On Cauchy problems for fuzzy differential equations

Existence and uniqueness theorems are proved for Cauchy problems of second-order fuzzy differential equations. 1. Introduction. In 1972, Chang and Zadeh [2] first introduced the concept of fuzzy derivative, followed up ten years later by Dubois and Prade [5], who used the extension principle in their approach. In the mean time, Puri and Ralescu [12] used the notion of H-differentiability to extend the differential of set-valued functions to that of fuzzy functions. This led Seikkala [13] to introduce the notion of fuzzy derivative as an extension of the Hukuhara derivative and the fuzzy integral, which was the same as that proposed by Dubois and Prade [3, 4]. Naturally, the investigation of fuzzy differential and integral equations, existence and uniqueness theorems for the solutions of fuzzy initial value problems, drew upon the interest of many researchers of the fuzzy domain. and the references therein. In this paper, using the properties of fuzzy integration [8, 9], we examine the existence and uniqueness of initial value problems of second-order fuzzy differential equations: