Order properties of spaces of non-absolutely integrable vector-valued functions and applications to differential equations

Abstract

We prove that if the space $Y$ of Henstock-Lebesgue integrable functions from a compact real interval to a Banach space $E$ is normed by the Alexiewicz norm and ordered by a regular cone $E_+$, then the $E_+$-valued functions of $Y$ form a regular order cone of $Y$. This property is shown to hold also when $Y$ is the space of Henstock-Kurzweil integrable functions if $E$ is weakly sequentially complete and $E_+$ is normal. As an application of the obtained results we prove existence and comparison results for least and greatest solutions of nonlocal implicit initial-value problems of discontinuous functional differential equations containing non-absolutely integrable functions.