- Email: [email protected]
On new solutions of fuzzy differential equations
Chaos, Solitons and Fractals 38 (2008) 112–119 www.elsevier.com/locate/chaos
On new solutions of fuzzy differential equations Y. Chalco-Cano a
a,*
, H. Roma´n-Flores
q
b
Departamento de Matema´tica, Universidad de Tarapaca´, Casilla 7D, Arica, Chile b Instituto de Investigacio´n, Universidad de Tarapaca´, Casilla 7D, Arica, Chile Accepted 23 October 2006
Communicated by Prof. L. Marek-Crnjac
Abstract We study fuzzy differential equations (FDE) using the concept of generalized H-differentiability. This concept is based in the enlargement of the class of differentiable fuzzy mappings and, for this, we consider the lateral Hukuhara derivatives. We will see that both derivatives are different and they lead us to different solutions from a FDE. Also, some illustrative examples are given and some comparisons with other methods for solving FDE are made. 2006 Elsevier Ltd. All rights reserved.
1. Introduction Fuzzy set theory is a powerful tool for modelling uncertainty and for processing vague or subjective information in mathematical models. Their main directions of development have been diverse and its applications to the very varied real problems, for instance, in the golden mean [8], particle systems [11], quantum optics and gravity [12], synnchronize hyperchaotic sytems [29], chaotic system [13,26,27], medicine [1,3], engineering problems [17]. Particularly, fuzzy differential equation is a topic very important as much of the theoretical point of view (see [9,14,19,20,22,24,28]) as well as of their applications, for example, in population models [15,16], civil engineering [23] and in modeling hydraulic [6]. Initially, the derivative for fuzzy valued mappings was developed by Puri and Ralescu [24], that generalized and extended the concept of Hukuhara differentiability (H-derivative) for set-valued mappings to the class of fuzzy mappings. Subsequently, using the H-derivative, Kaleva [19] started to develop a theory for FDE. In the last few years, many works have been done by several authors in theoretical and applied fields (see [6,7,9,19,22,25,28]). In some cases this approach suffers certain disadvantages since the diameter diam(x(t)) of the solution is unbounded as time t increases [9,14]. This problem demonstrates that this interpretation is not a good generalization of the associated crisp case and we assume that this problem is due to the fuzzification of the derivative utilized in the formulation of the FDE. In this direction, Bede and Gal in [4,5] introduce a more general definition of derivative for fuzzy mappings enlarging the class of differentiable fuzzy mappings. q
This work was partially supported by Fondecyt-Chile through Projects 1061244 and 1040303 and Dipog-UTA by Projects 4731-04 and 4731-05. * Corresponding author. E-mail address: [email protected] (Y. Chalco-Cano). 0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.10.043
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Following this idea, in this paper we define the fuzzy lateral H-derivative for a fuzzy mapping F : ða; bÞ ! Fn as the existence of one of the following limits: ð1Þ
lim
F ðt0 þ hÞ F ðt0 Þ F ðt0 Þ F ðt0 hÞ ¼ limþ ¼ F 0 ðt0 Þ h h h!0
lim
F ðt0 þ hÞ F ðt0 Þ F ðt0 Þ F ðt0 hÞ ¼ lim ¼ F 0 ðt0 Þ: h!0 h h
h!0þ
or ð2Þ
h!0
Note that the first equality (1) is the classic definition of the fuzzy H-derivative (or differentiability in the sense of Hukuhara). Adding the form (2), we enlarge the class of differentiable fuzzy mappings. Also, note that (1) and (2) are a generalization of the classical definitions of lateral derivative. This paper has been organized as follows: Section 2 contains the basic material to be used in the rest of the article, in Section 3 we discuss some properties of the forms (1) and (2) of lateral H-derivatives and the their relationships. In particular we shall show that there are fuzzy mappings which are differentiable in the first form (1) and are not differentiable in the second form (2), and vice versa. In Section 4, we shall see that this definition of fuzzy lateral H-derivative leads us to interpret a FDE in two different forms, generating new solutions for FDE. Now, we does not have unique solution, which it seems natural in the fuzzy context because the fuzzy lateral H-derivatives breaks the restriction associated to the H-difference of fuzzy sets and solves the deficiency produced when we only consider the first form. Finally, in Section 5, we study the connection between the solutions of a FDE found using properties 1 and 2 and the solutions found via differential inclusions. 2. Basic concepts We denote by Kn the family of all nonempty compact subsets of Rn , the n-dimensional Euclidean space. If A; B 2 Kn and k 2 R, then the operations of addition and scalar multiplication are defined as A þ B ¼ fa þ b=a 2 A; b 2 Bg
kA ¼ fka=a 2 Ag:
n
If A 2 K we define the ‘‘-neighbourhood of A00 as the set NðA; Þ ¼ fx 2 Rn =dðx; AÞ < g; where d(x, A) = infa2Akx ak and kÆk is the usual Euclidean norm on Rn . The Hausdorff separation q(A, B) of A; B 2 KðX Þ is defined by qðA; BÞ ¼ inff > 0=A NðB; Þg; and the Hausdorff metric on Kn is defined by hðA; BÞ ¼ maxfqðA; BÞ; qðB; AÞg: It is well known that ðK n ; hÞ is a complete and separable metric space (see [25]). A fuzzy set u in an universe set X is a mapping u : X ! [0, 1]. We think u as assigning to each element x 2 X a degree of membership, 0 6 u(x) 6 1. If u is a fuzzy set in Rn , we define ½ua ¼ fx 2 Rn =uðxÞ P ag the a-level of u, with 0 < a 6 1. For a = 0 the support of u is defined as ½u0 ¼ suppðuÞ ¼ fx 2 Rn =uðxÞ > 0g. Remark 1. The family {Lau/a 2 [0, 1]} satisfies the following properties: (a) [u]b [u]a [u]0, forTall 0 6 a 6 b. an (b) If an%a then ½ua ¼ 1 (i.e., the level-application is left-continuous). n¼1 ½u (c) (Representation Theorem of Negoita–Ralescu [21]). Let be M X and suppose that {Ba/a 2 [0, 1]} is a family of subsets of M verifying (a) and (b) and B0 = M. Then, there exists a fuzzy set u in X such that [u]a = Ba for all a 2 [0, 1]. Moreover, supfa=x 2 Ba g; if x 2 M uðxÞ ¼ 0; if x 62 M: A fuzzy set u is called compact if ½ua 2 Kn ; "a 2 [0, 1]. Also, u is called convex if [u]a is a convex set for all a 2 [0, 1]. We will denote by Fn the space of all compact and convex fuzzy sets on Rn . If u 2 F, then u is called a fuzzy number if the a-level set [u]a is a nonempty compact interval for all a 2 [0, 1]. The sum and the scalar multiplication operations on Fn are defined as
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ðu þ vÞðxÞ ¼ supfuðyÞ ^ vðx yÞg and ðk uÞðxÞ ¼ y2Rn
uðkxÞ if k 6¼ 0 v0 ðxÞ if k ¼ 0;
where v{0} is the characteristic function of {0}. It is well known that the following operations are true for all alevels: ½u þ va ¼ ½ua þ ½va and ½k ua ¼ k½ua ;
8a 2 ½0; 1:
Also, we can extend the Hausdorff metric h to Fn by means Dðu; vÞ ¼ sup hð½ua ; ½va Þ;
8u; v 2 Fn ;
a2½0;1
and it is well known that ðFn ; DÞis a complete metric space (see [25]). Also, it is clear that the application a 2 Rn ,!vfag 2 Fn is an isometric embedding. Finally, if u 2 Fn then we say that u 2 Rn if u = v{a} for some a 2 Rn .
