On interactive fuzzy boundary value problems

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Fuzzy Sets and Systems ••• (••••) •••–•••

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www.elsevier.com/locate/fss

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On interactive fuzzy boundary value problems

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Daniel Sánchez Ibáñez ∗ , Laécio Carvalho de Barros, Estevão Esmi

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Department of Applied Mathematics, IMECC, University of Campinas, Campinas, Brazil

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Received 4 December 2017; received in revised form 7 April 2018; accepted 24 July 2018

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Abstract

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In this paper we use the concept of interactivity between fuzzy numbers for the solution to a linear fuzzy boundary value problem (FBVP). We show that a solution of a FBVP, with non-interactive fuzzy numbers as boundary values, can be obtained by the Zadeh’s Extension Principle. In addition, we show it is possible to obtain a fuzzy solution by means of the extension principle based on joint possibility distributions for the case where the boundary values are given by interactive fuzzy numbers. Examples of FBVPs with both cases, interactive and non-interactive, are presented. Also from arithmetic operations for linearly correlated fuzzy numbers, we compare our solutions to the one proposed by Gasilov et al. We conclude that the fuzzy solution in the interactive case (when the boundary values are linearly correlated fuzzy numbers) is contained in the fuzzy solution for the non-interactive case. Finally, we present the fuzzy solution for a nonlinear FBVP with Gaussian fuzzy numbers as boundary values. © 2018 Published by Elsevier B.V.

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Keywords: Fuzzy boundary value problems; Interactive fuzzy numbers; Extension principle

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1. Introduction

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One of the most important classes of differential equation problems is the boundary value problems (BVPs) which arise in many areas of knowledge such as in biology, chemistry, medicine, engineering, etc. Since some parameters of these problems can be uncertain, many researchers have studied BVPs where one or more parameters and/or state variables are given by fuzzy numbers. These problems are called of fuzzy boundary values problems (FBVPs). In this paper, we focus on FBVPs where the boundary values are given by possibly interactive fuzzy numbers. Fuzzy boundary value problems have been studied since the early 2000s. O’Regan et al. investigated FBVPs based on Hukuhara derivatives by solving fuzzy integral equations given in terms of Aumann integrals [1]. However, O’Regan et al.’s approach can not be applied to a large class of FBVPs [2]. Khastan and Nieto associated FBVPs based on generalized Hukuhara derivatives with four classical BVPs [3]. Their approach may not produce fuzzy solution on the entire domain, that is, the solution obtained at a given instant t may not be a fuzzy number. Moreover,

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* Corresponding author.

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E-mail addresses: [email protected] (D. Sánchez Ibáñez), [email protected] (L.C. de Barros), [email protected] (E. Esmi). https://doi.org/10.1016/j.fss.2018.07.009 0165-0114/© 2018 Published by Elsevier B.V.

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Gasilov et al. argued that the solutions obtained using Khastan and Nieto’s method are difficult to interpret because the solutions of the four different problems may not reflect the nature of the studied phenomenon [4]. Other approaches to FBVPs are based on the theory of differential inclusion [5] and modification/adaptation of classical numerical methods for the fuzzy case, such as the undetermined coefficients method [6], the finite difference method [7], and the finite element method [8]. Gasilov et al. investigated FBVPs where the corresponding solutions are given by fuzzy sets in the class of real functions [4]. We show that the approach presented by Gasilov et al. in [4] corresponds to the application of Zadeh’s extension principle for the solution of an associated classical BVP. Next, we deal with FBVPs where the boundary values are interactive fuzzy numbers. Recall that the relation of interactivity between two fuzzy numbers arises in the presence a joint possibility distribution J for them. In this case, the solution is obtained in terms of the sup-J extension principle of the solution of an associated classical BVP. This paper is organized as follows. In Section 2, we review some basic concepts of fuzzy set theory and deterministic methods for solving classical BVPs. In Section 3, we determine solutions to fuzzy BVPs with interactive and non-interactive boundary values using extension principle. Finally, in Section 4, we employ our proposal to solve linear and nonlinear fuzzy BVPs with interactive and non-interactive boundary values in order to illustrate the effect of considering the existence of interactivity of the boundary values.

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2. Preliminary

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2.1. Solution for a deterministic boundary value problem

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Under certain conditions on the functions k(t), p(t) and f (t), the general solution of (1), using the superposition principle [9], is: x(t) = xP (t) + c1 x1 (t) + c2 x2 (t),

(2)

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where xP is a particular solution of (1) and x1 , x2 are linearly independent solutions of the associated homogeneous problem (i.e., when f (t) = 0). The scalar coefficients c1 and c2 are determined from the boundary values x0 and xT : x2 (T )(x0 − xP (0)) − x2 (0)(xT − xP (T )) c1 = , x1 (0)x2 (T ) − x2 (0)x1 (T ) x1 (0)(xT − xP (T )) − x1 (T )(x0 − xP (0)) c2 = . x1 (0)x2 (T ) − x2 (0)x1 (T )

