Research Article New Results on Multiple Solutions for Nth-Order Fuzzy Differential Equations under Generalized Differentiability

Transcription

1 Hindawi Publising Corporation Boundary Value Problems Volume 009, Article ID , 13 pages doi: /009/ Researc Article New Results on Multiple Solutions for Nt-Order Fuzzy Differential Equations under Generalized Differentiability A. Kastan, 1, F. Barami, 1, and K. Ivaz 1, 1 Department of Applied Matematics, University of Tabriz, Tabriz , Iran Researc Center for Industrial Matematics, University of Tabriz, Tabriz , Iran Correspondence sould be addressed to A. Kastan, kastan@gmail.com Received 30 April 009; Accepted 1 July 009 Recommended by Juan José Nieto We firstly present a generalized concept of iger-order differentiability for fuzzy functions. Ten we interpret Nt-order fuzzy differential equations using tis concept. We introduce new definitions of solution to fuzzy differential equations. Some examples are provided for wic bot te new solutions and te former ones to te fuzzy initial value problems are presented and compared. We present an example of a linear second-order fuzzy differential equation wit initial conditions aving four different solutions. Copyrigt q 009 A. Kastan et al. Tis is an open access article distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited. 1. Introduction Te term fuzzy differential equation was coined in 1987 by Kandel and Byatt 1 and an extended version of tis sort note was publised two years later. Tere are many suggestions to define a fuzzy derivative and in consequence, to study fuzzy differential equation 3. One of te earliest was to generalize te Hukuara derivative of a set-valued function. Tis generalization was made by Puri and Ralescu 4 and studied by Kaleva 5. It soon appeared tat te solution of fuzzy differential equation interpreted by Hukuara derivative as a drawback: it became fuzzier as time goes by 6. Hence, te fuzzy solution beaves quite differently from te crisp solution. To alleviate te situation, Hüllermeier 7 interpreted fuzzy differential equation as a family of differential inclusions. Te main sortcoming of using differential inclusions is tat we do not ave a derivative of a fuzzynumber-valued function. Te strongly generalized differentiability was introduced in 8 and studied in Tis concept allows us to solve te above-mentioned sortcoming. Indeed, te strongly

2 Boundary Value Problems generalized derivative is defined for a larger class of fuzzy-number-valued functions tan te Hukuara derivative. Hence, we use tis differentiability concept in te present paper. Under tis setting, we obtain some new results on existence of several solutions for Ntorder fuzzy differential equations. Higer-order fuzzy differential equation wit Hukuara differentiability is considered in 1 and te existence and uniqueness of solution for nonlinearities satisfying a Lipscitz condition is proved. Buckley and Feuring 13 presented two different approaces to te solvability of Nt-order linear fuzzy differential equations. Here, using te concept of generalized derivative and its extension to iger-order derivatives, we sow tat we ave several possibilities or types to define iger-order derivatives of fuzzy-number-valued functions. Ten, we propose a new metod to solve iger-order fuzzy differential equations based on te selection of derivative type covering all former solutions. Wit tese ideas, te selection of derivative type in eac step of derivation plays a crucial role.. Preliminaries In tis section, we give some definitions and introduce te necessary notation wic will be used trougout tis paper. See, for example, 6. Definition.1. Let X be a nonempty set. A fuzzy set u in X is caracterized by its membersip function u : X 0, 1. Tus, u x is interpreted as te degree of membersip of an element x in te fuzzy set u for eac x X. Let us denote by R F te class of fuzzy subsets of te real axis i.e., u : R 0, 1 satisfying te following properties: i u is normal, tat is, tere exists s 0 R suc tat u s 0 1, ii u is convex fuzzy set i.e., u ts 1 t r min{u s,u r }, for all t 0, 1,s,r R, iii u is upper semicontinuous on R, iv cl{s R u s > 0} is compact were cl denotes te closure of a subset. Ten R F is called te space of fuzzy numbers. Obviously, R R F. For 0 <α 1 denote u α {s R u s α} and u 0 cl{s R u s > 0}. Ifu belongs to R F, ten α-level set u α is a nonempty compact interval for all 0 α 1. Te notation u α [ u α, u α],.1 denotes explicitly te α-level set of u. One refers to u and u as te lower and upper brances of u, respectively. Te following remark sows wen u α, u α is a valid α-level set. Remark. see 6. Tesufficient conditions for u α, u α to define te parametric form of a fuzzy number are as follows: i u α is a bounded monotonic increasing nondecreasing left-continuous function on 0, 1 and rigt-continuous for α 0, ii u α is a bounded monotonic decreasing nonincreasing left-continuous function on 0, 1 and rigt-continuous for α 0, iii u α u α, 0 α 1.

