Fuzzy differential equations: An approach via fuzzification of the derivative operator

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Fuzzy differential equations: An approach via fuzzification of the derivative operator L.C. Barrosl

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, L.T. Gomes , P.A. Tonelli

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5 a Department of Applied Mathematics, IMECC, University of Campinas, Campinas, Brazil b IDepartment of Applied Mathematics, IME, University of São Paulo, São, Brazil

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Abstract

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In this paper we study fuzzy differential equations (FDEs) in terms of derivative for fuzzy functions, in a different way from the traditional Hukuhara derivative defined for set valued functions. The derivative we use is obtained by means of fuzzification of the classical derivative operator for standard functions. We discuss the relation of this approach to fuzzy differential inclusions (FDIs) and Hukuhara and Generalized Hukuhara derivatives. A theorem of existence of a solution is studied, with hypothesis similar to those assumed for FDIs. Some examples are explored in order to illustrate the theory. © 2013 Published by Elsevier B.V. Keywords: Analysis; Zadeh’s extension; Fuzzy derivative; Fuzzy differential equations

1. Introduction 17

Consider the Problem  x  (t) = F(t, x(t)) x(0) = x 0

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(1)

(1) For the standard case, where F : [0, T ] × R n → R n , the solution is a function x : [0, T ] → R n , T > 0, and x(·) ∈ E([0, T ]; R n ), where E([0, T ]; R n ) is a space of functions with some properties. (2) If (1) is to be interpreted within a fuzzy context, F : [0, T ] × FK (R n ) → FK (R n ) and the solution has a representation of type X : [0, T ] → FK (R n ), where FK (R n ) is the space of fuzzy subsets of R n with nonempty compact -levels (full definition is found in Section 2). There are in the literature at least two ways to construct the fuzzy solution: (i) From a notion of derivative for fuzzy valued function of a real variable (see [1]). (ii) Without using any notion of derivative for fuzzy functions. The derivative is that for standard functions and the solution X(·) for the fuzzy system (1) is a fuzzy subset of classical trajectories. ∗ Corresponding author. Tel.: +55 19 3521 2121.

E-mail addresses: [email protected] (L.C. Barros), [email protected] (L.T. Gomes), [email protected] (P.A. Tonelli). 0165-0114/$ - see front matter © 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.fss.2013.03.004 Please cite this article as: L.C. Barros, et al., Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.004

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The context (i) is formulated in terms of Hukuhara derivative. For more information about it, the reader can see [2–10] and various other works that can be found in the references of books. Case (ii) was initially studied by [11–13,5]. For [12] the -levels of solution X(·) are formed by solutions of differential inclusions (DIs). Nevertheless, the proof that X(t) is, in fact, a fuzzy subset of R n was made in [14], which proves also that the family of trajectories X(·) is a fuzzy subset of a specific space of functions. It is noteworthy that both in [11,12] no fuzzy derivative concept is used. Consequently, there is no explicit differential equation (DE) to be solved, i.e., there is no identity to be solved. The fuzzy solution is given by a family of DIs (in (1), for each t ∈ [0, T ], the right side is a fuzzy set and the left side a real number) and, therefore, there is not an explicit equation of the form X  (t) = F(t, X(t)) to be solved. In the standard case, considering the field in each state x ∈ R n , we find the solution trajectories x(·) ∈ E([0, T ]; R n ). In this paper, we are concerned with the fuzzy case—context (2). The idea is, from knowledge of the differential field in each state X ∈ FK (R n ), the solutions are trajectories X(·) ∈ F (E([0, T ]; R n )), where F (E([0, T ]; R n )) is a fuzzy space of functions (see Section 2). Our purpose is to use Zadeh’s extension principle to define the derivative for fuzzy functions and use it in dynamical systems. This approach is close to case (ii) in the sense that deriving the solution of system (1) corresponds to deriving each classical trajectory of its -levels, as it will be clear. On the other hand, it is also related to case (i), since we define a derivative for fuzzy functions. Hence, the study made here treats, in fact, fuzzy differential equations (FDEs). As advantages, we have a theory for FDEs in which the solution of system (1) does not have necessarily non-decreasing diameter (in opposite to the case of Hukuhara derivative) and hence it is possible to explore properties of periodicity and stability. In Example 6.3, a fuzzy initial value problem (FIVP) representing a decay model is solved, namely   X(t) = −X(t) D (2) X(0) ∈ FK (R)  , x  ], with x  , x  > 0 for all  ∈ [0, 1]. where  is a positive real number and X(0) is such that [X(0)] = [x 01 02 01 02 A solution X(·) to the problem using the approach suggested in this paper has the -levels   [X(·)] = {x(·) : x(t) = x 0 e−t , x 0 ∈ [x 01 , x 02 ]}

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whose attainable fuzzy sets are  −t  −t   −t e , x 02 e ] = [x 01 , x 02 ]e ,  ∈ [0, 1]. [X(t)] = [x 01

