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Existence and uniqueness of cauchy problem for fuzzy differential equations under dissipative conditions
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ELSEVIER Computers and Mathematics with Applications 51 (2006) 1483-1492 www.elsevier.com/locate/camwa
E x i s t e n c e and U n i q u e n e s s of Cauchy P r o b l e m for Fuzzy Differential Equations under Dissipative Conditions SHIM SONG AND CHENG WU Department of Automation, Tsinghua University Beijing 100084, P.R. China shij is~mail, t s i n g g u a , edu. cn XIAOPING XUE Department of Mathematics, Harbin Institute of Technology Harbin 150001, P.R. China (Received November 2005; accepted December 2005)
Abstract--The
existence theorem of Peano for the fuzzy differential equation, x'(t) = f(t,x(t)),
x(to) = xo,
does not hold in general except in the special case where the dimensional [1] or f is assumed to be continuous and bounded conditions which guarantee the existence theorem for Peano theorem of approximate solutions to above Cauchy problem. served.
fuzzy number space (E '~, D) is finite [2]. In this paper, the dissipative-type are specified based on the existence @ 2006 Elsevier Ltd. All rights re-
K e y w o r d s - - E x i s t e n c e and uniqueness theorem, Fuzzy differential equations, Dissipative-type conditions, Fuzzy number space (E '~, D).
1. I N T R O D U C T I O N Various i n v e s t i g a t o r s have s t u d i e d tile e x i s t e n c e t h e o r e m of P e a n o for fuzzy differential equations [1 7] on t h e m e t r i c space ( E ~, D) of n o r m a l , convex, u p p e r s e m i c o n t i n u o u s , a n d c o m p a c t l y s u p p o r t e d fuzzy sets in R ~. T h e m e t r i c space ( E ~, D) has a linear s t r u c t u r e b u t is not a v e c t o r space, it can be i m b e d d e d i s o m e t r i c a l l y and i s o m o r p h i e a l l y as a c o n v e x cone in a B a n a e h space C([0, 1], C ( S ' ~ - I ) ) , w h e r e S '~-1 is t h e unite sphere in R ~ [8].
K a l e v a [1] has s h o w n t h a t t h e
P e a n o t h e o r e m does n o t hold b e c a u s e t h e m e t r i c space ( E ~, D ) g e n e r a l l y is n o t locally c o m p a c t . It was s h o w n in [3,4] t h a t if f is c o n t i n u o u s and satisfies t h e L i p s c h i t z c o n d i t i o n or g e n e r a l i z e d L i p s c h i t z c o n d i t i o n w i t h respect x, t h e n t h e r e exists a u n i q u e s o l u t i o n to t h e C a u c h y p r o b l e m on This work is supported by Tile National Natural Science Foundation of China (Grant No. 60574077, 10571035) and Sasal Research Foundation of Tsinghua University (Grant No. JC2001029) and 985 Basic Research Foundation of Information Science and Technology of Tsinghua University 973 project (Grant No. 2002cb312205). The authors of this paper would like to thank E.S. Lee for his suggestions and help. 0898-1221/06/$ - see front matter @ 2006 Elsevier Ltd. All rights reserved. doi: 10.1016/j.camwa.2005.12.001
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S. SONG e~ al.
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(E *~, D). In addition, Nieto proved the existence theorem of Peano for fuzzy differential equation on (E ", D) if f is continuous and bounded. We also note that there are some papers that claim the Peano-like theorem by restricting to continuous set-valued functions f defined on some locally compact subsets of (E n, D) [11, or consider other metrics on the space (E ~, D) [9]. But unfortunately, Friedman, Ma and Kandel [7] had pointed out the mistakes of these papers. Nevertheless, the question of what conditions on f guarantee the existence of solution is still an open question. In [5], the existence theorem of the Cauchy problem for fuzzy differential equations is obtained under some compactness-type conditions. Moreover, in this paper, the existence and uniqueness theorems for Peano of the cauchy problem for fuzzy differential equations are obtained under the dissipative-type conditions by applying the existence theorem of approximate solutions to the Cauchy problem given in [5].
2. P R E L I M I N A R I E S Let Pk(R n) denote the family of all nonempty compact convex subsets of R n and define the addition and scalar multiplication in Pk(R n) as usual. Denote E n : { u : R n --~ [0, 1] [ u satisfies (i)-(iv) below}, where
(i) u is normal, i.e., there exists an x0 G/{~ such that u(x0) = 1, (ii) u is fuzzy convex, i.e.,
u(Ax + (1 -- A)y) > min{u(x),u(y)},
for any x , y C R n and 0 < A < 1,
(iii) u is upper semicontinuous, (iv) [ul ° =cl{x • R n l u(x) > 0} is compact. For 0 < a <_ 1, denote b] ° : {x ¢ R ~ l u(x) _> ~}.
Then, from (i)-(iv), it follows that the a-level set [u]a • Pk(R '~) for all 0 < a < 1. According to Zadeh's extension principle, we have addition and scalar multiplication in fuzzy number space E n as usual. Define D : E ~ x E "~ ~ [0, co) by the equation,
D(u,v)=
sup d([u]~,[v]'~), O
where d is the Hausdorff metric defined in Pk(R'~). Then, it is easy to see that D is a metric in E '~ and has the following properties [10]. (1) (E ~, D) is a complete metric space; (2) D(u + w,v + w) = D(u,v) for all u , v , w • En; (3) D(ku, kv) IklD(u,v) for all u,v • E '~ and k •/{. =
In the following, we recall some main concepts and properties of integrability and H-differentiability for the fuzzy-set functions [3,11]. In this paper, we denote
T = [to, to + a] c R(~ > 0) be a compact interval. The fuzzy-valued function F : T ~ E n is called strongly measurable, if for every a • [0, 1] the set-valued function F~ : T -~ Pk(R ~) defined by
f ~ (t)
=
IF (t)] ~ ,
is Lebesgue measurable, where Pk(R n) is endowed with the topology generated by the Hausdorff metric d.