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We begin by considering a second order linear non-homogeneous differential equation with boundary values x(0) = x0 and x(T ) = xT , where x0 , xT ∈ R, given by  x  (t) + k(t)x  + p(t)x = f (t), (1) x(0) = x0 , x(T ) = xT .

x(t) = xP (t) − xP (0)w1 (t) − xP (T )w2 (t) + x0 w1 (t) + xT w2 (t),

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where w1 (t) and w2 (t) are defined for x1 (0)x2 (T ) − x2 (0)x1 (T ) = 0 as follows [4]: x2 (T )x1 (t) − x1 (T )x2 (t) w1 (t) = , x1 (0)x2 (T ) − x2 (0)x1 (T ) x1 (0)x2 (t) − x2 (0)x1 (t) w2 (t) = . x1 (0)x2 (T ) − x2 (0)x1 (T )

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Thus, from (2), the general (deterministic) solution of (1) is given by

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In Section 3, we will use Equation (3) to define a fuzzy solution for second-order linear boundary values problems with boundary values given by fuzzy numbers as in (12). Before we introduce our approach to solve a FBVP, it is necessary first to review some concepts of fuzzy sets theory.

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2.2. Concepts of fuzzy sets theory

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Definition 2.1. [10,11] Let U be a universal set. A fuzzy subset A of U is given by its membership function μA : U −→ [0, 1], where μA (x) means the degree to which x ∈ U belongs to A.

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We denote the class of the fuzzy subsets of U by the symbol F(U ). Each fuzzy subset of U can be associated with a family of subsets of U , namely α-levels of A.

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Definition 2.2. [11] Let A be a fuzzy subset of U . The α-level of A is the classical subset of U defined by [A]α = {x ∈ U : μA (x) ≥ α} for 0 < α ≤ 1. When U is a topological space, the 0-level of A is defined as the closure of the support of A, that is, [A]0 = supp(A), where supp(A) = {x ∈ U : μA (x) > 0}.

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The 1-level of a fuzzy subset A is also called core of A and may be denoted by the symbol core(A) instead of [A]1 [11,12].

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Definition 2.3. [11,13] A fuzzy subset A of R is called a fuzzy number if each α-level of A is a non-empty bounded closed interval of R.

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[A]α = [aα− , aα+ ], ∀α ∈ [0, 1].

We denote the class of all fuzzy numbers by RF . Next, we review the definition of two well-known types of fuzzy numbers, namely triangular and Gaussian. Definition 2.4. A fuzzy number A is said to be triangular if the parametric representation of its α-levels is of the form [A]α = [(m − a0− )α + a0− , (m − a0+ )α + a0+ ], for all α ∈ [0, 1], where [A]0 = [a0− , a0+ ] and {m} = core(A). A triangular fuzzy number is denoted by the triple (a0− ; m; a0+ ). Definition 2.5. [11,14] A fuzzy number H with {m} = core(H ) is said to be Gaussian, or bell shaped, if the parametric representation of its α-levels is of the form ⎧   2

 ⎪ − σδ ⎪ ⎨ m − ln 1 2 , m + ln 1 2 , if α ≥ e ασ ασ + [H ]α = [h− α , hα ] =

2 ⎪ ⎪ − δ ⎩ [m − δ, m + δ] if α < e σ ,

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where σ denotes a spread of H and δ is a constant number that delimits the support of H . In this case, the Gaussian fuzzy number H is also denoted by the symbol G(σ ; m; δ).

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Zadeh’s extension principle is a method to extend functions to deal with fuzzy sets as arguments or inputs. Definition 2.6. [11,15] (Zadeh’s extension principle) The Zadeh’s extension of a given function f : X → Z is the function fˆ : F(X) → F(Z) defined for each fuzzy set A ∈ F(X) as the fuzzy number fˆ(A) ∈ F(Z), whose membership function is given by ⎧ ⎨ sup μA (x) if f −1 (z) = ∅, ∀ z ∈ Z, (5) μfˆ(A) (z) = x∈f −1 (z) ⎩0 if f −1 (z) = ∅,

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where f −1 (z) = {x ∈ X | f (x) = z}.

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In view of Definition 2.3, the α-levels of the fuzzy number A can be represented in the following parametric form:

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The next theorem establishes that the α-levels of the fuzzy number obtained by the Zadeh’s extension principle of a continuous function f at A coincides with the image of function restricted to the α-level of A.

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Theorem 2.7. [16,17] Let X and Z be topological spaces, f : X −→ Z a continuous function and A a fuzzy subset of X. So, for all α ∈ [0, 1], we have

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The notion of sup-J extension principle proposed by Carlsson et al. consists of a generalization of the Zadeh’s extension principle to deal with functions with two arguments given by fuzzy numbers. The sup-J extension principle is based on the notion of joint possibility distributions. Definition 2.8. [18] Let A, B ∈ RF and let J be a subset of R2 . The fuzzy subset J is called a joint possibility distribution of A and B if

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sup μJ (x, y) = μA (x) for all x ∈ R,

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and

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sup μJ (x, y) = μB (y) for all y ∈ R.