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This is not the only solution, that is, similar to the Generalized Hukuhara derivative, fuzzifying the derivative operator may give us more than one solution to the same FIVP. The solution X(t) to the same problem via Hukuhara derivative is such that [X(t)] = [x 1 (t), x 2 (t)] with  x 1 (t) = c1 et + c2 e−t (3) x 2 (t) = −c1 et + c2 e−t and

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c1 =

 − x  x 01 x  + x 02 02 and c2 = 01 2 2

which has increasing diameter. This is not expected from the modeling of a decay model, for which the classical model produces a solution with attracting point. Our approach offers a solution that presents decreasing diameter, stability and convergence, reflecting the rich behavior of classical ordinary differential equations. This is an example where the solutions of two approaches are different. However, in this paper we show that, under suitable assumptions, our approach reproduces the same solutions of those of Hüllermeier’s [12], that is, via FDIs. Furthermore, Mizukoshi et al. [15] have proved that certain conditions guarantee that the solution obtained from fuzzification of the deterministic solutions (where the FDE presents some fuzzy parameters) coincides with the solution via FDI. Kaleva [3] has shown that, under suitable assumptions, the solutions of FDEs via Hukuhara’s Please cite this article as: L.C. Barros, et al., Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.004

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derivative coincide with those via FDI, and hence with those via Mizukoshi et al. So, if specific conditions hold, all these mentioned approaches produce the same solutions. The paper is organized as follows. In Section 2 some basic definitions, terminology and notations are introduced. Section 3 briefly discusses the main results about Zadeh’s extension and fuzzy functions. Section 4 presents the fuzzy derivative with its principal properties. Section 5 treats FDE, presents a comparison with FDE via fuzzy FDI and establishes an existence theorem for fuzzy initial value problem (FIVP). Section 6 presents FDE where the field is given by Zadeh’s extension and an existence theorem is proposed for this case. The relation between FDE (as in this paper) and FDI [12,14] then becomes clear and some examples are solved. We conclude the paper with some brief comments in Section 7. 2. Preliminaries and fuzzy functions

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In this section we present some basic concepts and results, as well as some essential notations. Let E be a topological space. We will denote by FK (E) the family of fuzzy sets u of E whose -levels  {x ∈ E : u(x) ≥ } if  > 0  [u] = cl{x ∈ E : u(x) > 0} if  = 0 are nonempty compact subsets of E. Recall that if A and B are nonempty subsets of a metric space, the Pompeiu metric (also known as Hausdorff metric) is defined by d H ( A, B) = max{( A, B), ( A, B)}, where

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( A, B) = max dist(a, B). a∈A

It is known that the metric 21 23

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d∞ (u, v) = sup{d H ([u] , [v] ) : 0≤≤1}, makes the space metrizable (FK (E), d∞ ) into complete metric space [16], provided E is a complete space too. The following statements are essential for our purpose. Theorem 2.1 (Stacking Theorem, Negoita and Ralescu [17]). Let {A ⊂ R n : 0≤≤1} be a family of nonempty compact subsets satisfying the following: • A ⊆  A for all 0≤≤≤1; • A = ∞ i=1 A i for any nondecreasing sequence i →  ∈ (0, 1].  • ∈0,1] A = A0 .

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Then there is a fuzzy set u of R n , whose membership is an upper semicontinuous (usc, for short) function such that [u] = A . That is, there is a fuzzy set u ∈ FK (R n ) such that [u] = A .

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In this paper we deal with different types of fuzzy functions. We denote by E([0, T ]; R n ) = {x : [0, T ] → R n } a classical space of functions where its elements x(·) have some properties (e.g., it is continuous). A fuzzy valued function is a function that maps crisp or fuzzy elements into fuzzy subsets. The Hukuhara derivative, for instance, operates on such type of functions, namely X : [0, T ] → F (R n ). We call a fuzzy bunch of functions the fuzzy subset of a classical space of functions [18]. The concept of derivative presented in Section 4 is defined for this last of type of functions, X(·) ∈ F (E([0, T ]; R n )). Initially, let us observe that, for each fuzzy function W (·) ∈ F (E([0, T ]; R n )), we can define the attainable fuzzy sets in time t, W (t), whose -levels are

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[W (t)] = [W ] (t) = {w(t) : w(·) ∈ [W (·)] }. Please cite this article as: L.C. Barros, et al., Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.004

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Under suitable assumptions, it is possible to show that, if W (·) ∈ FK (E([0, T ]; R n )), then [W (t)] satisfy the Stacking Theorem 2.1. So, for each W (·) ∈ FK (E([0, T ]; R n )), we have a fuzzy valued function W : [0, T ] → FK (R n ) that, for each t ∈ [0, T ], associates the attainable fuzzy sets W (t). Other important concepts are continuity and semicontinuity of fuzzy valued functions [19]. Definition 2.2. A fuzzy valued function F :  → FK (R n ),  ∈ R m , is upper semicontinuous (usc) at t0 ∈  if for every  > 0 there exists a  = (t0 , ) > 0 such that ([F(t0 )] , [F(t)] ) <  if t − t0  < , t ∈ , for all  ∈ [0, 1].

