Compactness and Existence Results for Ordinary Differential Equations in Banach Spaces

Abstract

We prove that the Picard-Lindelöf operator

with a vector function $f$ is continuous and compact (condensing) in $C$, if $f$ satisfies only a mild boundedness condition, and if $f(s,\cdot )$ is continuous and compact (resp. condensing). This generalizes recent results of the second author and immediately leads to existence theorems for local weak solutions of the initial value problem for ordinary differential equations in Banach spaces.