# Important Notes for a Fuzzy Boundary Value Problem

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## Abstract

In this paper is studied a fuzzy Sturm-Liouville problem with the eigenvalue parameter in the boundary condition. Important notes are given for the problem. Integral equations are found of the problem.

## Abstract

In this paper is studied a fuzzy Sturm-Liouville problem with the eigenvalue parameter in the boundary condition. Important notes are given for the problem. Integral equations are found of the problem.

## 1 Introduction

Firstly, Zadeh introduced the concept of fuzzy numbers and fuzzy arithmetic [22]. The major application of fuzzy arithmetic is fuzzy differential equations. Fuzzy differential equations are suitable models to model dynamic systems in which there exist uncertainties or vaguness. Fuzzy differential equations can be examined by several approach, such as Hukuhara differentiability, generalized differentiability, the concept of differential inclusion etc [1], [3], [4], [5], [6, 7, 8], [9], [11], [13, 14], [15] [17], [19], [20].

In this paper is on a fuzzy Sturm-Liouville problem with the eigenvalue parameter in the boundary condition. Important notes are given for the problem. Integral equations are found of the problem.

## 2 Preliminaries

Definition 1. [18] A fuzzy number is a function u : $ℝ→0,1$satisfying the properties:u is normal, u is convex fuzzy set, u is upper semi-continuous on ℝ, $clxεℝux>0$is compact, where cl denotes the closure of a subset.

Let RF denote the space of fuzzy numbers.

Definition 2. [15] Let uɛF. The a-level set of u, denoted, $uα,0<α≤1,$is

$uα=xεℝux≥α.$

If a = 0; the support of u is defined

$u0=clxεℝux>0.$

The notation, $uα=u¯α,u¯α$denotes explicitly the a-level set of u. We refer to $u¯$and ū as the lower and upper branches of u,respectively.

The following remark shows when $u¯α,u¯α$ is a valid a-level set.

Remark 1. [10,15] The sufficient and necessary conditions for $u¯α,u¯α$to define the parametric form of a fuzzy number as follows:

$u¯α$is bounded monotonic increasing (nondecreasing) left-continuous function on (0;1] and right-continuous for a = 0,

$u¯α$is bounded monotonic decreasing (nonincreasing) left-continuous function on (0;1] and right-continuous fora = 0,

$u¯α≤u¯α,0≤α≤1.$

Definition 3. [15] For u,vεF and λ ∈R, the sum u+v and the product λ u are defined by $u+vα=uα+vα,$$λuα=λuα$where means the usual addition of two intervals (subsets) ofand λ [u]α means the usual product between a scalar and a subset of R.

The metric structure is given by the Hausdorff distance

$D:ℝF×ℝF→ℝ+∪0,$

by

$Du,v=supα∈0,1 maxu¯α−v¯α,u¯α−v¯α.$

Definition 4. [18] If A is a symmetric triangular numbers with supports $a¯,a¯,$the α–level sets of A is $Aα=$$a¯+a¯−a¯2α,a¯−a¯−a¯2α.$

Definition 5. [16] u,v 2F, $uα=u¯α,u¯α,vα=v¯α,v¯α,$the product uv is defined by

$uvα=uαvα,∀α∈0,1,$

where

$uαvα=u¯a,u¯αv¯a,v¯α=w¯a,w¯α,w¯a=minu¯av¯a,u¯av¯α,u¯αv¯a,u¯αv¯α,w¯a=maxu¯av¯a,u¯av¯α,u¯αv¯a,u¯αv¯α.$

Definition 6. [15, 21] Let u,v 2F.If there exists w 2F such that u = v+w; then w is called the Hukuhara difference of fuzzy numbers u and v;and it is denoted by w = u⊖v:

Definition 7. [2,15] Let f : [a;b]RF and t0 [a,b] :We say that f is Hukuhara differential at t0, if there exists an element f' 0 (t0) 2F such that for all h > 0 sufficiently small, $∃ft0+h⊖ft0, ft0⊖ft0−h$and the limits (in the metric D)

$limh→0ft0+h⊖ft0h=limh→0ft0⊖ft0−hh=f′t0.$

Definition 8. [12] If $p′x=0, rx=1$and $Ly=pxy˝+qxy$in the fuzzy differential equation $pxy′′+qxy+λrxy=0,px,p′x,qx,rx,$are continuous functions and positive, the fuzzydifferential equation

$Ly+λy=0$

is called a fuzzy Sturm-Liouville equation.

