The Cauchy problem for complex fuzzy differential equations

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ScienceDirect Fuzzy Sets and Systems 245 (2014) 18–29 www.elsevier.com/locate/fss

The Cauchy problem for complex fuzzy differential equations Daria Karpenko a , Robert A. Van Gorder b,∗ , Abraham Kandel c a Department of Mathematics, University of South Florida, United States b Department of Mathematics, University of Central Florida, United States c Department of Computer Science, University of South Florida, United States

Received 17 January 2013; received in revised form 20 June 2013; accepted 1 November 2013 Available online 8 November 2013

Abstract We discuss the existence of a solution to the Cauchy problem for fuzzy differential equations that accommodates the notion of fuzzy sets defined by complex-valued membership functions. We first propose definitions of complex fuzzy sets and discuss entailed results which parallel those of regular fuzzy numbers. We then give two existence results relevant to the Cauchy problem for fuzzy differential equations in the case of bounded integral operators. These results require either Hölder continuous or Lipschitz continuous response functions. © 2013 Elsevier B.V. All rights reserved. Keywords: Complex-valued grades of membership; Complex fuzzy differential equations; Cauchy problem; Existence theorem; Uniqueness theorem

1. Introduction The concept of complex fuzzy sets as sets with complex membership functions was first introduced by Ramot et al., who in [22] demonstrated the increased expressive power gained by endowing a set S with a complex membership function μS (x) = rS (x) · eiωS (x) , where rS (x) and ωS (x) are real-valued functions with rS solely responsible for the fuzzy information and ωS functioning as a phase term containing additional crisp information. In [25], Tamir et al. pointed out the limitations of the mixed fuzzy and crisp definition of [22] and generalized it by allowing a fuzzy phase term. As illustrated with examples in [25], the advantage of this augmented definition of complex fuzzy sets is its ability to accommodate fuzzy cycles. Since then, of note is [26], where Tamir and Kandel propose an axiomatic framework for complex fuzzy logic and demonstrate its application to complex economic systems. We draw the readers attention to the difference between complex fuzzy sets and fuzzy complex numbers (a distinct concept); compare Buckley [5] (fuzzy complex numbers) and Tamir and Kandel [25] (complex fuzzy sets). In this paper we provide a way of incorporating such complex fuzzy sets (henceforth referred to as complex fuzzy sets so as to avoid any confusion with regular fuzzy complex numbers) into the theory of fuzzy differential equations. * Corresponding author. Tel.: +1 4078236284; fax: +1 4078236253.

E-mail address: [email protected] (R.A. Van Gorder). 0165-0114/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.11.001

D. Karpenko et al. / Fuzzy Sets and Systems 245 (2014) 18–29

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This is accomplished by extending certain general results for metric spaces of fuzzy sets (see [8]) to spaces of complex fuzzy sets and using these to propose two Cauchy solution existence theorems for complex fuzzy real numbers. The classical Peano theorem for nonfuzzy Rn states that the first order Cauchy problem   x  (t) = f t, x(t) , x(t0 ) = x0 (1.1) has a solution if f is continuous. Much effort has been expended toward establishing Peano-like theorems for fuzzy differential equations and the Cauchy problem for fuzzy differential equations has been studied extensively: see [12] for results in differentiability and integrability properties and [13,18,20,24] for methods and results concerning Peanolike theorems for fuzzy differential equations. Kaleva in [13] concluded that, for fuzzy differential equations, just as for the classical case, continuity is a sufficient condition for the existence of a solution to the Cauchy problem; this conclusion, however, was disputed by Friedman et al. in [10]. Later, in Nieto [18], continuity and boundedness were claimed to be sufficient conditions for the existence of a solution to (1.1). However, Choudary and Donchev [7] showed that there was an error in the method of proof employed. In particular, they showed that the space of integral operators vital to the existence result of Nieto [18] is uniformly bounded, but not totally bounded. The existence proof of Nieto [18] relied on the total boundedness of this space. This is not to say that the claimed result is incorrect, only that the proof does not seem sufficient. Indeed, the result may very likely be correct, there is just no solid proof. Picard–Lindelöf type theorems for the Cauchy problem (1.1) were presented by Wu and Song [28]. For classical results about the spaces of fuzzy numbers in general, the reader is referred to [8]. In the other direction, for the generalization of the Cauchy problem for fuzzy differential equations to generalized metric spaces, Ref. [19] should be of interest. In the following section we review the basic results and definitions from the theory of fuzzy differential equations and provide their corresponding extensions to complex fuzzy sets. The polar representation of the complex membership function is considered separately from the Cartesian representation due to its special periodic properties. We conclude with the presentation of two existence results for the Cauchy problem on the domain of complex fuzzy numbers. The first of these is similar in form to the result of Nieto [18]. This theorem is for f appropriately Hölder continuous in x. With this assumption, we bypass the need for total boundedness of the space of operators considered by Nieto; rather, uniform boundedness is sufficient. The second theorem is a Picard–Lindelöf type result, which requires Lipschitz continuity of f in x. 2. Background Let PK (Rn ) denote the family of all nonempty convex compact spaces of Rn . The Hausdorff metric for A, B ∈ PK (Rn ) is defined as    d(A, B) = inf ε  A ⊂ N (B, ε) and B ⊂ N (A, e) , (2.1) where N(A, ε) = {x ∈ Rn | x − y < ε for some y ∈ A}. We consider complex fuzzy sets on Rn , i.e. complex membership functions referred to as “pure complex fuzzy” in [25] (an expansion of the original definition in [22]), where Tamir et al. define a Cartesian and a polar representation of complex grades of membership. We consider the Cartesian definition first. 2.1. Cartesian representation of complex grades of membership The complex membership function in [25], μ, is defined as μ(V , z) = μR (V ) + iμI (z), where V is to be interpreted as a set in a fuzzy set of sets and z as an element of V . This definition can be easily extended to Rn : For x ∈ Rn , let f (x) = u(x) + iv(x), where u, v : Rn → [0, 1]. For ease of notation, denote f by (u, v). Thus, f assigns to each x ∈ Rn a value in the unit square in C, representing a complex grade of membership. Note that u, v considered individually define non-complex fuzzy sets in Rn .

