Stieltjes-Type Integrals for Metric Semigroup-Valued Functions Defined on Unbounded Intervals (original) (raw)

Steklov operators and semigroups in weighted spaces of continuous real functions

Acta Mathematica Hungarica, 2008

We consider Steklov operators in weighted spaces of continuous functions on the whole real line and on a bounded interval. We study the connections of these operators with some second order degenerate parabolic problems establishing a general Voronovskaja type formula. * Supported by PRIN 20062007.

THE RIEMANN-STIELTJES INTEGRAL AND SOME OF ITS APPLICATIONS

The process of Riemann Integration which is taught in Real Analysis classes is a specific case of the Riemann-Stieltjes Integration. Thus many of the terms and properties used to describe Riemann Integration are discussed in this project and they are extended to the Riemann-Stieltjes integral. This project therefore provides a careful introduction to the theory of Riemann-Stieltjes integration, and explains the properties of this integral. After doing so, we present some applications in functional analysis, where we used the fact that continuous functions on a closed interval are Riemann-Stieltjes Integrable with respect to any function of bounded variation, and this was used in proving the Riesz Representation Theorem. To show versatility of the Riemann-Stieltjes Integral, we also present some applications in Probability Theory, where the integral generates a formula for the Expectation, regardless of its underlying distribution. Other applications considered are population growth, and Mechanics.

Bounded convergence theorem for abstract Kurzweil–Stieltjes integral

Monatshefte für Mathematik, 2015

In the theories of Lebesgue integration and of ordinary differential equations, the Lebesgue Dominated Convergence Theorem provides one of the most widely used tools. Available analogy in the Riemann or Riemann-Stieltjes integration is the Bounded Convergence Theorem, sometimes called also the Arzelà or Arzelà-Osgood or Osgood Theorem. In the setting of the Kurzweil-Stieltjes integral for real valued functions its proof can be obtained by a slight modification of the proof given for the σ-Young-Stieltjes integral by T.H. Hildebrandt in his monograph from 1963. However, it is clear that the Hildebrandt's proof cannot be extended to the case of Banach space-valued functions. Moreover, it essentially utilizes the Arzelà Lemma which does not fit too much into elementary textbooks. In this paper, we present the proof of the Bounded Convergence Theorem for the abstract Kurzweil-Stieltjes integral in a setting elementary as much as possible.

The Metric Integral of Set-Valued Functions

Set-Valued and Variational Analysis, 2017

This paper introduces a new integral of univariate set-valued functions of bounded variation with compact images in R d. The new integral, termed the metric integral, is defined using metric linear combinations of sets and is shown to consist of integrals of all the metric selections of the integrated multifunction. The metric integral is a subset of the Aumann integral, but in contrast to the latter, it is not necessarily convex. For a special class of segment functions equality of the two integrals is shown. Properties of the metric selections and related properties of the metric integral are studied. Several indicative examples are presented.

A class of semigroups regularized in space and time

Journal of Mathematical Analysis and Applications, 2006

We consider α-times integrated C-regularized semigroups, which are a hybrid between semigroups regularized in space (C-semigroups) and integrated semigroups regularized in time. We study the basic properties of these objects, also in absence of exponential boundedness. We discuss their generators and establish an equivalence theorem between existence of integrated regularized semigroups and well-posedness of certain Cauchy problems. We investigate the effect of smoothing regularized semigroups by fractional integration.  2005 Elsevier Inc. All rights reserved.