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In this case, A and B are called marginal possibility distributions of J .

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Remark 2.9. If J is a joint possibility distribution of A and B, then we can show, by Definition 2.8, that the relationships μJ (x, y) ≤ min{μA (x), μB (y)} and [J ]α ⊆ [A]α × [B]α hold true for all x, y ∈ R and for all α ∈ [0, 1] [18].

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The notion of interactivity between two fuzzy numbers, say A and B, arises when one has at hand a given joint possibility distribution of A and B [18].

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Definition 2.10. [18,19] Let J be a joint possibility distribution of the fuzzy numbers A and B. The fuzzy numbers A and B are said to be non-interactive if J is given by μJ (x, y) = min{μA (x), μB (y)} for all (x, y) ∈ R , 2

or, equivalently, [J ]α = [A]α × [B]α for all α ∈ [0, 1].

Otherwise, A and B are said to be interactive.

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Next, we present the extension principle of a function from R2 to R based on a given joint possibility distribution J between two fuzzy numbers. Definition 2.11. [18] Let A, B ∈ RF and f : R2 → R. Given a joint possibility distribution J of A and B, the sup-J extension of f or, the (interactive) extension of f at (A, B) with respect to J , is the fuzzy subset fJ (A, B) of R whose membership function is given by ⎧ ⎨ μJ (x, y) if f −1 (z) = ∅, sup (6) μfJ (A,B) (z) = (x,y)∈f −1 (z) ⎩0 if f −1 (z) = ∅,

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where f −1 (z) = {(x, y) : f (x, y) = z}.

Remark 2.12. If A and B are non-interactive, that is, if the corresponding joint possibility distribution J is defined as in (2.10), then the sup-J extension principle corresponds to the Zadeh’s extension principle of a function f : R2 → R [18,20]. In this case, we use the symbol fˆ(A, B) instead of fJ (A, B) to denote the Zadeh’s extension of f at (A, B).

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[fˆ(A)]α = f ([A]α ).

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The next corollary can be viewed as a consequence of Theorem 2.7.

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Corollary 2.13. [18,19] Let A, B ∈ RF and let f : R2 −→ R be a continuous function. If J is a joint possibility distribution of A and B, then we have that

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[A + B]α = [A]α + [B]α = [aα−

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∀ α ∈ [0, 1].

Moreover, the multiplication of a fuzzy number A by a scalar λ ∈ R is a fuzzy number λA such that its α-levels are defined as follows:  [λaα− , λaα+ ] if λ ≥ 0, ∀ α ∈ [0, 1]. (8) [λA]α = λ[A]α = [λaα+ , λaα− ] if λ < 0, Other forms of arithmetic operations between fuzzy numbers can be established using the notion of sup-J extension principle. Next, we present an arithmetic defined for the class of the completely correlated fuzzy numbers [18,21].

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μJ (x, y) = μA (x).χ{qu+r=v} (x, y) = μB (y).χ{qu+r=v} (x, y),

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where



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χ{qu+r=v} (x, y) =

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1 if qx + r = y 0 if qx + r = y

is the characteristic function of the line

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where

[A]α = [aα− , aα+ ].

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: qu + r = v}. In this case, for all α ∈ [0, 1], we have

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In Definition 2.14, if the value q is positive (negative), then the fuzzy numbers A and B are said to be completely positively (negatively) correlated. Under these conditions, from Equation (10) and Corollary 2.13, the addition of A and B is the fuzzy number A +L B whose α-levels are given by ⎧ − − + + ⎪ ⎨ [aα + bα , aα + bα ] if q ≥ 0, (11) [A +L B]α = (q + 1)[A]α + r = [aα− + bα+ , aα+ + bα− ] if −1 ≤ q < 0, ⎪ ⎩ + − − + [aα + bα , aα + bα ] if q < −1,

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[B]α = q[A]α + r,

where [A]α = [aα− , aα+ ] and [B]α = [bα− , bα+ ] for all α ∈ [0, 1] [18,21].

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[J ]α = {(x, qx + r) ∈ R2 : x = (1 − s)aα− + saα+ , s ∈ [0, 1]}

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Definition 2.14. Let A, B ∈ RF and let J be a joint possibility distribution of A and B. The fuzzy numbers A and B are said to be linearly correlated (or completely correlated) if there exist q, r ∈ R such that the membership of J is given by

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+ bα+ ],

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The usual arithmetic operations of addition, subtraction, multiplication, and division for fuzzy numbers are defined via Zadeh’s extension principle [11]. For example, let A, B ∈ RF with [A]α = [aα− , aα+ ] and [B]α = [bα− , bα+ ] for all α ∈ [0, 1], the usual sum of A and B is a fuzzy number A + B such that

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[fJ (A, B)]α = f ([J ]α ) for all α ∈ [0, 1].

