Fuzzy differential equations

Fuzzy Sets and Systems24 (1987) 301-317 North-Holland

FUZZY

DIFFERENTIAL

301

EQUATIONS

Osmo K A L E V A Tampere University of Technology, Department of Mathematics, P.O. Box 527, SF 33101 Tampere, Finland Received January 1985 Revised January 1986 This paper deals with fuzzy-set-valuedmappings of a real variable whose values are normal, convex, upper semicontinuous and compactly supported fuzzy sets in Rn. We study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzydifferential equation. Keywords: Fuzzy-set-valued mapping, Integration, Differentiation, Fuzzy differential equation.

1. Introduction A differential and integral calculus for fuzzy-set-valued, shortly fuzzy-valued, mappings was developed in recent papers of Dubois and Prade [6, 7, 8] and Puri and Ralescu [14]. The purpose of this paper is to study differential equations for fuzzy-valued mappings of a real variable. We restrict our analysis to mappings whose values are normal, convex, upper semicontinuous and compactly supported fuzzy sets in R n. T o make our analysis possible we have at first to generalize certain basic results of calculus, for instance the fundamental theorem of calculus, to fuzzy-valued mappings. Since a fuzzy-valued mapping is essentially a family of set-valued mappings we utilize results for set-valued mappings. Section 2 is devoted to notations and terminology and in Section 3 we discuss the measurability of fuzzy-valued functions. Then in Section 4 we define the integral of a fuzzy-valued function and establish some of its properties. The definition given here generalizes that of A u m a n n [1] for set-valued mappings. Furthermore, our definition is consistent with the definition of Dubois and Prade [6] under the commutativity condition. Recently Puri and Ralescu [15] have used the same definition for defining the expected value of a fuzzy random variable.1 For the concept of differentiability we adopt the H-differentiability of Puri and Ralescu [14], which generalizes the Hukuhara differentiability of set-valued mappings. In Section 5 we study the properties of differentiable mappings and finally in Section 6 we prove the existence and uniqueness of a solution to a fuzzy differential equation x ' ( t ) = f ( t , x ( t ) ) provided f satisfies a Lipschitz condition. 11 am indebted to one of the referees for bringing the reference [15] to my attention. 0165-0114/87/$3.50 (~) 1987, Elsevier Science Publishers B.V. (North-Holland)

O. Kaleva

302 2. Preliminaries

The symbol ~K(R") denotes the family of all nonempty compact convex subsets of R ". Define the addition and scalar multiplication in ~ r ( R ~) as usual. Then R~dstr6m [17] states that ~K(R ~) is a commutative semigroup under addition, which satisfies the cancellation law. Moreover, if a, ft • R and A, B • ~ K ( R n) then a ( A + B) = aA + aB,

a(ftA) = (aft)A,

1A = A

and if o~, ft/> 0 then ( a + f t ) A = a A + flA. In the following 0 denotes the closure of a set U c R ~. Let T = [a, b] c R be a compact interval and denote E ~ = {u : R"--> [0, 1 ] l u satisfies (i)-(iv) below}, where (i) u is normal i.e. there exists an x0 • R n such that U(Xo) = 1, (ii) u is fuzzy convex, (iii) u is upper semicontinuous, (iv) [u] ° = {x • R n ] u ( x ) > 0} is compact. For 0 < a ~< 1 denote [u] ~ = {x • R ~ ] u ( x ) 1> a}. Then from (i)-(iv) it follows that the a-level set [u] ~ • ~K(R ~) for all 0 ~< a ~< 1. If g : R ~ x R ~---) R" is a function then according to Zadeh's extension principle we can extend g to E n x E " ~ E" by the equation g(u, v ) ( z ) =

sup

z=g(x,y)

min(u(x), v ( y ) ) .

(2.1)

It is well known that [g(u, v ) ] ° ' = g ( [ u ] L [v] °~)

(2.2)

for all u, v • E n, 0 ~< a ~< 1 and g continuous (see [13]). For the addition Eq. (2.2) yields [u + v] ~ = [u]~+ [vl ~.

(2.3)

Recall that the real numbers can be embedded to E 1 by the correspondence c---~ ~(t) = {~

i f / = c, elsewhere.

Then analogously to (2.1) we can also generalize the multiplication by a real number and for any real number c we obtain [cu] °~= c[u]",

(2.4)

where 0 ~< a ~< 1 and u • E ~. In the sequel we need the following representation theorem. For the proof see [121.

