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 ${x}^{\text{'}}\left(t\right)=f(t,x\left(t\right)),$ $x\left({t}_{0}\right)={x}_{0}$, 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 for ODE, existence, uniqueness, etc. of solutions |