{ρ} (0)}$ , respectively. Then in the fuzzy Banach space J, FFIE (1) is equivalent to operator equation $Tx (t) \oplus Sx (t) = x (t), \forall t > 0 .$ (6)

Since the function a (t) is continuous on $ℝ^{+}$ and Condition (C3) holds, we infer that the mapping T is well define and the function Tx is continuous and bounded on $ℝ^{+}$ . Moreover, since the function θ is continuous on $ℝ^{+}$ , the function Sx is also continuous and bounded in view of Condition (C4). Hence, T and S define the operator T : J → J and $S : \bar{N_{ρ} (0)} \to J$ . We need to demonstrate that T and S satisfying in the all requirements of Theorem 11 on $\bar{N_{ρ} (0)}$ . Firstly, we show the validity of condition (i) of Theorem 12 for operator T on J. For arbitrary x, y ∈ J by Condition (C3) for all $t \in ℝ^{+}$ , we get $\begin{matrix} D (Tx (t), Ty (t)) & = & D (a (t) \oplus h (t, x (t), x (η (t))), \\ a (t) \oplus h (t, y (t), y (η (t)))) \\ = & D (a (t), a (t)) + D (h (t, x (t), \\ x (η (t))), h (t, y (t), y (η (t)))) \\ \leq & \frac{b (t) max (D (x (t), y (t)), D (x (η (t)), y (η (t))))}{K + max (D (x (t), y (t)), D (x (η (t)), y (η (t))))} \\ \leq & \frac{{MD}^{*} (x, y)}{K + D^{*} (x, y)}, \end{matrix}$ taking supremum over t from above inequality we have $D^{*} (Tx, Ty) \leq [M / (K + D^{*} (x, y))] D^{*} (x, y) .$ (7)

From (7) and since M ≤ K for all x, y ∈ J, we deduce that operator T satisfies in the first condition of Theorem 12 on J with function $ψ : ℝ^{+} \to ℝ^{+}$ defined by ψ (q) = Mq/(K + q). Next we prove that S is a continuous and compact operator on $\bar{N_{ρ} (0)}$ . To this end, let arbitrary fix ɛ > 0 and suppose $x, y \in \bar{N_{ρ} (0)}$ such that D^* (x, y) ≤ ɛ. we have $\begin{matrix} D (Sx (t), Sy (t)) & = & D ((FR) \int_{0}^{θ (t)} H (t, s, x (s), x (ϑ (s))) ds, \\ (FR) \int_{0}^{θ (t)} H (t, s, y (s), y (ϑ (s))) ds) \\ \leq & (L) \int_{0}^{θ (t)} D (H (t, s, x (s), x (ϑ (s))), \\ H (t, s, y (s), y (ϑ (s)))) ds \\ \leq & (L) \int_{0}^{θ (t)} [D (H (t, s, x (s), x (ϑ (s))), \tilde{0}) \\ + D (H (t, s, y (s), y (ϑ (s))), \tilde{0})] \\ \leq & 2 (L) \int_{0}^{θ (t)} τ (t, s) ds = 2 δ (t) \leq 2 Δ, \end{matrix}$ therefore we obtained $D (Sx (t), Sy (t)) \leq 2 Δ .$ (8)

Regarding to Condition (C4), we derive that there exists N > 0 such that δ (t) ≤ ɛ for t ≥ N. By this fact and inequality (8) we get $D (Sx (t), Sy (t)) \leq 2 ɛ .$ (9) In addition, if 0 ≤ t ≤ N, then by using similarly to above and taking $θ^{N} = sup {θ (t), 0 \leq t \leq N},$ $ω_{ρ}^{N} (H, ɛ) = sup {D (H (t, s, x_{1}, x_{2}), H (t, s, y_{1}, y_{2}))},$ where 0 ≤ t ≤ N, 0 ≤ s ≤ θ^N, - ρ ≤ x₁, x₂, y₁, y₂≤ρ, D (x₁, y₁) ≤ ɛ and D (x₂, y₂) ≤ ɛ, we obtain $\begin{matrix} D (Sx (t), Sy (t)) & \leq (L) \int_{0}^{θ (t)} D (H (t, s, x (s), x (ϑ (s))), \\ H (t, s, y (s), y (ϑ (s)))) ds \\ \leq (L) \int_{0}^{θ (t)} ω_{ρ}^{N} (H, ɛ) ds \leq θ^{N} ω_{ρ}^{N} (H, ɛ) . \end{matrix}$ (10) Obviously, we have in view of the continuity of θ that θ^N< ∞. Furthermore, since H (t, s, x, y) is fuzzy uniform continuous function on the set [0, N] × [0, θ^N] × [- ρ, ρ] × [- ρ, ρ] we see that $lim_{ɛ \to 0} ω_{ρ}^{N} (H, ɛ) = 0$ . Using this proven fact and (9),(10) we conclude that the operator S maps $\bar{N_{ρ} (0)}$ into J continuously.