Definition 9. [12] $yx,λ0α=y¯x,λ0,y¯x,λ0≠0,$we say that λ = λ0 is eigenvalue of (2.1) if the fuzzy differential equation (2.1) has the nontrivial solutions $y¯x,λ0≠0,y¯x,λ0≠0.$

## 3 Findings and Main Results

Consider the fuzzy boundary value problem

$τu=u˝+qxu,$
$τu+λu=0,x∈−1,1$
$u−1+u′−1=0,$
$λβu1+γu′1=0,$

where q(x) is continuous function and positive on [-1,1] , λ > 0 and β; γ > 0.

Let u1 (x;λ) and u2 (x;λ) be linearly independent solutions of the classical differential equation τu+λu=0. Then, the general solution of the fuzzy differential equation (3.1) is

$ux,λα=u¯αx,λ,u¯αx,λ,$
$u¯αx,λ=aαλu1x+λ+bαλu2x+λ,$

Also,

$u¯αx,λ=cαλu1x,λ+dαλu2x,λ.$
$φx,λα=φ¯αx,λ,φ¯αx,λ$

be the solution of the equation (3.1) satisfying the conditions

$u−1=1,u´−1=−1$

and

$χx,λα=χ¯αx,λ,χ¯αx,λ$

be the solution of the equation (3.1) satisfying the conditions

$u1=γ,u′1=−λβ$

where

$φ¯αx,λ+c¯1αλu1x,λ+c¯2αλu2x,λ,φ¯αx,λ+c¯1αλu1x,λ+c¯2αλu2x,λ$
$χ¯αx,λ=c¯3αλu1x,λ+c¯4αλu2x,λχ¯αx,λ=c¯3αλu1x,λ+c¯4αλu2x,λ.$

From here, yields

$Wφ¯α,χ¯αx,λ=c¯1αλc¯4αλ−c¯2αλc¯3αλWu1,u2x,λ$
$Wφ¯α,χ¯αx,λ=c¯1αλc¯4αλ−c¯2αλc¯3αλWu1,u2x,λ$

Also, since u1 (x;λ) and u2 (x;λ) are linearly independent solutions of the classical differential equation $τu+λu=0,$the solution of the equation is

$ux,λ=aλu1x,λ+bλu2x,λ.$

φ(x;λ) be the solution of the classical differential equation τu+lu = 0 satisfying the conditions $u−1=$ $1,u′−1=−1.$Using boundary conditions, we have

$aλu1−1,λ+bλu2−1,λ=1,$
$aλu′1−1,λ+bλu′2−1,λ=−1.$

From this, a(λ) , b(λ) are obtained as

$aλ=u′2−1,λ+u2−1,λWu1,u2−1,λ,bλ=−u1−1,λ−u′1−1,λWu1,u2−1,λ$

Then,

$φx,λ=1Wu1,u2−1,λu′2−1,λ+u2−1,λu1x,λ −u2−1,λ+u′1−1,λu2x,λ.$

Again,χ (x,λ) be the solution of the classical differential equation τu+lu = 0 satisfying the conditions $u1=γ,u1=−λβ.$Similarly, χ(x;λ) is obtained as

$χx,λ=1Wu1,u2−1,λλβu21,λ+γu2´1,λu1x,λ −λβu11,λ+γu1´1,λu2x,λ$

Thus,

$φx,λα=φ¯αx,λ,φ¯αx,λ=c1α,c2αφx,λ$

is the solution of the equation (3.1) satisfying the conditions (3.4) and

$χx,λα=χ¯αx,λ,χ¯αx,λ=c1α,c2αχx,λ$

is the solution of the equation (3.1) satisfying the conditions (3.5), where $c1α,c2α=1α.$We take $1α=α,2−α.$From here,