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3. Fuzzy boundary value problems (FBVPs) In this section we study the fuzzy boundary value problem obtained by replacing the boundary values x0 and xT with fuzzy numbers A and B in (1). More precisely, let us consider the following FBVP:  x  (t) + k(t)x  (t) + p(t)x(t) = f (t), (12) x(0) = A, x(T ) = B.

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3.1. Solution for a FBVP with non-interactive fuzzy numbers as boundary values

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In this subsection we consider the case where the boundary values A and B in the FBVP (12) are non-interactive fuzzy numbers. This assumption allows us to obtain a solution for (12) by applying the Zadeh’s extension principle to the solution of BVP (1) given in Equation (3) as we explain next. Let U be an open subset of R2 such that [A]0 × [B]0 ⊂ U and a solution x(·, x0 , xT ) of (1) exists with boundary values x0 , xT ∈ U . For each t fixed, we can define the operator St : U −→ R, given by St (x0 , xT ) = x(t, x0 , xT ). Since the boundary values A and B are non-interactive, we can use Zadeh’s extension principle to obtain a solution x : [0, T ] → RF for (12) given by x(t) = Sˆt (A, B) for all t ∈ [0, T ]. If St is a continuous function, then by Theorem 2.7, for all α ∈ [0, 1], we have:

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[Sˆt (A, B)]α = St ([A]α × [B]α ) = {St (x0 , xT ) : x0 ∈ [A]α = [aα− , aα+ ] and xT ∈ [B]α = [bα− , bα+ ]} = xP (t) − xP (0)w1 (t) − xP (T )w2 (t) + w1 (t)[aα− , aα+ ] + w2 (t)[bα− , bα+ ].

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˜ x(t) ˜ = xcr (t) + w1 (t)a˜ + w2 (t)b,

xcr (t) = xP (t) + c1 x1 (t) + c2 x2 (t)

x2 (T )(acr − xP (0)) − x2 (0)(bcr − xP (T )) x1 (t) x1 (0)x2 (T ) − x2 (0)x1 (T ) x1 (0)(bcr − xP (T )) − x1 (T )(acr − xP (0)) + x2 (t). x1 (0)x2 (T ) − x2 (0)x1 (T ) xcr (t) = xP (t) +

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˜ x(t) ˜ = xP (t) + (acr − xP (0))w1 (t) + (bcr − xP (T ))w2 (t) + w1 (t)a˜ + w2 (t)b.

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In other words, if A and B are triangular (or trapezoidal) fuzzy numbers, then the solution (14) proposed by Gasilov in [4] for the FBVP (12) coincides with the solution given in terms of Zadeh’s extension principle. Proof. Since the boundary values are triangular (or trapezoidal) fuzzy numbers of the form A = acr + a˜ and B = ˜ α = [bα− − bcr , bα+ − bcr ] = [bα− , bα+ ] − bcr . bcr + b˜ in [4], we have [a] ˜ α = [aα− − acr , aα+ − acr ] = [aα− , aα+ ] − acr and [b] Thus, the α-levels of (16) can be rewritten as follows:

+ w1 (t)([aα− , aα+ ] − acr ) + w2 (t)([bα− , bα+ ] − bcr )

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Theorem 3.1. Let A and B be triangular (or trapezoidal) fuzzy numbers as boundary values of the FBVP (12) and let [A]0 × [B]0 ⊂ U be an open subset in R2 . If for each (x0 , xT ) ∈ U there exists a solution x(·, x0 , xT ) of (1) and if x(t, ·, ·) is continuous on U for each t ∈ [0, T ], then for each fixed t ∈ [0, T ], the following equality holds

[x(t)] ˜ α = xP (t) + (acr − xP (0))w1 (t) + (bcr − xP (T ))w2 (t)

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The next theorem establishes that the solutions of (12) via Gasilov et al.’s approach and via Zadeh’s extension principle coincide.

˜ Sˆt (A, B) = x(t).

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By grouping the terms of Equation (15) and then replacing it in Equation (14), we obtain the solution of (12) proposed by Gasilov et al. [4]:

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is the deterministic solution of (1) with x0 = acr and xT = bcr , that is

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where the functions w1 (t), w2 (t) are defined in (4) and

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On the other hand, in Gasilov et al.’s proposal [4], the solution of (12) is defined for boundary values of the form ˜ with core(A) = acr , core(B) = bcr , and a˜ and b˜ representing the uncertain part of A = acr + a˜ and B = bcr + b, A and B given by triangular fuzzy numbers. Specifically, the solution proposed by Gasilov et al. is the function x˜ : [0, T ] → RF given by

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= xP (t) − xP (0)w1 (t) − xP (T )w2 (t) + w1 (t)[aα− , aα+ ] + w2 (t)[bα− , bα+ ]

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Note that Gasilov et al.’s approach is restricted to triangular (or trapezoidal) fuzzy numbers. However, the application of Zadeh’s extension principle holds for all fuzzy number. Moreover, our approach to obtain solutions of FBVPs via Zadeh’s extension principle is not restricted to linear equations as in (12) and, therefore, can be used whenever the associated solution x(t, ·, ·) exists for all (x0 , xT ) belongs to an open U containing [A]0 × [B]0 . In the next section we present a method to obtain a solution for the case where the boundary values of a FBVP present a certain type of interactivity.