Here, we shall to show that operator S is fuzzy compact on $\bar{N_{ρ} (0)}$ . To do this, we prove that any sequence {Sx_n} in $\bar{N_{ρ} (0)}$ has a Cauchy subsequence. Condition (C4) and Corollary 14 for all $t \in ℝ^{+}$ imply that $\begin{matrix} D ({Sx}_{n} (t), \tilde{0}) & = & D ((FR) \int_{0}^{θ (t)} H (t, s, x_{n} (s), x_{n} (ϑ (s))), \tilde{0}) \\ \leq & (L) \int_{0}^{θ (t)} D (H (t, s, x_{n} (s), x_{n} (ϑ (s))), \tilde{0}) ds \\ \leq & (L) \int_{0}^{θ (t)} τ (t, s) ds = δ (t) \leq Δ . \end{matrix}$ Taking the supremum from this inequality over t ≥ 0 for all $n \in ℕ$ , we get $D^{*} ({Sx}_{n} (t), \tilde{0}) = ∥ {Sx}_{n} ∥_{ϝ s} \leq Δ .$ (11) and this shows that {Sx_n} is a fuzzy uniformly bounded sequence in $S (\bar{N_{ρ} (0)})$ . Now, we prove that it is also equicontinuous. For arbitrary ɛ > 0, since $lim_{t \to \infty} \int_{0}^{θ (t)} τ (t, s) ds = 0$ , i.e. $lim_{t \to \infty} δ (t) = 0$ , there is a constant N > 0 such that |δ (t) | < ɛ/2, ∀ t ≥ N. Assume that t, t′ be two arbitrary nonnegative numbers. If 0 ≤ t, t′ ≤ N then we observe $\begin{matrix} D ({Sx}_{n} (t), {Sx}_{n} (t^{'})) \\ = & D ((FR) \int_{0}^{θ (t)} H (t, s, x_{n} (s), x_{n} (ϑ (s))) ds, \\ ((FR) \int_{0}^{θ (t^{'})} H (t^{'}, s, x_{n} (s), x_{n} (ϑ (s))) ds) \\ \leq & D (((FR) \int_{0}^{θ (t)} H (t, s, x_{n} (s), x_{n} (ϑ (s))) ds, \\ ((FR) \int_{0}^{θ (t)} H (t^{'}, s, x_{n} (s), x_{n} (ϑ (s))) ds) \\ + & D (((FR) \int_{0}^{θ (t)} H (t^{'}, s, x_{n} (s), x_{n} (ϑ (s))) ds, \\ ((FR) \int_{0}^{θ (t^{'})} H (t^{'}, s, x_{n} (s), x_{n} (ϑ (s))) ds) \\ \leq & (L) \int_{0}^{θ^{N}} D (H (t, s, x_{n} (s), x_{n} (ϑ (s))), \\ H (t^{'}, s, x_{n} (s), x_{n} (ϑ (s)))) \\ + & D (((FR) \int_{0}^{θ (t)} H (t^{'}, s, x_{n} (s), x_{n} (ϑ (s))) ds, \\ ((FR) \int_{0}^{θ (t^{'})} H (t^{'}, s, x_{n} (s), x_{n} (ϑ (s))) ds) . \end{matrix}$