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303

Theorem 2.1.

I f u • E ~ then (1) [u] ~ • ~K(R") for all 0 <~ o: <<-1, (2) [u] ~: c [u] °~1for 0 <<-o:1 <<-0{2 ~ 1, (3) if (ak) is a nondecreasing sequence converging to o: > 0 then

f-) [ulna. k>~l

Conversely, if ( A " I 0 ~< o: ~< 1} is a family of subsets of R" satisfying (1)-(3) then there exists a u E E n such that [ u ] ~ = A ~ for

0
and

[u] ° =

[,_J A ~ c A °. 0
Define D : E n × E" ~ R+ t_J {0} by the equation

O(u, v ) =

sup d([u]% [v]°~), 0~
where d is the Hausdorff metric defined in ~K(R"). Then it is easy to show that D is a metric in E". In addition, using results of [5, 15], we see that (E", D) is a complete metric space. Combining L e m m a 3 in [17] and Eq. (2.3) we obtain

D(u + w, v + w) = D(u, v)

(2.5)

for all u, v, w • E". Recall that the Hausdorff metric is defined as

d(A, B) = inf{e [A c N ( B , e), B = u ( a , e)}, where A, B • ~ r ( R " ) and N ( A , e) = {x • R" I Ilx -YI[ < e for some y • A } . Since AN(A, e) = N(AA, IZl e) for all Z • R, we see from the definition of D that

O(Zu, Xv) = I'~10(u, v)

(2.6)

for all u, v • E " , Z e R . Let (Ak) be a sequence in ~K(R") converging to A. Then T h e o r e m II-2 in [4] gives us an expression for the limit.

Theorem 2.2. If d(Ak, A )---> O as k---> ~ then A=

~~ k~l

[._J A m . m>-k

3. Measurability Definition 3.1. We say that a mapping F : T--> E n is strongly measurable if for all o: • [0, 1] the set-valued mapping F~ : T---> ~K(R n) defined by

F~(t) = [F(t)] ~

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O. Kaleva

is (Lebesgue) measurable, when by the Hausdorff metric d.

~K(Rn) is endowed

with the topology generated

In [15] Puri and Ralescu used a more general concept for measurability. They assumed, in our terminology, that for all or ~ [0, 1] the mapping F~ has a measurable graph, i.e. {(t, x) Ix 6 F~(t)} 6 M × ~ ( R " ) , where M denotes the o-algebra of measurable sets and ~ ( R " ) the Borel sets of R n"

However, taking into account Theorems III-2 and III-30 in [4], it turns out that, in the setting of this paper, this definition is equivalent to strong measurability. Lemma 3.1. If F is strongly measurable then it is measurable with respect to the

topology generated by D. Proof. Let e > 0 and u ~ E n be arbitrary. Then

T1 = {t I D(F(t), u) ~< e) = ~

{t I d(F,(t), [u] ") ~< e}.

o~e[O, 1]

But for all v 6 E n we have (see e.g. [10]) lim d([v] ~, [v]") = O, k---~a¢

whenever (ak) is a nondecreasing sequence converging to cr. Thus by the triangle inequality for the metric d we have

d(F~(t), [u] ~) ~< lim sup d(Fo<~(t), [u]°<~), where ~k,'~ c~ and consequently

{t I d(F~(t), [u] ~) <~e) = (-'] {t J d(F,~(t), [u] "~) ~< e). k~l

Thus

T1 = ~ {t I d(Fo
where {0 0 be arbitrary and to ~ T. By the continuity there exists a 6 > 0 such that

D(F(t), F(to)) < e whenever

I t - tol < 6.