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3.2. Solution for a FBVP with interactive fuzzy numbers as boundary values

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and r = w2 r0 . Thus, if q0 > 0, from (11), the interactive sum between C and D is given by: ⎧ [w1 aα− + w2 bα− , w1 aα+ + w2 bα+ ] if q ≥ 0 with w1 > 0 and w2 ≥ 0. ⎪ ⎪ ⎪ ⎪ [w a + + w b+ , w a − + w b− ] if q ≥ 0 with w < 0 and w < 0. ⎪ 1 α 2 α 1 α 2 α 1 2 ⎪ ⎪ ⎪ ⎪ − + w b− , w a + + w b+ ] if − 1 ≤ q < 0 with w > 0 and w < 0. ⎪ a [w 1 α 2 α 1 α 2 α 1 2 ⎪ ⎪ ⎪ ⎨ [w a + + w b+ , w a − + w b− ] if − 1 ≤ q < 0 with w < 0 and w > 0. 1 α 2 α 1 α 2 α 1 2 [C +L D]α = + + − − ⎪ [w1 aα + w2 bα , w1 aα + w2 bα ] if q < −1 with w1 > 0 and w2 < 0. ⎪ ⎪ ⎪ ⎪ ⎪ [w1 aα− + w2 bα− , w1 aα+ + w2 bα+ ] if q < −1 with w1 < 0 and w2 > 0. ⎪ ⎪ ⎪ ⎪ ⎪ [w b− , w b+ ] if w1 = 0 and w2 ≥ 0. ⎪ ⎪ 2 α 2 α ⎩ [w2 bα+ , w2 bα− ] if w1 = 0 and w2 < 0.

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where +L denotes the addition operator between two linearly correlated fuzzy numbers: C = w1 (t)A and D = w2 (t)B = w2 (t)(q0 A + r0 ) = q0 w2 (t)A + r0 w2 (t). In what follows, for notation convenience we simply use the symbols w1 and w2 to denote respectively w1 (t) and w2 (t). If w1 = 0, for every α ∈ [0, 1], we have w 2 q0 [D]α = w2 [B]α = w2 (q0 [A]α + r0 ) = [C]α + w2 r0 = q[C]α + r, w1 where q =

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where q0 , r0 ∈ R are constants associated with the underlying joint possibility distribution. Since the boundary values are interactive fuzzy numbers, we use the sup-J extension principle in order to produce a solution for FBVP (12) regarding this relation of interactivity. More precisely, let J be the joint possibility distribution of A and B given in Equation (9) and let St : U −→ R be a function defined as in Subsection 3.1, where U is an open set of R2 such that [J ]0 ⊂ U , and the solution of (12) is the function x : [0, T ] → RF given by x(t) = (St )J (A, B) for all t ∈ [0, T ]. If St is a continuous function, then Corollary 2.13 reveals that the α-levels of (St )J (A, B), for all α ∈ [0, 1], are given by [(St )J (A, B)]α = St ([J ]α ) = {St (z, q0 z + r0 ) : z ∈ [A]α } = {xP (t) − xP (0)w1 (t) − xP (T )w2 (t) + w1 (t)(z) + w2 (t)(q0 z + r0 ) : z ∈ [A]α }

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In this subsection we deal with the case where the boundary values A and B in FBVP (12) are linearly correlated fuzzy numbers. From Definition 2.14, this assumption implies that α-levels of A and B satisfy the following property:

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(19)

Note that in Equation (19), we assume that q0 > 0, that is, the fuzzy numbers A and B are positively correlated. A similar equation can be obtained if A and B are negatively correlated, that is, q0 < 0. Moreover, if q0 = 0 (or q = 0), then we have that B (or D) is a real number. The next theorem establishes an order relation between the solution of (12) with non-interactive boundary values and the solution of (12) with linearly correlated boundary values.

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Theorem 3.2. Let A, B ∈ RF be the boundary values of the FBVP (12) and let U be an open set of R2 such that [A]0 × [B]0 ⊂ U . If there exists a function x : [0, T ] × U → R such that x(·, x0 , xT ) is a solution of the BVP (1) for each (x0 , xT ) ∈ U and x(t, ·, ·) is continuous in U for each t ∈ [0, T ], then (20)

where (St )J and Sˆt are given by Equations (18) and (13), respectively.

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[(St )J (A, B)]α = {St (x0 , xT ) : (x0 , xT ) ∈ [J ]α } ⊆ {St (x0 , xT ) : (x0 , xT ) ∈ [A]α × [B]α } = [Sˆt (A, B)]α for all α ∈ [0, 1].

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[(St )J (A, B)]α ⊆ [Sˆt (A, B)]α = [x(t)] ˜ α for all α ∈ [0, 1].

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[(St )L (A, B)]α ⊆ [x(t)] ˜ α,

4. Examples

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Let us consider the following linear and non-homogeneous FBVP [4]:  x  − 4x  + 4x = 1 − 2t 2 , x(0) = A = (2; 3; 4), x(1) = B = (1; 2; 2.5).

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(21)

In addition, let us assume that x(0) and x(1) are non-interactive triangular fuzzy numbers. This FBVP can be associated with the following classical BVP:  x  − 4x  + 4x = 1 − 2t 2 , (22) x(0) = x0 , x(1) = xT . By Section 2.1, the solution of (22) is given by

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4.1. A linear FBVP with non-interactive fuzzy boundary values

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Here we discuss examples of linear and non-linear FBVPs whose boundary values are possibly interactive. First, we study the case where the boundary values are non-interactive fuzzy numbers. In this case, we present the solution proposed by Gasilov et al. [4] that corresponds to the solution obtained by means of the Zadeh’s extension principle according to Theorem 3.1. Second, we deal with the case where the boundary values are interactive fuzzy numbers. As expected from Theorem 3.2, the solution proposed by Gasilov contains the solution obtained by the extension principle using a joint possibility distribution. In the nonlinear case we assume that the boundary values are linearly correlated. This assumption allows us to obtain a fuzzy solution for this problem using the extension principle.

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for all α ∈ [0, 1]. The next section presents some examples of FBVPs with interactive and non-interactive boundary values.

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In particular, if the boundary values are linearly correlated fuzzy numbers then we have

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In view of Theorems 3.1 and 3.2, we have that the interactive solution is contained in the solution x(t) ˜ proposed by Gasilov [4], that is,

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For each t ∈ [0, 1], we obtain the operator

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(24)

In this example, we assume that x(0) and x(2) are linearly correlated triangular fuzzy numbers. In order to produce a fuzzy solution by means of the extension principle, we need first to investigate the following associated BVP given by  x  + 16x = 0, (25) x(0) = x0 , x(2) = xT , whose solution is

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4.2. A linear FBVP with interactive fuzzy boundary values In this subsection, we consider the following linear and homogeneous case [4]:  x  + 16x = 0, x(0) = A = (−1; 0; 1), x(2) = B = (−0.5; 0; 0.5).

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[Sˆt (A, B)]α = {St (x0 , xT ) : x0 ∈ [A]α = [α + 2, −α + 4],

α xT ∈ [B]α = α + 1, − + 2.5 } 2 (23) 1 1 2 = − (t + 1) + (1 − t)e2t + 2te2(t−1) + [α + 2, −α + 4](1 − t)e2t 2 2 α + α + 1, − + 2.5 te2(t−1) 2 for all α ∈ [0, 1]. The solution (23) coincides with the one proposed by Gasilov in [4], corroborating Theorem 3.1. Fig. 1 exhibits the fuzzy solution (23).

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1 1 Sˆt (A, B) = − (t + 1)2 + (1 − t)e2t + 2te2(t−1) + A(1 − t)e2t + Bte2(t−1) , 2 2 whose α-levels are

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1 1 St (x0 , xT ) = − (t + 1)2 + (1 − t)e2t + 2te2(t−1) + x0 (1 − t)e2t + xT te2(t−1) 2 2 for all x0 , xT ∈ R. Thus, the fuzzy solution Sˆt (A, B) of (21), obtained by means of the Zadeh’s extension principle, is given by

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Fig. 1. The right image exhibits the fuzzy solution x of the FBVP (21) as in Equation (23). The left image exhibits the level curves of the solution x in the tx-plane where the greatest and least membership values are represented respectively by the black and white colors. The black and white dashed-lines illustrate respectively the 0-level and 0.5-level of x(t) for all t .

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sin(8 − 4t) sin(4t) St (x0 , xT ) = x0 + xT sin(8) sin(8) is continuous. Since the boundary values A and B are linearly correlated fuzzy numbers, there exist q and r such that their joint possibility distribution J is given by Equation (9). In addition, from Equation (11), the sup-J extension principle of St at (A, B) can be rewritten as follows: sin(8 − 4t) sin(4t) +L B , (26) sin(8) sin(8) which is the fuzzy solution of the FBVP (24). Note that if we assume that A and B are non-interactive, then Gasilov et al.’s fuzzy solution is obtained by replacing the sum +L by the standard (non-interactive) sum in (26). Fig. 2 illustrates the fuzzy solutions (26) of this FBVP for the cases where the boundary values are interactive as well as non-interactive. In contrast to Gasilov et al.’s proposal which deals only with linear FBVPs, in the next subsection we solve a nonlinear FBVP with interactive fuzzy boundary values. (St )J (A, B) = A

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4.3. A nonlinear FBVP with fuzzy boundary values

Let us consider the simplified problem of finding the shape of a cable in suspension that is fastened at each end and carries a distributed load (see [9]). We assume that the extreme values are uncertain and given by fuzzy numbers. Thus, the FBVP that describes the shape x(t) of the cable, for t ∈ [0, T ], is given by    x = β 1 + (x  )2 , (27) x(0) = H0 , x(T ) = HT

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sin(8 − 4t) sin(4t) + xT . sin(8) sin(8) Moreover, for each t ∈ [0, 2], the operator St defined for all x0 , xT ∈ R by

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Fig. 2. The left and right gray-scale images represent the top-view of the fuzzy solution of FBVP (24) for the cases where the boundary values are linearly correlated and are non-interactive, respectively. The black and white dashed-lines correspond to the endpoints of the α-levels of the solution for α = 0, 0.5.

