The Cauchy problem for fuzzy differential equations. (English) Zbl 0696.34005

Kaleva, Osmo

Fuzzy Sets Syst. 35, No. 3, 389-396 (1990).

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 \({\mathbb{R}}^ n\) and assume that \(f: [t_ 0,t_ 0+a]\times U\to U\) is continuous. We show that the Cauchy problem has a solution if and only if U is locally compact.

MSC:

34A12

Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations

Keywords:

Cauchy problem; Banach space; fuzzy differential equations

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