Relations among Gauge and Pettis integrals for $cwk(X)$-valued multifunctions

TL;DR

This paper explores the relationships among various gauge and Pettis integrals for multifunctions with weakly compact convex values in Banach spaces, establishing new decomposition theorems and integrability characterizations.

Contribution

It introduces new decomposition theorems for gauge integrable multifunctions and extends known results on integrability in Banach spaces.

Findings

Existence of variationally Henstock integrable selections.

Decomposition theorems for gauge integrable multifunctions.

Characterizations of Henstock and ${ m H}$-integrable multifunctions.

Abstract

The aim of this paper is to study relationships among "gauge integrals" (Henstock, Mc Shane, Birkhoff) and Pettis integral of multifunctions whose values are weakly compact and convex subsets of a general Banach space, not necessarily separable. For this purpose we prove the existence of variationally Henstock integrable selections for variationally Henstock integrable multifunctions. Using this and other known results concerning the existence of selections integrable in the same sense as the corresponding multifunctions, we obtain three decomposition theorems. As applications of such decompositions, we deduce characterizations of Henstock and ${\mathcal H}$ integrable multifunctions, together with an extension of a well-known theorem of Fremlin.