3. The fuzzy derivative It is well-known that the H-derivative (differentiability in the sense of Hukuhara) for fuzzy mappings was initially introduced by Puri and Ralescu [24] and it is based in the H-difference of sets, as follows. Definition 1. Let be u; v 2 Fn . If there exists w 2 Fn such that u = v + w, then w is called the H-difference of u and v and it is denoted by u v. Definition 2 [24]. Let be T = (a, b) and consider a fuzzy mapping F : ða; bÞ ! Fn . We say that F is differentiable at t0 2 T if there exists an element F 0 ðt0 Þ 2 Fn such that the limits lim
h!0þ
F ðt0 þ hÞ F ðt0 Þ F ðt0 Þ F ðt0 hÞ and limþ h h h!0 0
exist and are equal to F (t0). Here the limit is taken in the metric space ðFn ; DÞ. Note that this definition of derivative is very restrictive; for instance in [5] the authors showed that if F(t) = c Æ g(t), where c is a fuzzy number and g : ½a; b ! Rþ is a function with g 0 (t) < 0, then F is not differentiable. To avoid this difficulty, the authors of Ref. [5] introduce a more general definition of derivative for fuzzy mappings enlarging the class of differentiable fuzzy mappings by considering a lateral type of H-derivatives. In this paper we consider the following definition: Definition 3. Let be F : ða; bÞ ! Fn and t0 2 (a, b). We say that F is differentiable at t0 if: (1) it exists an element F 0 ðt0 Þ 2 Fn such that, for all h > 0 sufficiently near to 0, there are F(t0 + h) F(t0), F(t0) F(t0 h) and the limits (in the metric D) lim
h!0þ
F ðt0 þ hÞ F ðt0 Þ F ðt0 Þ F ðt0 hÞ ¼ limþ ¼ F 0 ðt0 Þ h h h!0
ð1Þ
or (2) it exists an element F 0 ðt0 Þ 2 Fn such that, for all h < 0 sufficiently near to 0, there are F(t0 + h) F(t0), F(t0) F(t0 h) and the limits lim
h!0
F ðt0 þ hÞ F ðt0 Þ F ðt0 Þ F ðt0 hÞ ¼ lim ¼ F 0 ðt0 Þ: h!0 h h
ð2Þ
Remark 2. Note that if F is differentiable in the sense (1) and (2) simultaneously, then for h > 0 sufficiently small we have F(t0 + h) = F(t0) + u1, F(t0) = F(t0 h) + u2, F(t0 h) = F(t0) + v1 and F(t0) = F(t0 + h) + v2, with u1, u2, v1, v2 2 Fn . Thus, F(t0) = F(t0) + (u2 + v1), i.e., u2 + v1 = v{0}, it implies two possibilities: u2 = v1 = v{0} if F 0 (t0) = v{0}; or u2 = v{a} = v1, with a 2 Rn , if F 0 ðt0 Þ 2 Rn . Therefore, if there exists F 0 (t0) in the first form (second form) with F 0 ðt0 Þ 62 Rn , then doesn’t exist F 0 (t0) in the form second (first form, respectively).
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Remark 3. In this paper we consider the two cases (1) and (2). In Ref. [5] the authors consider four cases: the case (i) in [5] is coincident with Eq. (1); the case (iii) in [5] is equivalent to Eq. (2); in the other cases, the derivative is trivial because it is reduced to crisp element (more precisely, F 0 2 Rn . For details see Theorem 7 in [5]). Example 1. In [5] was studied the derivative of the fuzzy mapping F(t) = c Æ g(t), where c is a fuzzy number (non crisp) and g : ða; bÞ ! Rþ is a differentiable function on (a, b). In this case, if g 0 (t0) > 0, then F is differentiable in the first form (1) and we have F 0 (t0) = c Æ g 0 (t0). But, from Remark 2, F is not differentiable in the second form (2). Analogously, if g 0 (t0) < 0, then F is differentiable in the second form (2) and F 0 (t0) = c Æ g 0 (t0), but F is not differentiable in the first form (1). The principal properties of the H-derivative in the first form (1), some of which still holds for the second form (2) [5], are well known and can by found in [19]. Next we shall write some properties for the second form (2) whose demonstrations are similar to the previous ones. Theorem 1. If F is differentiable in the second form (2), then it is continuous. Theorem 2. If F, G are differentiable in the second form (2) at the point t and k 2 R then (F + G) 0 (t) = F 0 (t) + G 0 (t) and (k F) 0 (t) = k Æ F 0 (t). Theorem 3. Let F be continuous in T = [a, b]. Then the function GðtÞ ¼ G 0 (t) = F(t), for all t 2 T.