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305

But by the definition of D we have d(F~(t), F~(to)) < e for all It - t0l < 6, so Fo~ is continuous with respect to the Hausdorff metric. Therefore F21(U) is open, and hence measurable, for each open U in ~K(Rn). [] If F maps T into E 1 then Fo~(t) is a compact interval, i.e. F~(t) = [~,°~(t),/~(t)]. We have the following:

Lemma 3.3. Let F:T---~E 1 be strongly measurable and [U~(t),/~ ~(t)] for o: ~ [0, 1]. Then Z ~ and I~o~are measurable.

denote

F,(t)=

Proof. Let c~ e [0, 1] be fixed. Then Fo~ is measurable and closed valued. Consequently it has a Castaing representation (see [4]), i.e. there exists a sequence (g~) of measurable selections such that for all t e T,

F~(t) = {g'~(t) l k = 1, 2 . . . . }. But from the definition of F,(t) it follows that )~o,= infg~ and # 6 = sup g~, which proves the lemma. []

4. Integrability A mapping F : T ~ E n is called integrably bounded if there exists an integrable function h such that Ilxll ~
or

[fTF(t) dt]~= fTF~(t) at ={f/(t)

d t l f : T - - ~ R n is a measurable selection for F~}

for all 0 < ct ~< 1. A strongly measurable and integrably bounded mapping F : T ~ E n is said to be integrable over T if ~7-F(t) dt ~ E n. Note that this definition is exactly the same as employed by [15]. In their terminology the integral is called the expected random variable. In the following instead of the integrals STF(t) dt, J'Tf(t) dt, F, J"f, etc. When the integral is taken over a subset S c T we The following theorem due to Puri and Ralescu shows that are integrable.

Puri and Ralescu value of a fuzzy etc. we will write will write ~s F. certain mappings

Theorem 4.1. If F : T ~ E n is strongly measurable and integrably bounded then F is integrable.

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O. Kaleva

Proof. See [15].

Remark 4.1. Since for all u ~ E n, lim d([u] °, [u] ~k) = 0, k----~oo

whenever (irk) is a nonincreasing sequence converging to zero (see [10]), then we have (see [15, Theorem 2.5 and Remark])

f o) =o. Since

°,

we conclude that [f F] ° = f Fo. Remark 4.2. If F : T - - - ~ E 1 is integrable then in the view of Lemma 3.3 f F is obtained by integrating the re-level curves, that is

where [F(t)] ~ = [),~(t), ~ ( t ) l ,

c~ e [0, 11.

Remark 4.3. Let F be integrable, cr e [0, 1] and {o~l n = 1, 2 , . . . } be a Castaing representation for Fo~. Since J" F~ is convex and closed and includes f cr~ for all n = 1, 2 . . . . then clearly

where ~--d(A) denotes the closed convex hull of A. Corollary 4.1. I f F : T --~ E " is c o n t i n u o u s then it is integrable. Proof. By Lemma

3.2, F is strongly measurable. Since Fo is continuous, Fo(t) e ~ r ( R n) for all t e T and T is compact then (._Jt~rFo(t) is compact (see [2, Theorem 3 p. 116]). Thus F is integrably bounded and the conclusion follows from Theorem 4.1. [] The rest of this section is devoted to elementary properties of the integral. Theorem 4.2. L e t F : T ~ E n be integrable a n d c ~ T. T h e n =

+

F.

Fuzzy differential equations

307

Proof. Clearly the integrability of F implies that F is integrable over any subinterval of T. Now let or e [0, 1] and f be a measurable selection for F~. Since f b f = ~ f + j'~f then we clearly have

On the other hand let z = Sc gl + ~ g2, where gl is a measurable selection for F~ in [a, c] and g2 is a measurable selection for F~ in [c, b]. Then f, defined by

f(t)

~g,(t) = tg2(t)

if t ~ [a, c], if t e (c, b],

is a measurable selection for F~ in T and

Hence

and the theorem is proved.

[]

Corollary 4.2. If F: T---~ E n is continuous then G(t) = f t F is Lipschitz continuous on T. Proof. Let s, t e T and assume that s > t. Then by Theorem 4.2 and Eq. (2,5) we have D

([ f]) (ff ,o) F,

F =D

,

where 0(t) = ( ;

ift = 0 , elsewhere.

Since (_J,,rFo(t) is compact (see the proof of Corollary 4.1) then there exists an M > 0 such that IIx II ~
which was to be proved.

[]

Theorem 4.3. Let F, G : T--~ E" be integrable and )~ ~ R. Then (i) f F + G = I F + I G , (ii) f ZF = )~ f F, (iii) D(F, G) is integrable, (iv) o ( f F, f G) <~f D(F, G).

