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 (Theorem 3.2, Theorem 4.2 and Theorem 5.3). As applications of such decompositions, we deduce characterizations of Henstock (Theorem 3.3) and H (Theorem 4.3) integrable multifunctions, together with an extension of a well-known theorem of Fremlin [22, Theorem 8].
Relations among Gauge and Pettis integrals for cwk(X)-valued multifunctions
CANDELORO, DomenicoMembro del Collaboration Group
;SAMBUCINI, Anna RitaMembro del Collaboration Group
2018
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 (Theorem 3.2, Theorem 4.2 and Theorem 5.3). As applications of such decompositions, we deduce characterizations of Henstock (Theorem 3.3) and H (Theorem 4.3) integrable multifunctions, together with an extension of a well-known theorem of Fremlin [22, Theorem 8].I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.