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Definition 2.3. A fuzzy valued function F :  → FK (R n ),  ∈ R m , is lower semicontinuous (lsc) at t0 ∈  if for every  > 0 there exists a  = (t0 , ) > 0 such that ([F(t)] , [F(t0 )] ) <  if t − t0  < , t ∈ , for all  ∈ [0, 1].

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We say that F is usc (lsc) if it is usc (lsc) at every t ∈ . If F is usc (lsc), the multifunctions [F] :  → Kn (where Kn is the space of all nonempty compact subsets of R n ) are clearly usc (lsc). The converse implication is not necessarily true, unless [F] are uniformly usc (lsc) in  ∈ [0, 1]. As a result of these definitions, F is d∞ -continuous if and only if it is usc and lsc. 3. Zadeh’s extension Initially, Zadeh’s extension was defined as in Eq. (4) below. However, motivated by applications in neural network field, this formula was generalized to (5) (see [20]). For studies evolving FDE via DIs [12,14] and FDE as we develop it in Section 5, there is a close relation between these two approaches: the differential field, in the first case, maps a crisp element into a fuzzy set while in the second one the field maps a fuzzy set into a fuzzy set. In Section 6 this theme will be further studied. Definition 3.1. Let U and V be two topological spaces and f : U → V a function. For each u ∈ F (U ) we define Zadeh’s extension of f as  f (u) ∈ F (V ) with membership function ⎧ ⎨ sup u (s) if f −1 (y)  ∅ s∈ f −1 (y)  (4) f (u) (y) = ⎩ 0 if f −1 (y) = ∅, for all y ∈ V .

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This expression means that f (x) belongs to  f (u) with the same membership that x belongs to u, provided that f is an injective function. The proof of the following results can be found in [21,22]. Theorem 3.2 (Barros et al. [21], Nguyen[22]). Let f : R n → R n be a function.

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(a) If f is surjective, then a necessary and sufficient condition for [ f (u)] = f ([u] )

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to hold is that sup{u(s) : s ∈ f −1 (x)} be attainable for all x ∈ R n . (b) If f is continuous, then  f : FK (R n ) → FK (R n ) is well-defined and [ f (u)] = f ([u] ) for all  ∈ [0, 1]. Please cite this article as: L.C. Barros, et al., Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.004

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The results of Theorem 3.2 can be generalized from R n to any Hausdorff space (see [23]). Particularly, we are interested in the base space of absolutely continuous functions, AC([0, T ]; R n ), i.e., the fuzzy space F (AC([0, T ]; R n )). A more general definition of Zadeh’s extension can be found in [20].

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Definition 3.3 (Huang and Wu [20]). Let U and V be two topological spaces and f : U → F (V ) a function. For each u ∈ F (U ) we define Zadeh’s extension of f as  f (u) ∈ F (V ) with membership function

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 f (u) (y) = sup { f (x) (y) ∧ u (x)},

(5)

x∈U

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for all y ∈ V .

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ˆ Theorem 3.4  (Huang and Wu [20]). Let f : R → F (R) be a d∞ -continuous function. Then f is d∞ -continuous and [ fˆ(u)] = x∈[u] [ f (x)] .

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Remark 3.5. It is easy to see that if F : FK (R n ) → FK (R n ) is a d∞ -continuous function, then F| R n is usc and lsc and so is the set valued function G : R n → Kn , G(x) = [F| R n (x)] . It is clear that if f : R n → FK (R n ) is d∞ -continuous, then the mapping g(x) = [  f (x)] is also usc and lsc.

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4. Derivative of fuzzy functions

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Before introducing the theme of fuzzy differential equations, we will define the derivative for fuzzy functions. We will use D to represent the operator derivative, i.e., D : AC([0, T ]; R n ) → L ∞ ([0, T ]; R n ) w  Dw = w

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where w is the derivative in the sense of distributions (see [24]). Thus, there exists Dw(t) a.e., in [0, T ].  , Definition 4.1 (Derivative of fuzzy function). Let W ∈ F (AC([0, T ]; R n )). The derivative of W is given by DW whose membership function is ⎧ ⎨ sup W (w) if D −1 (y)  ∅ w∈D −1 (y) (6) D(W  ) (y) = ⎩ 0 if D −1 (y) = ∅,

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 is Zadeh’s extension of operator D. for all y ∈ L ∞ ([0, T ]; R n ). In other words, D

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Example 4.2. Let A be the symmetrical triangular fuzzy number with support [−a, a], a > 0. The fuzzy function W (·) ∈ F (AC([0, T ]; R n )) such that [W (·)] = {w(·) : w(t) = t, ∈ [ A] }

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for each  ∈ [0, 1] has attainable sets W (t) = At.