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where H0 , HT ∈ RF and β is a constant value that represents the ratio between the horizontal component of tension and the weight per unit length of the cable. Proceeding as in the previous examples, we consider the classical BVP associated with (27), with boundary values h0 , hT ∈ R, given by   x  = β 1 + (x  )2 , (28) x(0) = h0 , x(T ) = hT .

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c1 + c1 +

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and J is a joint possibility distribution of H0 and HT . If we assume that t + c2 ∈ [0, T ], then c2 must be a real number, and, in this case, we have that (St )J (H0 , HT ) and the fuzzy number c1 = c1 (H0 , HT ) are linearly correlated. In addition, from system (30), one can note that H0 and HT are linearly correlated, so we have that (St )J (H0 , HT ) is linearly correlated with c1 (H0 , HT ) = H0 − β1 cosh (βc2 ). Thus, (St )J (H0 , HT ) and H0 are linearly correlated with 1

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1 (St )J (H0 , HT ) = H0 + cosh (β (t + c2 )) − cosh (βc2 ) . β β

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(30)

In order to obtain a fuzzy solution of (27), for each t ∈ [0, T ], we can apply the extension principle in (29). That is, the fuzzy solution of (27) is given by (St )J (H0 , HT ) where St (h0 , hT ) = c1 (h0 , hT ) +

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(31)

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Note that the right side of Equation (31) corresponds to Zadeh extension of the operator st (h0 ) = h0 + β1 cosh(β(t +

c2 )) − β1 cosh(βc2 ) at H0 and, therefore, we obtain that (St )J (H0 , HT ) = sˆt (H0 ). Thus, we can interpret that the fuzzy solution (St )J (H0 , HT ) = sˆt (H0 ) can be seen as the fuzzy number H0 moving along the function r(t) =

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c2 )) − β1 cosh(βc2 ), for t ∈ [0, T ]. Fig. 3 exhibits the fuzzy solution of (27), for t ∈ [0, 1], β = 0.5, c2 = −0.11, and the boundary values given by the Gaussian fuzzy numbers H0 = G(0.2; 1; 0.4) and HT = G(0.2; 1.2; 0.4). We finish this subsection by presenting a theorem that connects the fuzzy solution of an FBVP with a parametrized family of solutions for classical BVPs.

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(32)

where f : [0, T ] × R2 → R and A, B ∈ RF . Moreover, let us suppose that A and B are interactive fuzzy numbers such that the corresponding joint possibility distribution J is given by [J ]α = {(a, b)|b = g(a), for all a ∈ [A]α }, for all α ∈ [0, 1] where g : R → R is a bijective function. If the classical BVP  x  = f (t, x, x  ), (33) x(0) = a, x(T ) = b has a unique continuous solution x(·) = S(·, a, b) for all (a, b) ∈ R2 and if the operator St (·, ·) = S(t, ·, ·) is continuous for each t ∈ [0, T ], then we have

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where (St )J (A, B) is the fuzzy solution of (32) via the sup-J extension principle.

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Proof. The proof is an immediate consequence of Corollary 2.13 since St (a, b) is a continuous function with respect to (a, b) ∈ R2 and b = g(a). In particular case where A and B are linearly correlated fuzzy numbers, we have that g(x) = qx + r, for some q, r ∈ R. 2

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This theorem also indicates that the fuzzy solution of (32), via the extension principle, can be seen as the union of classical solutions of (33) and it depends only on A. Example (27) corroborates this last observation.