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308

Proof. Let c~ e [0, 1]. Since Fo~ and G~ are compact-convex-valued it follows from Debreu [5] that the integrals j" Fo, and f G~ are in fact Bochner integrals. Hence applying Eq. (2.3) we obtain

which proves (i). A similar reasoning yields (ii); instead of Eq. (2.3) use Eq. (2.4). Recall that the Hausdorff metric can be written in the form (see [4])

d(a, B)=max(sup inf [Ix - y l l , sup inf Ilx - y l l ] /. \xEA

y~B

x~B

(4.1)

y~A

Now for (iii) let { o ~ l n = 1, 2 . . . . } (resp. {p~ln = 1, 2 . . . . }) be a Castaing representation for F~ (resp. G~). Applying Eq. (4.1) we get

d ( f ~(t), G~(t)) = max(sup inf Ho~(t) - p~(t)l I, sup inf lip~(t)- o~(t)l[], / \n~l

k~l

n~l

k~l

which is measurable. Thus

D(F(t), G(t)) = sup d(F~(t), G~(t)), k~l

where {trk I k = 1, 2, . . . } is dense in [0, 1], is measurable. Furthermore

O(F(t), G(t)) <~O(F(t), O) + O(G(t), 0) ~ hi(t) + h2(t), where h~ and h2 are integrable bounds for F and G respectively. Thus Proposition 7 in [16, p. 82] gives us (iii). Finally, from [5, p. 366] we deduce

and consequently

o(fF, fo)~

sup fd(F~, G~)<~ f sup d(F~, G ~ ) = fD(F, G).

a:6[O, 1] J

Theorem 4.4. If F : T--->E"

(t, o0--*diam

J a'~[O, 1]

is integrable then the real function

]

F , t e T , o~e[O, 1],

is nondecreasing w.r.t, t on T and nonincreasing w.r.t, oc on [0, 1]. Proof. Let q, t2 e T with ta < t2. According to Theorem 4.2

I171 --

E ,?I

which proves the first assertion. The second is trivial.

[]

[]

Fuzzy differential equations

309

Example 4.1. Let A • E ~ and define F : [0, t]---> E ~ by F(s) = A for all 0 ~
~F = tA. Clearly tA c fto F. Conversely, let o: • [0, 1] and choose any f f • .f F~. Then J ' f can be expressed as a limit of a sum

Sn = ~ (ti -- ti-1)f(ri), i=l

where {(r/, [ti-1, ti)) I i = 1 . . . . . n} is a belated partition of [0, t) with measure/~n (see [11]). Since f(zi) • [A] ~ for all i = 1 . . . . . n and [A] ~ is convex it follows that Sn • # n [ A ] ~ for all n. As we pass to the limit then #~/~t and consequently l i m ~ d(t[A]% #~[A] ~) = 0. It follows that f f • t[A] ~ and hence [.toF c t A . Example 4.2. Define F : T ~ E" by the equation

F~(t) = FI [a~(t), bT(t)],

tr • [0, 1],

i=1

where aT, bT: T---> R are integrable, a~'(t)<-b~'(t) for all t • T and for each t • T aT(t), bT(t) are left continuous and aT(t) (resp. bT(t)) is nondecreasing (resp. nonincreasing) w . r . t . a . By T h e o r e m 2.1, F(t) • E ~. Now for all a~ • [0, 1] we have

fFo=~[fa~,

fb~]

and again by T h e o r e m 2.1 we see that F is integrable. The condition (3) follows from Lebesgue's convergence theorem, (1) and (2) are trivially valid.

5. Differentiability Let x, y • E ~. If there exists a z • E n such that x = y + z then we call z the H-difference of x and y, denoted x - y. The following definition is due to Puri and Ralescu [14]. Definition 5.1. A mapping F : T---> E n is differentiable at to • T if there exists a F'(to) • E n such that the limits lim F(to + h ) - F(to)

h--,o+

h

and

lim F(to) - F ( t o - h)

h--,o+

h

exist and equal to F'(to). H e r e the limit is taken in the metric space (E n, D). At the end points of T we consider only the one-sided derivatives.

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O. Kaleva

Remark 5.1. From the definition it directly follows that if F is differentiable then the multivalued mapping Fo~ is Hukuhara differentiable for all 0: e [0, 1] (for the definition see [9]) and DF~(t) = [F'(t)] ~.