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1485
A f u n c t i o n F : T --* E ~ is called i n t e g r a b l y b o u n d e d , if there exists a n i n t e g r a b l e f u n c t i o n h such t h a t II~ll -< h(t) for all x • Fo(t). DEFINITION 2.1. Let F : T ~ E ~. T h e integral of F over T, denoted by f r F(t) dt, is defined
levehvise by the equation, [fTF(t)dt]~
= / T F ~ ( t ) dt
= { /rf(t)
dt t f : T--~ Rn is a measurable selection for F~,} ,
for all 0 < c~ <_ 1. A s t r o n g l y m e a s u r a b l e a n d i n t e g r a b l y b o u n d e d f u n c t i o n F : T ~ E n is said to b e i n t e g r a b l e over T if f r F ( t ) dt • E ~. F r o m [3], we know t h a t i f F : T ~ E ~ is c o n t i n u o u s , t h e n it is integrable. Let x, y • E n. If there exists a z C E ~ such t h a t x = y + z, t h e n we call z t h e H-difference of x a n d y, it is d e n o t e d b y x - y. DEFINITION 2.2. A function F : T ---+E ~ is differentiable at tl • T, if there exists a F ' ( t l ) • E "~ such that the limits, lira F ( t l + h ) -
h---*+O
F(tl)
h
and lim F ( t l ) - F ( t l - h) h~+0 h '
exist and are equal to F ' ( t l ) . Here, t h e limits are t a k e n in t h e m e t r i c space ( E ~, D). At t h e e n d p o i n t of T, we consider o n l y one-sided derivatives. If F : T --~ E n is differentiable at t l E T, t h e n we say t h a t F ' ( Q ) is t h e fuzzy derivative of F(t) at t h e p o i n t tl. T h e i n t e g r a b i l i t y a n d H - d i f f e r e n t i a b i l i t y properties of t h e fuzzy-valued f u n c t i o n s c a n be referred
to [3]. REMARK 2.1. We d e n o t e a c o n t i n u o u s fuzzy-valued f u n c t i o n f : T x t 2 -+ E ~ b y f C C[T×f~, E n ] , where t2 C E n is a n o p e n set. In a d d i t i o n , if F : T ~ E ~ is differential a n d F'(t) is c o n t i n u o u s , t h e n we d e n o t e F C C 1 [T, E~]. PROPOSITION 2.1. A s s u m e that f E C[T x f~,ET~]. A function x : T ~ f~ is a solution to the pro blem x' - f (t, x), x (to) = xo if and only if it is continuo us a n d satisfies the integral eq uation,
x(t) = x 0 +
f:
/ ( s , x ( s ) ) ds,
for all t • T [3]. As preliminaries, we recall further t h e e m b e d d i n g t h e o r e m of fuzzy n u m b e r space ( E ~, D ) , which is a g e n e r a l i z a t i o n of t h e classical R £ d s t r h m e m b e d d i n g t h e o r e m in [12]. PROPOSITION 2.2. There exists a real Banach space X such that E n can be embedding as a
convex cone C with vertex 0 into X . The following conditions hold true [1]. (i) The embedding j is isometric. (ii) Addition in X induces addition in E ~. (iii) Multiplication by nonnegative real number in X induces the corresponding operation in
E n" (iv) C - C is dense in X . (v) C is closed.
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REMARK 2.2. If x : T ~ E n is differentiable at tl C T. Then, it is easy to see that (jx) (t) = j (x ( t ) ) : T -~ X is Frechet differentiable at tl • T and ( j x ) ' ( t l ) = j ( x ' ( t l ) ) , where j is the embedding in Proposition 2.2. Finally, in this section, we will state several comparison theorem on classical ordinary differential equation [13], and some well-known results on abstract Banach space. PROPOSITION 2.3. Let G C R 2 be open set and g E C[G, R], (to, uo) E G. Suppose that r(t) is the m a x i m u m solution to the initial value problem, u'=g(t,u),
u(to) = u o ,
(2.1)
and its largest interval of existence of right solution is [to, to + a). I f [to, tl] C [to, to + a), then there exists an Co > 0 such that the m a x i m u m solution r(t, ~) to the initial value problem, ~' - g (t, ~ ) + ~,
~ ( t o ) = ~o + ~,
exist on [to, tl] whenever 0 < ~ < ~o, and ~(t, ~) ~niformly converges to ~(t) on [to, t~] as c -~ +0. PROPOSITION 2.4. Let G C R 2 be an open set, g E C[G, R], (to, uo) • G. Suppose that the m a x i m u m solution to the initial value probIem (2.1) is r(t) and its largest interval of existence of right solution is [to, to + a ). If re(t) E C[(to, to + a ), R] satisfies ( t, re(t)) c G for all t E [to, to + a), m(to) <_ uo, and
D + . ~ (t) < g (t,.~ (t)),
Vt • [to, to + ~) \ r,
where D+ is one of the four Dini derivatives, F at most is a countable set on [t0,t0 + a). Then, we must have [13] rn (t) _< r (t), for all t • [to, to + a). For the Banach space X which appears in Proposition 2.2, we denote C[T, X] the space of a11 abstract continuous functions from T into X , its norm is given by
Ilxll = max IIx(t)II, tET
Vx • c [ r , X].
For H C C[T, X], we denote U(t) = {z(t) l x • H } C X . PROPOSITION 2.5 ASCOLI-ARZELA. A set H C C[T, X] is relative compact if and only if H is equicontinuous and for any t • T, H (t) is relative compact set in X. PROPOSITION 2.6. MEAN VALUE THEOREM. Suppose x • C[T, X], F C T be at m o s t countable set, and for any t • T \ P , x ( t ) is differential and there exists M > 0 such that IIx'(t)[I _< M. Then, we have IIx (to +p) - x(to)[I <_ Mp. 3.
EXISTENCE THEOREMS
PROBLEM DISSIPATIVE-TYPE
AND OF
UNIQUENESS CAUCHY
UNDER
THE
CONDITIONS
In this section, by using existence theorcm of approximate solutions and the embedding results on fuzzy number space (E ~, D), we give the existence and uniqueness theorem under dissipativetype conditions for the Cauchy problem of the fuzzy differential equations. Let p, q be two positive real numbers. Denote B(xo, q) = {x C E ~ ] D ( x , x0) _< q}, Ro = [to,to + p ] x B ( x o , q ) . At first, we recall the existence theorem in [5], for the approximate solutions to the Cauehy problem of fuzzy differential equations.
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Existence and Uniqueness LEMMA 3.1. Suppose f E C[Ro, E~], here Ro = [to,to + p ] x B(xo,q), B(xo, q) = {x ~ E n IN(x, xo) < q},
and D ( f (t, x ) , O) < M,
for all (t, x) E Ro.