3 Boundary Value Problems 3 For u, v R F and λ R, tesumu v and te product λ u are defined by u v α u α v α, λ u α λ u α, for all α 0, 1, were u α v α means te usual addition of two intervals subsets of R and λ u α means te usual product between a scalar and a subset of R. Te metric structure is given by te Hausdorff distance: D : R F R F R {0},. by D u, v sup max { u α v α, u α v α }. α 0,1.3 Te following properties are wellknown: i D u w, v w D u, v, for all u, v, w R F, ii D k u, k v k D u, v, for all k R,u,v R F, iii D u v, w e D u, w D v, e, for all u, v, w, e R F, and R F,D is a complete metric space. Definition.3. Let x, y R F. If tere exists z R F suc tat x y z, ten z is called te H-difference of x, y anditisdenotedx y. In tis paper te sign stands always for H-difference and let us remark tat x y / x 1 y in general. Usually we denote x 1 y by x y, wile x y stands for te H-difference. 3. Generalized Fuzzy Derivatives Te concept of te fuzzy derivative was first introduced by Cang and Zade 14 ; itwas followed up by Dubois and Prade 15 wo used te extension principle in teir approac. Oter metods ave been discussed by Puri and Ralescu 4, Goetscel and Voxman 16, Kandel and Byatt 1,. Laksmikantam and Nieto introduced te concept of fuzzy differential equation in a metric space 17. Puri and Ralescu in 4 introduced H-derivative differentiability in te sense of Hukuara for fuzzy mappings and it is based on te H- difference of sets, as follows. Hencefort, we suppose I T 1,T for T 1 <T,T 1,T R. Definition 3.1. Let F : I R F be a fuzzy function. One says, F is differentiable at t 0 I if tere exists an element F t 0 R F suc tat te limits F t 0 F t 0 F t 0 F t 0 lim, lim exist and are equal to F t 0. Here te limits are taken in te metric space R F,D.

4 4 Boundary Value Problems Te above definition is a straigtforward generalization of te Hukuara differentiability of a set-valued function. From 6, Proposition 4..8, it follows tat Hukuara differentiable function as increasing lengt of support. Note tat tis definition of derivative is very restrictive; for instance, in 9, te autors sowed tat if F t c g t, were c is a fuzzy number and g : a, b R is a function wit g t < 0, ten F is not differentiable. To avoid tis difficulty, te autors 9 introduced a more general definition of derivative for fuzzy-number-valued function. In tis paper, we consider te following definition 11. Definition 3.. Let F : I R F and fix t 0 I. One says F is 1 -differentiable at t 0, if tere exists an element F t 0 R F suc tat for all >0sufficiently near to 0, tere exist F t 0 F t 0,F t 0 F t 0, and te limits in te metric D F t 0 F t 0 F t 0 F t 0 lim lim F t F is -differentiable if for all <0sufficiently near to 0, tere exist F t 0 F t 0,F t 0 F t 0 and te limits in te metric D F t 0 F t 0 F t 0 F t 0 lim lim F t If F is n -differentiable at t 0, we denote its first derivatives by D 1 n F t 0,forn 1,. Example 3.3. Let g : I R and define f : I R F by f t c g t, for all t I. Ifg is differentiable at t 0 I, ten f is generalized differentiable on t 0 I and we ave f t 0 c g t 0. For instance, if g t 0 > 0, f is 1 -differentiable. If g t 0 < 0, ten f is -differentiable. Remark 3.4. In te previous definition, 1 -differentiability corresponds to te H-derivative introduced in 4, sotisdifferentiability concept is a generalization of te H-derivative and obviously more general. For instance, in te previous example, for f t c g t wit g t 0 < 0, we ave f t 0 c g t 0. Remark 3.5. In 9, te autors consider four cases for derivatives. Here we only consider te two first cases of 9, Definition 5. In te oter cases, te derivative is trivial because it is reduced to crisp element more precisely, F t 0 R. For details, see 9, Teorem 7. Teorem 3.6. Let F : I R F be fuzzy function, were F t α f α t,g α t for eac α 0, 1. i If F is (1)-differentiable, ten f α and g α are differentiable functions and D 1 1 F t α f α t,g α t. ii If F is ()-differentiable, ten f α and g α are differentiable functions and D 1 F t α g α t,f α t. Proof. See 11. Now we introduce definitions for iger-order derivatives based on te selection of derivative type in eac step of differentiation. For te sake of convenience, we concentrate on te second-order case.