$W¯αx,λ=α2φx,λχ′x,λ−χx,λφ′x,λ$
$W¯αx,λ=2−α2φx,λχ′x,λ−χx,λφ′x,λ$

are obtained. Computing the value $φx,λχ′x,λ−χx,λφ′x,λ,$we have

$Wu1,u2x,λWu1,u2−1,λWu1,u21,λu1−1,λ+u1´−1,λλβu21,λ+γu2´1,λ−u2−1,λ+u2´−1,λλβu11,λ+γu1´1,λ$

Considering the 3.7), the value is $c¯1αλc¯4αλ−c¯2αλc¯3αλ$

$α2Wu1,u2−1,λWu1,u2−1,λu1−1,λ+u1´−1,λλβu21,λ+γu2´1,λ−u2−1,λ+u2´−1,λλβu11,λ+γu1´1,λ$

and the value $c¯1αλc¯4αλ−c¯2αλc¯3αλ$is

$2−α2Wu1,u2−1,λWu1,u2−1,λu1−1,λ+u1´−1,λλβu21,λ+γu2´1,λ−u2−1,λ+u2´−1,λλβu11,λ+γu1´1,λ$

Consequently,

$Wφ¯α,χ¯αx,λ=α22−α2Wφ¯α,χ¯αx,λ.$

Theorem 1. The Wronskian functions $Wφ¯α,χ¯α$(x;λ) and W $φ¯α,χ¯α$(x;λ) are independent of variable x for x ∈ (1;1), where functions $φα¯,χα¯,φ¯α,χ¯α$are the solution of the fuzzy boundary value problem (3.1)-(3.3).

Proof. Derivating of equations W $φ¯α,χ¯α$(x;λ) and W $φ¯α,χ¯α$(x;λ) according to variable x and using the functions $φx,λα,χx,λα$are the solutions of the equation (3.1)

$W´φ¯α,χ¯αx,λ=0 and W´φ¯α,χ¯αx,λ=0.$

are obtained. The proof is complete.

$W¯αλ=Wφ¯α,χ¯αx,λ=W¯αλ=Wφ¯α,χ¯αx,λ.$

Theorem 2. The eigenvalues of the fuzzy boundary value problem (3.1)-(3.3) if and only if are consist of the zeros of functions$W¯αλ$and$W¯αλ.$

Proof. Let be λ = λ0 is the eigenvalue. We show that $W¯αλ0=0$and $W¯αλ0=0.$We assume that $W¯αλ0≠0 or W¯αλ0≠0 .$Let be $W¯αλ0≠0.$Then, the functions $φ¯αx,λ0$and $χ¯αx,λ0$are linearly independent. So, the general solution of the equation (3.1)

$ux,λ0α=u¯αx,λ0,u¯αx,λ0,$
$u¯αx,λ0=aαλ0φ¯αx,λ0+bαλ0χ¯αx,λ0,$
$u¯αx,λ0=cαλ0φ¯αx,λ0+dαλ0χ¯αx,λ0.$

Using the boundary condition (3.2) and using the solution function $φx,λ0α=φ¯αx,λ0,φ¯αx,λ0$satisfies the boundary condition (3.2),

$bαλ0χ¯α−1,λ0+χ¯′α−1,λ0=0,$
$dαλ0χ¯α−1,λ0+χ¯′α−1,λ0=0$

are obtained. Again, using (3.4), (3.10), we have

$bαλ0W¯αλ0=0,dαλ0W¯αλ0=0.$

From this, since $W¯αλ0≠0,bαλ0=0$is obtained. Similarly, using the boundary condition (3.3), we obtained $aαλ0=0.$Thus, $u¯αx.λ0=0,λ0$is not an eigenvalue. That is, we have a contradiction. Similarly, if$W¯αλ0≠0 ,u¯αx,λ0=0$is obtained. λ0 is not an eigenvalue.