By Condition (C4), Corollary 6, uniform continuity of the function |δ (t) | on [0, N] and uniform fuzzy continuity of the function H in [0, N] × [0, θ^N] × [- ρ, ρ] × [- ρ, ρ], we infer that $lim_{t \to t^{'}} D ({Sx}_{n} (t), {Sx}_{n} (t^{'})) = 0 .$ (12)

If t, t′ ∈ [N, ∞), then we have $\begin{matrix} D ({Sx}_{n} (t), {Sx}_{n} (t^{'})) \\ = D ((FR) \int_{0}^{θ (t)} H (t, s, x_{n} (s), x_{n} (ϑ (s))) ds, \\ ((FR) \int_{0}^{θ (t^{'})} H (t^{'}, s, x_{n} (s), x_{n} (ϑ (s))) ds) \\ = D ((FR) \int_{0}^{θ (t)} H (t, s, x_{n} (s), x_{n} (ϑ (s))) ds, \tilde{0}) \\ + D ((FR) \int_{0}^{θ (t^{'})} H (t^{'}, s, x_{n} (s), x_{n} (ϑ (s))) ds, \tilde{0}) \\ \leq (L) \int_{0}^{θ (t)} D (H (t, s, x_{n} (s), x_{n} (ϑ (s))) ds, \tilde{0}) \\ + (L) \int_{0}^{θ (t^{'})} D (H (t^{'}, s, x_{n} (s), x_{n} (ϑ (s))) ds, \tilde{0}) \\ \leq (L) \int_{0}^{θ (t)} τ (t, s) ds + (L) \int_{0}^{θ (t^{'})} τ (t^{'}, s) ds \\ = | δ (t) | + | δ (t^{'}) | \leq \frac{ɛ}{2} + \frac{ɛ}{2} = ɛ \end{matrix}$ (13)

So, the inequality (12) holds true again. If t < N < t′ with similar way we obtain $\begin{matrix} D ({Sx}_{n} (t), {Sx}_{n} (t^{'})) \\ = D ((FR) \int_{0}^{θ (t)} H (t, s, x_{n} (s), x_{n} (ϑ (s))) ds, \\ ((FR) \int_{0}^{θ (t^{'})} H (t^{'}, s, x_{n} (s), x_{n} (ϑ (s))) ds) \\ \leq D ({Sx}_{n} (t), {Sx}_{n} (N)) + D ({Sx}_{n} (N), {Sx}_{n} (t^{'})), \end{matrix}$ (14) if t → t′, then t → N and N → t′. Hence, inequality (13) implies that $lim_{t \to t^{'}} D ({Sx}_{n} (t), {Sx}_{n} (t^{'})) = 0 .$ (15)

