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Abstract In this paper, we consider the existence theorems of solution for fuzzy functional differential equations under the compactness-type conditions and dissipative type conditions, via the properties of the embedding mapping from fuzzy number to Banach space. Keywords Fuzzy number · Fuzzy functional differential equations · Initial value problems · Compactness-type conditions · Dissipative-type conditions

1 Introduction The fuzzy differential equation was first introduced by Kandel and Byatt [1]. Since 1987, the Cauchy problems for fuzzy differential equations have been extensively investigated by several authors [2–11] on the metric space (E n , D) of normal fuzzy convex set with the distance D given by the maximum of the Hausdorff distance between the corresponding level sets. In particular, Kaleva [2] studied the initial value problem  x (t) = f (t, x(t)), x(t0 ) = x0 where f : T × E n → E n is a continuous fuzzy mapping, T = [a, b], E n is a fuzzy number space and x0 ∈ E n . The result was obtained as follows Theorem 1. Let f : T × E n → E n be continuous and assume that there exist a k > 0 such that D( f (t, x), f (t, y)) ≤ k D(x, y) for all t ∈ T, x, y ∈ E n . Then the above initial value problem has a unique solution on T . Y. Shao (B) · H. Zhang · G. Xue College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou 730030, China e-mail: [email protected] B.-Y. Cao and H. Nasseri (eds.), Fuzzy Information & Engineering and Operations Research & Management, Advances in Intelligent Systems and Computing 211, DOI: 10.1007/978-3-642-38667-1_22, © Springer-Verlag Berlin Heidelberg 2014

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Since then, many authors studied the existence and uniqueness of solutions for the initial value problem for fuzzy differential equations under kinds of conditions and obtained several meaningful results. However, those research findings were not satisfactory. Until 1997, Nieto [3] proved the initial value problem for fuzzy differential equations has solutions if f is a continuous and bounded function. It is well known that the Lipschtz conditions can not be educed if f is a continuous and bounded function. That is to say, the results in [3] were perfect complement for the Theorem 1. What’s more, Wu and Song [5, 6] and Song, Wu and Xue [7] changed the initial value problem of fuzzy differential equations into a abstract differential equations on a closed convex cone in a Banach space by the operator j that is the isometric embedding from (E n , D) onto its range in the Banach space X . They established the relationship between a solution and its approximate solutions to fuzzy differential equations. Furthermore, they obtained the local existence theorem under the compactness-type and dissipative-type conditions. Park and Han [8] obtained the global existence and uniqueness of fuzzy solution of fuzzy differential equation using the the properties of Hasegawa’s function and successive approximation. There exists an extensive theory for function differential equations wit