Let λ0 be zero of$W¯αλ$and$W¯αλ.$Then,

$φ¯αx,λ0=k1χ¯αx,λ0,φ¯αx,λ0=k2χ¯αx,λ0.$

That is, the functions $φ¯α,χ¯α$and $φ¯α,χ¯α$are linearly dependent. Also, since $χx,λα$satisfies the boundary condition (3.3), $χ¯αx,λ0$and $χ¯αx,λ0$satisfy the boundary condition (3.3). In addition, from (3.14) the functions $φ¯α$x;λ0) and $φ¯αx,λ0$satisfy the boundary condition (3.3). So, $φx,λ0α$satisfies the boundary condition (3.3). Hence, [φ(x;λ0)]a is the solution of the boundary value problem (3.1)-(3.3) for λ = λ0. Thus, λ = λ0 is the eigenvalue. The proof is complete.

Lemma 1. Let λ = s2. The lower and the upper solutions $φ¯αx,λ,φ¯αx,λ$satisfy the following integral equations for k=0 and k=1:

$φ¯αx,λk=Cossx+1k−1sSinsx+1k +1s∫−1xSinsx−ykqyφ¯αx,λkdy,$
$φ¯αx,λk=Cossx+1k−1sSi⁡nsx+1k+1s∫−1xSi⁡nsx−ykqyφ¯αx,λkdy.$

Proof. Since

$φx,λα=φ¯αx,λ,φ¯αx,λ$

is the solution of the equation (3.1), the equation

$φ¯αy,λ,φ¯αy,λ′​′+qyφ¯αy,λ,φ¯αy,λ+λφ¯αy,λ,φ¯αy,λ=0$

is provided. Using the Hukuhara differentiability and fuzzy arithmetic,

$φ¯′​′αy,λ,φ¯′​′αy,λ+qyφ¯αy,λ,qyφ¯αy,λ+λφ¯αy,λ,λφ¯αy,λ=0$

is obtained. From here, yields

$φ¯′​′αy,λ+qyφ¯αy,λ+λφ¯αy,λ=0,φ¯′​′αy,λ+qyφ¯αy,λ+λφ¯αy,λ=0.$

Substituing the identity $qyφ¯αy,λ=−λφ¯αy,λ−φ¯′​′ay,λ$in the right side of (3.15)

$∫−1xSinsx−yqyφ¯αy,λdy=−s2∫−1xSinsx−yφ¯αy,λdy −∫−1xSinsx−yφ″¯αx,λdy$

On integrating by parts twice and using (3.4)

$∫−1xSinsx−yφ¯′​′αx,λdy=Sinsx+1−sφ¯αx,λ+sCossx+1 +s2∫−1xSinsx−yφ¯αy,λdy$

is obtained. Substituing this back into the previous equality yields

$∫−1xSinsx−yqyφ¯αy,λdy=Sinsx+1−sφ¯αx,λ+sCossx+1$

From here, we have

$φ¯αx,λ=Cossx+1−1sSinsx+1+∫−1xSinsx−yqyφ¯αx,λdy.$

Similarly $φ¯αx,λ$is found. Derivating in these equations according to x, the derivative equations are obtained.

Lemma 2. Let λ = s2. The lower and the upper solutions $χ¯αx,λ,χ¯αx,λ$satisfy the following integral equations for k=0 and k=1:

$χ_αx,λk=−βCossx+1k+λαsSinsx+1k−1s∫x1Sinsx−ykqyχ_αx,λkdy$
$χ¯αx,λk=−βCossx+1k+λαsSinsx+1k−1s∫x1Sinsx−ykqyχ¯αx,λkdy.$

Proof. Substituing the identity $qyχ¯αy,λ=−λχ¯αy,λ−χ¯′​′ay,λ$in the right side of (3.17), integrating by parts twice and using (3.5) yields (3.17) for k=0. Similarly, the equation (3.18) is found for k=0. Derivating in these equations according to x, the derivative equations are obtained.

Acknowledgement 1

This paper was supported by Scientific Research Projects Coordination Unit of Giresun University (The project number is FEN-BAP-A-230218-52). Thanks to Scientific Research Projects Coordination Unit of Giresun University.

## References

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B. Bede A note on ”two-point boundary value problems associated with non-linear fuzzy differential equations” Fuzzy Sets and Systems 157 (2006) 986-989.