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Now, for u : Rn → [0, 1], α-level sets are classically defined as follows:    [u]α = x ∈ Rn  u(x)  α, α ∈ (0, 1] ,    [u]0 = x ∈ Rn  u(x) > 0 . We use the above to define (α, β)-level sets for f = (u, v): [f ](α,β) = [u]α ∩ [v]β .

(2.2)

That is, we take [f ](α,β) to be the set of all x ∈ Rn satisfying simultaneously u(x)  α and v(x)  β when 0 < α, β  1 and consider the corresponding closures in the case of α or β equal to 0. Consider the following set of conditions as an alternative definition of [f ](α,β) :    [f ](α,β) = x ∈ Rn  u(x)  α, v(x)  β, 0 < α, β  1 , (2.3)    (2.4) [f ](α,0) = x ∈ Rn  u(x)  α, v(x) > 0, 0 < α  1 ,    (0,β) [f ] (2.5) = x ∈ Rn  u(x) > 0, v(x)  β, 0 < β  1 ,    [f ](0,0) = x ∈ Rn  u(x) > 0, v(x) > 0 . (2.6) Observe that (2.3) and (2.6) are equivalent to definition (2.2) for the corresponding α, β, but that (2.4) and (2.5) are not: (2.2) may not yield closed sets in the case when exactly one of α, β is equal to 0, but (2.4) and (2.5) would yield the respective closures of those sets. We shall revisit this point later. Proceeding with the essential definitions, take E n to be the set of all w : Rn → [0, 1] satisfying all of the following conditions: (i) w is normal, i.e. there exists x0 ∈ Rn such that w(x0 ) = 1; (ii) w is fuzzy convex, i.e. for all x1 , x2 ∈ Rn , λ ∈ [0, 1]:     u λx1 + (1 − λ)x2  min u(x1 ), u(x2 ) ; (iii) w is upper semi-continuous; (iv) [w]0 is compact. Condition (i) defines [w]1 to be nonempty. Since [w]α ⊂ [w]β whenever β  α, we have [w]1 ⊂ [w]α for all α, so [w]α is always nonempty. Condition (ii) yields convexity of [w]α for all α. Condition (iii), by definition of upper semi-continuity, guarantees that [w]α is closed for all α. And since [w]α ⊂ [w]0 for all α, condition (iv) guarantees that [w]α is always bounded. Thus, conditions (iii) and (iv) establish compactness for all [w]α . It follows that for all w ∈ E n and α ∈ [0, 1], we have [w]α ∈ PK (Rn ) (see [8], 6.1 for rigorous proofs). The above guarantees that [u]α ∩ [v]β is always compact and convex for u, v ∈ E n . Observe further that [u]1 ∩ 1 [v] ⊂ [u]α ∩ [v]β ⊂ [u]0 ∩ [v]0 for all α, β ∈ [0, 1]. Therefore in order to ensure that [u]α ∩ [v]β be nonempty, it is sufficient that [u]1 ∩ [v]1 be nonempty, meaning that there should exist some x0 ∈ Rn such that x0 ∈ [u]1 , i.e., u(x0 ) = 1, and x0 ∈ [v]1 , i.e. v(x0 ) = 1. With that in mind, we define the following set:    Eˆ 2n = (u, v) ∈ E n × E n  ∃x0 ∈ Rn s.t. u(x0 ) = v(x0 ) = 1 . (2.7) Then for f = (u, v) ∈ Eˆ 2n , [f ](α,β) = [u]α ∩ [v]β ∈ PK (Rn ) for all α, β ∈ [0, 1]. We have used definition (2.2) for [f ](α,β) here; however, observe that for f = (u, v) ∈ Eˆ 2n the compactness of the [f ](α,β) sets guarantees the complete equivalence of definition (2.2) and the set of definitions (2.3)–(2.6). We now use the well known fact that E n is closed under addition and scalar multiplication (see [8], 6.1) to establish a similar result for Eˆ 2n . For functions u, v ∈ E n , addition and scalar multiplication can be defined via level sets as follows:    [u + v]α = [u]α + [v]α = x + y  x ∈ [u]α , y ∈ [v]α ,    [cu]α = c[u]α = cx  x ∈ [u]α .

