Peano's theorem fails for infinite-dimensional Fréchet spaces
type: Item
instance of: scholarly article
title: Peano's theorem fails for infinite-dimensional Fréchet spaces
zbMATH Open document ID: 0820.47048
DOI: 10.1007/BF01085989
publication date: 1994-11-08T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 34G20
zbMATH DE Number: 652163
zbMATH Keywords: Peano's theorem
zbMATH Keywords: nonnormable Fréchet space
author: S. Shkarin
reviewed by: Gheorghe Moroşanu
MaRDI profile type: Publication
cites work: On the nonexistence of solutions of differential equations in nonreflexive spaces
cites work: Q4722576
cites work: Q5736344
full work available at URL: https://doi.org/10.1007/bf01085989
OpenAlex ID: W2020019707
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 652163