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B. Bede Note on ”Numerical solutions of fuzzy differential equations by predictor-corrector method” Inf. Sci. 178 (2008) 1917-1922.

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B. Bede S.G. Gal Almost periodic fuzzy-number-valued functions Fuzzy Sets and Systems 147 (2004) 385-403.

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B. Bede S. G. Gal Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations Fuzzy Sets and Systems 151 (2005) 581-599.

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• Export Citation
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B. Bede I.J. Rudas A.L. Bencsik First order linear fuzzy differential equations under generalized differentiability Inform. Sci. 177 (2007) 1648-1662.

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• Export Citation
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J.J. Buckley T. Feuring Fuzzy differential equations Fuzzy Sets and Systems 110 (2000) 43-54.

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• Export Citation
• [7]

J.J. Buckley T. Feuring Fuzzy initial value problem for Nth-order linear differential equations Fuzzy Sets and Systems 121 (2001) 247-255.

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• Export Citation
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Y. Chalco-Cano H. Roman-Flores On new solutions of fuzzy differential equations Chaos Solutions and Fractals 38 (2008) 112-119.

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• Export Citation
• [9]

Y. Chalco-Cano H. Roman-Flores Comparation between some approaches to solve fuzzy differential equations Fuzzy Sets and Systems 160 (2009) 1517-1527.

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• Export Citation
• [10]

D. Dubois H. Prade Operations on fuzzy numbers International Journal of Systems Sciences 9 (1978) 613-626.

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• [11]

N. Gasilov ¸ S.E. Amrahov A.G. Fatullayev A geometric approach to solve fuzzy linear systems of differential equations Appl. Math. & Inf. Sci. 5 (2011) 484-495.

• [12]

H. Gültekin Çitil N. Altını¸sık On the eigenvalues and the eigenfunctions of the Sturm-Liouville fuzzy boundary value problem Journal of Mathematical and Computational Science 7(4) (2017) 786-805.

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E. Hüllermeir An approach to modeling and simulation of uncertain dynamical systems International Journal of Uncertanity Fuzziness and Knowledge-Based Systems 5 (1997) 117-137.

• Crossref
• Export Citation
• [14]

A. Khastan F. Bahrami K. Ivaz New Results on Multiple Solutions for Nth-order Fuzzy Differential Equations under Generalized Differentiability Boundary Value Problems (2009) 1-13.

• Crossref
• Export Citation
• [15]

A. Khastan J.J. Nieto A boundary value problem for second order fuzzy differential equations Nonlinear Analysis 72 (2010) 3583-3593.

• Crossref
• Export Citation
• [16]

V. Lakshmikantham R. N. Mohapatra Theory of Fuzzy Differential Equations and Inclusions Taylor&Francis London New York 2003.

• [17]

V. Lakshmikantham J.J. Nieto Differential Equations in Metric Spaces: An introduction and application to fuzzy differential equations Dynamics of Continuous. Discrete and Impulsive Systems Series A: Mathematical Analysis 10 (2003) 991-1000.

• [18]

H.-K. Liu Comparison results of two-point fuzzy boundary value problems International Journal of Computational and Mathematical Sciences 5(1) (2011) 1-7.

• [19]

J.J. Nieto R. Rodriguez-Lopez Euler polygonal method for metric dynamical systems Inform. Sci. 177 (2007) 42564270.

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J.J. Nieto A. Khastan K. Ivaz Numerical solution of fuzzy differential equations under generalized differentiability Nonlinear Analysis: Hybrid Syst. 3 (2009) 700-707.

• [21]

M.L. Puri D.A. Ralescu Differentials for fuzzy functions Journal of Mathematical Analysis and Applications 91 (1983) 552-558.

• Crossref
• Export Citation
• [22]

L.A. Zadeh Fuzzy Sets Information and Control 8 (1965) 338-353.

• Crossref
• Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

• [1]

B. Bede A note on ”two-point boundary value problems associated with non-linear fuzzy differential equations” Fuzzy Sets and Systems 157 (2006) 986-989.