(5.1)

Here DF~ denotes the Hukuhara derivative of F~. The converse result doesn't hold, since the existence of Hukuhara differences [ x ] ~ - [ y ] % a~e [0, 1], does not imply the existence of H-difference x - y . However, for the converse result we have the following:

Theorem 5.1. Let F: T---~ E n satisfy the assumptions: (a) for each t • T there exists a fl > 0 such that the H-differences F(t + h) - F(t) and F(t) - F ( t - h ) exist for all 0 <~h < fl; (b) the set-valued mappings F~, 0:6[0, 1], are uniformly Hukuhara differentiable with derivative DFo,, i.e. for each t ~ T and e > 0 there exists a 6 > 0 such that d((Fo,(t + h) - Fo~(t))/h, DFo,(t)) < e

(5.2)

d((Fo,(t) - F~(t - h))/h, DFo~(t)) < e

(5.3)

and for all 0 <~h < 6 and 0: ~ [0, 1]. Then F is differentiable and the derivative is given by Eq. (5.1).

Proof. Consider the family {DF~(t)[0:e[O, 1]}. By definition DF~(t) is a compact, convex and nonempty subset of R n. If 0:1 ~ 0:2 then by assumption (a), goq(t

-1-

h) - Fo, l(t) ~ Foc2(t q- h)

-

F~2(t )

for 0 ~ h < fl

and consequently Dro~(t) = D f ~2(t).

(5.4)

Let 0: > 0 and (Ok) be a nondecreasing sequence converging to 0:. For e > 0 choose h > 0 such that Eq. (5.2) holds true. Then the triangle inequality yields 1 d(DF~(t), DFo,,(t)) <- 2e + -~ d(F~(t + h) - Fo,(t), Fo,,(t + h) - Fo~k(t)). By assumption (a), the rightmost term goes to zero as k ~ oo and hence lim d(DF~(t), DF~,(t))= O. k----~ oo

Now by Theorem 2.2 and Eq. (5.4) we have O F o~(t) = 0 OFo,~(t). k~l

Fuzzy differential equations

311

If Oc = 0 t h e n using E q . (3.1) in [10] we d e d u c e as b e f o r e that lim

d(DFo(t), DF~,(t))= O,

k.-.~ oo

w h e r e (0~k) is a n o n i n c r e a s i n g s e q u e n c e converging to zero, and c o n s e q u e n t l y OFo(t) = I,] DFo~k(t). k>-i

T h e n f r o m T h e o r e m 2.1 it follows that t h e r e is an e l e m e n t u 6 E ~ such that [u] °~= DF~,(t)

for cr • [0, 1].

F u r t h e r m o r e , let t • T, e > 0 and 6 > 0 be as in (b). T h e n D ( ( F ( t + h) - F ( t ) ) / h , u) = sup d((F~(t +

h) -

Fo~(t))/h, DF~(t)) < e

0
for all 0 ~< h < ~ a n d similarly for D ( ( F ( t ) - F ( t - h ) ) / h , u). H e n c e F ' ( t ) = u and we h a v e the t h e o r e m . []

Theorem 5.2. L e t F : T---> E ~ be differentiable.

Denote

F~(t) = [f~(t), go~(t)],

o: • [0, 1]. Then f~ a n d g~ are differentiable and [ F ' ( t ) ] '~= [ f ' ( t ) , g ' ( t ) ] .

Proof. N o w [F(t + h) - F ( t ) ] ~ = [fo~(t + h) - f~(t), g~(t + h) - g~(t)] and similarly for [ F ( t ) the t h e o r e m . []

F(t-

h)] ~. Dividing by h and passing to the limit gives

Theorem 5.3. L e t F : T--> E ~ be differentiable on T. I f tl, t 2 E T with t I ~ t 2 then there exists a C • E ~ such that F(t2) -- F ( t l ) + C.

Proof. F o r each s • [tl, t2] t h e r e exists a 6 ( s ) > 0 such that the H-differences F ( s + h) - F ( s ) and F ( s ) - F ( s - h) exist for all 0 ~< h < 6(s). T h e n we can find a finite s e q u e n c e tl = s l < s 2 < ' - ' < s n = t2 such that the family {Is,= ( s i - 6(si), si + 6 ( s 3 ) I i = 1 , . . . , n } covers [tl, t2] a n d Is, M Is,+~ =k O. Pick a vi • Is, fq Is .... i = 1 . . . . . n - 1, such that s~ < v~ < si+~. T h e n F(Si+l) = F ( v i ) + n l = F(si) + B2 + 81 = F(si) + Ci,

i = 1. . . . .

n - 1,

for s o m e B~, B2, Ci • E ~. H e n c e n--1

F(t2) = F ( t O + ~

Ci

=

F ( t l ) + C.

