Existence and Uniqueness of Cauchy Problem for ... - Semantic Scholar
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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.
1484
(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 ~}.
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 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 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) 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) 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)
computers &
Available online at www.sciencedirect.com
-~,=-~= @ 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.
1484
(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 ~}.
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 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 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) 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) 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)
