Relations among Henstock, McShane and Pettis integrals for multifunctions with compact convex values

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

“…For more detailed properties of the integrals involved and for all that is unexplained in this paper we refer to [12,18,19,26,[35][36][37][38].…”

Gauge integrals and selections of weakly compact valued multifunctions

“…The second one describes each H-integrable multifunction as the sum of a Birkhoff integrable multifunction and of an H-integrable function and the third one proves that each variationally Henstock integrable multifunction is the sum of a variationally Henstock integrable selection of the multifunction and a Birkhoff integrable multifunction that is also variationally Henstock integrable. As applications of such decomposition results, characterizations of Henstock (Theorem 3.4) and H (Theorem 4.3) integrable multifunctions, being extensions of the result given by Fremlin in a remarkable paper [18,Theorem 8] and of more recent results given in [16] and [5], are presented.…”

“…Some pioneering and highly influential ideas and notions around the matter were inspired by problems arising in Control Theory and Mathematical Economics. Furthermore this topic is interesting also from the point of view of measure and integration theory, as showed in the papers [1,2,6,9,10,[15][16][17]26]. Inspired by the papers [5,8,16,27], 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.…”

“…For the definitions and properties of Pettis and of Henstock-Kurzweil-Pettis integral for functions and multifunctions we refer the reader to [10,15,16,20,[23][24][25][26].…”

Multi-integrals of finite variation

“…In [23] was proved that a Banach space valued function is McShane integrable if and only if it is Pettis and Henstock integrable. That result has been then generalized to compact valued multifunctions Γ (see [20]), weakly compact valued multifunctions (see [6]) and bounded valued multifunctions (see [8]). Di Piazza and Marraffa [16] presented an example of a Pettis and variationally Henstock integrable function that is not variationally McShane integrable (= Bochner integrable in virtue of [18,Lemma 2]).…”