5 Boundary Value Problems 5 For a given fuzzy function F, we ave two possibilities Definition 3. to obtain te derivative of F ot t: D 1 1 again two possibilities: D 1 respectively. F t and D 1 F t. Ten for eac of tese two derivatives, we ave 1 D 1 1 F t,d 1 D 1 1 F t, and D 1 1 D 1 F t,d 1 D 1 F t, Definition 3.7. Let F : I R F and n, m 1,. One says say F is n, m -differentiable at t 0 I, if D 1 n F exists on a neigborood of t 0 as a fuzzy function and it is m -differentiable at t 0. Te second derivatives of F are denoted by D n,mf t 0 for n, m 1,. Remark 3.8. Tis definition is consistent. For example, if F is 1, and, 1 -differentiable simultaneously at t 0, ten F is 1 - and -differentiable around t 0. By remark in 9, F is a crisp function in a neigborood of t 0. Teorem 3.9. Let D 1 1 F : I R F or D 1 F : I R F be fuzzy functions, were F t α f α t,g α t. i If D 1 1 F is (1)-differentiable, ten f α and g α are differentiable functions and D 1,1 F t α f α t,g α t. ii If D 1 1 F is ()-differentiable, ten f α and g α are differentiable functions and D 1, F t α g α t,f α t. iii If D 1 F is (1)-differentiable, ten f α and g α are differentiable functions and D,1 F t α g α t,f α t. iv If D 1 F is ()-differentiable, ten f α and g α are differentiable functions and D, F t α f α t,g α t. Proof. We present te details only for te case i, since te oter cases are analogous. If >0andα 0, 1, we ave [ D 1 1 F t D 1 1 F t ] α [ f α t f α t,g α t g α t ], 3.4 and multiplying by 1/, we ave [ D 1 1 ] α F t D 1 1 F t [ f α t f α t, g α t g α t ]. 3.5 Similarly, we obtain [ D 1 1 F t D 1 1 F t ] α [ f α t f α t, g α t g α t ]. 3.6

6 6 Boundary Value Problems Passing to te limit, we ave [ D 1,1 F t ] α [ f α t,g α t ]. 3.7 Tis completes te proof of te teorem. Let N be a positive integer number, pursuing te above-cited idea, we write D N k 1,...,k N F t 0 to denote te Nt-derivatives of F at t 0 wit k i 1, fori 1,...,N.Now we intend to compute te iger derivatives in generalized differentiability sense of te H-difference of two fuzzy functions and te product of a crisp and a fuzzy function. Lemma If f, g : I R F are Nt-order generalized differentiable at t I in te same case of differentiability, ten f g is generalized differentiable of order N at t and f g N t f N t g N t. (Te sum of two functions is defined pointwise.) Proof. By Definition 3. te statement of te lemma follows easily. Teorem Let f, g : I R F be second-order generalized differentiable suc tat f is (1,1)- differentiable and g is (,1)-differentiable or f is (1,)-differentiable and g is (,)-differentiable or f is (,1)-differentiable and g is (1,1)-differentiable or f is (,)-differentiable and g is (1,)-differentiable on I.IfteH-difference f t g t exists for t I, ten f g is second-order generalized differentiable and ( f g ) t f t 1 g t, 3.8 for all t I. Proof. We prove te first case and oter cases are similar. Since f is 1 -differentiable and g is -differentiable on I, by 10, Teorem 4, f g t is 1 -differentiable and we ave f g t f t 1 g t. Bydifferentiation as 1 -differentiability in Definition 3. and using Lemma 3.10, weget f g t is 1,1 -differentiable and we deduce ( f g ) t ( f t 1 g t ) f t 1 g t. 3.9 Te H-difference of two functions is understood pointwise. Teorem 3.1. Let f : I R and g : I R F be two differentiable functions (g is generalized differentiable as in Definition 3.). i If f t f t > 0 and g is (1)-differentiable, ten f g is (1)-differentiable and ( f g ) t f t g t f t g t. 3.10

