On summability, integrability and impulsive differential equations in Banach spaces

TL;DR

This paper investigates summability and integrability of functions in Banach spaces, providing conditions for various types of integrability and analyzing the solvability of impulsive differential equations.

Contribution

It introduces generalized summability and iteration methods to establish integrability criteria and studies impulsive differential equations in Banach spaces.

Findings

Derived necessary and sufficient conditions for integrability of step and right regulated mappings.

Established solvability conditions for impulsive differential equations in Banach spaces.

Extended classical integrability concepts to more general settings in functional analysis.

Abstract

We shall first study summability of families in normed spaces indexed with well ordered sets of real numbers extended by infinity. Obtained results and a generalized iteration method are applied to derive necessary and sufficient conditions for HK, HL, Bochner and Riemann integrability of step mappings and for right regulated mappings from an interval of reals extended by infinity to a Banach space. Finally, solvability of impulsive differential equations are studied.