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To determine the derivative of W using Definition 4.1, we need to explicit the membership function of A and W : ⎧

⎪ + 1 if − a≤ < 0 ⎪ ⎪ ⎨a A ( ) = − + 1 if 0≤ < a ⎪ ⎪ a ⎪ ⎩ 0 otherwise

(7)

Please cite this article as: L.C. Barros, et al., Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.004

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and

⎧

⎪ +1 if w(t) = t with − a≤ < 0 ⎪ ⎪ ⎨a W (w) = − + 1 if w(t) = t with 0≤ < a ⎪ ⎪ a ⎪ ⎩ 0 otherwise

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Formula (6) states that DW  (y)  0 only if there exists w such that Dw = y and W (w)  0. In this example, it happens only if w(t) = t with ∈ [ A] , that is, y = . DW  ( ) = sup W (w) Dw=

=

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sup W ( t)

D( t )=

= W ( t) ⎧

⎪ +1 if − a≤ < 0 ⎪ ⎪ ⎨a = − + 1 if 0≤ < a ⎪ ⎪ a ⎪ ⎩ 0 otherwise = A ( ).

(9)

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 (·) is composed of constant functions such that, at each instant t, the derivative This means that the support of DW of W (·) is always the fuzzy number A.

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Example 4.3. Consider A the same symmetrical triangular fuzzy number of the previous example and the fuzzy function W (·) ∈ F (AC([0, T ]; R n )) such that [W (·)] = {w(·) : w(t) = , ∈ [ A] }

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for each  ∈ [0, 1]. Its attainable sets are W (t) = A.

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The membership function of W is ⎧

⎪ +1 if w(t) = with − a≤ < 0 ⎪ ⎪ ⎨a W (w) = − + 1 if w(t) = with 0≤ < a ⎪ ⎪ a ⎪ ⎩ 0 otherwise

(10)

We have that DW  (y)  0 only if there exists w such that Dw = y and W (w)  0, that is, only if w(t) = with

∈ [ A] . Then y = 0: DW  (0) = sup W (w) Dw=0

= sup W ( ) D =0

= sup W ( )

(11)



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= 1.  (·), while any other function has membership degree zero So, the function y = 0 has membership degree one to DW to this fuzzy subset. In other words, the derivative of the constant function of this example is the crisp number zero. Please cite this article as: L.C. Barros, et al., Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.004

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 ) with the same membership that w + k belongs to W, for some Formula (6) establishes that w belongs to D(W  ] = D([W ] ) is k ∈ R n . Since the operator D is not continuous for uniform norm (sup norm) [25], the equality [ DW not immediate.  ] = D([W ] ) is true. We will proceed by investigating sufficient conditions for which [ DW

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Lemma 4.4. For D defined as above, D −1 (g) is a closed nonempty subset of AC([0, T ]; R n ) with respect to the uniform norm for each g ∈ L ∞ ([0, T ]; R n ).

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Proof. The lemma holds because D −1 (g) is an affine subspace of finite dimension (dimension one), the Banach space AC([0, T ]; R n ). Hence, D −1 (g) is closed [25]. 

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Theorem 4.5. Let W ∈ FK (AC([0, T ]; R n )). Then  )] = D([W ] ). [ D(W

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Proof. According to Lemma 4.4, D −1 (g)  ∅ and it is closed. Thus, [W ]0 ∩ D −1 (g) is compact, since it is a closed subset of compact [W ]0 .  )] ⊂ D([W ] ) we will distinguish two cases:  ∈ (0, 1] and later  = 0. For inclusion [ D(W  )] , then For  ∈ (0, 1], let g ∈ [( DW  )(g) = ≤( DW

sup h∈D −1 (g)

W (h) =

sup h∈[W ]0 ∩D −1 (g)

W (h) = W ( f )

for some f , since W is an upper semicontinuous function (i.e., the membership of W is usc) and [W ]0 ∩ D −1 (g) is compact. So, W ( f )≥. That is, f ∈ [W ] ∩ D −1 (g). Hence g ∈ D([W ] ). For  = 0,



 ] = [ DW D([W ] ) ⊆ D([W ]0 ). ∈(0,1]

∈(0,1]

Consequently,

 ]0 = [ DW 21 23

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 ] = [ DW

∈(0,1]



D([W ] ) ⊆ D([W ]0 ) = D([W ]0 ).