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For f, g ∈ Eˆ 2n , where f = (uf , vf ) and g = (ug , vg ), and c is a scalar, let f + g = (uf + ug , vf + vg ), cf = (cuf , cvf ). From the closure of E n under the respective operations, it is clear that cf, f + g ∈ E n × E n . We need to show that there exists x0 ∈ Rn such that (uf + ug )(x0 ) = (vf + vg )(x0 ) = 1, i.e., that [f + g](1,1) is nonempty, and also that [cf ](1,1) is likewise nonempty. We know that there exist xf , xg ∈ Rn such that uf (xf ) = vf (xf ) = 1 and ug (xg ) = vg (xg ) = 1. Then [cf ](1,1) = [cuf ]1 ∩ [cvf ]1       = cx  x ∈ [uf ]1 ∩ cx  x ∈ [vf ]1       = cx  uf (x) = 1 ∩ cx  vf (x) = 1 . Clearly cxf ∈ [cf ](1,1) . Also, [f + g](1,1) = [uf + ug ]1 ∩ [ug + vg ]1     = [uf ]1 + [ug ]1 ∩ [vf ]1 + [vg ]1       = x + y  x ∈ [uf ]1 , y ∈ [ug ]1 ∩ x + y  x ∈ [vf ]1 , y ∈ [vg ]1       = x + y  uf (x) = ug (y) = 1 ∩ x + y  vf (x) = vg (y) = 1 . From the above one can see that uf (xf ) = ug (xg ) = 1 and vf (xf ) = vg (xg ) = 1. Thus, x0 = xf + xg ∈ [f + g](1,1) as required. We have thus shown that Eˆ 2n is closed under addition and scalar multiplication. As in [18], let us define D : E n × E n → [0, ∞) by     D(u1 , u2 ) = sup d [u1 ]α , [u2 ]α  α ∈ [0, 1] , (2.8) where d is the Hausdorff metric (refer to (2.1)). D defines a linear metric on E n in the sense that for all u, v ∈ E n , λ ∈ R, D(u + w, v + w) = D(u, v)

and D(λu, λv) = |λ|D(u, v).

(E n , D) is a complete metric space which can be embedded isomorphically as a cone in a Banach space [18]. However,

D is not a suitable metric for our space of interest, Eˆ 2n , as we quickly see that linearity is violated. Instead, let us consider the product metric on E 2n = E n × E n , D  . For f1 = (u1 , v1 ) ∈ E 2n and f2 = (u2 , v2 ) ∈ E 2n , we define D  : E 2n × E 2n → [0, ∞) by the relation     D  (f1 , f2 ) = D  (u1 , v1 ), (u2 , v2 ) = max D(u1 , u2 ), D(v1 , v2 ) . (2.9) Then, D  is a linearity preserving metric for E 2n . Since Eˆ 2n ⊂ E 2n , D  is also a metric for Eˆ 2n . Hence, (Eˆ 2n , D  ) is a complete metric space. Now, as (Eˆ 2n , D  ) is a metric space and D  preserves linearity, by the Arens–Eells theorem [1] there exists an embedding Eˆ 2n → B where B is a Banach space. The existence of such an embedding shall prove useful in Section 3, where we explore the Cauchy problem for functions X : R → Eˆ 2n . (For further information about embeddings between compact-convex and level-continuous fuzzy sets and a Banach space of real continuous functions, see [23].) ˆ It will also prove useful to define a zero element in Eˆ 2n . Recall from [18] that on E n we define 0ˆ ∈ E n by 0(x) =1 2n 2n ˆ ˆ ˆ ˆ when x = 0 and 0(x) = 0 otherwise. The zero element on E then reads 02 (x) = (0(x), 0(x)) ∈ E . We have 0ˆ 2 (0) = (1, 1), verifying that 0ˆ 2 ∈ Eˆ 2n . More generally for p ∈ N, if we define    p (2.10) Eˆ pn = (u1 , . . . , up ) ∈ ×i=1 E n  ∃x0 ∈ Rn s.t. u1 (x0 ) = · · · = up (x0 ) = 1 , p p and, for f1 = (f11 , . . . , f1 ), f2 = (f21 , . . . , f2 ) ∈ Eˆ pn , let   p p    D  (f1 , f2 ) = max D f11 , f21 , . . . , D f1 , f2 ,

