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1. A note on integration by parts for abstract Perron-Stieltjes integrals

Creator:
Schwabik, Štefan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
integration by parts, Kurzweil-Stieltjes integral, and Perron-Stieltjes integral
Language:
English
Description:
Integration by parts results concerning Stieltjes integrals for functions with values in Banach spaces are presented. The background of the theory is the Kurzweil approach to integration based on Riemann type integral sums, which leads to the Perron integral.
Rights:
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15. General integration and extensions. I

Creator:
Schwabik, Štefan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
abstract integration, extension of integral, and Kurzweil-Henstock integration
Language:
English
Description:
A general concept of integral is presented in the form given by S. Saks in his famous book Theory of the Integral. A special subclass of integrals is introduced in such a way that the classical integrals (Newton, Riemann, Lebesgue, Perron, Kurzweil-Henstock\dots ) belong to it. \endgraf A general approach to extensions is presented. The Cauchy and Harnack extensions are introduced for general integrals. The general results give, as a specimen, the Kurzweil-Henstock integration in the form of the extension of the Lebesgue integral.
Rights:
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16. General integration and extensions. II

Creator:
Schwabik, Štefan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
abstract integration, extension of integral, and Kurzweil-Henstock integration
Language:
English
Description:
This work is a continuation of the paper (Š. Schwabik: General integration and extensions I, Czechoslovak Math.\ J. 60 (2010), 961--981). Two new general extensions are introduced and studied in the class $\frak T$ of general integrals. The new extensions lead to approximate description of the Kurzweil-Henstock integral based on the Lebesgue integral close to the results of S. Nakanishi presented in the paper (S. Nakanishi: A new definition of the Denjoy's special integral by the method of successive approximation, Math.\ Jap. 41 (1995), 217--230).
Rights:
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23. Henstock-Kurzweil and McShane product integration; descriptive definitions

Creator:
Slavík, Antonín and Schwabik, Štefan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Henstock-Kurzweil product integral, McShane product integral, and Bochner product integral
Language:
English
Description:
The Henstock-Kurzweil and McShane product integrals generalize the notion of the Riemann product integral. We study properties of the corresponding indefinite integrals (i.e. product integrals considered as functions of the upper bound of integration). It is shown that the indefinite McShane product integral of a matrix-valued function $A$ is absolutely continuous. As a consequence we obtain that the McShane product integral of $A$ over $[a,b]$ exists and is invertible if and only if $A$ is Bochner integrable on $[a,b]$.
Rights:
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35. Linear Stieltjes integral equations in Banach spaces: II. Operator valued solutions

Creator:
Schwabik, Štefan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
linear Stieltjes integral equations, generalized linear differential equation, and equation in Banach space
Language:
English
Description:
This paper is a -continuation of [9]. In [9] results concerning equations of the form x(t) = x(a) + ∫ ţa d[A(s)]x(s) + f(t) - f(a) were presented. The Kurzweil type Stieltjes integration in the setting of [6] for Banach space valued functions was used. Here we consider operator valued solutions of the homogeneous problem Ф(t) = I+ ∫td d[A(s)]Ф(s) as well as the variation-of-constants formula for the former equation.
Rights:
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38. McShane equi-integrability and Vitali’s convergence theorem

Creator:
Kurzweil, Jaroslav and Schwabik, Štefan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
McShane integral, Vitali convergence theorem, and equi-integrability
Language:
English
Description:
The McShane integral of functions f : I → ! defined on an m-dimensional interval I is considered in the paper. This integral is known to be equivalent to the Lebesgue integral for which the Vitali convergence theorem holds. For McShane integrable sequences of functions a convergence theorem based on the concept of equi-integrability is proved and it is shown that this theorem is equivalent to the Vitali convergence theorem.
Rights:
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39. On the strong McShane integral of functions with values in a Banach space

Creator:
Schwabik, Štefan and Guoju, Ye
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Bochner integral and strong McShane integral
Language:
English
Description:
The classical Bochner integral is compared with the McShane concept of integration based on Riemann type integral sums. It turns out that the Bochner integrable functions form a proper subclass of the set of functions which are McShane integrable provided the Banach space to which the values of functions belong is infinite-dimensional. The Bochner integrable functions are characterized by using gauge techniques. The situation is different in the case of finite-dimensional valued vector functions.
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40. Operator-valued functions of bounded semivariation and convolutions

Creator:
Schwabik, Štefan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Kurzweil-Henstock integration, convolution, and Banach space
Language:
English
Description:
The abstract Perron-Stieltjes integral in the Kurzweil-Henstock sense given via integral sums is used for defining convolutions of Banach space valued functions. Basic facts concerning integration are preseted, the properties of Stieltjes convolutions are studied and applied to obtain resolvents for renewal type Stieltjes convolution equations.
Rights:
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