On bounded lower \(\Lambda\)-variation
type: Item
instance of: scholarly article
title: On bounded lower \(\Lambda\)-variation
zbMATH Open document ID: 1318.26019
author: Piotr Kasprzak
author: Daria Bugajewska
published in: Journal of Mathematical Analysis and Applications
publication date: 2014-10-31T00:00:00Z
reviewed by: Mirosława Zima
Mathematics Subject Classification ID: 26A45
Mathematics Subject Classification ID: 45D05
zbMATH DE Number: 6363013
zbMATH Keywords: bounded \(\Lambda\)-variation
zbMATH Keywords: bounded variation in the mean
zbMATH Keywords: convolution operator
zbMATH Keywords: integral equations
zbMATH Keywords: linear differential equation
zbMATH Keywords: nonautonomous superposition operator
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.09.072
OpenAlex ID: W2044942543
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DOI: 10.1016/J.JMAA.2014.09.072
review text: The authors introduce a new notion of the lower \(\Lambda\)-variation of a real-valued function \(f\) on an interval \(I\) NEWLINE\[NEWLINE\underline{\text{var}}_{\Lambda}f:=\inf\{\text{var}_{\Lambda}g:f=g\;\text{ a. e. on } I\},NEWLINE\]NEWLINE where \(\text{var}_{\Lambda}g\) stands for the \(\Lambda\)-variation of \(g\). They prove that the vector space NEWLINE\[NEWLINE\underline{\Lambda\text{BV}}[0,1]:=\{f\in L^1[0,1]:\underline{\text{var}}_{\Lambda}f<+\infty\}NEWLINE\]NEWLINE endowed with the norm \(|f|_{\Lambda}:=\|f\|_1+\underline{\text{var}}_{\Lambda}f\) forms a Banach space. It is proved that in each equivalence class \([f]\) there is a representative, whose \(\Lambda\)-variation coincides with \(\underline{\text{var}}_{\Lambda}f\). Such representatives of the class are called good ones. Several topological and geometrical properties of the space \(\underline{\Lambda\text{BV}}[0,1]\) are provided. Applications to the linear differential equation of first order and nonlinear integral equations are given.
description: scientific article; zbMATH DE number 6363013