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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 $x'(t)=f(t,x(t)),$ $x(t\sb 0)=x\sb 0$, 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 ${\bbfR}\sp n$ and assume that $f: [t\sb 0,t\sb 0+a]\times U\to U$ is continuous. We show that the Cauchy problem has a solution if and only if U is locally compact.

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