Rt a
F is differentiable in the second form and
Theorem 4. Let F be differentiable in T = [a, b] with respect to the second form and assume that the derivative F 0 is integrable over T. Then for each s 2 T we have Z s F ðsÞ ¼ F ðaÞ þ F 0: a
In the special case when F is a fuzzy number valued mapping, we have the following result. Theorem 5. Let F : T ! F be a function and denote [F(t)]a = [fa(t), ga(t)], for each a 2 [0, 1]. Then (i) If F is differentiable in the first form (1), then fa and ga are differentiable functions and ½F 0 ðtÞa ¼ ½fa0 ðtÞ; g0a ðtÞ:
ð3Þ
(ii) If F is differentiable in the second form (2), then fa and ga are differentiable functions and ½F 0 ðtÞa ¼ ½g0a ðtÞ; fa0 ðtÞ: Proof. (i) See demonstration of Theorem 5.2 in [19]. (ii) If h < 0 and a 2 [0, 1], then we have
½F ðt þ hÞ F ðtÞa ¼ ½fa ðt þ hÞ fa ðtÞ; ga ðt þ hÞ ga ðtÞ and, multiplying by 1/h we have ½F ðt þ hÞ F ðtÞa 1 ¼ ½fa ðt þ hÞ fa ðtÞ; ga ðt þ hÞ ga ðtÞ h h g ðt þ hÞ ga ðtÞ fa ðt þ hÞ fa ðtÞ ¼ a ; : h h Similarly we obtain
½F ðtÞ F ðt hÞa g ðtÞ ga ðt hÞ fa ðtÞ fa ðt hÞ ; : ¼ a h h h
ð4Þ
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Passing to the limit we have ½F 0 ðtÞa ¼ ½g0a ðtÞ; fa0 ðtÞ; and the proof is now complete.
h
4. Solving fuzzy differential equations In this section, we study the fuzzy initial value problem x0 ðtÞ ¼ F ðt; xðtÞÞ;
xð0Þ ¼ x0 ;
ð5Þ
where F : ½0; a F ! F is a continuous fuzzy mapping and x0 is a fuzzy number. Following Kaleva [20], we observe that the relations (3) and (4) in Theorem 5 give us an useful procedure to solve the fuzzy differential equation (5). In fact, denote ½xðtÞa ¼ ½ua ðtÞ; va ðtÞ;
½x0 a ¼ ½u0a ; v0a
and ½F ðt; xðtÞÞa ¼ ½fa ðt; ua ðtÞ; va ðtÞÞ; ga ðt; ua ðtÞ; va ðtÞÞ: Then, we have the following alternatives for solving the problem (5): 0
Case I. If we consider x (t) by using the derivative in the first form (1), then from (3) we have ½x0 ðtÞa ¼ ½u0a ðtÞ; v0a ðtÞ. Now, we proceed as follows: (i) Solve the differential system ( u0a ðtÞ ¼ fa ðt; ua ðtÞ; va ðtÞÞ; ua ð0Þ ¼ u0a v0a ðtÞ ¼ ga ðt; ua ðtÞ; va ðtÞÞ; va ð0Þ ¼ v0a for ua and va; (ii) Ensure that [ua(t), va(t)] and [u0a (t), v0a (t)] are valid level sets; (iii) By using the Representation Theorem (Remark 1c)), we construct a fuzzy solution x(t) such that ½xðtÞa ¼ ½ua ðtÞ; va ðtÞ;
for all a 2 ½0; 1:
Case II. If we consider x 0 (t) by using the derivative in the second form (2), then from (4) we have [x 0 (t)]a = [v0a (t), u0a (t)] and, consequently, now we proceed as follows: (i) Solve the differential system ( u0a ðtÞ ¼ ga ðt; ua ðtÞ; va ðtÞÞ; ua ð0Þ ¼ u0a v0a ðtÞ ¼ fa ðt; ua ðtÞ; va ðtÞÞ; va ð0Þ ¼ v0a for ua and va; (ii) Ensure that [ua(t), va(t)] and [v0a (t), u0a (t)] are valid level sets; (iii) By using the Representation Theorem again, we obtain a new fuzzy solution x(t) such that [x(t)]a = [ua (t), va(t)], for all a 2 [0, 1].
Remark 4. We note that above Case II is an extension of the procedure used by Kaleva in [20] for solving fuzzy differential equations, where the derivative is considered in the first form (1) only (which is coincident with our Case I). Thus, from Case II we obtain new solutions for FDE. Example 2. Let us consider the fuzzy malthusian problem 0 x ðtÞ ¼ kxðtÞ xð0Þ ¼ X 0 ;
ð6Þ
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where k > 0 and, as in [9], the initial condition X0 is a symmetric triangular fuzzy number with support [a, a]. That is, ½X 0 a ¼ ½að1 aÞ; að1 aÞ ¼ ð1 aÞ½a; a: If we consider x 0 (t) in the first form (1), we have solve the following differential system: 0 ua ðtÞ ¼ kva ðtÞ; ua ð0Þ ¼ að1 aÞ v0a ðtÞ ¼ kua ðtÞ;
va ð0Þ ¼ að1 aÞ:
The solutions of this system are ua ðtÞ ¼ að1 aÞekt
and
va ðtÞ ¼ að1 aÞekt ;
and we see that ua(t) 6 va(t) for all t P 0. Therefore, the fuzzy function x(t) solving (6) has level sets ½xðtÞa ¼ ½að1 aÞekt ; að1 aÞekt ; for all t P 0. Now, if we consider x 0 (t) in the second form (2), we solve the following differential system: 0 ua ðtÞ ¼ kua ðtÞ; ua ð0Þ ¼ að1 aÞ v0a ðtÞ ¼ kva ðtÞ;
va ð0Þ ¼ að1 aÞ;
and the solutions of this system are ua ðtÞ ¼ að1 aÞekt
and
va ðtÞ ¼ að1 aÞekt ;
and we see that ua(t) 6 va(t) for all t P 0. Therefore, the fuzzy function x(t) solving (6) in this case has level sets ½xðtÞa ¼ ½að1 aÞekt ; að1 aÞekt ; for all t P 0. Remark 5. Note that the solution of the FDE (6) considering the derivative x 0 in the first form (1), has the property that diam(supp x(t)) = 2aekt is unbounded as t ! 1, demonstrating that this interpretation does not generalize in an appropriate way the crisp case and gives counterintuitive result. However, if in (6) the derivative x 0 is interpreted in the second form (2), then the result is much more intuitive for (6) since now diam(supp x(t)) = 2aekt ! 0 as t ! 1. From Remark 5, we see that the solution of a FDE is dependent of the election of the derivative: in the first form or in the second form. Thus, as in the above example, the solution can be adequately chosen. On the other hand, it is clear that in this new procedure the unicity of the solution is lost, but it is a expected situation in the fuzzy context. Nevertheless, we can speak of the existence and unicity of two solutions (one solution for each lateral derivative), as is showed in the following result. Theorem 6. Let F : T Fn ! Fn be continuous and assume that there exists a k > 0 such that DðF ðt; uÞ; F ðt; vÞÞ 6 kDðu; vÞ for all t 2 T, u; v 2 Fn . Then the problem x0 ðtÞ ¼ F ðt; xðtÞÞ;
xð0Þ ¼ x0 ;
ð7Þ
has two unique solutions on T. Proof. If in the problem (7) we consider the derivative x 0 in the first form (1), then from Theorem 6.1 in [19], there exists an unique solution on T. In the same way, if we consider the derivative x 0 in the second form (2) then, analogously to the demonstration of Theorem 6.1 in [19], we can prove that there exists an unique solution on T. This proves that there exists an unique solution for each lateral direction, and the proof is now complete. h
5. Constructing solutions via differential inclusions We consider the fuzzy differential equation x0 ðtÞ ¼ F ðt; xðtÞÞ;
xð0Þ ¼ x0 ;
ð8Þ
where F : ½0; a F ! F is obtained by Zadeh’s extension principle from a continuous function f : ½0; a R ! R. Now F(t, x) can be computed levelwise, i.e., for each a 2 [0, 1]
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½F ðt; xÞa ¼ f ðt; ½xa Þ for all t 2 [0, a] and x 2 F. Following the idea of Hullermeier [18] (see also [2,9,10]) we interpret the fuzzy initial value problem (8) as a family of differential inclusions: y 0a ðtÞ ¼ f ðt; y a ðtÞÞ;
y a ð0Þ 2 ½x0 a ;
a 2 ½0; 1:
ð9Þ
Under suitable assumptions, the attainable sets Aa ðtÞ ¼ fy a ðtÞ=y a is a solution of ð9Þg; are a-levels of a fuzzy set x(t), which we call a solution of (8). The connection between the fuzzy solutions and solutions via differential inclusion has been studied recently by Kaleva [19] and, supposing that f is nondecreasing, the following result has been obtained: Theorem 7 [20]. If f is nondecreasing with respect to the second argument then, using the first form (1), the fuzzy solution of (8) and the solution via differential inclusions are identical. Now, as an extension of the above theorem to the class of differentiable functions with respect to the second form (2) we present the following result: Theorem 8. If f is nonincreasing with respect to the second argument then, using the derivative in the second form, the fuzzy solution of (8) and the solution via differential inclusions are identical. Proof. For each a 2 [0, 1], we consider [x(t)]a = [ua(t), va(t)]. Since f is continuous, it follows that: ½F ðt; xÞa ¼ f ðt; ½xðtÞa Þ ¼ f ðt; ½ua ðtÞ; va ðtÞÞ: and f(t,[ua(t), va(t)]) is a closed bounded interval. As f is nonincreasing, we have that f ðt; ½ua ðtÞ; va ðtÞÞ ¼ ½f ðt; va ðtÞÞ; f ðt; ua ðtÞÞ: Consequently, from (9), we have the differential system ( v0a ðtÞ ¼ f ðt; va ðtÞÞ; va ð0Þ ¼ v0a : u0a ðtÞ ¼ f ðt; ua ðtÞÞ; ua ð0Þ ¼ u0a
ð10Þ
Now, if x 0 (t) is consider in the second form, we have that ½x0 ðtÞa ¼ ½v0a ðtÞ; u0a ðtÞ ¼ ½f ðt; va ðtÞÞ; f ðt; ua ðtÞÞ; Obtaining the differential system (10), and the proof is complete.
h
6. Conclusions In this work we introduce the concept of lateral fuzzy H-derivatives for fuzzy mappings (see Section 3), enlarging the class of differentiable fuzzy mappings. Subsequently, by using the laterel H-derivatives, we obtain two different interpretations of a FDE generating new solutions for a FDE (see Section 4). Different studies and many examples have been developed by several authors by using the fuzzy derivative x 0 in the first form (which is coincident with the classic H-derivative). For instance, Nieto in [2] has studied the linear case x 0 (t) + x(t) = c(t) considering the derivative in the first form (1). Also, these results can be generalized in a natural way, with some small variations, for the second case (2). On the other hand, fuzzy differential inclusions (FDI) have been used by several authors for solving FDE (for instance, see [2,9,18]) and it presents the advantage that the solutions obtained via FDI seem to be more intuitive than other solution using H-derivative (first form, see [9,14]). It is worthy to stress also that this interpretation has a disadvantage because we cannot speak about the fuzzy derivative. Instead, using fuzzy H-derivative in the second form (2), we overcome this difficulty and, moreover, the fuzzy solution and the solution via differential inclusions are identical (see Section 5). References [1] Abbod MF, Von Keyserlingk DG, Linkens DA, Mahfouf M. Survey of utilisation of fuzzy technology in medicine and healthcare. Fuzzy Set Syst 2001;120:331–49.
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On new solutions of fuzzy differential equations Y. Chalco-Cano a
a,*
, H. Roma´n-Flores
q
b
Departamento de Matema´tica, Universidad de Tarapaca´, Casilla 7D, Arica, Chile b Instituto de Investigacio´n, Universidad de Tarapaca´, Casilla 7D, Arica, Chile Accepted 23 October 2006
Communicated by Prof. L. Marek-Crnjac
Abstract We study fuzzy differential equations (FDE) using the concept of generalized H-differentiability. This concept is based in the enlargement of the class of differentiable fuzzy mappings and, for this, we consider the lateral Hukuhara derivatives. We will see that both derivatives are different and they lead us to different solutions from a FDE. Also, some illustrative examples are given and some comparisons with other methods for solving FDE are made. 2006 Elsevier Ltd. All rights reserved.
1. Introduction Fuzzy set theory is a powerful tool for modelling uncertainty and for processing vague or subjective information in mathematical models. Their main directions of development have been diverse and its applications to the very varied real problems, for instance, in the golden mean [8], particle systems [11], quantum optics and gravity [12], synnchronize hyperchaotic sytems [29], chaotic system [13,26,27], medicine [1,3], engineering problems [17]. Particularly, fuzzy differential equation is a topic very important as much of the theoretical point of view (see [9,14,19,20,22,24,28]) as well as of their applications, for example, in population models [15,16], civil engineering [23] and in modeling hydraulic [6]. Initially, the derivative for fuzzy valued mappings was developed by Puri and Ralescu [24], that generalized and extended the concept of Hukuhara differentiability (H-derivative) for set-valued mappings to the class of fuzzy mappings. Subsequently, using the H-derivative, Kaleva [19] started to develop a theory for FDE. In the last few years, many works have been done by several authors in theoretical and applied fields (see [6,7,9,19,22,25,28]). In some cases this approach suffers certain disadvantages since the diameter diam(x(t)) of the solution is unbounded as time t increases [9,14]. This problem demonstrates that this interpretation is not a good generalization of the associated crisp case and we assume that this problem is due to the fuzzification of the derivative utilized in the formulation of the FDE. In this direction, Bede and Gal in [4,5] introduce a more general definition of derivative for fuzzy mappings enlarging the class of differentiable fuzzy mappings. q
This work was partially supported by Fondecyt-Chile through Projects 1061244 and 1040303 and Dipog-UTA by Projects 4731-04 and 4731-05. * Corresponding author. E-mail address: [email protected] (Y. Chalco-Cano). 0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.10.043
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Following this idea, in this paper we define the fuzzy lateral H-derivative for a fuzzy mapping F : ða; bÞ ! Fn as the existence of one of the following limits: ð1Þ
lim
F ðt0 þ hÞ F ðt0 Þ F ðt0 Þ F ðt0 hÞ ¼ limþ ¼ F 0 ðt0 Þ h h h!0
lim
F ðt0 þ hÞ F ðt0 Þ F ðt0 Þ F ðt0 hÞ ¼ lim ¼ F 0 ðt0 Þ: h!0 h h
h!0þ
or ð2Þ
h!0
Note that the first equality (1) is the classic definition of the fuzzy H-derivative (or differentiability in the sense of Hukuhara). Adding the form (2), we enlarge the class of differentiable fuzzy mappings. Also, note that (1) and (2) are a generalization of the classical definitions of lateral derivative. This paper has been organized as follows: Section 2 contains the basic material to be used in the rest of the article, in Section 3 we discuss some properties of the forms (1) and (2) of lateral H-derivatives and the their relationships. In particular we shall show that there are fuzzy mappings which are differentiable in the first form (1) and are not differentiable in the second form (2), and vice versa. In Section 4, we shall see that this definition of fuzzy lateral H-derivative leads us to interpret a FDE in two different forms, generating new solutions for FDE. Now, we does not have unique solution, which it seems natural in the fuzzy context because the fuzzy lateral H-derivatives breaks the restriction associated to the H-difference of fuzzy sets and solves the deficiency produced when we only consider the first form. Finally, in Section 5, we study the connection between the solutions of a FDE found using properties 1 and 2 and the solutions found via differential inclusions. 2. Basic concepts We denote by Kn the family of all nonempty compact subsets of Rn , the n-dimensional Euclidean space. If A; B 2 Kn and k 2 R, then the operations of addition and scalar multiplication are defined as A þ B ¼ fa þ b=a 2 A; b 2 Bg
kA ¼ fka=a 2 Ag:
n
If A 2 K we define the ‘‘-neighbourhood of A00 as the set NðA; Þ ¼ fx 2 Rn =dðx; AÞ < g; where d(x, A) = infa2Akx ak and kÆk is the usual Euclidean norm on Rn . The Hausdorff separation q(A, B) of A; B 2 KðX Þ is defined by qðA; BÞ ¼ inff > 0=A NðB; Þg; and the Hausdorff metric on Kn is defined by hðA; BÞ ¼ maxfqðA; BÞ; qðB; AÞg: It is well known that ðK n ; hÞ is a complete and separable metric space (see [25]). A fuzzy set u in an universe set X is a mapping u : X ! [0, 1]. We think u as assigning to each element x 2 X a degree of membership, 0 6 u(x) 6 1. If u is a fuzzy set in Rn , we define ½ua ¼ fx 2 Rn =uðxÞ P ag the a-level of u, with 0 < a 6 1. For a = 0 the support of u is defined as ½u0 ¼ suppðuÞ ¼ fx 2 Rn =uðxÞ > 0g. Remark 1. The family {Lau/a 2 [0, 1]} satisfies the following properties: (a) [u]b [u]a [u]0, forTall 0 6 a 6 b. an (b) If an%a then ½ua ¼ 1 (i.e., the level-application is left-continuous). n¼1 ½u (c) (Representation Theorem of Negoita–Ralescu [21]). Let be M X and suppose that {Ba/a 2 [0, 1]} is a family of subsets of M verifying (a) and (b) and B0 = M. Then, there exists a fuzzy set u in X such that [u]a = Ba for all a 2 [0, 1]. Moreover, supfa=x 2 Ba g; if x 2 M uðxÞ ¼ 0; if x 62 M: A fuzzy set u is called compact if ½ua 2 Kn ; "a 2 [0, 1]. Also, u is called convex if [u]a is a convex set for all a 2 [0, 1]. We will denote by Fn the space of all compact and convex fuzzy sets on Rn . If u 2 F, then u is called a fuzzy number if the a-level set [u]a is a nonempty compact interval for all a 2 [0, 1]. The sum and the scalar multiplication operations on Fn are defined as
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ðu þ vÞðxÞ ¼ supfuðyÞ ^ vðx yÞg and ðk uÞðxÞ ¼ y2Rn
uðkxÞ if k 6¼ 0 v0 ðxÞ if k ¼ 0;
where v{0} is the characteristic function of {0}. It is well known that the following operations are true for all alevels: ½u þ va ¼ ½ua þ ½va and ½k ua ¼ k½ua ;
8a 2 ½0; 1:
Also, we can extend the Hausdorff metric h to Fn by means Dðu; vÞ ¼ sup hð½ua ; ½va Þ;
8u; v 2 Fn ;
a2½0;1
and it is well known that ðFn ; DÞis a complete metric space (see [25]). Also, it is clear that the application a 2 Rn ,!vfag 2 Fn is an isometric embedding. Finally, if u 2 Fn then we say that u 2 Rn if u = v{a} for some a 2 Rn .
3. The fuzzy derivative It is well-known that the H-derivative (differentiability in the sense of Hukuhara) for fuzzy mappings was initially introduced by Puri and Ralescu [24] and it is based in the H-difference of sets, as follows. Definition 1. Let be u; v 2 Fn . If there exists w 2 Fn such that u = v + w, then w is called the H-difference of u and v and it is denoted by u v. Definition 2 [24]. Let be T = (a, b) and consider a fuzzy mapping F : ða; bÞ ! Fn . We say that F is differentiable at t0 2 T if there exists an element F 0 ðt0 Þ 2 Fn such that the limits lim
h!0þ
F ðt0 þ hÞ F ðt0 Þ F ðt0 Þ F ðt0 hÞ and limþ h h h!0 0
exist and are equal to F (t0). Here the limit is taken in the metric space ðFn ; DÞ. Note that this definition of derivative is very restrictive; for instance in [5] the authors showed that if F(t) = c Æ g(t), where c is a fuzzy number and g : ½a; b ! Rþ is a function with g 0 (t) < 0, then F is not differentiable. To avoid this difficulty, the authors of Ref. [5] introduce a more general definition of derivative for fuzzy mappings enlarging the class of differentiable fuzzy mappings by considering a lateral type of H-derivatives. In this paper we consider the following definition: Definition 3. Let be F : ða; bÞ ! Fn and t0 2 (a, b). We say that F is differentiable at t0 if: (1) it exists an element F 0 ðt0 Þ 2 Fn such that, for all h > 0 sufficiently near to 0, there are F(t0 + h) F(t0), F(t0) F(t0 h) and the limits (in the metric D) lim
h!0þ
F ðt0 þ hÞ F ðt0 Þ F ðt0 Þ F ðt0 hÞ ¼ limþ ¼ F 0 ðt0 Þ h h h!0
ð1Þ
or (2) it exists an element F 0 ðt0 Þ 2 Fn such that, for all h < 0 sufficiently near to 0, there are F(t0 + h) F(t0), F(t0) F(t0 h) and the limits lim
h!0
F ðt0 þ hÞ F ðt0 Þ F ðt0 Þ F ðt0 hÞ ¼ lim ¼ F 0 ðt0 Þ: h!0 h h
ð2Þ
Remark 2. Note that if F is differentiable in the sense (1) and (2) simultaneously, then for h > 0 sufficiently small we have F(t0 + h) = F(t0) + u1, F(t0) = F(t0 h) + u2, F(t0 h) = F(t0) + v1 and F(t0) = F(t0 + h) + v2, with u1, u2, v1, v2 2 Fn . Thus, F(t0) = F(t0) + (u2 + v1), i.e., u2 + v1 = v{0}, it implies two possibilities: u2 = v1 = v{0} if F 0 (t0) = v{0}; or u2 = v{a} = v1, with a 2 Rn , if F 0 ðt0 Þ 2 Rn . Therefore, if there exists F 0 (t0) in the first form (second form) with F 0 ðt0 Þ 62 Rn , then doesn’t exist F 0 (t0) in the form second (first form, respectively).
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Remark 3. In this paper we consider the two cases (1) and (2). In Ref. [5] the authors consider four cases: the case (i) in [5] is coincident with Eq. (1); the case (iii) in [5] is equivalent to Eq. (2); in the other cases, the derivative is trivial because it is reduced to crisp element (more precisely, F 0 2 Rn . For details see Theorem 7 in [5]). Example 1. In [5] was studied the derivative of the fuzzy mapping F(t) = c Æ g(t), where c is a fuzzy number (non crisp) and g : ða; bÞ ! Rþ is a differentiable function on (a, b). In this case, if g 0 (t0) > 0, then F is differentiable in the first form (1) and we have F 0 (t0) = c Æ g 0 (t0). But, from Remark 2, F is not differentiable in the second form (2). Analogously, if g 0 (t0) < 0, then F is differentiable in the second form (2) and F 0 (t0) = c Æ g 0 (t0), but F is not differentiable in the first form (1). The principal properties of the H-derivative in the first form (1), some of which still holds for the second form (2) [5], are well known and can by found in [19]. Next we shall write some properties for the second form (2) whose demonstrations are similar to the previous ones. Theorem 1. If F is differentiable in the second form (2), then it is continuous. Theorem 2. If F, G are differentiable in the second form (2) at the point t and k 2 R then (F + G) 0 (t) = F 0 (t) + G 0 (t) and (k F) 0 (t) = k Æ F 0 (t). Theorem 3. Let F be continuous in T = [a, b]. Then the function GðtÞ ¼ G 0 (t) = F(t), for all t 2 T.
Rt a
F is differentiable in the second form and
Theorem 4. Let F be differentiable in T = [a, b] with respect to the second form and assume that the derivative F 0 is integrable over T. Then for each s 2 T we have Z s F ðsÞ ¼ F ðaÞ þ F 0: a
In the special case when F is a fuzzy number valued mapping, we have the following result. Theorem 5. Let F : T ! F be a function and denote [F(t)]a = [fa(t), ga(t)], for each a 2 [0, 1]. Then (i) If F is differentiable in the first form (1), then fa and ga are differentiable functions and ½F 0 ðtÞa ¼ ½fa0 ðtÞ; g0a ðtÞ:
ð3Þ
(ii) If F is differentiable in the second form (2), then fa and ga are differentiable functions and ½F 0 ðtÞa ¼ ½g0a ðtÞ; fa0 ðtÞ: Proof. (i) See demonstration of Theorem 5.2 in [19]. (ii) If h < 0 and a 2 [0, 1], then we have
½F ðt þ hÞ F ðtÞa ¼ ½fa ðt þ hÞ fa ðtÞ; ga ðt þ hÞ ga ðtÞ and, multiplying by 1/h we have ½F ðt þ hÞ F ðtÞa 1 ¼ ½fa ðt þ hÞ fa ðtÞ; ga ðt þ hÞ ga ðtÞ h h g ðt þ hÞ ga ðtÞ fa ðt þ hÞ fa ðtÞ ¼ a ; : h h Similarly we obtain
½F ðtÞ F ðt hÞa g ðtÞ ga ðt hÞ fa ðtÞ fa ðt hÞ ; : ¼ a h h h
ð4Þ
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116
Passing to the limit we have ½F 0 ðtÞa ¼ ½g0a ðtÞ; fa0 ðtÞ; and the proof is now complete.
h
4. Solving fuzzy differential equations In this section, we study the fuzzy initial value problem x0 ðtÞ ¼ F ðt; xðtÞÞ;
xð0Þ ¼ x0 ;
ð5Þ
where F : ½0; a F ! F is a continuous fuzzy mapping and x0 is a fuzzy number. Following Kaleva [20], we observe that the relations (3) and (4) in Theorem 5 give us an useful procedure to solve the fuzzy differential equation (5). In fact, denote ½xðtÞa ¼ ½ua ðtÞ; va ðtÞ;
½x0 a ¼ ½u0a ; v0a
and ½F ðt; xðtÞÞa ¼ ½fa ðt; ua ðtÞ; va ðtÞÞ; ga ðt; ua ðtÞ; va ðtÞÞ: Then, we have the following alternatives for solving the problem (5): 0
Case I. If we consider x (t) by using the derivative in the first form (1), then from (3) we have ½x0 ðtÞa ¼ ½u0a ðtÞ; v0a ðtÞ. Now, we proceed as follows: (i) Solve the differential system ( u0a ðtÞ ¼ fa ðt; ua ðtÞ; va ðtÞÞ; ua ð0Þ ¼ u0a v0a ðtÞ ¼ ga ðt; ua ðtÞ; va ðtÞÞ; va ð0Þ ¼ v0a for ua and va; (ii) Ensure that [ua(t), va(t)] and [u0a (t), v0a (t)] are valid level sets; (iii) By using the Representation Theorem (Remark 1c)), we construct a fuzzy solution x(t) such that ½xðtÞa ¼ ½ua ðtÞ; va ðtÞ;
for all a 2 ½0; 1:
Case II. If we consider x 0 (t) by using the derivative in the second form (2), then from (4) we have [x 0 (t)]a = [v0a (t), u0a (t)] and, consequently, now we proceed as follows: (i) Solve the differential system ( u0a ðtÞ ¼ ga ðt; ua ðtÞ; va ðtÞÞ; ua ð0Þ ¼ u0a v0a ðtÞ ¼ fa ðt; ua ðtÞ; va ðtÞÞ; va ð0Þ ¼ v0a for ua and va; (ii) Ensure that [ua(t), va(t)] and [v0a (t), u0a (t)] are valid level sets; (iii) By using the Representation Theorem again, we obtain a new fuzzy solution x(t) such that [x(t)]a = [ua (t), va(t)], for all a 2 [0, 1].