∈(0,1]

The last equality holds because D is a closed operator.  )] . Now, the inclusion D([W ] ) ⊂ [ D(W  If g ∈ D([W ] ), there exists f ∈ [W ] such that D(g) = f . Thus,  )(g) = ( DW

sup h∈D −1 (g)

 ] W (h)≥W ( f )≥ ⇒ g ∈ [ DW

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Example 4.6. We will use Theorem 4.5 to find the derivative of the fuzzy function of Example 4.2. For each -level [W (·)] = {w(·) : w(t) = t, ∈ [ A] } we obtain  (·)] = D[W (·)] [ DW = {Dw(·) : w(t) = t, ∈ [ A] } = {y(·) : y(t) = , ∈ [ A] }

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= [ A] .  (·) = A. Thus, we found the same answer of Example 4.2, DW Please cite this article as: L.C. Barros, et al., Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.004

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 (t) coincides with Chang and Zadeh’s definition [26] and with Dubois’ [18]. For It is important to note that DW them, the -levels of the derivative of a fuzzy function W , at a point t, are given by the set of all derivatives of the functions w which belong to the -levels of W , calculated at t. Finding adequate subspaces of AC([0, T ]; R n ) and L ∞ ([0, T ]; R n ) is necessary for investigating qualitative prop and whether D  is well defined or not. erties of D Example 4.7. In the space C 1([0, T ]; R n ), endowed with the norm x1 = sup0≤t ≤T {|x(t)|+|x (t)|} and C 0([0, T ]; R n )  : FK (C 1 ([0, T ]; R n )) → FK (C 0 ([0, T ]; R n )) is well-defined endowed with the usual supremum norm, the operator D 1 n and for each W ∈ FK (C ([0, T ]; R )) we have  )] = D([W ]) [ D(W for all  ∈ [0, 1]. The result follows from Theorem 3.2(b), since D is a continuous function for these spaces.

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Another possibility of D being a continuous operator is as follows. Theorem 4.8. Consider the subset of AC([0, T ]; R n ):

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Z T (R n ) = {x(·) ∈ C 0 ([0, T ]; R n ) : ∃ x  (·) ∈ L ∞ ([0, T ]; R n )}, with Z T (R n ) having the topology of uniform norm and L ∞ ([0, T ]; R n ) with the weak*-topology. Thus,

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 : FK (Z T (R n )) → FK (L ∞ ([0, T ]; R n )), D  is Zadeh’s extension of the derivative D, is well defined, i.e., for each W ∈ FK (Z T (R n )), the -level [ DW  ] where D ∞ n    is a compact subset of FK (L ([0, T ]; R )) and [ DW ] = D[W ] . Proof. The result follows from Theorem 3.2(b) because

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D : Z T (R n ) → L ∞ ([0, T ]; R n ) is a continuous linear operator (see [24, p. 104]). 

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 is a fuzzy function, we will denote by DW  (t) the fuzzy set in R n whose -levels are Since DW  (t)] = D[W ] (t) = {Dw(t) : w(·) ∈ [W (·)] } [ DW

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for each t ∈ [0, T ].

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Remark 4.9. It is not hard to prove that if K is a compact subset of a metrizable space of functions on [0, T ] to R n , then the attainable set K (t) of R n is compact too.

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 (t) the derivative of W : [0, T ] → F (R n ) at a point t. From Remark 4.9, for We call the fuzzy sets DW  (t) is a fuzzy subset of R n , since L ∞ ([0, T ]; R n ) supplied with weak*-topology is metrizable each t ∈ [0, T ], DW (see [14]).

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5. Fuzzy differential equation Consider the fuzzy initial value problem (FIVP)   X(t) = F(t, X(t)) D

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X(0) = X 0

(13)

where F : R × FK (R n ) → FK (R n ) and X 0 ∈ FK (R n ). A solution for (13) is a fuzzy function X(·) ∈ FK (AC([0, T ]; R n )) that satisfies (13) a.e. in [0, T ]. Please cite this article as: L.C. Barros, et al., Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.004

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From Theorem 4.5, in -levels, (13) is equivalent to the family of initial value problems (IVPs)  D[X] (t) = [F(t, X(t))] [X(0)] = [X 0 ]

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D[X] (t) = F[(t, X(t))] ⇔ {Dx(t) : x ∈ [X] } = [F(t, X(t))] . Thus, X(·) ∈ FK (AC([0, T ]; R n )) is a solution for (13), if and only if, the classical trajectories x(·) that compose the -levels [X(·)] are given by the family of solutions of DIs  x  (t) ∈ [F(t, X(t))] (15) x(0) ∈ [X 0 ] and [F(t, X(t))] ⊂ D[X] (t).