Then, for any given ~ > O, there exists x~ E C 1[[to, to + a~], B(xo, q)] (where a~ = rain{p, q / ( M + ¢)}) such that jx'~(t)=jf(t,x~(t))+y~(t),
z~(to)=Xo,
Iiy~(t)il_
vtE[to,to+a~],
where
y~(t) e c[[to, to + ~,], x], and j is the isometric embedding from (E ~, D) onto its r a n g e in the Banaeh space X as in [9]. DEFINITION 3.1. Assume that V : [to, to + p] x B(xo, q) x B(xo, q) -~ [0, oc)
is a continuous function provided
(i) V(t, z, x) - o, (3.1)
V ( t , x , y ) > O,
x • y,
for all t E [to, to + p], x, y E B(xo, q) and that lim V(t, xn,yn) = 0 implies n---~ oo
lim D(xm y**) = 0,
(3.2)
n---*oo
whenever {x,~} C B(xo, q), {y,,} C B(xo,q). (ii)
]V(t,x,y) - V(t, xl,yl)l <_ L . ( D ( x , x l ) + D(y, yl)),
(3.3)
for every t E [to, to + p], x, y, xt, Yl C B(xo, q), where L > 0 is a constant. Then, V is called a function of L - D type. DEFINITION 3.2. A s s u m e that f E C[Ro, E~], and there exists a function V of L - D that D+ V(t, x, y) <_g(t, V(t, x, y))
type such (3.4)
for all t E [to,to + p ] , x , y E B(xo,q), where .
1
D+V (t, x, y) = h~n~o -~ IV (t + h, x + h f (t, x), y + h f (t, y)) - V (t, x, y)]. g C C[[to, to + p] x [0, oo), R], g(t, O) -- O, and the maximum solution of the scalar differential equation, u' = g (t, u ) , u (to) = 0, (3.5) is u = 0 on [to, to + p]. Then, f is said to satisfy Lyapunov dissipative-type conditions.
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THEOREM 3.2. Assume that
f • C[Ro, E"I,
D ( f (t, z) , O) < M ,
V(t,x)•Ro,
and f satisfies Lyapunov dissipative-type conditions. Then, the Cauchy problem, x' (t) = f (t,x (t)), has unique solution x •
C 1 [[to,
x (to) = xo,
(3.6)
to ~-O~], g(x0, q)], where a < min{p, q/M}.
PROOF. Take e > 0 such that a = min{p, q / ( M + ¢)}. According to L e m m a 3.1, there exists e c ' [[to, to + ~], B(~o, q)],
~,~
such that
jx~,~(t) = j f ( t , x , ~ ( t ) ) + y , ~ ( t ) ,
x~(to) = x 0 , (3.7)
E
bly~ (t)ll ~ - ,
t e [to,to + ~ ]
n
(n = 1 , 2 , . . . )
where j is the isometric embedding from (E ~, D) onto its range in the Banach space X,
~ ( t ) • c[[to, to + ~], x]. For fixed natural numbers m and n, let
re(t)
=
V(t, xn(t),xm(t)),
It is easy to see that re(to) = 0. From
(3.3),
t • [to, to + a].
we have
m(t + h) - re(t) < L[D(xn(t + h),xn(t) + hf(t, xn(t))) +D(x,~(t + h), x,~(t) + hi(t, Xm(t)))] +V(t + h,x,~(t) + hf(t, xn(t)),x,~(t) + hf(t,x,c(t))) - V ( t , xn(t),x,~(t)), whenever t • [to, to + ~],
t+ h
•
[to, to + a].
So, by (3.4) and (3.7) we have
(~ ~) + g(t, re(t)),
D+m(t) <_ L n + m
t • [to, to + c~],
(3.8)
where D+m(t) is the Dini derivative [13]. Applying Proposition 2.4, it is known that
.~(t) < r~,.~(t),
t • [to, to +
~],
where rn,m(t) is the m a x i m u m solution of the initial value problem, =
,
u(to)
=
o.
According to Proposition 2.3, and noting that u - 0 is the m a x i m u m solution to the initial value problem (3.5), we get lim V (t, xn ( t ) , x m (t)) -- 0. m-~oo
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From this, using Definition 3.1, we know that {x,~(t)} is a basic fuzzy-valued functions sequence in the sense of uniform convergence on [to, to + a]. In addition, noting t h a t (E n, D) is complete, we deduce that {x~(t)} uniform converges to some continuous fuzzy-valued function x(t) on [to, to + a]. Furthermore, we have x(t) E C 1 [[to, to + a], B(xo, q)] and x(t) is a solution of the initial value problem (3.6) (see Theorem 3.2 in [41). Next, we prove the uniqueness. Suppose y(t) is another solution of initial value problem (3.6) on [to, to + a]. Let rrtl(t ) = V ( t , x ( t ) , y ( t ) ) . Then, similar to the proof of (3.8), we have D + m l ( t ) < g(t, ml(t)). By proposition 2.4, and noting that the m a x i m u m solution of the initial value problem (3.5) is u = 0, we get m~(t) = 0. T h a t i s , x(t) - y(t). The proof is concluded. COROLLARY 3.3. A s s u m e that f E C[Ro,En], D ( f ( t , x ) , O ) t E [to, to + p], x, y, E B(xo, q), we have
< M, V(t,x)
c Ro, and for all
lira 1 [D (x + hf (t,x),y + hf (t,y)) - D (x,y)] < g ( t , D ( x , y ) ) ,
(3.9)
h-++O h
where g C C[[to, to + c~] x [0, oo), R], g(t, O) - O, and the m a x i m u m solution of the initial value problem (3.5) is u - 0 on [to, to + c~]. Then, the conclusions of Theorem 3.2 hold true also. PROOF. In the proof of Theorem 3.2, we take V(t, x, y) --- D ( x , y), then this proof is obtained immediately. REMARK 3.1. We take V ( t , x , y ) = D ( x , y ) p when 1 _< p < +oo, then V is a function of L - D type. Using this function, some particular forms of Theorem 3.2 is well known. DEFINITION 3.3. A s s u m e that g(t, u) : (to, to + p] x [0, oo) --* R is a real function. If for any given ~ > 0, there exists 6 > O, 6i > 0 (i - 1 , 2 , . . . ) , ti E (to,to + p), ti ~ to, and 'ui C C[[ti, to + p], [0, oo)] such that ui (t~) > 6 (ti - to),
D+ui (t) > g (t, ui (t)) + 6i,
0 < ui (t) <_ z,
(3.10)
Vt E (ti,to + p ] ,
then g is called a function of U1 kind. THEOREM 3.4. A s s u m e that f C C[Ro,E~], D ( f ( t , x ) , O ) <_ M , V ( t , x ) E Ro, and there exists a function g of U1 kind such that lira l [ D ( x + h f ( t , x ) h-,+o -h
'Y
+hf(t,y))
-
D(x,y)]
'
(3.11)
for all t E (to, to + p], x, y, E B(xo, q). Then, the Cauchy problem (3.6) has unique solution on [to, to + a), where a = min{p, q / ( M + 1)}. PROOF. Applying L e m m a 3.1, there exists x~ E C[[t0, to + a], B(xo, q)] such that jx',~(t) = j f ( t , x,~(t)) + y , ( t ) , 1 []yn(t)[] <_ - , n
x~(to) = xo, (3.12)
tE[to,to+~]
(n = 1 , 2 , . . . ) ,
where j is the isometric embedding from (E 7~, D) onto its range in the Banach space X , yn(t) E
c[[t0, to + ~], x].
