## Differential and Integral Equations

### 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.

#### Article information

Source
Differential Integral Equations, Volume 22, Number 1/2 (2009), 135-156.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356038558

Mathematical Reviews number (MathSciNet)
MR2483016

Zentralblatt MATH identifier
1240.26012

#### Citation

Carl, S.; Heikkilä, S.; Ye, Guoju. Order properties of spaces of non-absolutely integrable vector-valued functions and applications to differential equations. Differential Integral Equations 22 (2009), no. 1/2, 135--156. https://projecteuclid.org/euclid.die/1356038558