1,721,109 research outputs found
Henstock-Kurzweil-Pettis integrability of compact valued multifunctions with values in an arbitrary Banach space
No full textThe aim of this paper is to describe Henstock-Kurzweil-Pettis (HKP for short) integrable compact valued multifunctions. Such characterizations are known in case of functions. It is also known that each HKP-integrable compact valued multifunction can be represented as a sum of a Pettis integrable multifunction and of an HKP-integrable function. Invoking to that decomposition, we present a pure topological characterization of integrability. Having applied the above results, we obtain two convergence theorems, that generalize results known for HKP-integrable functions. We emphasize also the special role played in the theory by weakly sequentially complete Banach spaces and by spaces possessing the Schur property
A decomposition theorem for compact-valued Henstock integral
No full textWe prove that if X is a separable Banach space, then a measurable multifunction Γ : [0, 1] → ck(X) is Henstock integrable if and only if Γ can be represented as Γ = G + f, where G : [0, 1] → ck(X) is McShane integrable and f is a Henstock integrable selection of
Relations among Henstock, McShane and Pettis integrals for multifunctions with compact convex values
No full textFremlin [Ill J Math 38:471-479, 1994] proved that a Banach space valued function is McShane integrable if and only if it is Henstock and Pettis integrable. In this paper we prove that the result remains valid also in case of multifunctions with compact convex values being subsets of an arbitrary Banach space (see Theorem 3.4). Di Piazza and Musial [Monatsh Math 148:119-126, 2006] proved that if X is a separable Banach space, then each Henstock integrable multifunction which takes as its values convex compact subsets of X is a sum of a McShane integrable multifunction and a Henstock integrable function. Here we show that such a decomposition is true also in case of an arbitrary Banach space (see Theorem 3.3). We prove also that Henstock and McShane integrable multifunctions possess Henstock and McShane (respectively) integrable selections (see Theorem 3.1)
Set valued Kurzweil-Henstock-Pettis integral
Full text linkIt is shown that the obvious
generalization of the Pettis integral of a multifunction obtained
by replacing the Lebesgue integrability of the support functions
by the Kurzweil--Henstock integrability, produces an integral
which can be described -- in case of multifunctions with (weakly)
compact convex values -- in terms of the Pettis set-valued
integral
A characterization of the weak Radon-Nikodym property by finitely additive interval functions
No full textA characterization of Banach spaces possessing the weak
Radon-Nikodym property is given in terms of finitely additive
interval functions. Due to that characterization several
Banach space valued set functions that are only finitely additive
can be represented as integrals
A Decomposition Theorem for the Fuzzy Henstock Integral
No full textWe study the fuzzy Henstock and the fuzzy McShane integrals for fuzzy-number valued functions. The main purpose
of this paper is to establish the following decomposition theorem: a fuzzy-number valued function is fuzzy Henstock integrable if and only if
it can be represented as a sum of a fuzzy McShane integrable fuzzy-number valued function and
of a fuzzy Henstock integrable fuzzy number valued function generated by a Henstock integrable function
Differentiation of an additive interval measure with values in a conjugate Banach space
Full text linkWe present a complete characterization of finitely additive interval measures with values in conjugate Banach spaces which can be represented as Henstock-Kurzweil-Gelfand integrals. If the range space has the weak Radon-Nikodym property (WRNP), then we precisely describe when these integrals are in fact Henstock-Kurzweil-Pettis integrals
A variational Henstock integral characterization of the Radon-Nikodym property
No full textA characterization of Banach spaces
possessing the Radon-Nikodym property is given in terms of
finitely additive interval functions. We prove that a Banach space
X has the RNP if and only if each X-valued finitely additive
interval function possessing absolutely continuous variational
measure is a variational Henstock integral of an X-valued
function. Due to that characterization several X-valued set
functions that are only finitely additive can be represented as
integrals
Decompositions of Weakly Compact Valued Integrable Multifunctions
Full text linkWe give a short overview on the decomposition property for integrable multifunctions, i.e., when an "integrable in a certain sense" multifunction can be represented as a sum of one of its integrable selections and a multifunction integrable in a narrower sense. The decomposition theorems are important tools of the theory of multivalued integration since they allow us to see an integrable multifunction as a translation of a multifunction with better properties. Consequently, they provide better characterization of integrable multifunctions under consideration. There is a large literature on it starting from the seminal paper of the authors in 2006, where the property was proved for Henstock integrable multifunctions taking compact convex values in a separable Banach space X. In this paper, we summarize the earlier results, we prove further results and present tables which show the state of art in this topi
On Denjoy type extensions of the Pettis integral
Full text linksummary:In this paper two Denjoy type extensions of the Pettis integral are defined and studied. These integrals are shown to extend the Pettis integral in a natural way analogous to that in which the Denjoy integrals extend the Lebesgue integral for real-valued functions. The connection between some Denjoy type extensions of the Pettis integral is examined
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