S. SONG et al.
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In what follows, we verify that {x~(t)} is a basic fuzzy-valued functions sequence. For any given e > 0, because g is a function of U1 kind, we know that there exists an 5 > 0, 5i > 0, ti E (to, to + p ) and such ui(t) that has the properties of Definition 3.3. Take 77 > 0 and natural number no such that 1 1 - + -- + r/< Tt
5,
(3.13)
m
whenever m, n _> no. Let m (t) = D (x~ (t), x,~ (t)) =
tljx~
(t) - j x m
(t) II,
then, we have
m(t + h) - re(t) _< D(x,~(t + h),xn(t) + hf(t,x~(t))) + D(x,~(t + h),
xm(t) ÷ hf(t, xm(t))) + D(xn(t) + hf(t, x,~(t)), xm(t) + hf(t, xm(t))) - D(zn(t), xm(t)). Consequently, by (3.11) and (3.12), we have
m~+(t)~g(t,m(t))+(~+l),
tE[to,to+c~].
(3.14)
In addition, by Iljz~(t)l L = Iljf(t,z,~(t)) + y,~(t)l I _< M + 1. It is known that D(xn(t),xo) = Iljz,~(t) - j x o II -< (M + 1 ) I t - t0l. From the continuity of the fuzzy-valued function f , there exists an r > 0 such that
D(f(t,x), whenever
D(x, xo) <_r.
f(t, xo)) <_ ~,
Take t , E (to, to + a) such that
D(x,(t),xo) <_ (M + l) l t - t o 1%r, whenever t E [to,
tn].
Hence, we have
Iljx~ (t) - jx~ (t)II <- IlJf( t, xn (t)) - jr(t, xo)II ÷ Iljf(t,x,~(t))-jf(t, xo)ll + IlYn(t)-
Y,,~(t)ll
Thus, we get
.~(t)=~(t)-~(to)<_(¼-±+~)(t-to),.~
t e [t0,t~].
Take n a t u r a l number i such t h a t t i < t n. Choose again n I ~ ~0 such t h a t 1 1 -+-<5i, m whenever n, m _ h i . Therefore, we have
m(t') ~- ( 1 + -m1 +77)
< 5(t~ - to) < ~,(t,).
(t'-t°)
(3.1~)
Existence and Uniqueness
1491
Assume t h a t re(t) < u~(t) holds true no everywhere. Let t* = sup{{ I re(t) < u~(t), for t E [ti, t ] } . Then, re(t*) = u~(t*) > O, t~ < t* <_ to + c~ by
(3.14),
we have
~'+(t*) <_ g(t*,.~(t*)) +
+
1)
<_ g(t*, re(t*)) + ~i
(3.16)
< D+u~(t*). On the other hand, we know re(t) > u~(t) whenever t E (t*, to + c~]. So, we have rn~(t*) =
lim h--++O
>
lim h---*+0
.~(t* + h) - .~(t*) h ~,(t* + h) - ~,(t*) h
= D+ui(t*). This is a contradiction with (3.16). This implies re(t) < ui(t) < ~
whenever t C [ti,to + c~].
Letting t~ ~ 0, we know re(t) <_ e whenever t E (t0,to + c~]. By the continuity of fuzzy-valued function re(t), it follows that re(t) < e whenever t E [t0,t0 + c~]. Therefore, {xn(t)} is a basic fuzzy-valued functions sequence in the sense of uniform convergence on [to, to + c~]. Noting that (E ~, D) is a complete space, there exists a continuous fuzzy-valued function x(t) on [to, to + (~] such that {x~(t)} uniform converges to x(t) on [t0,to + c~]. This means t h a t x(t) is a solution of the initial value problem (3.6) on [to, to + c~], by using Theorem 3.2 in [4] again. Finally, we prove the uniqueness. Suppose y(t) is another solution of initial value problem (3.6) on [to, to + c~]. Let 'n(t) = D ( x ( t ) , y(t)) = IlJx(t) - jy(t)II. Then similar to preceding proof, we have
n~+(t) < g(t,n(t)). For any given s > 0, referring to the proof of (3.15), it is known that there exists an ta E (to, t 0 + a ) such that 5 n(t) <_ -~ (t - to) whenever t E [to, t6]. Take natural number i such that ti < ts. Then, we have
n(td
<
5 ~(t~ -to)
<
~(td.
Referring to the proof of the inequality re(t) <_ s whenever t E [to, to + c~], we know n(t) <_ s whenever [to, to + a]. Letting s ~ 0, we get ~(t)
-
o,
i.e. 1
• (t) _= y(t). The proof is completed.
REMARK 3.2. The function of [/1 kind is very universal. For example, g l ( t , u ) constant), 92(t, u) = u / ( t -- to), etc., are the functions of U1 kind, respectively.
-- L u ( L is a
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REFERENCES 1. O. Kaleva, The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems 35,389-396, (1990). 2. J.J. Nieto, The Cauchy problem for continuous fuzzy differential equations, Fuzzy Sets and Systems 102, 259-262, (1999). 3. O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24, 301-317, (1987). 4. C.X. Wu, S.J. Song and E.S. Lee, Approximate solutions, existence, and uniqueness of the Cauchy problem of fuzzy differential equations, J. Math. Anal. Appl. 202, 629-644, (1996). 5. C.X. Wu and S.J. Song, Existence theorem to the Cauchy problem of fuzzy differential equations under compactness-type conditions, Information Science 108, 123-134, (1998). 6. Z. Ding and M. Ma, Existence of the solutions of fuzzy differential equations with parameter, Info~nation Sciences 99, 205-217, (1997). 7. M. l:¥iedman, M. Ma and A. Kandel, On the validity of Peano theorem for fuzzy differential equations, Fuzzy Sets and Systems 86, 331-334, (1997). 8. M. Ma, On embedding problems of fuzzy number space: Part 5, Fuzzy Sets and Systems 55, 313-318, (1993). 9. P.E. Kloeden, Remarks on Peano-like theorems for fuzzy differential equations, Fuzzy Sets and Systems 44, 161 163, (1991). 10. M.L. Puri and D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114, 409-422, (1986). 11. M.L. Purl and D.A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl. 91, 552-558, (1983). 12. H. Rgdstr6m, An embedding theorem for spaces of convex set, Proc. Amer. Math. Soe. 3, 165-169, (1952). 13. V. Lakshmikantham and S. Leela, Differential and integral inequalities, Volumes I and II, Academic Press, New York, (1969).