7 Boundary Value Problems 7 ii If f t f t < 0 and g is ()-differentiable, ten f g is ()-differentiable and ( f g ) t f t g t f t g t Proof. See 10. Teorem Let f : I R and g : I R F be second-order differentiable functions (g is generalized differentiable as in Definition 3.7). i If f t f t > 0,f t f t > 0, and g is (1,1)-differentiable ten f g is (1,1)-differentiable and ( f g ) t f t g t f t g t f t g t. 3.1 ii If f t f t < 0,f t f t < 0 and g is (,)-differentiable ten f g is (,)-differentiable and ( f g ) t f t g t f t g t f t g t Proof. We prove i, and te proof of anoter case is similar. If f t f t > 0andg is 1 differentiable, ten by Teorem 3.1 we ave ( f g ) t f t g t f t g t Now by differentiation as first case in Definition 3.,sinceg t is 1 -differentiable and f t f t > 0, ten we conclude te result. Remark By 9, Remark 16,letf : I R,γ R F and define F : I R F by F t γ f t, for all t I. Iff is differentiable on I, ten F is differentiable on I, witf t γ f t. By Teorem 3.1, iff t f t > 0, ten F is 1 -differentiable on I. Alsoiff t f t < 0, ten F is -differentiable on I. Iff t f t 0, by 9, Teorem 10, we ave F t γ f t. We can extend tis result to second-order differentiability as follows. Teorem Let f : I R be twice differentiable on I, γ R F and define F : I R F by F t γ f t, for all t I. i If f t f t > 0 and f t f t > 0, ten F t is (1,1)-differentiable and its second derivative, D 1,1 F, is F t γ f t, ii If f t f t > 0 and f t f t < 0, ten F t is (1,)-differentiable wit D 1, F γ f t, iii If f t f t < 0 and f t f t > 0, ten F t is (,1)-differentiable wit D,1 F γ f t, iv If f t f t < 0 and f t f t < 0, ten F t is (,)-differentiable wit D, F γ f t.

8 8 Boundary Value Problems Proof. Cases i and iv follow from Teorem To prove ii, sincef t f t > 0, by Remark 3.14, F is 1 -differentiable and we ave D 1 1 F γ f t on I.Also,sincef t f t < 0, ten D 1 1 F is -differentiable and we conclude te result. Case iii is similar to previous one. Example If γ is a fuzzy number and φ : 0, 3 R, were φ t t 3t 3.15 is crisp second-order polynomial, ten for F t γ φ t, 3.16 we ave te following i for 0 < t < 1: φ t φ t < 0andφ t φ t < 0 ten by iv, F t is - differentiable and its second derivative, D, F is F t γ, ii for 1 <t<3/: φ t φ t > 0andφ t φ t < 0 ten by ii, F t is 1- differentiable wit D 1,F γ, iii for 3/ <t<: φ t φ t < 0andφ t φ t > 0 ten by iii, F t is -1 differentiable and D,1F γ, iv for <t<3: φ t φ t > 0andφ t φ t > 0 ten by i, F t is 1-1 -differentiable and D 1,1F γ, v for t 1, 3/, : we ave φ t φ t 0, ten by 9, Teorem 10 we ave F t γ φ t, again by applying tis teorem, we get F t γ. 4. Second-Order Fuzzy Differential Equations In tis section, we study te fuzzy initial value problem for a second-order linear fuzzy differential equation: y t a y t b y t σ t, y 0 γ 0, y 0 γ 1, 4.1 were a, b > 0, γ 0,γ 1 R F, and σ t is a continuous fuzzy function on some interval I. Te interval I can be 0,A for some A>0orI 0,. In tis paper, we suppose a, b > 0. Our strategy of solving 4.1 is based on te selection of derivative type in te fuzzy differential equation. We first give te following definition for te solutions of 4.1. Definition 4.1. Let y : I R F be a fuzzy function and n, m {1, }. One says y is an n, m solution for problem 4.1 on I, ifd 1 nyd n,my exist on I and D n,my t a D 1 ny t b y t σ t,y 0 γ 0,D 1 ny 0 γ 1.