[]

i=1

A s an i m m e d i a t e c o n s e q u e n c e we have:

Corollary 5.1. I f F : T--> E n is differentiable on T then f o r each te~ [0, 1] the real function t--> d i a m [ F ( t ) ] ~ / s nondecreasing on T.

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Theorem 5.4. If F : T ~ E ~ is differentiable then it is continuous. Proof. Let t, t + h e T with h > 0. Then by Eqs. (2.5), (2.6) and the triangle inequality we have O ( F ( t + h), F ( t ) ) = D ( F ( t + h) - V(t), ()) <<-h D ( ( F ( t + h) - F ( t ) ) / h , F ' ( t ) ) + h D ( F ' ( t ) , 0),

where h is so small that the H-difference F ( t + h ) - F ( t ) exists. By the differentiability the right-hand side goes to zero as h ~ 0 + and hence F is right continuous. The left continuity is proved similarly. [] A direct consequence of Eqs. (2.5) and (2.6) is also: Theorem 5.5. If F, G: T---~E ~ are differentiable and ~ R

then (F + G ) ' ( t ) =

F'(t) + G'(t) and (XF)'(t) = ;~F'(t). Theorem 5.6. Let F : T---~E n be continuous. G ( t ) = S~ F is G ' ( t ) = F(t).

differentiableand

Then f o r all t 6 T the integral

ProoL Note that according to Corollary 4.1 F is integrable. Now for h > 0, Theorem 4.2 gives G ( t + h) - G(t) =

~Jtt+h

F.

Let e > 0 be arbitrary. Then by Example 4.1, Eq. (2.6), Theorem 4.3 and the continuity of F we have O ( ( G ( t + h) - G ( t ) ) / h , V(t)) =

-hlD(~r+h-Jt F(s) ds, J,['t+hF ( t ) dS)

1 ~t+h ~--< D ( F ( s ) , F(t)) ds < e h ~t

for all h > 0 sufficiently small. Hence limh--,0+ ( G ( t + h) - G ( t ) ) / h = F ( t ) and similarly limh--,0÷ ( G ( t ) - G ( t - h ) ) / h = F(t), which proves the theorem. [] Theorem 5.7. Let F : T--* E n be differentiable and assume that the derivative F' is integrable over T. Then for each s ~ T we have F ( s ) = F ( a ) + fa~F '. Proof. Let tr~ [0, 1] be fixed. We shall prove that Fo~(s) = F~(a) +

DFo,

(5.5)

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313

where DF~ is the Hukuhara derivative of F~, from which the theorem follows by Eq. (2.5) and Remark 5.1. Recall that the supporting functional 6(-, K ) : R " ~ R of K • ~K(R") is defined by 6(a, K) = sup{a • k I k • K}, where a • k denotes the usual scalar product of a and k. If K1, Ka • ~K(R") then Theorem 11-18 in [4] gives us the equation

d(K,, K2)

=

sup I6(a, K1) - 6(a, K2)I.

(5.6)

Ilall=l

Now let t, t + h • T with h > 0 so small that the H-difference F(t + h ) - F(t) exists. Then by Theorem 11-17 in [4] we have

6(x, Fo~(t + h) - Fo~(t)) = 6(x, F~(t + h)) - ~(x, F~(t)) for all x • R ~, a~ • [0, 1] and consequently 6(x, (Fo,(t + h) - F~(t))/h) = (6(x, F~(t + h)) - 6(x, Fo,(t)))/h.

(5.7)

Then by the differentiability of F~ and Eqs. (5.6) and (5.7) we obtain that 6(x, F~(t)) is right differentiable and the right derivative equals to 6(x, DF~(t)), where x is an arbitrary element of the surface of the unit ball S in R ~. Applying a similar reasoning for h < 0 we conclude that for all x • S , 6(x, F~(t)) is differentiable on T and

d 6(x, Fo~(t)) = 6(x, DF,(t)). dt Since DF~(t) is compact and convex it can be expressed as an intersection of all closed half spaces containing it, i.e.