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-~,=-~= @ o , - ~ .
mathematics with applications
ELSEVIER Computers and Mathematics with Applications 51 (2006) 1483-1492 www.elsevier.com/locate/camwa
E x i s t e n c e and U n i q u e n e s s of Cauchy P r o b l e m for Fuzzy Differential Equations under Dissipative Conditions SHIM SONG AND CHENG WU Department of Automation, Tsinghua University Beijing 100084, P.R. China shij is~mail, t s i n g g u a , edu. cn XIAOPING XUE Department of Mathematics, Harbin Institute of Technology Harbin 150001, P.R. China (Received November 2005; accepted December 2005)
Abstract--The
existence theorem of Peano for the fuzzy differential equation, x'(t) = f(t,x(t)),
x(to) = xo,
does not hold in general except in the special case where the dimensional [1] or f is assumed to be continuous and bounded conditions which guarantee the existence theorem for Peano theorem of approximate solutions to above Cauchy problem. served.
fuzzy number space (E '~, D) is finite [2]. In this paper, the dissipative-type are specified based on the existence @ 2006 Elsevier Ltd. All rights re-
K e y w o r d s - - E x i s t e n c e and uniqueness theorem, Fuzzy differential equations, Dissipative-type conditions, Fuzzy number space (E '~, D).
1. I N T R O D U C T I O N Various i n v e s t i g a t o r s have s t u d i e d tile e x i s t e n c e t h e o r e m of P e a n o for fuzzy differential equations [1 7] on t h e m e t r i c space ( E ~, D) of n o r m a l , convex, u p p e r s e m i c o n t i n u o u s , a n d c o m p a c t l y s u p p o r t e d fuzzy sets in R ~. T h e m e t r i c space ( E ~, D) has a linear s t r u c t u r e b u t is not a v e c t o r space, it can be i m b e d d e d i s o m e t r i c a l l y and i s o m o r p h i e a l l y as a c o n v e x cone in a B a n a e h space C([0, 1], C ( S ' ~ - I ) ) , w h e r e S '~-1 is t h e unite sphere in R ~ [8].
K a l e v a [1] has s h o w n t h a t t h e
P e a n o t h e o r e m does n o t hold b e c a u s e t h e m e t r i c space ( E ~, D ) g e n e r a l l y is n o t locally c o m p a c t . It was s h o w n in [3,4] t h a t if f is c o n t i n u o u s and satisfies t h e L i p s c h i t z c o n d i t i o n or g e n e r a l i z e d L i p s c h i t z c o n d i t i o n w i t h respect x, t h e n t h e r e exists a u n i q u e s o l u t i o n to t h e C a u c h y p r o b l e m on This work is supported by Tile National Natural Science Foundation of China (Grant No. 60574077, 10571035) and Sasal Research Foundation of Tsinghua University (Grant No. JC2001029) and 985 Basic Research Foundation of Information Science and Technology of Tsinghua University 973 project (Grant No. 2002cb312205). The authors of this paper would like to thank E.S. Lee for his suggestions and help. 0898-1221/06/$ - see front matter @ 2006 Elsevier Ltd. All rights reserved. doi: 10.1016/j.camwa.2005.12.001
Typeset by A3/~-TEX
S. SONG e~ al.
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(E *~, D). In addition, Nieto proved the existence theorem of Peano for fuzzy differential equation on (E ", D) if f is continuous and bounded. We also note that there are some papers that claim the Peano-like theorem by restricting to continuous set-valued functions f defined on some locally compact subsets of (E n, D) [11, or consider other metrics on the space (E ~, D) [9]. But unfortunately, Friedman, Ma and Kandel [7] had pointed out the mistakes of these papers. Nevertheless, the question of what conditions on f guarantee the existence of solution is still an open question. In [5], the existence theorem of the Cauchy problem for fuzzy differential equations is obtained under some compactness-type conditions. Moreover, in this paper, the existence and uniqueness theorems for Peano of the cauchy problem for fuzzy differential equations are obtained under the dissipative-type conditions by applying the existence theorem of approximate solutions to the Cauchy problem given in [5].
2. P R E L I M I N A R I E S Let Pk(R n) denote the family of all nonempty compact convex subsets of R n and define the addition and scalar multiplication in Pk(R n) as usual. Denote E n : { u : R n --~ [0, 1] [ u satisfies (i)-(iv) below}, where
(i) u is normal, i.e., there exists an x0 G/{~ such that u(x0) = 1, (ii) u is fuzzy convex, i.e.,
u(Ax + (1 -- A)y) > min{u(x),u(y)},
for any x , y C R n and 0 < A < 1,
(iii) u is upper semicontinuous, (iv) [ul ° =cl{x • R n l u(x) > 0} is compact. For 0 < a <_ 1, denote b] ° : {x ¢ R ~ l u(x) _> ~}.
Then, from (i)-(iv), it follows that the a-level set [u]a • Pk(R '~) for all 0 < a < 1. According to Zadeh's extension principle, we have addition and scalar multiplication in fuzzy number space E n as usual. Define D : E ~ x E "~ ~ [0, co) by the equation,
D(u,v)=
sup d([u]~,[v]'~), O
where d is the Hausdorff metric defined in Pk(R'~). Then, it is easy to see that D is a metric in E '~ and has the following properties [10]. (1) (E ~, D) is a complete metric space; (2) D(u + w,v + w) = D(u,v) for all u , v , w • En; (3) D(ku, kv) IklD(u,v) for all u,v • E '~ and k •/{. =
In the following, we recall some main concepts and properties of integrability and H-differentiability for the fuzzy-set functions [3,11]. In this paper, we denote
T = [to, to + a] c R(~ > 0) be a compact interval. The fuzzy-valued function F : T ~ E n is called strongly measurable, if for every a • [0, 1] the set-valued function F~ : T -~ Pk(R ~) defined by
f ~ (t)
=
IF (t)] ~ ,
is Lebesgue measurable, where Pk(R n) is endowed with the topology generated by the Hausdorff metric d.