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Remark 5.1.  (a) According to the approach of [12,14,27], the solutions of (1) are constructed from a family ( ) of classical  trajectories x : [0, T ] → R n , i.e., X(t) = {x(t)|x(·) ∈ }. These trajectories are solutions of the DIs  x  (t) ∈ [F(t, x(t))] (17) x(0) ∈ [X 0 ] , where F : R × R n → FK (R n ) and its -levels are connected, for each  ∈ [0, 1]. The relations between (15) and (17) will be useful to demonstrate an existencetheorem for (13). (b) For this theorem, we will suppose that the field has the property [F(t, X)] = x∈[X ] [F(t, x)] , ∀ ∈ [0, 1]. The idea is to show that, under certain conditions, the solution X(·) of (17) satisfies (13) too. For this purpose, we have  to prove that if z ∈ [F(s, X(s))] = x(s)∈[X (s)] [F(s, x(s))] or z ∈ [F(s, x(s))] , for some x(s) ∈ [X(s)] , there exists x(·) ∈ [X(·)] such that x  (s) = z. Boundedness assumption Diamond [14] and Aubin and Cellina [24]. Let  be an open subset of R × R n such that (0, x 0 ) ∈  and H a mapping from  into the compact and convex subsets of R n . The boundedness assumption is said to hold if there exist b, T, M > 0 such that: • the set Q = [0, T ] × (x 0 + (b + M T )B n ) ⊂ , where B n is the unit ball of R n ; • H maps Q into the ball of radius M. Theorem 5.2. Let X 0 ∈ FK (R n ) and  be an open set in R × R n containing {0} × supp X 0 and F : R × n  FK (R n ) → F K (R ) a fuzzy function such that F(t, x) = F| is continuous with [F(t, x)] compact, convex and [F(t, X)] = x∈[X ] [F(t, x)]. Also, suppose that the boundedness assumption holds. Then, there exists a solution X(·) ∈ FK (Z T (R n )) for problem (13). Moreover, [X(t)] are compact and connected in R n , for all  ∈ [0, 1]. Proof. The idea is to show that (15) has a solution X(·) and then to demonstrate that X(·) satisfies (16). Under weaker conditions than those stated for the theorem (upper semicontinuity instead of continuity), [14] proves that Problem (17) has a solution X(·) ∈ FK (Z T (R n )). Now, (15) holds, since for all x(·) ∈ [X(·)] and a.e. t ∈ [0, T ], we have x  (t) ∈ [F(t, x(t))] ⊂ F(t, [X(t)] ).

35

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for all  ∈ [0, 1]. Now,

5 7

9

(18)

To prove (16) we have to show that if z ∈ [F(s, x)] , with x ∈ [X(s)] , then there exists x(·) ∈ [X(·)] such that = z (Remark 5.1(b)). In order to do this, let y(·) ∈ [X(·)] such that y(s) = x. Since F is lsc, we meet the

x  (s)

Please cite this article as: L.C. Barros, et al., Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.004

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hypothesis of Michael’s Selection Theorem and Corollary 1 [24, pp. 82 and 83] and hence there exists a continuous selection z(t) of F(t, y(t)), t ∈ [0, T ] and z(s) = z. Finally, consider the curve

t

x(t) = x 0 +

z() d, 0

5 7

for some x 0 ∈ [X 0 ] . Of course that x(·) ∈ [X(·)] , since x  (t) = z(t) for all t ∈ [0, T ] and in particular x  (s) = z(s) = z. Hence, [F(t, x(t))] ⊂ D[X] (t).

9

(19)

From (18) and (19), we have that the theorem holds. The proof that [X(t)] is compact and connected in R n , for all  ∈ [0, 1], is in [24, Theorem 2, p. 106]. 

11

Corollary 5.3. Considering R instead of R n in Theorem 5.2, the solution of problem (15) exists and for each  ∈ [0, 1], the -level of the solution, [X(t)] , is a closed and bounded interval.

13

Proof. The proof is immediate since the compact and connected subsets of R are the closed and finite intervals. 

15

The next section is devoted to the particular case where the differential field F of the FDE is given by Zadeh’s extension. Although it is a restricted case, it includes a large number of applications. 6. Field given by Zadeh’s extension

17 19 21

Let us now consider the FIVP (13) with F given by Zadeh’s extension of a d∞ -continuous function f : R × R n → FK (R n ). Kaleva [3] and Mizukoshi [15] have studied Zadeh’s extension with the purpose of comparing FDEs via Hukuhara derivative with FDIs. Kaleva used the particular case where the field is given by the extension of a function of kind f : R n → R n . We use the more general case where f : R n → F (R n ), which includes more applications than f : Rn → Rn .

23

Corollary 6.1. Let f : R × R → FC (R) be d∞ -continuous, where FC (R) is the space of compact and convex fuzzy subsets of R. Then the FIVP (13), with differential field  f , has a solution.

25

f (t, X)] = Proof. From Theorem 3.4,  f is d∞ -continuous and [ f (t, x)] ⊂ [  of Theorem 5.2 are satisfied and therefore the FIVP (13) has a solution. 

27

f and X 0 ∈ FK (R); if X(·) is a solution When f : R × R → R is continuous, F : R × FK (R) → FK (R) with F =  of (13), then [X(t)] = [x 1 (t), x 2 (t)] is a closed and finite interval (according to Corollary 5.3). Therefore, the solution by DIs is a solution to the approach by fuzzifying the derivative operator. In other words, the fuzzy solution constructed from (17) is a solution to (13). We will proceed by illustrating our study by means of some examples.