(2.11)

we can conclude that there exists an embedding Eˆ pn → B where B is a Banach space. We use this fact in the following discussion of the polar case.

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2.2. Polar representation of complex grades of membership The polar representation of the membership function as presented in [25], μ(V , z) = r(V )eiσ φ(z) , where σ is a scaling factor, does not translate directly to and from the respective Cartesian representation. Therefore the two representations of the corresponding extension to Rn are not equivalent as defined, which will be seen below (recall that the Cartesian function maps into the unit square [0, 1]2 ). Thus, depending on the application, one may be more appropriate to use than the other. For x ∈ Rn , the polar form of f is defined as follows: f (x) = r(x)e2πφ(x)i , where r, φ : Rn → [0, 1], and we denote f by (r, φ). The scaling factor is taken to be 2π , allowing the range of f to be the entire unit circle. Because e2πiφ is periodic, we take the value of φ giving the maximum distance from e0 , φ = 0.5, to be the “maximum” membership value. Now, while [r]α can be defined just as [u]α above, the corresponding level sets for φ, denoted [φ]β , must be defined differently to account for the periodicity:   [φ]β = x ∈ Rn : φ(x) ∈ [β, 1 − β], β ∈ (0, 0.5] ,   [φ]0 = x ∈ Rn : 0 < φ(x) < 1 , [φ]β = [φ]1−β ,

for all β ∈ [0, 1].

We can then define the level sets [f ]α,β as [f ]α,β = [r]α ∩ [φ]β ,

(2.12)

or by the relations   [f ]α,β = x ∈ Rn : r(x)  α, φ(x) ∈ [β, 1 − β] ,   [f ]α,0 = x ∈ Rn : r(x)  α > 0, 0 < φ(x) < 1 ,   [f ]0,β = x ∈ Rn : r(x) > 0, φ(x) ∈ [β, 1 − β], β ∈ (0, 0.5] ,   [f ]0,0 = x ∈ Rn : r(x) > 0, 0 < φ(x) < 1 ,

(2.13) (2.14) (2.15) (2.16)

together with [f ]α,β = [f ]α,1−β ,

for all α, β ∈ [0, 1]

(2.17)

(cf. the two definitions of [f ](α,β) , (2.2) and (2.3)–(2.6)). Given this polar definition of complex fuzzy membership, Eˆ 2n is no longer a suitable set of functions to consider. It is clear that for φ ∈ E n , [φ]β ⊂ [φ]β for all β ∈ [0, 0.5]. However, [φ]β need not be compact or convex. In order to address this issue, we define F n to be the set of all w : Rn → [0, 1] satisfying all of the following conditions: (i) there exists x0 ∈ Rn such that w(x0 ) = 0.5; (ii) w is monotone; (iii) w is upper semi-continuous on K1 and lower semi-continuous on K2 where    K1 = x ∈ Rn  0 < w(x)  0.5 ,    K2 = x ∈ Rn  1 > w(x)  0.5 ; (iv) K1 ∪ K2 is compact.

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Now, take    Eˆ ∗2n = (r, φ) ∈ E n × F n  ∃x0 ∈ Rn such that r(x0 ) = 1 and φ(x0 ) = 0.5

(2.18)

(cf. the definition of Eˆ 2n given in (2.7)). Observe now that for f = (r, φ) ∈ Eˆ ∗2n , definition (2.12) is equivalent to the set of definitions (2.13)–(2.17). We claim that Eˆ ∗2n is embeddable into a Banach space. To see this, notice that for w ∈ F n , we may write  w− (x), x ∈ K1 , w(x) = (2.19) w+ (x), x ∈ K2 , where for some z− , z+ ∈ E n 1 w− = z− 2

and

1 w+ = (2 − z+ ). 2 ι

ι

Thus, there exists an embedding ι such that F n → E n × E n by w → (z− , z+ ), which implies there exists an embedding id,ι

E n × F n → E n × E n × E n ,

(2.20)

where id is the canonical identity map. Now, if w(x0 ) = 0.5, we can choose z− , z+ so that z− (x0 ) = z+ (x0 ) = 1, hence Eˆ ∗2n → Eˆ 3n .