Existence and Uniqueness
1485
A f u n c t i o n F : T --* E ~ is called i n t e g r a b l y b o u n d e d , if there exists a n i n t e g r a b l e f u n c t i o n h such t h a t II~ll -< h(t) for all x • Fo(t). DEFINITION 2.1. Let F : T ~ E ~. T h e integral of F over T, denoted by f r F(t) dt, is defined
levehvise by the equation, [fTF(t)dt]~
= / T F ~ ( t ) dt
= { /rf(t)
dt t f : T--~ Rn is a measurable selection for F~,} ,
for all 0 < c~ <_ 1. A s t r o n g l y m e a s u r a b l e a n d i n t e g r a b l y b o u n d e d f u n c t i o n F : T ~ E n is said to b e i n t e g r a b l e over T if f r F ( t ) dt • E ~. F r o m [3], we know t h a t i f F : T ~ E ~ is c o n t i n u o u s , t h e n it is integrable. Let x, y • E n. If there exists a z C E ~ such t h a t x = y + z, t h e n we call z t h e H-difference of x a n d y, it is d e n o t e d b y x - y. DEFINITION 2.2. A function F : T ---+E ~ is differentiable at tl • T, if there exists a F ' ( t l ) • E "~ such that the limits, lira F ( t l + h ) -
h---*+O
F(tl)
h
and lim F ( t l ) - F ( t l - h) h~+0 h '
exist and are equal to F ' ( t l ) . Here, t h e limits are t a k e n in t h e m e t r i c space ( E ~, D). At t h e e n d p o i n t of T, we consider o n l y one-sided derivatives. If F : T --~ E n is differentiable at t l E T, t h e n we say t h a t F ' ( Q ) is t h e fuzzy derivative of F(t) at t h e p o i n t tl. T h e i n t e g r a b i l i t y a n d H - d i f f e r e n t i a b i l i t y properties of t h e fuzzy-valued f u n c t i o n s c a n be referred
to [3]. REMARK 2.1. We d e n o t e a c o n t i n u o u s fuzzy-valued f u n c t i o n f : T x t 2 -+ E ~ b y f C C[T×f~, E n ] , where t2 C E n is a n o p e n set. In a d d i t i o n , if F : T ~ E ~ is differential a n d F'(t) is c o n t i n u o u s , t h e n we d e n o t e F C C 1 [T, E~]. PROPOSITION 2.1. A s s u m e that f E C[T x f~,ET~]. A function x : T ~ f~ is a solution to the pro blem x' - f (t, x), x (to) = xo if and only if it is continuo us a n d satisfies the integral eq uation,
x(t) = x 0 +
f:
/ ( s , x ( s ) ) ds,
for all t • T [3]. As preliminaries, we recall further t h e e m b e d d i n g t h e o r e m of fuzzy n u m b e r space ( E ~, D ) , which is a g e n e r a l i z a t i o n of t h e classical R £ d s t r h m e m b e d d i n g t h e o r e m in [12]. PROPOSITION 2.2. There exists a real Banach space X such that E n can be embedding as a
convex cone C with vertex 0 into X . The following conditions hold true [1]. (i) The embedding j is isometric. (ii) Addition in X induces addition in E ~. (iii) Multiplication by nonnegative real number in X induces the corresponding operation in
E n" (iv) C - C is dense in X . (v) C is closed.
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REMARK 2.2. If x : T ~ E n is differentiable at tl C T. Then, it is easy to see that (jx) (t) = j (x ( t ) ) : T -~ X is Frechet differentiable at tl • T and ( j x ) ' ( t l ) = j ( x ' ( t l ) ) , where j is the embedding in Proposition 2.2. Finally, in this section, we will state several comparison theorem on classical ordinary differential equation [13], and some well-known results on abstract Banach space. PROPOSITION 2.3. Let G C R 2 be open set and g E C[G, R], (to, uo) E G. Suppose that r(t) is the m a x i m u m solution to the initial value problem, u'=g(t,u),
u(to) = u o ,
(2.1)
and its largest interval of existence of right solution is [to, to + a). I f [to, tl] C [to, to + a), then there exists an Co > 0 such that the m a x i m u m solution r(t, ~) to the initial value problem, ~' - g (t, ~ ) + ~,
~ ( t o ) = ~o + ~,
exist on [to, tl] whenever 0 < ~ < ~o, and ~(t, ~) ~niformly converges to ~(t) on [to, t~] as c -~ +0. PROPOSITION 2.4. Let G C R 2 be an open set, g E C[G, R], (to, uo) • G. Suppose that the m a x i m u m solution to the initial value probIem (2.1) is r(t) and its largest interval of existence of right solution is [to, to + a ). If re(t) E C[(to, to + a ), R] satisfies ( t, re(t)) c G for all t E [to, to + a), m(to) <_ uo, and
D + . ~ (t) < g (t,.~ (t)),
Vt • [to, to + ~) \ r,
where D+ is one of the four Dini derivatives, F at most is a countable set on [t0,t0 + a). Then, we must have [13] rn (t) _< r (t), for all t • [to, to + a). For the Banach space X which appears in Proposition 2.2, we denote C[T, X] the space of a11 abstract continuous functions from T into X , its norm is given by
Ilxll = max IIx(t)II, tET
Vx • c [ r , X].
For H C C[T, X], we denote U(t) = {z(t) l x • H } C X . PROPOSITION 2.5 ASCOLI-ARZELA. A set H C C[T, X] is relative compact if and only if H is equicontinuous and for any t • T, H (t) is relative compact set in X. PROPOSITION 2.6. MEAN VALUE THEOREM. Suppose x • C[T, X], F C T be at m o s t countable set, and for any t • T \ P , x ( t ) is differential and there exists M > 0 such that IIx'(t)[I _< M. Then, we have IIx (to +p) - x(to)[I <_ Mp. 3.
EXISTENCE THEOREMS
PROBLEM DISSIPATIVE-TYPE
AND OF
UNIQUENESS CAUCHY
UNDER
THE
CONDITIONS
In this section, by using existence theorcm of approximate solutions and the embedding results on fuzzy number space (E ~, D), we give the existence and uniqueness theorem under dissipativetype conditions for the Cauchy problem of the fuzzy differential equations. Let p, q be two positive real numbers. Denote B(xo, q) = {x C E ~ ] D ( x , x0) _< q}, Ro = [to,to + p ] x B ( x o , q ) . At first, we recall the existence theorem in [5], for the approximate solutions to the Cauehy problem of fuzzy differential equations.