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5. Final remarks

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The main contribution of this article is the study of second order (linear and non-linear) boundary value problems with boundary values given by interactive fuzzy numbers. Our analysis is based mainly on the idea of the generalization of the classical Zadeh’s extension principle, namely sup-J extension principle. We obtain fuzzy solutions by means of the sup-J extension principle of deterministic solutions of the associated BVPs. In the case where the boundary values of the underlying FBVP are non-interactive fuzzy numbers, the fuzzy solutions are obtained using Zadeh’s extension principle. In the particular case of linear FBVPs, the fuzzy solutions can be written in terms of interactive (non-interactive) sums since the boundary values are interactive (or non-interactive) fuzzy numbers and the general deterministic solution corresponds to the addition of a particular solution and a linear combination of two linear independent solution of the corresponding homogeneous BVP. Gasilov et al. argued that their approach to obtain a fuzzy solution for a linear FBVP coincides to Zadeh’s extension of an associated BVP, but they did not provide a proof of this claim. In Theorem 3.1, we prove that the claim of Gasilov et al. holds true. However, in contrast to this approach, our strategy of producing fuzzy solutions in terms of the extension principle is not restricted to particular forms of boundary values such as triangular or trapezoidal fuzzy numbers. We also study linear FBVPs with boundary values given by interactive fuzzy numbers. In this case, we obtain a fuzzy solution by applying the extension principle of the deterministic solution of an associated BVP with respect to the underlying joint possibility distribution. In particular, we focus on a particular case of interactivity, namely linear correlation. Furthermore, we show that the interactive fuzzy solution (when the boundary values are interactive fuzzy numbers) is contained in the non-interactive fuzzy solution (as corroborated by Fig. 2). In Subsection 4.3, we investigate an example of a nonlinear FBVP with boundary values given by fuzzy numbers. We argue that if we assume that one of the parameters of the general solution, which depends on the boundary values, is a real number (since the dimensional analysis reveals that it is a time quantity), then the boundary values are linearly correlated fuzzy numbers. This interactive relation between the boundary values implies that the fuzzy solution given in terms of the sup-J extension principle coincides with Zadeh’s extension of a function defined on R. Subsequently, we present a theorem that connects the fuzzy solution of a FBVP via sup-J extension principle with the union of classical solutions of parametrized BVPs.

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This research was partially supported by CONICYT of Chile, CNPq under grant no. 306546/2017-5, and FAPESP under grant no. 2016/26040-7. References

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[1] D. O’Regan, V. Lakshmikantham, J.J. Nieto, Initial and boundary value problems for fuzzy differential equations, Nonlinear Anal., Theory Methods Appl. 54 (3) (2003) 405–415. [2] B. Bede, A note on “two-point boundary value problems associated with non-linear fuzzy differential equations”, Fuzzy Sets Syst. 157 (7) (2006) 986–989. [3] A. Khastan, J.J. Nieto, A boundary value problem for second order fuzzy differential equations, Nonlinear Anal., Theory Methods Appl. 72 (9) (2010) 3583–3593. ¸ Amrahov, A.G. Fatullayev, Solution of linear differential equations with fuzzy boundary values, Fuzzy Sets Syst. 257 (2014) [4] N. Gasilov, S.E. 169–183. [5] D. Li, M. Chen, X. Xue, Two-point boundary value problems of uncertain dynamical systems, Fuzzy Sets Syst. 179 (1) (2011) 50–61. [6] X. Guo, D. Shang, X. Lu, Fuzzy approximate solutions of second-order fuzzy linear boundary value problems, Bound. Value Probl. 2013 (1) (2013) 212. [7] T. Allahviranloo, K. Khalilpour, A numerical method for two-point fuzzy boundary value problems, World Appl. Sci. J. 13 (10) (2011) 2137–2147. [8] D. Sánchez, L.C. Barros, BVP with fuzzy boundary values: solution based in the finite element method, in: Proceedings of the XXXVII Brazilian National Conference on Computational and Applied Mathematics, SBMAC, 2017 (in Portuguese). [9] D.L. Powers, Boundary Value Problems: and Partial Differential Equations, Academic Press, 2009. [10] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (3) (1965) 338–353. [11] L.C. Barros, R.C. Bassanezi, W.A. Lodwick, First Course in Fuzzy Logic, Fuzzy Dynamical Systems, and Biomathematics, Springer, 2016. [12] L.T. Gomes, L.C. Barros, B. Bede, Fuzzy Differential Equations in Various Approaches, Springer, 2015. [13] G. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic, vol. 4, Prentice Hall, New Jersey, 1995. [14] W. Pedrycz, F. Gomide, Fuzzy Systems Engineering: Toward Human-Centric Computing, John Wiley & Sons, 2007. [15] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning—I, Inf. Sci. 8 (3) (1975) 199–249. [16] L.C. Barros, R.C. Bassanezi, P.A. Tonelli, On the continuity of the Zadeh’s extension, in: Proc. Seventh IFSA World Congress, vol. 2, 1997, pp. 3–8. [17] H.T. Nguyen, A note on the extension principle for fuzzy sets, J. Math. Anal. Appl. 64 (2) (1978) 369–380. [18] C. Carlsson, R. Fullér, P. Majlender, Additions of completely correlated fuzzy numbers, in: Proceedings of the 2004 IEEE International Conference on Fuzzy Systems, vol. 1, IEEE, 2004, pp. 535–539. [19] V. Cabral, L.C. Barros, Fuzzy differential equation with completely correlated parameters, Fuzzy Sets Syst. 265 (2015) 86–98. [20] E. Esmi, P. Sussner, G.B.D. Ignácio, L.C. Barros, A parametrized sum of fuzzy numbers with applications to fuzzy initial value problems, Fuzzy Sets Syst. 331 (2018) 85–104. [21] L.C. Barros, F. Santo Pedro, Fuzzy differential equations with interactive derivative, Fuzzy Sets Syst. 309 (2017) 64–80.

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