(2.21)

Since, as shown in the previous section, Eˆ 3n is embeddable into a Banach space, then so is Eˆ ∗2n . The following results, therefore, apply equally to the space Eˆ 2n in the Cartesian case and to the space Eˆ ∗2n in the polar case. 3. The Cauchy problem for complex fuzzy differential equations In this section, we will concern ourselves with the question of the existence of a solution to a Cauchy problem for complex fuzzy differential equations. For brevity, we shall let E = Eˆ 2n when dealing with the Cartesian complex form, and E = Eˆ ∗2n when dealing with the polar complex form. The analysis for each case will be symbolically identical, since for each case there exists a Banach space B such that E → B, hence we shall simply refer to E for the remainder of the section. By complex fuzzy differential equation, we mean a differential equation for which the solution X is a continuous map X : I → E . The analysis shall proceed in a more or less similar fashion to Nieto [18]. We define differentiability as in [12] in terms of the Hukuhara difference. For x, y ∈ E, if there exists z ∈ E such that x + z = y, we write x − y = z and call z the difference of x and y. Let I = [a, b] ⊂ R be a compact interval. We call a mapping F : I → E differentiable at t0 ∈ I if there exists some F  (t0 ) ∈ E such that the following limits exist and are equal to F  (t0 ): lim

h→0+

F (t0 + h) − F (t0 ) h

and

lim

h→0+

F (t0 ) − F (t0 − h) . h

Henceforth, by differentiation we shall implicitly mean differentiability in the Hukuhara sense. Again, take I = [a, b] ⊂ R to be a compact interval, and let F : I → E be a continuous mapping. We then define G : I → E by t G(t) =

F (τ ) dτ,

for all t ∈ I.

(3.1)

a

Note that d G(t) = G (t) = F (t), dt

for all t ∈ I.

(3.2)

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We then have the mean value theorem       D  F (b), F (a)  (b − a) sup D  F  (t), 0ˆ 2  t ∈ I ,

(3.3)

or, in a more useful form,       D  G(b), G(a)  (b − a) sup D  F (t), 0ˆ 2  t ∈ I .

(3.4)

D

This is in complete analogy to [18], where we have in place of D. Consider the mapping H : I × E → E which we take to be continuous in both arguments. We then define the Cauchy problem   X  (t) = H t, X(t) , t ∈ I, X(a) = xa . (3.5) Then, we say that X : I → E is a solution to the Cauchy problem (3.5) if and only if X is continuous and X satisfies the integral equation t X(t) = xa +

  H τ, X(τ ) dτ.

(3.6)

a

This is completely analogous to the case of continuous maps X : I → E n considered in [12,13], where we map I on to E as opposed to simply E n . 3.1. Existence theorem For arbitrary H , solutions to (3.6) might not exist. So, let us consider only H such that there exists a constant H¯ for which the bound   (3.7) D  H (t, X), 0ˆ 2 < H¯ holds for all t ∈ I and all X ∈ E . Such bounded H will permit the existence of solutions. Let C(I, E) denote the set of all continuous maps from I to E and let dC denote a metric on C(I, E) defined as    dC (X1 , X2 ) = sup D  X1 (t), X2 (t) . (3.8) t∈I

It follows that (C(I, E), dC ) is a complete metric space. We next define the integral operator O : C(I, E) → C(I, E) by t OY (t) = xa +

  H τ, Y (τ ) dτ.

(3.9)

a

Recall that X ∈ C(I, E) is a solution to the Cauchy problem (3.5) if and only if X is a solution to the integral equation (3.6). But X satisfies the integral equation (3.6) if and only if X satisfies the operator equation X = OX. In other words, X is a solution to the Cauchy problem (3.5) if and only if X is a fixed point of the operator O. In the theorems that follow, we assume that all relevant Hukuhara differences taken exist. Recall that for two convex sets J1 and J2 where there exists a translation J2 → J3 such that J3 ⊂ J1 , the Hukuhara difference is given by JH where J1 = J2 + JH . See [11]. We remind that reader that the mapping H is Hölder continuous if there exists a constant M > 0 and a constant 0 < μ  1 such that D  (H (y1 ), H (y2 ))  M(D  (y1 , y2 ))μ for all y1 , y2 , under the metric D  . Theorem 1. Let H : I × E → E be Hölder continuous and bounded, with Hölder constant M and index 0 < μ  1. Then, there exists a solution to the Cauchy problem (3.5) on I. Proof. For H bounded by H¯ as in (3.7), note that the operator O defined in (3.9) is compact. To see this, let N ⊂ C(I, E) be a bounded subset. Let ON = {OY | Y ∈ N } be the action of the operator O on the subset N . For any t1 , t2 ∈ I, t1  t2 and any Y ∈ N we have from (3.4) that