29 31



x∈[X ] [ f (t, x)]

 . Thus the conditions

Example 6.2 (Kaleva [3]). Consider the FIVP  33

x  (t) = x(t)2 X 0 ∈ F (R)

(20)

where X 0 is a triangular fuzzy number with the -levels [X 0 ] = [1 + , 3 − ]. Please cite this article as: L.C. Barros, et al., Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.004

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)

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11

The solution from the method of DI (20) has the -levels [X(·)] = {x(·) : x  (t) = (x(t))2 , x 0 ∈ [1 + , 3 − ]} 

x0 , x 0 ∈ [1 + , 3 − ] . = x(·) : x(t) = 1 − x0t

3

5

7

The attainable fuzzy sets, X(t), are given by the -levels

 x0  , x 0 ∈ [1 + ,  − 3] [X(t)] = 1 − x0t   3− 1+ , = , 1 − (1 + )t 1 − (3 − )t

(21)

because the function x 0 /(1 − x 0 t) is continuous with respect to x 0 , if 0 ≤ t < 1/3. Kaleva [3] showed that the solution of (20) via Hukuhara derivative is the same as (21). To check that X(·) is also a solution to   X(t) = X(t)2 D X 0 ∈ F (R)

(22)

we use Theorem 4.5:  X(·)] = D[X(·)] [D

= D x(·) : x(t) =

x0 , x 0 ∈ [1 + , 3 − ] 1 − x0t



= {Dx(·) : Dx(t) = x(t)2 , x 0 ∈ [1 + , 3 − ]} 

x0 2 , x 0 ∈ [1 + , 3 − ] = x(·) : x(t) = 1 − x0 t = [X(·)2 ] . 9

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 X(t) = X(t). Also, Hence at time t, D

 x0  , x 0 ∈ [1 + , 3 − ] [X(0)] = x(0) : x(t) = 1 − x0 t = {x(0) : x(0) = x 0 , x 0 ∈ [1 + , 3 − ]} = {x 0 , x 0 ∈ [1 + , 3 − ]} = [X 0 ] .

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11 13

15

Therefore X(·) is a solution via DIs, via fuzzification of the classical derivative operator and, according to [15], via fuzzification (with respect to the initial condition) of the solution to the classical problem. Furthermore, its attainable sets are solution to the FIVP (20) via Hukuhara derivative. Example 6.3. Consider the FIVP   X(t) = −X(t) D X(0) ∈ FK (R)

17

(25)

with  > 0. Please cite this article as: L.C. Barros, et al., Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.004

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Since F(X) = −X is Zadeh’s extension of f (x) = −x, the solution via DIs is a solution to (25). It suffices, therefore, to solve  x  (t) ∈ −x(t)  , x  ]. x(0) ∈ [x 01 02

As solution, for each -level we obtain   [X(·)] = {x(·) : x  (t) = −x(t), x 0 ∈ [x 01 , x 02 ]}   = {x(·) : x(t) = x 0 e−t , x 0 ∈ [x 01 , x 02 ]}

5

with attainable sets  −t  −t   −t [X(t)] = [x 01 e , x 02 e ] = [x 01 , x 02 ]e ,  ∈ [0, 1].

7

To verify that X(·) is indeed a solution to (25), see that  X(·)] = D[X(t)] [D   = {Dx(t) : x(t) = x 0 e−t , x 0 ∈ [x 01 , x 02 ]}   = {−x 0 e−t , x 0 ∈ [x 01 , x 02 ]}   = {−x(t) : x(t) = x 0 e−t , x 0 ∈ [x 01 , x 02 ]}

= [−X(t)] , 9

11

for every  ∈ [0, 1]. Clearly, the attainable sets [X(t)] ,  ∈ [0, 1] have decreasing diameter, in contrast to the Hukuhara derivative, which always produces solutions with non-decreasing diameter. Example 6.4. Consider the FIVP   X(t) = −X(t) D x(0) = x 0

13 15

with  ∈ FK (R). Suppose also that [] = [1 , 2 ] and 1 > 0, for all  ∈ [0, 1]. We recall that F(X) = −X =  f (X), where  f is Zadeh’s extension of f (x) = −x (see [20]). So, according to Corollary 5.3, (26) has a solution given by the solution of  x  (t) ∈ [−x(t)] = [−2 , −1 ]x(t) x(0) = x 0 ∈ R.

17

19

(26)

(27)

Now, G  (x) = [−2 , −1 ]x is a typical parametrized set valued function (see [24]) and the solution of (27) has attainable set given by  x  (t) = min{ x(t), ∈ [−2 , −1 ]}, x(0) = x 0 x  (t) = max{ x(t), ∈ [−2 , −1 ]}, x(0) = x 0 .