9 Boundary Value Problems 9 Let y be an n, m -solution for 4.1. To find it, utilizing Teorems 3.6 and 3.9 and considering te initial values, we can translate problem 4.1 to a system of secondorder linear ordinary differential equations ereafter, called corresponding n, m -system for problem 4.1. Terefore, four ODEs systems are possible for problem 4.1, as follows: 1, 1 -system y t; α ay t; α by t; α σ t; α, y t; α ay t; α by t; α σ t; α, y 0; α γ α 0, y 0; α γ α 0, y 0; α γ α 1, y 0; α γ α 1, 4. 1, -system y t; α ay t; α by t; α σ t; α, y t; α ay t; α by t; α σ t; α, y 0; α γ α 0, y 0; α γ α 0, y 0; α γ α 1, y 0; α γ α 1, 4.3, 1 -system y t; α ay t; α by t; α σ t; α, y t; α ay t; α by t; α σ t; α, y 0; α γ α 0, y 0; α γ α 0, y 0; α γ 1,y 0; α γ 1, 4.4, -system y t; α ay t; α by t; α σ t; α, y t; α ay t; α by t; α σ t; α, y 0; α γ α 0, y 0; α γ α 0, y 0; α γ α 1,y 0; α γ α

10 10 Boundary Value Problems Teorem 4.. Let n, m {1, } and y y, y be an n, m -solution for problem 4.1 on I.Teny and y solve te associated n, m -systems. Proof. Suppose y is te n, m -solution of problem 4.1. According to te Definition 4.1, ten D 1 ny and D n,my exist and satisfy problem 4.1. By Teorems 3.6 and 3.9 and substituting y, y and teir derivatives in problem 4.1,wegette n, m -system corresponding to n, m solution. Tis completes te proof. Teorem 4.3. Let n, m {1, } and f α t and g α t solve te n, m -system on I, for every α 0, 1. Let F t α f α t,g α t. IfF as valid level sets on I and Dn,mF exists, ten F is an n, m -solution for te fuzzy initial value problem 4.1. Proof. Since F t α f α t,g α t is n, m -differentiable fuzzy function, by Teorems 3.6 and 3.9 we can compute DnF 1 and Dn,mF according to f α,g α,f α,g α. Due to te fact tat f α,g α solve n, m -system, from Definition 4.1, it comes tat F is an n, m -solution for 4.1. Te previous teorems illustrate te metod to solve problem 4.1. We first coose te type of solution and translate problem 4.1 to a system of ordinary differential equations. Ten, we solve te obtained ordinary differential equations system. Finally we find suc a domain in wic te solution and its derivatives ave valid level sets and using Stacking Teorem 5 we can construct te solution of te fuzzy initial value problem 4.1. Remark 4.4. We see tat te solution of fuzzy differential equation 4.1 depends upon te selection of derivatives. It is clear tat in tis new procedure, te unicity of te solution is lost, an expected situation in te fuzzy context. Noneteless, we can consider te existence of four solutions as sown in te following examples. Example 4.5. Let us consider te following second-order fuzzy initial value problem y t σ 0, y 0 γ 0, y 0 γ 1, t 0, 4.6 were σ 0 γ 0 γ 1 are te triangular fuzzy number aving α-level sets α 1, 1 α. If y is 1,1 -solution for te problem, ten [ y t ] [ ] α y t; α, y t; α, [ y t ] [ ] α y t; α, y t; α, 4.7 and tey satisfy 1,1 -system associated wit 4.1. On te oter and, by ordinary differential teory, te corresponding 1,1 -system as only te following solution: ( ) ( ) t t y t; α α 1 t 1, y t; α 1 α t We see tat y t α y t; α, y t; α are valid level sets for t 0and ( ) t y α 1, 1 α t

11 Boundary Value Problems 11 By Teorem 3.15, y is 1,1 -differentiable for t 0. Terefore, y defines a 1,1 -solution for t 0. For 1, -solution, we get te following solutions for 1, -system: y t; α α 1 ( t ) t 1, y t; α 1 α ( t ) t 1, 4.10 were y t as valid level sets for t 0, 1. How ever-also y t α α 1, 1 α t / t 1 were y is 1, -differentiable. Ten y gives us a 1, -solution on 0, 1.,1 -system yields y t; α α 1 ( t ) t 1, y t; α 1 α ( t ) t 1, 4.11 were y t as valid level sets for t 0, 3 1. We can see y is a,1 -solution on 0, 3 1 Finally, - -system gives ( ) ( ) t t y t; α α 1 t 1, y t; α 1 α t 1, 4.1 were y t as valid level sets for all t 0, 1, and defines a, -solution on 0, 1. Ten we ave an example of a second-order fuzzy initial value problem wit four different solutions. Example 4.6. Consider te fuzzy initial value problem: y t y t σ 0, y 0 γ 0, y 0 γ 1 t 0, 4.13 were σ 0 is te fuzzy number aving α-level sets α, α and γ 0 α γ 1 α α 1, 1 α. To find 1,1 -solution, we ave y t; α α 1 sin t sin t cos t, y t; α α 1 sin t sin t cos t, 4.14 were y t as valid level sets for t 0andy t σ 0 1 sin t sin t cos t.fromteorem 3.15, y is 1, -differentiable on 0,π/, ten by Remark 3.8, y is not 1, 1 -differentiable on 0,π/. Hence, no 1,1 -solution exists for t>0. For 1, -solutions we deduce y t; α α 1 sin t sin t cos t, y t; α α 1 sin t sin t cos t, 4.15 we see tat y t as valid level sets and is 1,1 -differentiable for t>0. Since te 1, -system as only te above solution, ten 1, -solution does not exist.