D. Karpenko et al. / Fuzzy Sets and Systems 245 (2014) 18–29

        D  OY (t2 ), OY (t1 )  (t2 − t1 ) sup D  H t, Y (t) , 0ˆ 2  t ∈ I  (t2 − t1 )H¯ , which shows that ON is equi-continuous. Meanwhile, for fixed t1 , we have   D  OY (t1 ), OY (t)  |t1 − t|H¯ ,

25

(3.10)

(3.11)

for all t ∈ I and all Y ∈ N . As such, ON is uniformly bounded in E . This is in direct analogy to [7]. Note that we do not say that this implies that ON is totally bounded in E , as this is discounted in [7]. Now, let us make use of Hölder continuity. Let t ∈ [a − , a + ], > 0 small, and take Y, Z ∈ N . Then by the Hölder continuity   D  (OY, OZ) = D  O(Y − Z), 0ˆ 2  t

      H τ, Y (τ ) − H τ, Z(τ ) dτ, 0ˆ 2 =D a

 t              D  H τ, Y (τ ) − H τ, Z(τ ) , 0ˆ 2 dτ    a

  t      μ   dτ   M  D  Y (τ ), Z(τ )   a μ   M D  (Y, Z) . Appealing to the Arzelà–Ascoli theorem for real-valued continuous functions on compact Hausdorff spaces (see [2–4,9]) we see that, due to the Hölder continuity and the fact that ON is uniformly bounded, ON is a relatively compact subset of C(I, E). Thus, O is compact. X is a solution to the Cauchy problem (3.5) if and only if X is a fixed point of the operator O. In the metric space (C(I, E), dC ) consider the closed neighborhood Nρ = {Y ∈ C(I, E) | dC (Y, 0ˆ 2 )  ρ} where ρ = (b − a)H¯ . Then, ONρ ⊂ Nρ (and O is a contraction map) as for X ∈ C(I, E) we have from (3.11) that     D  OX(t), OX(a) = D  OX(t), 0ˆ 2  |t − a|H¯  |b − a|H¯ , (3.12) and, upon setting 0ˆ 2 : I → E (where 0ˆ 2 (t) = 0ˆ 2 for all t ∈ I) we have dC (OX, O0ˆ 2 )  |b − a|H¯ .

(3.13)

As O is compact, it has a fixed point X ∈ Nρ . But if X is a fixed point for O then it is a solution to the Cauchy problem (3.5) and we are done. 2 Remark. Note that we have bypassed the error in Nieto [18] pointed out by Choudary and Donchev [7]. In assuming the Hölder continuity, we remove the need for total boundedness. We have used the standard modification for Arzelà–Ascoli theorem in the presence of Hölder continuity. Thus, under the Hölder condition provided, we have demonstrated the existence of a solution to the Cauchy problem (3.5) on I. This may be compared to the proof by Kaleva for the real fuzzy sets. 3.2. Existence and uniqueness Theorem 1 grants the existence of a solution. In some cases, uniqueness is possible. This is particularly of interest as far as applications are concerned, since most real-world applications rely on there being a single solution to an initial value problem. In the following result, we consider the case where H is Lipschitz continuous, so as to obtain a uniqueness result. Theorem 2. Let H : I × E → E be Lipschitz continuous and bounded, with Lipschitz constant L. Then, there exists a unique solution to the Cauchy problem (3.5) on a neighborhood of a ∈ I.

