On Peano's theorem in Banach spaces

TL;DR

This paper demonstrates that in certain infinite-dimensional Banach spaces, there exist continuous functions for which the associated autonomous differential equations have no solutions, highlighting limitations of classical existence results.

Contribution

It establishes the non-existence of solutions for autonomous differential equations in infinite-dimensional Banach spaces with specific structural properties.

Findings

Existence of continuous functions with no solutions in certain Banach spaces

Extension of Peano's theorem failure to infinite-dimensional settings

Identification of Banach space conditions affecting differential equation solvability

Abstract

We show that if $X$ is an infinite-dimensional separable Banach space (or more generally a Banach space with an infinite-dimensional separable quotient) then there is a continuous mapping $f\colon X\to X$ such that the autonomous differential equation $x'=f(x)$ has no solution at any point.