Peano's theorem fails for infinite-dimensional Fréchet spaces
In 1975 A. N. Godunov proved that Peano's theorem does not hold for any infinite-dimensional Banach space. The first result of this paper says that if \(E\) is a nonnormable Fréchet space and \(\tau\) is an arbitrary topology consistent with the duality \((E,E')\) then there exists a continuous function \(f: E\to E\) such that the equation \(x'(t)= f(x(t))\) has no solution on any interval. The second result says that in a Fréchet space \(E\) we have: 1. Peano's theorem holds for \(E\Leftrightarrow E\) is finite- dimensional; 2. Peano's theorem holds for \(E\) with weak topology \(\Leftrightarrow E\) is normable and reflexive.
scientific article; zbMATH DE number 652163