Using (12), (13) and (15), we deduce that {Sx_n} is a fuzzy equicontinuous sequence in J. Furthermore, a result of the Arzela-Ascoli theorem yields that the sequence {Sx_n} has a uniformly convergent fuzzy subsequence on the compact subset [0, N] of $ℝ$ . Denoting this subsequence by {Sz_n}, we show that {Sz_n} is Cauchy in J. Since $lim_{m, n \to \infty} D ({Sz}_{m} (t), {Sx}_{n} (t)) = 0, \forall 0 \leq t \leq N$ , then for given ɛ > 0 there exists an $n_{1} \in ℕ$ such that for all m, n ≥ n₁ we get $\begin{matrix} sup_{0 \leq t \leq N} (L) \int_{0}^{θ (t)} D (H (t, s, z_{m} (s), z_{m} (ϑ (s))), \\ H (t, s, x_{n} (s), x_{n} (ϑ (s)))) ds < \frac{ɛ}{2}, \end{matrix}$ thus, if m, n ≥ n₁ we obtain $\begin{matrix} D^{*} ({Sz}_{m}, {Sx}_{n}) & = sup_{0 \leq t < \infty} D ({Sz}_{m} (t), {Sx}_{n} (t)) \\ \leq sup_{0 \leq t \leq N} D ({Sz}_{m} (t), {Sx}_{n} (t)) \\ + sup_{N \leq t < \infty} D ({Sz}_{m} (t), {Sx}_{n} (t)) \\ \leq sup_{0 \leq t \leq N} (L) \int_{0}^{θ (t)} D (H (t, s, z_{m} (s), z_{m} (ϑ (s))), \\ H (t, s, x_{n} (s), x_{n} (ϑ (s)))) ds \\ + sup_{N \leq t} (L) \int_{0}^{θ (t)} D (H (t, s, z_{m} (s), z_{m} (ϑ (s))), \\ H (t, s, x_{n} (s), x_{n} (ϑ (s)))) ds \\ \leq \frac{ɛ}{2} + 2 sup_{N \leq t} (L) \int_{0}^{θ (t)} τ (t, s) ds \leq \frac{ɛ}{2} \\ + 2 sup_{N \leq t} | δ (t) | \leq \frac{3 ɛ}{2}, \end{matrix}$ (16) and this shows that the fuzzy subsequence ${{Sz}_{n}} \subset S (\bar{N_{ρ} (0)}) \subset J$ is a Cauchy sequence in closed ball $S (\bar{N_{ρ} (0)})$ . Since J is a fuzzy Banach space, {Sz_n} converges to a point in J. Also, since $S (\bar{N_{ρ} (0)})$ is a closed subset of J, therefore, {Sz_n} converges to a point in $S (\bar{N_{ρ} (0)})$ . Thus, $S (\bar{N_{ρ} (0)})$ is relatively fuzzy compact and consequently S is a continuous and aompact operator on $\bar{N_{ρ} (0)}$ . Eventually, we demonstrate that if for all $y \in \bar{N_{ρ} (0)}$ , Tx ⊕ Sy = x, then $x \in \bar{N_{ρ} (0)}$ . To finish, suppose that x ∈ J be arbitrary such that Tx ⊕ Sy = x. For some y ∈ J, we have $\begin{matrix} ∥ x (t) ∥_{ϝ} = D (x (t), \tilde{0}) \\ = & ∥ Tx (t) \oplus Sy (t) ∥_{ϝ} \leq ∥ Tx (t) ∥_{ϝ} + ∥ Sy (t) ∥_{ϝ} \\ = & ∥ a (t) \oplus h (t, x (t), x (η (t))) ∥_{ϝ} \\ + ∥ (FR) \int_{0}^{θ (t)} H (t, s, y (s), y (ϑ (s))) ds ∥_{ϝ} \\ \leq & ∥ a (t) ∥_{ϝ} + ∥ h (t, x (t), x (η (t))) ∥_{ϝ} \\ + (L) \int_{0}^{θ (t)} ∥ H (t, s, y (s), y (ϑ (s))) ∥_{ϝ} ds \\ \leq & ∥ a (t) ∥_{ϝ} + D (h (t, x (t), x (η (t))), h (t, \tilde{0}, \tilde{0})) \\ + ∥ h (t, \tilde{0}, \tilde{0}) ∥_{ϝ} + (L) \int_{0}^{θ (t)} τ (t, s) ds . \end{matrix}$

Using Conditions (C3),(C4) and Corollary 14, from above inequality we obtain $\begin{matrix} D (x (t), \tilde{0}) & \leq & ∥ a (t) ∥_{ϝ} + ∥ h (t, \tilde{0}, \tilde{0}) ∥_{ϝ} + Δ \\ + & \frac{b (t) max (∥ x (t) ∥_{ϝ}, ∥ x (η (t)) ∥_{ϝ})}{K + max (∥ x (t) ∥_{ϝ}, ∥ x (η (t)) ∥_{ϝ})} \\ = & p + ρ + Δ + \frac{∥ x ∥_{ϝ s}}{K + ∥ x ∥_{ϝ s}} \\ M \leq p + ρ + Δ + M = ρ . \end{matrix}$

Therefore, for all $t \in ℝ^{+}$ we obtained $D (x (t), \tilde{0}) = ∥ x (t) ∥_{ϝ} \leq ρ .$ (17)