• Crossref
• Export Citation
• [2]

B. Bede Note on ”Numerical solutions of fuzzy differential equations by predictor-corrector method” Inf. Sci. 178 (2008) 1917-1922.

• Crossref
• Export Citation
• [3]

B. Bede S.G. Gal Almost periodic fuzzy-number-valued functions Fuzzy Sets and Systems 147 (2004) 385-403.

• Crossref
• Export Citation
• [4]

B. Bede S. G. Gal Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations Fuzzy Sets and Systems 151 (2005) 581-599.

• Crossref
• Export Citation
• [5]

B. Bede I.J. Rudas A.L. Bencsik First order linear fuzzy differential equations under generalized differentiability Inform. Sci. 177 (2007) 1648-1662.

• Crossref
• Export Citation
• [6]

J.J. Buckley T. Feuring Fuzzy differential equations Fuzzy Sets and Systems 110 (2000) 43-54.

• Crossref
• Export Citation
• [7]

J.J. Buckley T. Feuring Fuzzy initial value problem for Nth-order linear differential equations Fuzzy Sets and Systems 121 (2001) 247-255.

• Crossref
• Export Citation
• [8]

Y. Chalco-Cano H. Roman-Flores On new solutions of fuzzy differential equations Chaos Solutions and Fractals 38 (2008) 112-119.

• Crossref
• Export Citation
• [9]

Y. Chalco-Cano H. Roman-Flores Comparation between some approaches to solve fuzzy differential equations Fuzzy Sets and Systems 160 (2009) 1517-1527.

• Crossref
• Export Citation
• [10]

D. Dubois H. Prade Operations on fuzzy numbers International Journal of Systems Sciences 9 (1978) 613-626.

• Crossref
• Export Citation
• [11]

N. Gasilov ¸ S.E. Amrahov A.G. Fatullayev A geometric approach to solve fuzzy linear systems of differential equations Appl. Math. & Inf. Sci. 5 (2011) 484-495.

• [12]

H. Gültekin Çitil N. Altını¸sık On the eigenvalues and the eigenfunctions of the Sturm-Liouville fuzzy boundary value problem Journal of Mathematical and Computational Science 7(4) (2017) 786-805.

• [13]

E. Hüllermeir An approach to modeling and simulation of uncertain dynamical systems International Journal of Uncertanity Fuzziness and Knowledge-Based Systems 5 (1997) 117-137.

• Crossref
• Export Citation
• [14]

A. Khastan F. Bahrami K. Ivaz New Results on Multiple Solutions for Nth-order Fuzzy Differential Equations under Generalized Differentiability Boundary Value Problems (2009) 1-13.

• Crossref
• Export Citation
• [15]

A. Khastan J.J. Nieto A boundary value problem for second order fuzzy differential equations Nonlinear Analysis 72 (2010) 3583-3593.

• Crossref
• Export Citation
• [16]

V. Lakshmikantham R. N. Mohapatra Theory of Fuzzy Differential Equations and Inclusions Taylor&Francis London New York 2003.

• [17]

V. Lakshmikantham J.J. Nieto Differential Equations in Metric Spaces: An introduction and application to fuzzy differential equations Dynamics of Continuous. Discrete and Impulsive Systems Series A: Mathematical Analysis 10 (2003) 991-1000.

• [18]

H.-K. Liu Comparison results of two-point fuzzy boundary value problems International Journal of Computational and Mathematical Sciences 5(1) (2011) 1-7.

• [19]

J.J. Nieto R. Rodriguez-Lopez Euler polygonal method for metric dynamical systems Inform. Sci. 177 (2007) 42564270.

• [20]

J.J. Nieto A. Khastan K. Ivaz Numerical solution of fuzzy differential equations under generalized differentiability Nonlinear Analysis: Hybrid Syst. 3 (2009) 700-707.

• [21]

M.L. Puri D.A. Ralescu Differentials for fuzzy functions Journal of Mathematical Analysis and Applications 91 (1983) 552-558.

• Crossref
• Export Citation
• [22]

L.A. Zadeh Fuzzy Sets Information and Control 8 (1965) 338-353.

• Crossref
• Export Citation

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