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Existence and Uniqueness LEMMA 3.1. Suppose f E C[Ro, E~], here Ro = [to,to + p ] x B(xo,q), B(xo, q) = {x ~ E n IN(x, xo) < q},
and D ( f (t, x ) , O) < M,
for all (t, x) E Ro.
Then, for any given ~ > O, there exists x~ E C 1[[to, to + a~], B(xo, q)] (where a~ = rain{p, q / ( M + ¢)}) such that jx'~(t)=jf(t,x~(t))+y~(t),
z~(to)=Xo,
Iiy~(t)il_
vtE[to,to+a~],
where
y~(t) e c[[to, to + ~,], x], and j is the isometric embedding from (E ~, D) onto its r a n g e in the Banaeh space X as in [9]. DEFINITION 3.1. Assume that V : [to, to + p] x B(xo, q) x B(xo, q) -~ [0, oc)
is a continuous function provided
(i) V(t, z, x) - o, (3.1)
V ( t , x , y ) > O,
x • y,
for all t E [to, to + p], x, y E B(xo, q) and that lim V(t, xn,yn) = 0 implies n---~ oo
lim D(xm y**) = 0,
(3.2)
n---*oo
whenever {x,~} C B(xo, q), {y,,} C B(xo,q). (ii)
]V(t,x,y) - V(t, xl,yl)l <_ L . ( D ( x , x l ) + D(y, yl)),
(3.3)
for every t E [to, to + p], x, y, xt, Yl C B(xo, q), where L > 0 is a constant. Then, V is called a function of L - D type. DEFINITION 3.2. A s s u m e that f E C[Ro, E~], and there exists a function V of L - D that D+ V(t, x, y) <_g(t, V(t, x, y))
type such (3.4)
for all t E [to,to + p ] , x , y E B(xo,q), where .
1
D+V (t, x, y) = h~n~o -~ IV (t + h, x + h f (t, x), y + h f (t, y)) - V (t, x, y)]. g C C[[to, to + p] x [0, oo), R], g(t, O) -- O, and the maximum solution of the scalar differential equation, u' = g (t, u ) , u (to) = 0, (3.5) is u = 0 on [to, to + p]. Then, f is said to satisfy Lyapunov dissipative-type conditions.
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THEOREM 3.2. Assume that
f • C[Ro, E"I,
D ( f (t, z) , O) < M ,
V(t,x)•Ro,
and f satisfies Lyapunov dissipative-type conditions. Then, the Cauchy problem, x' (t) = f (t,x (t)), has unique solution x •
C 1 [[to,
x (to) = xo,
(3.6)
to ~-O~], g(x0, q)], where a < min{p, q/M}.
PROOF. Take e > 0 such that a = min{p, q / ( M + ¢)}. According to L e m m a 3.1, there exists e c ' [[to, to + ~], B(~o, q)],
~,~
such that
jx~,~(t) = j f ( t , x , ~ ( t ) ) + y , ~ ( t ) ,
x~(to) = x 0 , (3.7)
E
bly~ (t)ll ~ - ,
t e [to,to + ~ ]
n
(n = 1 , 2 , . . . )
where j is the isometric embedding from (E ~, D) onto its range in the Banach space X,
~ ( t ) • c[[to, to + ~], x]. For fixed natural numbers m and n, let
re(t)
=
V(t, xn(t),xm(t)),
It is easy to see that re(to) = 0. From
(3.3),
t • [to, to + a].
we have
m(t + h) - re(t) < L[D(xn(t + h),xn(t) + hf(t, xn(t))) +D(x,~(t + h), x,~(t) + hi(t, Xm(t)))] +V(t + h,x,~(t) + hf(t, xn(t)),x,~(t) + hf(t,x,c(t))) - V ( t , xn(t),x,~(t)), whenever t • [to, to + ~],
t+ h
•
[to, to + a].
So, by (3.4) and (3.7) we have
(~ ~) + g(t, re(t)),
D+m(t) <_ L n + m
t • [to, to + c~],
(3.8)
where D+m(t) is the Dini derivative [13]. Applying Proposition 2.4, it is known that
.~(t) < r~,.~(t),
t • [to, to +
~],
where rn,m(t) is the m a x i m u m solution of the initial value problem, =
,
u(to)
=
o.
According to Proposition 2.3, and noting that u - 0 is the m a x i m u m solution to the initial value problem (3.5), we get lim V (t, xn ( t ) , x m (t)) -- 0. m-~oo
Existence and Uniqueness
1489
From this, using Definition 3.1, we know that {x,~(t)} is a basic fuzzy-valued functions sequence in the sense of uniform convergence on [to, to + a]. In addition, noting t h a t (E n, D) is complete, we deduce that {x~(t)} uniform converges to some continuous fuzzy-valued function x(t) on [to, to + a]. Furthermore, we have x(t) E C 1 [[to, to + a], B(xo, q)] and x(t) is a solution of the initial value problem (3.6) (see Theorem 3.2 in [41). Next, we prove the uniqueness. Suppose y(t) is another solution of initial value problem (3.6) on [to, to + a]. Let rrtl(t ) = V ( t , x ( t ) , y ( t ) ) . Then, similar to the proof of (3.8), we have D + m l ( t ) < g(t, ml(t)). By proposition 2.4, and noting that the m a x i m u m solution of the initial value problem (3.5) is u = 0, we get m~(t) = 0. T h a t i s , x(t) - y(t). The proof is concluded. COROLLARY 3.3. A s s u m e that f E C[Ro,En], D ( f ( t , x ) , O ) t E [to, to + p], x, y, E B(xo, q), we have
< M, V(t,x)
c Ro, and for all
lira 1 [D (x + hf (t,x),y + hf (t,y)) - D (x,y)] < g ( t , D ( x , y ) ) ,
(3.9)
h-++O h
where g C C[[to, to + c~] x [0, oo), R], g(t, O) - O, and the m a x i m u m solution of the initial value problem (3.5) is u - 0 on [to, to + c~]. Then, the conclusions of Theorem 3.2 hold true also. PROOF. In the proof of Theorem 3.2, we take V(t, x, y) --- D ( x , y), then this proof is obtained immediately. REMARK 3.1. We take V ( t , x , y ) = D ( x , y ) p when 1 _< p < +oo, then V is a function of L - D type. Using this function, some particular forms of Theorem 3.2 is well known. DEFINITION 3.3. A s s u m e that g(t, u) : (to, to + p] x [0, oo) --* R is a real function. If for any given ~ > 0, there exists 6 > O, 6i > 0 (i - 1 , 2 , . . . ) , ti E (to,to + p), ti ~ to, and 'ui C C[[ti, to + p], [0, oo)] such that ui (t~) > 6 (ti - to),
D+ui (t) > g (t, ui (t)) + 6i,
0 < ui (t) <_ z,
(3.10)
Vt E (ti,to + p ] ,
then g is called a function of U1 kind. THEOREM 3.4. A s s u m e that f C C[Ro,E~], D ( f ( t , x ) , O ) <_ M , V ( t , x ) E Ro, and there exists a function g of U1 kind such that lira l [ D ( x + h f ( t , x ) h-,+o -h
'Y
+hf(t,y))
-
D(x,y)]
'
(3.11)
for all t E (to, to + p], x, y, E B(xo, q). Then, the Cauchy problem (3.6) has unique solution on [to, to + a), where a = min{p, q / ( M + 1)}. PROOF. Applying L e m m a 3.1, there exists x~ E C[[t0, to + a], B(xo, q)] such that jx',~(t) = j f ( t , x,~(t)) + y , ( t ) , 1 []yn(t)[] <_ - , n
x~(to) = xo, (3.12)
tE[to,to+~]
(n = 1 , 2 , . . . ) ,
where j is the isometric embedding from (E 7~, D) onto its range in the Banach space X , yn(t) E
c[[t0, to + ~], x].
