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The Cauchy problem for fuzzy differential equations. (English) Zbl 0696.34005
Summary: The classical Peano theorem states that in finite dimensional spaces the Cauchy problem x ' (t)=f(t,x(t)), x(t 0 )=x 0 , has a solution provided f is continuous. In addition, Godunov has shown that each Banach space in which the Peano theorem holds true is finite dimensional. For differential inclusions, the existence of a solution to the Cauchy problem is also guaranteed under various assumptions on the right-hand side. In this paper, we study the Cauchy problem for fuzzy differential equations. To be more specific, let U be a subspace of normal, convex, upper semicontinuous, compactly supported fuzzy sets defined in n and assume that f:[t 0 ,t 0 +a]×UU is continuous. We show that the Cauchy problem has a solution if and only if U is locally compact.

34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions