Gauge integrals and selections of weakly compact valued multifunctions

Abstract

Abstract.

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

“…The next theorems 5.5 extends [6,Theorems 4.3,4.4]. In fact we can remove the hypothesis S vH (Γ) = ∅ thanks to Theorem 5.1 and [6, Proposition 3.6].…”

“…Let Γ (t) := conv{0, g(t)}. Then, Γ is vH-integrable (see [6,Example 4.7]) but it is not vMS-integrable ( [6,Theorem 3.7] or [6,Example 4.7]) and possesses at least one vH-integrable selection by Theorem 5.1 . Let now f ∈ S vH (Γ) and consider the multifunction G = Γ − f .…”

“…In particular, comparison of different generalizations of Lebesgue integral is, in our opinion, one of the milestones of the modern theory of integration. Inspired by [6,7,10,12,13,19,24,39], we continue in this paper the study on this subject and we examine relationship 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.…”

“…The main results of the paper are the existence of variationally Henstock integrable selections (Theorem 5.1), which solves the problem of the existence of variationally Henstock integrable selection for a cwk(X)-valued variationally Henstock integrable multifunction ( [6,Question 3.11]) and three decomposition theorems (Theorem 3.2, Theorem 4.2 and Theorem 5.3). The first one says that each Henstock integrable multifunction is the sum of a McShane integrable multifunction and a Henstock integrable function.…”

Multi-integrals of finite variation

“…Proof. It is an immediate consequence of Theorem 3.6 if we proceed analogously to [5,Proposition 3.6].…”