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On a generalization of Henstock-Kurzweil integrals

Jan Malý, Kristýna Kuncová (2019)

Mathematica Bohemica

We study a scale of integrals on the real line motivated by the M C α integral by Ball and Preiss and some recent multidimensional constructions of integral. These integrals are non-absolutely convergent and contain the Henstock-Kurzweil integral. Most of the results are of comparison nature. Further, we show that our indefinite integrals are a.e. approximately differentiable. An example of approximate discontinuity of an indefinite integral is also presented.

On belated differentiation and a characterization of Henstock-Kurzweil-Ito integrable processes

Tin-Lam Toh, Tuan-Seng Chew (2005)

Mathematica Bohemica

The Henstock-Kurzweil approach, also known as the generalized Riemann approach, has been successful in giving an alternative definition to the classical Itô integral. The Riemann approach is well-known for its directness in defining integrals. In this note we will prove the Fundamental Theorem for the Henstock-Kurzweil-Itô integral, thereby providing a characterization of Henstock-Kurzweil-Itô integrable stochastic processes in terms of their primitive processes.

On Denjoy type extensions of the Pettis integral

Kirill Naralenkov (2010)

Czechoslovak Mathematical Journal

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.

On Denjoy-Dunford and Denjoy-Pettis integrals

José Gámez, José Mendoza (1998)

Studia Mathematica

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 Denjoy-Dundord...

On Henstock-Kurzweil method to Stratonovich integral

Haifeng Yang, Tin Lam Toh (2016)

Mathematica Bohemica

We use the general Riemann approach to define the Stratonovich integral with respect to Brownian motion. Our new definition of Stratonovich integral encompass the classical Stratonovich integral and more importantly, satisfies the ideal Itô formula without the “tail” term, that is, f ( W t ) = f ( W 0 ) + ∫ 0 t f ' ( W s ) ∘ d W s . Further, the condition on the integrands in this paper is weaker than the classical one.

On indefinite BV-integrals

Donatella Bongiorno, Udayan B. Darji, Washek Frank Pfeffer (2000)

Commentationes Mathematicae Universitatis Carolinae

We present an example of a locally BV-integrable function in the real line whose indefinite integral is not the sum of a locally absolutely continuous function and a function that is Lipschitz at all but countably many points.

On Kurzweil-Henstock equiintegrable sequences

Štefan Schwabik, Ivo Vrkoč (1996)

Mathematica Bohemica

For the Kurzweil-Henstock integral the equiintegrability of a pointwise convergent sequence of integrable functions implies the integrability of the limit function and the relation m abfm(s)s = abm fm(s)s. Conditions for the equiintegrability of a sequence of functions pointwise convergent to an integrable function are presented. These conditions are given in terms of convergence of some sequences of integrals.

On Kurzweil-Stieltjes integral in a Banach space

Giselle A. Monteiro, Milan Tvrdý (2012)

Mathematica Bohemica

In the paper we deal with the Kurzweil-Stieltjes integration of functions having values in a Banach space X . We extend results obtained by Štefan Schwabik and complete the theory so that it will be well applicable to prove results on the continuous dependence of solutions to generalized linear differential equations in a Banach space. By Schwabik, the integral ∫ a b d [ F ] g exists if F : [ a , b ] → L ( X ) has a bounded semi-variation on [ a , b ] and g : [ a , b ] → X is regulated on [ a , b ] . We prove that this integral has sense also if F is regulated on [ a , b ] ...

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