Thus, the -levels of the attainable sets of the solution of (26) are the intervals 21





[X(t)] = x 0 [e−2 t , e−1 t ] for each  ∈ [0, 1]. Please cite this article as: L.C. Barros, et al., Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.004

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3

(, )

9

(, )

1

(, )

= 2 + (1 − )1 and 2

= 1 + (1 − )2

(, )

where (, )

(, ) c1

 = =

(, )

15

17

2

 x 0 and

(, ) c2

=

(, )

1

(, )

/2 2

21 23 25

+1

x0 . (, )

given by (29). To The solution X(·) is such that [X(·)] = {x(·) : x(·) = x i (·); i = 1, 2; ∈ [0, 1]}, with x i check that X(·) is really a solution, see that the set of the derivatives of all functions x(·) equals the set of functions multiplied by [− ] , for each  ∈ [0, 1], at every t ∈ [0, T ]. Actually, system (28) is based on the solution via Hukuhara. As a consequence, the attainable sets of [X(·)] are the -levels of the solution obtained using Hukuhara derivative, namely [X(t)] = [x 1 (t), x 2 (t)] with ⎧ √   √  ⎨ x  (t) = c e 1 2 t + c e− 1 2 t 1 1 2   √ √ (30)  ⎩ x  (t) =  / −c e 1 2 t + c e− 1 2 t 1 2 2 1 2 where

19

(29)

(, )

1 /2 ,  (, ) (, ) 1 /2 − 1

(, )

13

13

with ∈ [0, 1], that is, 1 and 2 are the affine combinations of the endpoints 1 and 2 . Solving (28) gives us ⎧   (, ) (, ) (, ) (, ) ⎪ ⎪ ⎨ x 1(, ) (t) = c1(, ) e 1 2 t + c2(, ) e− 1 2 t     (, ) (, ) (, ) − (, ) ⎪ (, ) (, ) t (, ) t (  ,

) 1 2 1 2 ⎪ + c2 e −c1 e ⎩ x 2 (t) = k

k

11

–

such that

(, )

7

)

Another solution: At the same time, it is not hard to verify that X(·) below is also a solution to (26). To construct the solution, the idea is to solve the following system: ⎧ (, )  (, ) (, ) (x 1 ) (t) = −2 x 2 (t) ⎪ ⎪ ⎪ ⎪ ⎨ (, )  (, ) (, ) (x 2 ) (t) = −1 x 1 (t) (28)  (0) = x ⎪ ⎪ x 0 ⎪ 1 ⎪ ⎩ x 2 (0) = x 0 for all ∈ [0, 1] and i

5

(

c1

      1 /2 − 1 1 /2 + 1  x 0 and c2 = x0. = 2 2

As happens with the Generalized Hukuhara derivative, there is more than one answer to the same FIVP. Actually, the attainable sets of both solutions found are solutions to the correspondent FIVP via Generalized Hukuhara derivative, the first one with decreasing diameter and the second one with increasing diameter. From the modeling of a decay phenomenon, it is not expected that the uncertainty grows, but that the amount of the substance tends (certainly) to zero. In the point of view of modeling, this is one reason why the Hukuhara derivative is considered defective and the first solution of this example is considered desirable. 7. Final comments

27 29

This paper has introduced a new approach to FDEs by using Zadeh’s extension applied to standard derivative operators. We have also shown that this approach is close to the FDIs introduced by Hüllermeier [12] and studied by Diamond [14,27], in the sense that both make use of the derivative of classical functions. The main difference is that Please cite this article as: L.C. Barros, et al., Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.004

FSS 6295 14

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we use this derivative to define a fuzzy derivative and hence we have a theory for FDEs. Based on the theory of DIs, we were able to prove an existence theorem for solutions of FIVPs (Theorem 5.2). By determining the attainable sets, the approach presented can also be related to the Generalized Hukuhara derivative, which allows us to find more than one solution to the same FIVP, as shown in Example 6.4. In contrast to traditional FDEs (via Hukuhara derivative), the rich behavior of crisp DEs can be reproduced in our FDEs, since, under certain hypothesis, the solution via Hüllermeier’s approach is one solution to the FDE via fuzzification of the derivative. For instance, Examples 6.3 and 6.4 presented convergent functions, suggesting a future study on stability, attraction and periodicity of solutions of FDEs using the approach of this paper. Acknowledgments The authors wish to thank Professor Luiz Antonio Barrera San Martin for his collaboration and they would also like to thank CNPq for financial help (Grants 306872/2009-9 and 140798/2010-2). They deeply appreciate the valuable comments and suggestions from anonymous reviewers. References

11 13 15

[1] [2] [3] [4] [5] [6]

17 [7]

19 [8]

21 [9]

23 25

[10] [11] [12]

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[13] [14]

31

[15]

33

[16] [17] [18] [19] [20] [21]

35 37 39 41 43 45

[22] [23] [24] [25] [26] [27]

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Please cite this article as: L.C. Barros, et al., Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.03.004