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Proof. Let [a − , a + ] × [xa − δ, xa + δ] ⊂ I × E be a compact domain on which H is defined. Then for O to be well-defined, D  (OY − xa , 0ˆ 2 ) < δ for all Y ∈ [xa − δ, xa + δ]. Yet  t

    D (OY − xa , 0ˆ 2 ) = D H τ, Y (τ ) dτ, 0ˆ 2 a

 t            D H τ, Y (τ ) , 0ˆ 2 dτ    a

 H , hence we require H < δ, i.e. we must pick > 0 such that < δ/H . Define Z0 (t) = xa , t Zk+1 (t) = xa +

  H τ, Zk (τ ) dτ.

a

Then, observe  



D O(Zm − Zn ), 0ˆ 2 = D 

 t

     H τ, Zm (τ ) H τ, Zn (τ ) dτ, 0ˆ 2



a

 t            ˆ   D H τ, Zm (τ ) − H τ, Zn (τ ) , 02 dτ    a

  t        L D Zm (τ ) − Zn (τ ), 0ˆ 2 dτ    a

 L D  (Zm − Zn , 0ˆ 2 ). If < 1/L, the mapping is a contraction. In such a case, O is a contraction and, by the Banach fixed point theorem, the operator O has a unique fixed point. Thus, there exists a unique function Z ∗ ∈ C([a − , a + ] × [xa − δ, xa + δ]) such that OZ ∗ = Z ∗ . One may construct this function by Z ∗ (t) = limk→∞ Zk (t). This function is the unique solution to the Cauchy problem (3.5) on the interval [a − , a + ] where < min{δ/H , 1/L}. 2 3.3. Examples Here we give some examples in order to demonstrate when the theory discussed previously is useful. For notational simplicity, we shall abbreviate the relevant norms and metrics by | · | where needed. 3.3.1. An autonomous Hölder continuous example in Eˆ 2 Consider the nonlinear Cauchy problem for X ∈ Eˆ 2 given by       X  (t) = f X(t) = X(t) + i 3 X(t), X(0) = X0 .

(3.14)

Let X, Y ∈ E = Eˆ 2 for this case. Please observe that √     √ f (X) − f (Y ) =  |X| − |Y | + i 3 X − 3 Y   √ √   3 3   |X| − |Y | +  X − Y   |X − Y | + 3 |X − Y |  2 3 |X − Y |,

(3.15)

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so f is Hölder continuous with M = 2 and μ = 1/3. Furthermore, since X ∈ Eˆ 2 by assumption, | Re X|  1 and | Im X|  1, hence |f (X)|  2. So, f is Hölder continuous (with index μ = 1/3) and bounded. By Theorem 1, there exists a local solution to (3.14). 3.3.2. A non-autonomous Hölder continuous example in Eˆ 2 Consider the nonlinear Cauchy problem for X ∈ Eˆ 2 on t ∈ [0, b] given by     X  (t) = f t, X(t) = X(t) + 2πiX(t) + t,

X(0) = X0 .

(3.16)

Let X, Y ∈ E = Eˆ 2 , and note that     f (X) − f (Y ) =  |X| − |Y | + 2πi(X − Y )     |X| − |Y | + 2π |X − Y |    |X| − |Y | + 2π|X − Y |  (1 + 2π) |X − Y |,

(3.17)

so f is Hölder continuous with M = 1 + 2π and μ = 1/2. Furthermore, since X ∈ Eˆ 2 by assumption, | Re X|  1 and | Im X|  1, hence |f (X)|  1 + 2π + b. So, f is Hölder continuous (with index μ = 1/2) and bounded. By Theorem 1, there exists a local solution to (3.16). What we see here is that a non-autonomous Cauchy problem still will have a solution under Theorem 1. The explicit functions of t need to be bounded. For instance, if we replaced t with 1/t 2 in (3.16), we would no longer have that f was bounded near the origin. 3.3.3. A Lipschitz continuous example in Eˆ 2 Consider the nonlinear Cauchy problem for X ∈ Eˆ 2 given by   X  (t) = f X(t) =

    X(t)2 + 9 − i sin X(t) ,

X(0) = X0 .

(3.18)

Let X, Y ∈ E = Eˆ 2 , and note that          f (X) − f (Y ) =  |X|2 + 9 − |Y |2 + 9 − i sin |X| − sin |Y |            |X|2 + 9 − |Y |2 + 9 + sin |X| − sin |Y |      ||X| + |Y || |X| − |Y | + |X| − |Y | |X|2 + 9 + |Y |2 + 9    2|X| − |Y |, 

(3.19)

so f is Lipschitz continuous with√Lipschitz constant M = 2. Furthermore, since X ∈ Eˆ 2 by assumption, | Re X|  1 and | Im X|  1, hence |f (X)|  10 + 1. So, f is Lipschitz continuous and bounded. By Theorem 2, there exists a unique local solution to (3.18). 3.3.4. A Lipschitz continuous example in Eˆ 4 Let X1 , X2 ∈ Eˆ 2 . Consider the nonlinear Cauchy problem for X = (X1 , X2 )T ∈ Eˆ 4 given by     X1 (t) = f1 X1 (t), X2 (t) = iX1 (t) − 2 sin X2 (t) ,   X2 (t) = f2 X1 (t), X2 (t) = X1 (t) + 3iX2 (t) − cos(t),

X1 (0) = X10 ,

X2 (0) = X20 .