DF~(t) = ["1 nx, xeS

where Hx = {z • R n Ix • z <~6(x, DFo,(t))}. Thus DFo~(t) equals to the derivative of the set-valued mapping F~ defined by Bradley and Datko [3]. The equality (5.5) now follows from [3, Theorem 3.5]. []

Example 5.1. Let A • E n be fixed and r: T---~R n a differentiable function. Consider a mapping F:T---~ E n defined by F ( t ) = ~ + A, where as usual the membership function of r(t) equals 1 at r(t) and zero elsewhere. Thus F(t) is a fixed fuzzy set moving along a differentiable curve r in R". Then clearly F is differentiable and F'(t) = r'(~). Furthermore, as a continuous mapping it is integrable and f F = "f~ + (b - a)A. Theorem 5.7 allows us to derive a mean value theorem for fuzzy mappings.

Theorem 5.8. Let F : T--~ E" be continuously differentiable on T. Then D(F(b ), F(a)) ~ (b - a) sup D(F'(t), 8). teT

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O. Kaleva

Proof. By means of Theorems 5.7 and 4.3 we obtain []

Rolle's theorem holds also true for a fuzzy mapping F:T--+ E 1.

Theorem 5.9. Let F: T-->E 1 be differentiable on T. If F ( a ) = F(b) then there exists a to ~ T such that F'(to) = O. Proof. By Theorem 5.3, F ( t ) = F ( a ) + R ( t ) for some R ( t ) ~ E 1. Since F ( b ) = F(a) then diam F~(b)= diam F~(a) for all cee [0, 1] and hence by Corollary 5.1 diamF,,~) is constant on T. It follows that diamR~,(t)= 0 and consequently R(t) = ~(t~ for some r(t) e R. Since F is differentiable then also r is differentiable on T. Furthermore r(a) = r(b) = 0 and so there exists a to e T such that r'(to) = O. But by Example 5.1, we have that F'(to) = O. []

6. Fuzzy differential equations Let ac > 0 and assume that f : T × E"--+ E" is continuous. Consider the initial value problem x'(t) = f ( t , x(t)),

x(a) =x0.

(6.1)

From Theorems 5.4, 5.6 and 5.7 it immediately follows:

Lemma 6.1. A mapping x : T--> E n is a solution to the problem (6.1) if and only if it is continuous and satisfies the integral equation x(t) = Xo +

f ( s , x(s)) ds

(6.2)

for all t ~ T. Note that we cannot extend Lemma 6.1 for t < a . In fact, if x(t) satisfies (6.2) then diam[x(t)]~>-diam[xo]% ol ~ [0, 1], and hence in the view of Corollary 5.1, x(t) is not in general differentiable at t < a. If f is Lipschitz continuous then the problem (6.1) has a unique solution on T. Furthermore, the solution depends continuously on the initial value. It would be interesting to know whether the mere continuity of f implies the existence of a solution to (6.1).

Theorem 6.1. Let f : T × E'--+ E ~ be continuous and assume that there exists a k > 0 such that D ( f ( t , x), f ( t , y)) ~ kD(x, y) for all t ~ T, x, y ~ E ~. Then the problem (6.1) has a unique solution on T.

Fuzzy differential equations

315

Proof. Denote by C(J, E ~) the set of all continuous mappings from J to E ", where J is an interval in R. We metricize C(J, E ~) by setting H(~, ~p) = sup D(~(t), ~p(t)) t~J

for ~, ~p e C(J, E"). Since (E", D) is a complete metric space, a standard proof applies to show that also C(J, E ") is complete. Now let (tl, y) • T x E" be arbitrary and r / > 0 be such that r/k < 1. We shall show that the initial value problem

x'(t) = f ( t , x(t)),

X(tl) = y,

(6.3)

has a unique solution on the interval I = [h, t~ + r/]. For ~ • C(I, E ~) define G~ on I by the equation

G~(t) = y +

~(s)) ds.

Then by Corollary 4.2 G~ • C(I, E"). Furthermore, by Theorem 4.3 and the Lipschitz continuity of f we have

H(G~, G~p) = sup O

(s, ~(s)) ds,

(s, lp(s)) ds

tel

f f

tl+~

<<-

D(f(s,

f(s,

Jl 1 tl+rl

v,,(s)) ds

nkH( ,

for all ~, ~ • C(I, E~). Hence by Banach's contraction mapping theorem G has a unique fixed point, which by Lemma 6.1 is the desired solution to the problem (6.3). Express T as a union of a finite family of intervals Ik with the length of each interval less than r/. The preceding paragraph guarantees the existence of a unique solution to (6.3) on each interval Ik. Piecing these solutions together gives us the unique solution to (6.1) on the whole interval T. []

Corollary 6.1. Let f be as in Theorem 6.1. Denote by x(t, Xo) the solution to the initial value problem (6.1) with initial value Xo. Then there exists a real number q such that H(x(., Xo), x(., Yo)) <- qO(xo, Yo) for any Xo, Yo • E ~.