Taking the supremum over t from (17), we get $∥ x ∥_{ϝ s} \leq ρ,$ (18) for all y ∈ J and in particular for all $y \in \bar{N_{ρ} (0)}$ . Thus, the third condition of Theorem 12 holds. Now, applying Theorem 12 to the fuzzy operator equation Tx ⊕ Sy = x yields that the FFIE (1) has a solution in $\bar{N_{ρ} (0)}$ . Therefore, all the solutions of the FFIE (1) existing in J belong to $\bar{N_{ρ} (0)}$ . In other words, the solution which its existence is proved through Theorem 15 is a global solution for functional fuzzy integral equation (1). Indeed, fuzzy operator equation Tx ⊕ Sy = x for each Y ∈ J has at least one global solution x ∈ J. □

Remark 16. Notice that the aim of this article was merely providing some conditions using Krasnoselskii-Burton theorems such that under these conditions the functional fuzzy integral equation (1) has at least one global solution in $C (ℝ^{+}, R_{ϝ})_{B}$ . Indeed, we have not mentioned any methods or techniques to find this solution. However, as we know, there are many variant numerical and analytical methods to obtain the exact solution of Equation. (1) with depend to the kind of the functional fuzzy integral equation.

Remark 17. So far, the fuzzy fixed point theorems in fuzzy space have been studied by many researchers. Application of this theorems to analytical and numerical solution of fuzzy differential and fuzzy integral equations is investigated in some of those studies. Meanwhile, in all of literature, fuzzy differential or fuzzy integral equations are interpreted as a compact mapping or contraction mapping. Indeed, in any of the previous articles the linear combination of these mappings have not been considered. The main advantage of this this paper is that, we combined Krasnoselskii and Burton fixed point theorems to establish some conditions which under these conditions the functional fuzzy integral equation (1), which is a linear combination of a compact and contraction mappings, has at least one solution in $C (ℝ^{+}, R_{ϝ})_{B}$ . Also, it is clear that the Equation. (1) is a general form of fuzzy integral equations. In fact, if we impose some circumstances to it, the fuzzy integral equations such as linear and nonlinear Fredholm, Volterra, Hammerstein or Urysohn will be obtained.

4 Numerical experiments

In this section we present two illustrative examples to demonstrate and confirm the validity of the theoretical results.

Example 18. Consider the following FFIE $\begin{matrix} x (t, r) & = a (t, r) \oplus h (t, x (t, r), x (η (t), r)) \\ \oplus (FR) \int_{0}^{θ (t)} H (t, s, x (s, r), x (ϑ (s), r)) ds, \end{matrix}$ (19) where $η (t) = t^{2}, θ (t) = 2 \sqrt{t}, ϑ (s) = 3 s + 1$ and $\begin{matrix} h (t, x (t, r), x (η (t), r)) & = & (h_{-}^{r}, h_{+}^{r}) (t, x (t), x (η (t))) = t ⊙ x (t, r) \oplus x^{2} (η (t), r) \oplus 1 \\ = & ({tx}_{-}^{r} (t) + (x_{-}^{r})^{2} (η (t)) + 1, {tx}_{+}^{r} (t) + (x_{+}^{r})^{2} (η (t)) + 1), \end{matrix}$ $\begin{matrix} H (t, s, x (s, r), x (ϑ (s), r)) & = & (H_{-}^{r}, H_{+}^{r}) (t, s, x (s), x (ϑ (s))) = (t^{2} + s) \oplus s ⊙ (x (s, r) \oplus x^{2} (ϑ (s))) \\ = & (t^{2} + s + s (x_{-}^{r} (s) + (x_{-}^{r})^{2} (ϑ (s))), t^{2} + s + s (x_{+}^{r} (s) + (x_{+}^{r})^{2} (ϑ (s)))), \end{matrix}$ $\begin{matrix} a (t, r) & = & (a_{-}^{r}, a_{+}^{r}) (t), \\ a_{-}^{r} (t) & = & - 1 - (2 - r + 2 r^{2}) t - (\frac{8}{3} r + 16 r^{2}) t^{3 / 2} - (r + 36 r^{2}) t^{2} - 2 t^{5 / 2} - r^{2} t^{4}, \\ a_{+}^{r} (t) & = & - 1 - (8 - 7 r + 2 r^{2}) t - \frac{8}{3} (\frac{107}{4} - 25 r + 6 r^{2}) t^{3 / 2} + (142 + 145 r - 36 r^{2}) t^{2} - (2 - r)^{2} t^{4} . \end{matrix}$

We shall to check the assumptions of equation (1) and conditions of Theorem (15).