(3.20)

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Define f (X) = f ((X1 , X2 ))T = (f1 (X1 , X2 ), f2 (X1 , X2 )). Let X = (X1 , X2 ), Y = (Y1 , Y2 ) ∈ E = Eˆ 4 , and note that           f (X) − f (Y ) =  i(X1 − Y1 ) − 2 sin |X2 | − sin |Y2 | , X1 − Y1 + 3i(X2 − Y2 ) T           i(X1 − Y1 ) − 2 sin |X2 | − sin |Y2 |  + X1 − Y1 + 3i(X2 − Y2 )    |X1 − Y1 | + 2|X2 | − |Y2 | + |X1 − Y1 | + 3|X2 − Y2 |  2|X1 − Y1 | + 5|X2 − Y2 | = 5|X − Y |,

(3.21)

so f is Lipschitz continuous with Lipschitz constant M = 5. Furthermore, since X ∈ Eˆ 2 by assumption, | Re X1 |, | Re X2 |  1 and | Im X1 |, | Im X2 |  1, hence |f (X)|  |f1 (X1 , X2 )| + |f2 (X1 , X2 )|  3 + 5 = 8. So, f is Lipschitz continuous and bounded. By Theorem 2, there exists a unique local solution to (3.20). 4. Discussion and conclusions We have presented two existence theorems for solutions to a Cauchy problem for fuzzy differential equations which accommodates the notion of a complex fuzzy number. The first of these theorems states that a solution to the relevant Cauchy problem exists provided the response function is Hölder continuous, with index 0 < μ  1. The choice of the Hölder restriction allows us to bypass the error in Nieto [18] pointed out by Choudary and Donchev [7]. In particular, the Hölder restriction forced the existence of a solution without the need for a totally bounded space of operators. The second theorem is a Picard–Lindelöf type result which demonstrates the existence of a unique local solution to the relevant Cauchy problem. In this latter case, the response function is assumed to be Lipschitz. Note that these results would imply their real-valued counterparts. Of course, one may wonder what types of results are possible when either the Hölder or Lipschitz conditions are removed. While the general boundedness result claimed in Nieto [18] would be nice, Choudary and Donchev [7] demonstrate that known methods of proof have yet to crack this general problem. For the case of real fuzzy sets, some results using compactness conditions [27] and dissipative conditions [15,20] are present in the literature. For the most part, applications of complex fuzzy sets in the literature have been for static situations. The present results permit us to consider dynamic cases, where the solution to a mathematical problem defined on complex fuzzy sets can evolve in time. While the present results are completely mathematical, we should remind the reader that complex fuzzy sets have been proposed for use in a variety of applications. Chen et al. [6] and Man et al. [16] consider an application of complex fuzzy sets to adaptive neuro-fuzzy inference systems, while Li et al. [14] consider neuro-fuzzy function approximation with complex fuzzy sets. Applications for fuzzy controllers and microelectromechanical devices using complex fuzzy sets were discussed in Moses et al. [17]. Ramot et al. [22] considered an example where a solar activity was modeled using complex fuzzy sets. Moreover, Ramot et al. [22] also considered a signal processing example, demonstrating that the theory of complex fuzzy sets can be valuable in a number of different settings. In addition to direct applications to science or engineering, complex fuzzy rules have also been used in logic. Complex fuzzy logic was designed to exploit the properties of complex fuzzy sets, while holding traditional fuzzy logic as a special case. Rules constructed in the complex fuzzy logic framework are strongly related. This relation leads to a unique dependence between rules, which allows the construction of a novel system termed the complex fuzzy logic system (CFLS). This was discussed in Ramot et al. [21]. As one application, Ramot et al. [21] provide a voter turnout model. Finally, Tamir et al. [25] consider a stock performance model which applied complex fuzzy sets. Again, in these applications, the models employing complex fuzzy sets are usually time-static, or solved in some way that doesn’t involve the time evolution of a system involving complex fuzzy sets. Hence, the present results are a step toward allowing one to more fully study dynamic versions of these applications. The understanding of the Cauchy problem relating complex fuzzy sets is also a step toward being able to model more interesting applications, which involve the time evolution of an initial state. Acknowledgements R.A.V.G. was supported in part by NSF grant # 1144246. The authors appreciate the comments of the reviewers, which have led to improvement in the paper.

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