12 1 Boundary Value Problems For,1 -solutions we get y t; α α 1 sin t sin t cos t, y t; α α 1 sin t sin t cos t, 4.16 we see tat te fuzzy function y t as valid level sets for t 0, ln 1 and define a,1 -solution for te problem on 0, ln 1. Finally, to find, -solution, we find y t; α α 1 sin t sin t cos t, y t; α α 1 sin t sin t cos t, 4.17 tat y t as valid level sets for t 0andy is, -differentiable on 0,π/. We ten ave a linear fuzzy differential equation wit initial condition and two solutions. 5. Higer-Order Fuzzy Differential Equations Selecting different types of derivatives, we get several solutions to fuzzy initial value problem for second-order fuzzy differential equations. Teorem 4. as a crucial role in our strategy. To extend te results to Nt-order fuzzy differential equation, we can follow te proof of Teorem 4. to get te same results for derivatives of iger order. Terefore, we can extend te presented argument for second-order fuzzy differential equation to Nt-order. Under generalized derivatives, we would expect at most N solutions for an Nt-order fuzzy differential equation by coosing te different types of derivatives. Acknowledgments We tank Professor J. J. Nieto for is valuable remarks wic improved te paper. Tis researc is supported by a grant from University of Tabriz. References 1 A. Kandel and W. J. Byatt, Fuzzy differential equations, in Proceedings of te International Conference on Cybernetics and Society, pp , Tokyo, Japan, A. Kandel and W. J. Byatt, Fuzzy processes, Fuzzy Sets and Systems, vol. 4, no., pp , J. J. Buckley and T. Feuring, Fuzzy differential equations, Fuzzy Sets and Systems, vol. 110, no. 1, pp , M. L. Puri and D. A. Ralescu, Differentials of fuzzy functions, Journal of Matematical Analysis and Applications, vol. 91, no., pp , O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, vol. 4, no. 3, pp , P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientific, Singapore, E. Hüllermeier, An approac to modelling and simulation of uncertain dynamical systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 5, no., pp , B. Bede and S. G. Gal, Almost periodic fuzzy-number-valued functions, Fuzzy Sets and Systems, vol. 147, no. 3, pp , 004.

13 Boundary Value Problems 13 9 B. Bede and S. G. Gal, Generalizations of te differentiability of fuzzy-number-valued functions wit applications to fuzzy differential equations, Fuzzy Sets and Systems, vol. 151, no. 3, pp , B. Bede, I. J. Rudas, and A. L. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Information Sciences, vol. 177, no. 7, pp , Y. Calco-Cano and H. Román-Flores, On new solutions of fuzzy differential equations, Caos, Solitons & Fractals, vol. 38, no. 1, pp , D. N. Georgiou, J. J. Nieto, and R. Rodríguez-López, Initial value problems for iger-order fuzzy differential equations, Nonlinear Analysis: Teory, Metods & Applications, vol. 63, no. 4, pp , J. J. Buckley and T. Feuring, Fuzzy initial value problem for Nt-order linear differential equations, Fuzzy Sets and Systems, vol. 11, no., pp , S. S. L. Cang and L. A. Zade, On fuzzy mapping and control, IEEE Transactions on Systems Man Cybernetics, vol., pp , D. Dubois and H. Prade, Towards fuzzy differential calculus part III: differentiation, Fuzzy Sets and Systems, vol. 8, no. 3, pp. 5 33, R. Goetscel Jr. and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, vol. 18, no. 1, pp , V. Laksmikantam and J. J. Nieto, Differential equations in metric spaces: an introduction and an application to fuzzy differential equations, Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 10, no. 6, pp , 003.