Existence and uniqueness of cauchy problem for fuzzy differential equations under dissipative conditions

x'(t) = f ( t ,x( t ) ) , x(to) = xo, does not hold in general except in the special case where the fuzzy number space (E '~, D) is finite dimensional [1] or f is assumed to be continuous and bounded [2]. In this paper, the dissipative-type conditions which guarantee the existence theorem for Peano are specified based on the existence theorem of approximate solutions to above Cauchy problem. @ 2006 Elsevier Ltd. All rights reserved. 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 inves t iga to rs have s tud ied t i le ex is tence t h e o r e m of P e a n o for fuzzy different ial equat ions [1 7] on the me t r i c space (E ~, D) of normal , convex, uppe r semicont inuous , and 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 me t r i c space ( E ~, D) has a l inear s t ruc tu re bu t is not a vec tor space, it can be i m b e d d e d i somet r ica l ly and i somorph iea l ly as a convex cone in a B a n a e h space C([0, 1], C ( S ' ~ I ) ) , where S '~-1 is the uni te sphere in R ~ [8]. K a l e v a [1] has shown t h a t t he P e a n o t h e o r e m does no t hold because the me t r i c space ( E ~, D) genera l ly is no t local ly compac t . I t was shown in [3,4] t h a t if f is con t inuous and satisfies the Lipsch i tz cond i t ion or genera l ized Lipschi tz cond i t ion wi th respect x, t h e n there exists a un ique so lu t ion to t he C a u c h y p rob lem 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. Typeset by A3/~-TEX doi: 10.1016/j.camwa.2005.12.001