Remark 4. We note that above Case II is an extension of the procedure used by Kaleva in [20] for solving fuzzy differential equations, where the derivative is considered in the first form (1) only (which is coincident with our Case I). Thus, from Case II we obtain new solutions for FDE. Example 2. Let us consider the fuzzy malthusian problem 0 x ðtÞ ¼ kxðtÞ xð0Þ ¼ X 0 ;
ð6Þ
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where k > 0 and, as in [9], the initial condition X0 is a symmetric triangular fuzzy number with support [a, a]. That is, ½X 0 a ¼ ½að1 aÞ; að1 aÞ ¼ ð1 aÞ½a; a: If we consider x 0 (t) in the first form (1), we have solve the following differential system: 0 ua ðtÞ ¼ kva ðtÞ; ua ð0Þ ¼ að1 aÞ v0a ðtÞ ¼ kua ðtÞ;
va ð0Þ ¼ að1 aÞ:
The solutions of this system are ua ðtÞ ¼ að1 aÞekt
and
va ðtÞ ¼ að1 aÞekt ;
and we see that ua(t) 6 va(t) for all t P 0. Therefore, the fuzzy function x(t) solving (6) has level sets ½xðtÞa ¼ ½að1 aÞekt ; að1 aÞekt ; for all t P 0. Now, if we consider x 0 (t) in the second form (2), we solve the following differential system: 0 ua ðtÞ ¼ kua ðtÞ; ua ð0Þ ¼ að1 aÞ v0a ðtÞ ¼ kva ðtÞ;
va ð0Þ ¼ að1 aÞ;
and the solutions of this system are ua ðtÞ ¼ að1 aÞekt
and
va ðtÞ ¼ að1 aÞekt ;
and we see that ua(t) 6 va(t) for all t P 0. Therefore, the fuzzy function x(t) solving (6) in this case has level sets ½xðtÞa ¼ ½að1 aÞekt ; að1 aÞekt ; for all t P 0. Remark 5. Note that the solution of the FDE (6) considering the derivative x 0 in the first form (1), has the property that diam(supp x(t)) = 2aekt is unbounded as t ! 1, demonstrating that this interpretation does not generalize in an appropriate way the crisp case and gives counterintuitive result. However, if in (6) the derivative x 0 is interpreted in the second form (2), then the result is much more intuitive for (6) since now diam(supp x(t)) = 2aekt ! 0 as t ! 1. From Remark 5, we see that the solution of a FDE is dependent of the election of the derivative: in the first form or in the second form. Thus, as in the above example, the solution can be adequately chosen. On the other hand, it is clear that in this new procedure the unicity of the solution is lost, but it is a expected situation in the fuzzy context. Nevertheless, we can speak of the existence and unicity of two solutions (one solution for each lateral derivative), as is showed in the following result. Theorem 6. Let F : T Fn ! Fn be continuous and assume that there exists a k > 0 such that DðF ðt; uÞ; F ðt; vÞÞ 6 kDðu; vÞ for all t 2 T, u; v 2 Fn . Then the problem x0 ðtÞ ¼ F ðt; xðtÞÞ;
xð0Þ ¼ x0 ;
ð7Þ
has two unique solutions on T. Proof. If in the problem (7) we consider the derivative x 0 in the first form (1), then from Theorem 6.1 in [19], there exists an unique solution on T. In the same way, if we consider the derivative x 0 in the second form (2) then, analogously to the demonstration of Theorem 6.1 in [19], we can prove that there exists an unique solution on T. This proves that there exists an unique solution for each lateral direction, and the proof is now complete. h
5. Constructing solutions via differential inclusions We consider the fuzzy differential equation x0 ðtÞ ¼ F ðt; xðtÞÞ;
xð0Þ ¼ x0 ;
ð8Þ
where F : ½0; a F ! F is obtained by Zadeh’s extension principle from a continuous function f : ½0; a R ! R. Now F(t, x) can be computed levelwise, i.e., for each a 2 [0, 1]
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½F ðt; xÞa ¼ f ðt; ½xa Þ for all t 2 [0, a] and x 2 F. Following the idea of Hullermeier [18] (see also [2,9,10]) we interpret the fuzzy initial value problem (8) as a family of differential inclusions: y 0a ðtÞ ¼ f ðt; y a ðtÞÞ;
y a ð0Þ 2 ½x0 a ;
a 2 ½0; 1:
ð9Þ
Under suitable assumptions, the attainable sets Aa ðtÞ ¼ fy a ðtÞ=y a is a solution of ð9Þg; are a-levels of a fuzzy set x(t), which we call a solution of (8). The connection between the fuzzy solutions and solutions via differential inclusion has been studied recently by Kaleva [19] and, supposing that f is nondecreasing, the following result has been obtained: Theorem 7 [20]. If f is nondecreasing with respect to the second argument then, using the first form (1), the fuzzy solution of (8) and the solution via differential inclusions are identical. Now, as an extension of the above theorem to the class of differentiable functions with respect to the second form (2) we present the following result: Theorem 8. If f is nonincreasing with respect to the second argument then, using the derivative in the second form, the fuzzy solution of (8) and the solution via differential inclusions are identical. Proof. For each a 2 [0, 1], we consider [x(t)]a = [ua(t), va(t)]. Since f is continuous, it follows that: ½F ðt; xÞa ¼ f ðt; ½xðtÞa Þ ¼ f ðt; ½ua ðtÞ; va ðtÞÞ: and f(t,[ua(t), va(t)]) is a closed bounded interval. As f is nonincreasing, we have that f ðt; ½ua ðtÞ; va ðtÞÞ ¼ ½f ðt; va ðtÞÞ; f ðt; ua ðtÞÞ: Consequently, from (9), we have the differential system ( v0a ðtÞ ¼ f ðt; va ðtÞÞ; va ð0Þ ¼ v0a : u0a ðtÞ ¼ f ðt; ua ðtÞÞ; ua ð0Þ ¼ u0a
ð10Þ
Now, if x 0 (t) is consider in the second form, we have that ½x0 ðtÞa ¼ ½v0a ðtÞ; u0a ðtÞ ¼ ½f ðt; va ðtÞÞ; f ðt; ua ðtÞÞ; Obtaining the differential system (10), and the proof is complete.
h
6. Conclusions In this work we introduce the concept of lateral fuzzy H-derivatives for fuzzy mappings (see Section 3), enlarging the class of differentiable fuzzy mappings. Subsequently, by using the laterel H-derivatives, we obtain two different interpretations of a FDE generating new solutions for a FDE (see Section 4). Different studies and many examples have been developed by several authors by using the fuzzy derivative x 0 in the first form (which is coincident with the classic H-derivative). For instance, Nieto in [2] has studied the linear case x 0 (t) + x(t) = c(t) considering the derivative in the first form (1). Also, these results can be generalized in a natural way, with some small variations, for the second case (2). On the other hand, fuzzy differential inclusions (FDI) have been used by several authors for solving FDE (for instance, see [2,9,18]) and it presents the advantage that the solutions obtained via FDI seem to be more intuitive than other solution using H-derivative (first form, see [9,14]). It is worthy to stress also that this interpretation has a disadvantage because we cannot speak about the fuzzy derivative. Instead, using fuzzy H-derivative in the second form (2), we overcome this difficulty and, moreover, the fuzzy solution and the solution via differential inclusions are identical (see Section 5). References [1] Abbod MF, Von Keyserlingk DG, Linkens DA, Mahfouf M. Survey of utilisation of fuzzy technology in medicine and healthcare. Fuzzy Set Syst 2001;120:331–49.
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