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Fuzzy Sets and Systems 24 (1987) 301-317 3 01 North-Holland FUZZY DIFFERENTIAL EQUATIONS Osmo KALEVA 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-valued mappings of a real variable whose values are normal, convex, upper semicontinuous and compactly supported fuzzy sets in R n. We study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation. Keywords: Fuzzy-set-valued mapping, Integration, Differentiation, Fuzz y 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. To 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 Aumann [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. 1 1 am indebted to one of the referees for bringing the reference [15] to my attention.

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