S. SONG et al.
1490
In what follows, we verify that {x~(t)} is a basic fuzzy-valued functions sequence. For any given e > 0, because g is a function of U1 kind, we know that there exists an 5 > 0, 5i > 0, ti E (to, to + p ) and such ui(t) that has the properties of Definition 3.3. Take 77 > 0 and natural number no such that 1 1 - + -- + r/< Tt
5,
(3.13)
m
whenever m, n _> no. Let m (t) = D (x~ (t), x,~ (t)) =
tljx~
(t) - j x m
(t) II,
then, we have
m(t + h) - re(t) _< D(x,~(t + h),xn(t) + hf(t,x~(t))) + D(x,~(t + h),
xm(t) ÷ hf(t, xm(t))) + D(xn(t) + hf(t, x,~(t)), xm(t) + hf(t, xm(t))) - D(zn(t), xm(t)). Consequently, by (3.11) and (3.12), we have
m~+(t)~g(t,m(t))+(~+l),
tE[to,to+c~].
(3.14)
In addition, by Iljz~(t)l L = Iljf(t,z,~(t)) + y,~(t)l I _< M + 1. It is known that D(xn(t),xo) = Iljz,~(t) - j x o II -< (M + 1 ) I t - t0l. From the continuity of the fuzzy-valued function f , there exists an r > 0 such that
D(f(t,x), whenever
D(x, xo) <_r.
f(t, xo)) <_ ~,
Take t , E (to, to + a) such that
D(x,(t),xo) <_ (M + l) l t - t o 1%r, whenever t E [to,
tn].
Hence, we have
Iljx~ (t) - jx~ (t)II <- IlJf( t, xn (t)) - jr(t, xo)II ÷ Iljf(t,x,~(t))-jf(t, xo)ll + IlYn(t)-
Y,,~(t)ll
Thus, we get
.~(t)=~(t)-~(to)<_(¼-±+~)(t-to),.~
t e [t0,t~].
Take n a t u r a l number i such t h a t t i < t n. Choose again n I ~ ~0 such t h a t 1 1 -+-<5i, m whenever n, m _ h i . Therefore, we have
m(t') ~- ( 1 + -m1 +77)
< 5(t~ - to) < ~,(t,).
(t'-t°)
(3.1~)
Existence and Uniqueness
1491
Assume t h a t re(t) < u~(t) holds true no everywhere. Let t* = sup{{ I re(t) < u~(t), for t E [ti, t ] } . Then, re(t*) = u~(t*) > O, t~ < t* <_ to + c~ by
(3.14),
we have
~'+(t*) <_ g(t*,.~(t*)) +
+
1)
<_ g(t*, re(t*)) + ~i
(3.16)
< D+u~(t*). On the other hand, we know re(t) > u~(t) whenever t E (t*, to + c~]. So, we have rn~(t*) =
lim h--++O
>
lim h---*+0
.~(t* + h) - .~(t*) h ~,(t* + h) - ~,(t*) h
= D+ui(t*). This is a contradiction with (3.16). This implies re(t) < ui(t) < ~
whenever t C [ti,to + c~].
Letting t~ ~ 0, we know re(t) <_ e whenever t E (t0,to + c~]. By the continuity of fuzzy-valued function re(t), it follows that re(t) < e whenever t E [t0,t0 + c~]. Therefore, {xn(t)} is a basic fuzzy-valued functions sequence in the sense of uniform convergence on [to, to + c~]. Noting that (E ~, D) is a complete space, there exists a continuous fuzzy-valued function x(t) on [to, to + (~] such that {x~(t)} uniform converges to x(t) on [t0,to + c~]. This means t h a t x(t) is a solution of the initial value problem (3.6) on [to, to + c~], by using Theorem 3.2 in [4] again. Finally, we prove the uniqueness. Suppose y(t) is another solution of initial value problem (3.6) on [to, to + c~]. Let 'n(t) = D ( x ( t ) , y(t)) = IlJx(t) - jy(t)II. Then similar to preceding proof, we have
n~+(t) < g(t,n(t)). For any given s > 0, referring to the proof of (3.15), it is known that there exists an ta E (to, t 0 + a ) such that 5 n(t) <_ -~ (t - to) whenever t E [to, t6]. Take natural number i such that ti < ts. Then, we have
n(td
<
5 ~(t~ -to)
<
~(td.
Referring to the proof of the inequality re(t) <_ s whenever t E [to, to + c~], we know n(t) <_ s whenever [to, to + a]. Letting s ~ 0, we get ~(t)
-
o,
i.e. 1
• (t) _= y(t). The proof is completed.
REMARK 3.2. The function of [/1 kind is very universal. For example, g l ( t , u ) constant), 92(t, u) = u / ( t -- to), etc., are the functions of U1 kind, respectively.
-- L u ( L is a
1492
S. SONG et al.
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