(i) The functions $η, θ, ϑ : ℝ^{+} \to ℝ^{+}$ are continuous and $lim_{t \to \infty} η (t) = \infty$ .

(ii) $a : ℝ^{+} \to R_{ϝ}$ is a fuzzy bounded continuous function, therefore there exists p > 0 such that p = ∥ a ∥ _ϝs.

(iii) Since $h : ℝ^{+} \times R_{ϝ} \times R_{ϝ} \to R_{ϝ}$ is a fuzzy continuous function, then exists ρ > 0 such that ρ = ∥ h ∥ _ϝs. In addition, it can be found a bounded function $b : ℝ^{+} \to ℝ^{+}$ and constants M, K where |b| ≤ M and M ≤ K such that the Condition (C3) satisfies.

(iv) The function $H : ℝ^{+} \times ℝ^{+} \times R_{ϝ} \times R_{ϝ} \to R_{ϝ}$ is a fuzzy continuous function, hence, the continuous real function $τ : ℝ^{+} \times ℝ^{+} \to ℝ^{+}$ exists such that all requirements of Condition (C4) hold true.

Since all the above functions satisfy the assumptions of Equation. (1) and the Conditions (C1)-(C4) for all t ≥ 0 and 0 ≤ r ≤ 1 hold, then according to Theorem 15 we conclude that the Equation. (19) has at least one solution in the fuzzy space $C (ℝ^{+}, R_{ϝ})_{B}$ . It should be noted that the exact solution of Equation. (19) is that $x (t, r) = (x_{-}^{r}, x_{+}^{r}) (t) = (rt, (2 - r) t) .$

Example 19. We consider the integral equation (19) with

$\begin{array}{l} η (t) = 2 t + 1, θ (t) = t^{2} + 1, ϑ (s) = s^{3} a n d \\ h (t, x (t, r), x (η (t), r)) = t ⊙ x^{2} (t, r) ⊙ x (η (t), r), \\ H (t, s, x (s, r), x (ϑ (s), r)) = s t \oplus \sqrt{s ⊙ x (s, r)} \oplus x^{2} (ϑ (s)), \\ a_{-}^{r} (t) = κ [t - κ^{2} t^{3} (1 + 2 t)] + \frac{1}{14} [- 2 κ^{2} (1 + t^{2})^{7} \\ - 7 (1 + t^{2})^{2} (t + t^{3} + \sqrt{κ})], \\ a_{+}^{r} (t) = - \frac{t}{2} - t^{3} - \frac{t^{5}}{2} + ς [t - ς^{2} t^{3} (1 + 2 t) \\ - \frac{1}{7} ς (1 + t^{2})^{7}] - \frac{1}{2} (1 + t^{2})^{2} \sqrt{ς}, \end{array}$

where κ = r² + r and ς = 4 - r - r³. If we check Conditions (C1)-(C4) for this equations for all t ≥ 0 and 0 ≤ r ≤ 1 like as previous example, we found out that all conditions hold true. Therefore, Theorem 15 implies that there exists at least one solution for this functional fuzzy integral equation in $C (ℝ^{+}, R_{ϝ})_{B}$ . Of course the exact solution is given by x (t, r) = (r²t + rt, 4t-rt - r³t).

5 Concluding remark

In this paper, we try to study the existence of a global solution of nonlinear functional fuzzy integral equation by applying Krasnoselskii-burton fixed point theorem in fuzzy Banach space. First, comparing with crisp space some important concepts such as continuity, convexity and compactness in fuzzy space were re-defined, and next we have reorganized two very important fixed point theorems in mathematical analysis, that is, Krasnoselskii and Burton theorems for fuzzy metric and fuzzy Banach spaces. In theorem 15, we have presented some conditions which under them the FFIE (1) has at least one global solution in $C (ℝ^{+}, R_{ϝ})_{B}$ . As was observed in the article, to prove this theorem, the properties of fuzzy metric space, fuzzy integrals and Arzela-Ascoli theorem are used. Finally, the validity of our theoretical results is illustrated by two examples.

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