Displaying similar documents to “On Henstock-Dunford and Henstock-Pettis integrals.”

Characterizations of Kurzweil-Henstock-Pettis integrable functions

L. Di Piazza, K. Musiał (2006)

Studia Mathematica

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We prove that several results of Talagrand proved for the Pettis integral also hold for the Kurzweil-Henstock-Pettis integral. In particular the Kurzweil-Henstock-Pettis integrability can be characterized by cores of the functions and by properties of suitable operators defined by integrands.

On Denjoy type extensions of the Pettis integral

Kirill Naralenkov (2010)

Czechoslovak Mathematical Journal

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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.

The Henstock-Kurzweil-Pettis integrals and existence theorems for the Cauchy problem

Mieczysław Cichoń, Ireneusz Kubiaczyk, Sikorska-Nowak, Aneta Sikorska-Nowak, Aneta (2004)

Czechoslovak Mathematical Journal

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In this paper we prove an existence theorem for the Cauchy problem x ' ( t ) = f ( t , x ( t ) ) , x ( 0 ) = x 0 , t I α = [ 0 , α ] using the Henstock-Kurzweil-Pettis integral and its properties. The requirements on the function f are not too restrictive: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose that the function f satisfies some conditions expressed in terms of measures of weak noncompactness.

On Denjoy-Dunford and Denjoy-Pettis integrals

José Gámez, José Mendoza (1998)

Studia Mathematica

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The two main results of this paper are the following: (a) If X is a Banach space and f : [a,b] → X is a function such that x*f is Denjoy integrable for all x* ∈ X*, then f is Denjoy-Dunford integrable, and (b) There exists a Dunford integrable function f : [ a , b ] c 0 which is not Pettis integrable on any subinterval in [a,b], while ʃ J f belongs to c 0 for every subinterval J in [a,b]. These results provide answers to two open problems left by R. A. Gordon in [4]. Some other questions in connection with...

The McShane, PU and Henstock integrals of Banach valued functions

Luisa Di Piazza, Valeria Marraffa (2002)

Czechoslovak Mathematical Journal

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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...

Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals

A. Sikorska-Nowak (2007)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We prove an existence theorem for the equation x' = f(t,xₜ), x(Θ) = φ(Θ), where xₜ(Θ) = x(t+Θ), for -r ≤ Θ < 0, t ∈ Iₐ, Iₐ = [0,a], a ∈ R₊ in a Banach space, using the Henstock-Kurzweil-Pettis integral and its properties. The requirements on the function f are not too restrictive: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose that the function f satisfies some conditions expressed in terms of the measure of weak noncompactness. ...