BV continuous solutions of an evolution inclusion with maximal monotone operator and nonconvex-valued perturbation. Existence theorem
type: Item
instance of: scholarly article
title: BV continuous solutions of an evolution inclusion with maximal monotone operator and nonconvex-valued perturbation. Existence theorem
zbMATH Open document ID: 1482.34144
author: A. A. Tolstonogov
published in: Set-Valued and Variational Analysis
publication date: 2021-05-06T00:00:00Z
review text: This well-written and illuminating paper is devoted to a thorough study of evolution inclusions governed by set-valued perturbations of time-dependent maximal monotone operators in a separable Hilbert space. These set-valued perturbations are assumed to have nonempty, closed and not necessarily convex point images, and the dependence of the maximal monotone operators on time is described with the help of the pseudo-distance introduced by \textit{A. A. Vladimirov} [Nonlinear Anal., Theory Methods Appl. 17, No. 6, 499--518 (1991; Zbl 0756.34064)]. This pseudo-distance is assumed to be of bounded variation as a function of time with an upper bound given by a nonatomic positive Radon measure \(\mu\). A solution of the evolution inclusion is a continuous function of bounded variation, the differential measure of which is absolutely continuous with respect to \(\mu\), so that the values of the density of this differential measure with respect to \(\mu\) belong to the right-hand side of the inclusion almost everywhere. Using a variety of auxiliary results, which are of independent interest, the author establishes the existence of such solutions and describes several properties of the solution sets.
reviewed by: Simeon Reich
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 47H05
zbMATH DE Number: 7345309
zbMATH Keywords: functions of bounded variation
zbMATH Keywords: differential measure
zbMATH Keywords: non-atomic Radon measure
zbMATH Keywords: density of a measure
zbMATH Keywords: maximal monotone operator
zbMATH Keywords: perturbation
MaRDI profile type: Publication
full work available at URL: s11228-020-00535-3
OpenAlex ID: W3009305506
cites work: Q5517160
cites work: Q4147129
cites work: Q3179421
cites work: Nonstationary dissipative evolution equations in a Hilbert space
cites work: BV solutions to evolution problems with time-dependent domains
cites work: Perturbed evolution problems with continuous bounded variation in time and applications
cites work: Measurable relations
cites work: Q4328334
cites work: On property (Q) and other semi-continuity properties of multifunctions
cites work: Q4148368
cites work: Differential inclusions in nonsmooth mechanical problems. Shocks and dry friction
cites work: A chain rule involving vector functions of bounded variation
cites work: Perturbed evolution problems with absolutely continuous variation in time and applications
cites work: Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators
DOI: 10.1007/S11228-020-00535-3
Recommended article: Existence and relaxation of solutions for evolution differential inclusions with maximal monotone operators
Recommended article: BV solutions to evolution problems with time-dependent domains
Recommended article: Distances between maximal monotone operators
Recommended article: Perturbed evolution problems with continuous bounded variation in time and applications
Recommended article: Set-valued perturbation for time dependent subdifferential operator
Recommended article: Solutions of nonlinear evolution inclusions
Recommended article: Multivalued perturbations of subdifferential type evolution equations in Hilbert spaces
Recommended article: On a time and state dependent maximal monotone operator coupled with a sweeping process with perturbations
Recommended article: Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators
Recommended article: Q4585377
description: scientific article; zbMATH DE number 7345309