Search results for " Henstock"
showing 10 items of 22 documents
Henstock–Kurzweil–Pettis integrability of compact valued multifunctions with values in an arbitrary Banach space
2013
Abstract The aim of this paper is to describe Henstock–Kurzweil–Pettis (HKP) integrable compact valued multifunctions. Such characterizations are known in case of functions (see Di Piazza and Musial (2006) [16] ). It is also known (see Di Piazza and Musial (2010) [19] ) 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 …
P-adic Henstock integral in the problem of representation of functions by multiplicative transforms
2005
We introduce a path integral of Henstock type and use it to obtain inversion formulas for multiplicative integral transformations. The problem considered is a generalization of the problem of reconstruction of coefficients of a convergent orthogonal series from its sum.
Representation of quasi-measure by a Henstock-Kurzweil type integral on a compact zero-dimensional metric space
2009
A derivation basis is introduced in a compact zero-dimensional metric space X. A Henstock-Kurzweil type integral with respect to this basis is defined and used to represent the so-called quasi-measure on X.
Relations among Henstock, McShane and Pettis integrals for multifunctions with compact convex values
2013
Fremlin (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 Banac…
Gauge integrals and selections of weakly compact valued multifunctions
2016
In the paper Henstock, McShane, Birkhoff and variationally multivalued integrals are studied for multifunctions taking values in the hyperspace of convex and weakly compact subsets of a general Banach space X. In particular the existence of selections integrable in the same sense of the corresponding multifunctions has been considered.
Approximation by step functions of Banach space valued nonabsolute integrals.
2008
The approximation of Banach space valued nonabsolutely integrable functions by step functions is studied. It is proved that a Henstock integrable function can be approximated by a sequence of step functions in the Alexiewicz norm, while a Henstock-Kurzweil-Pettis and a Denjoy-Khintchine-Pettis integrable function can be only scalarly approximate in the Alexiewicz norm by a sequence of step functions. In case of Henstock-Kurzweil-Pettis and Denjoy-Khintchine-Pettis integrals the full approximation can be done if and only if the range of the integral is norm relatively compact. It is also proved that if the target Banach space X does not contain any isomorphic copy of c_0, then the range of t…
The McShane, PU and Henstock integrals of Banach valued functions
2002
Some relationships between the vector valued Henstock and McShane integrals are investigated. An integral for vector valued functions, defined by means of partitions of the unity (the PU-integral) is studied. In particular it is shown that a vector valued function is McShane integrable if and only if it is both Pettis and PU-integrable. Convergence theorems for the Henstock variational and the PU integrals are stated. The families of multipliers for the Henstock and the Henstock variational integrals of vector valued functions are characterized.
Radon-Nikodym derivatives of finitely additive interval measures taking values in a Banach space with basis
2011
Let X be a Banach space with a Schauder basis {en}, and let Φ(I)= ∑n en ∫I fn(t)dt be a finitely additive interval measure on the unit interval [0, 1], where the integrals are taken in the sense of Henstock–Kurzweil. Necessary and sufficient conditions are given for Φ to be the indefinite integral of a Henstock–Kurzweil–Pettis (or Henstock, or variational Henstock) integrable function f:[0, 1] → X.
Strongly measurable Kurzweil-Henstock type integrable functions and series
2008
We give necessary and sufficient conditions for the scalar Kurzweil-Henstock integrability and the Kurzweil-Henstock-Pettis integrability of functions $f:[1, infty) ightarrow X$ defined as $f=sum_{n=1}^infty x_n chi_{[n,n+1)}$. Also the variational Henstock integrability is considered
A new result on impulsive differential equations involving non-absolutely convergent integrals
2009
AbstractIn this paper we obtain, as an application of a Darbo-type theorem, global solutions for differential equations with impulse effects, under the assumption that the function on the right-hand side is integrable in the Henstock sense. We thus generalize several previously given results in literature, for ordinary or impulsive equations.