Proof. Let Ik, k = 1. . . . . m, be as in the proof of the preceding theorem. Choose any Ik = [tk, tk+l] and denote u = x(tk, Xo), v = x(tk, Yo) and Gu~(t) = u +

~(s)) ds.

O. Kaleva

316

Then in the space

C(Ik, E n) we

have

H(Gux(., Xo), G,x(., Xo)) = D(u, v) and

H(G,,x(', Xo), G~x(', Xo)) ~ H(G,,x(', Xo), G~,x(', Xo)) + ~ H(Gi,-lx( ", Xo), G~x(', Xo)) i=2

-< (1 + c + c 2 + . . .

+ cn-1)O(u, Y ) ,

where c = r/k < 1 is as in the preceding theorem. Since G,x(., Yo), as n ~ ~ we have

G~x(., Xo)

converges to

H(Gux(., Xo), G,x(., Yo)) <~l~cD( u, v). Now a simple induction argument gives the result with q = (1 - c ) - ' . Example 6.1. Let

A, B:T---~E 1 be

continuous. Define f : T

[]

× E1---~E1 by

f(t, x) = A(t)x + B(t), where the multiplication in E 1 is defined by Eq. (2.1). If Ix] ~ = [x~, x~] then

Ao~(t) =

[a~'(t), a~(t)] and

[A(t)x] °~= [min(a~(t)x~, a~(t)x~, a~(t)x~, a[(t)x~), max(a~(t)x~, a~(t)x~, a~(t)x~, a~(t)xzO]. Now by the proof of Corollary 4.1 the functions la~[ and la~l are bounded on T by a constant k independent of o~. Then a straightforward calculation gives that f(t, x) satisfies the assumptions of T h e o r e m 6.1 and hence the initial value problem

x'(t) =A(t)x(t) + B(t),

x(a)

=Xo,

has a unique solution on T.

References [1] R.J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965) 1-12. [2] C. Berge, Espaces Topologiques; Fonctions Multivoques (Dunod, Paris, 1959). [3] M. Bradley and R. Datko, Some analytic and measure theoretic properties of set-valued mappings, SIAM J. Control Optim. 15 (1977) 625-635. [4] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions (Springer, Berlin, 1977). [5] G. Debreu, Integration of correspondences, in: Proc. Fifth Berkeley Syrup. Math. Statist. Probab., Vol. 2, Part 1 (Univ. California Press, Berkeley, CA, 1967) 351-372. [6] D. Dubois and H. Prade, Towards fuzzy differential calculus - Part 1, Fuzzy Sets and Systems 8 (1982) 1-17. [7] D. Dubois and H. Prade, Towards fuzzy differential calculus - Part 2, Fuzzy Sets and Systems 8 (1982) 105-116.

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[8] D. Dubois and H. Prade, Towards fuzzy differential calculus - Part 3, Fuzzy Sets and Systems 8 (1982) 225-234. [9] M. Hukuhara, Integration des applications measurables dont la valeur est un compact convexe, Funkcialaj. Ekvacioj. 10 (1967) 205-223. [10] O. Kaleva, On the convergence of fuzzy sets, Fuzzy Sets and Systems 17 (1985) 53-65. Ill] E.J. McShane, Stochastic differential equations, J. Mulitvariate Anal. 5 (1975) 121-177. [12] C.V. Negoita and D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis (Wiley, New York, 1975). [13] H.T. Nguyen, A note on the extension principle for fuzzy sets, J. Math. Anal, Appl. 64 (1978) 369-380. [14] M.L. Puri and D.A. Ralescu, Differentials for fuzzy functions, J. Math. Anal. Appl. 91 (1983) 552-558. [15] M.L. Puri and D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409-422. [16] H.L. Royden, Real Analysis (Macmillan, London, 1968). [17] H. R~dstr6m, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952) 165-169.