A result on the singular perturbation theory for differential inclusions in Banach spaces
type: Item
instance of: scholarly article
title: A result on the singular perturbation theory for differential inclusions in Banach spaces
zbMATH Open document ID: 0971.34047
DOI: 10.12775/TMNA.2000.001
author: Alessandra Andreini
author: Paolo Nistri
author: Mikhail Kamenskii
published in: Topological Methods in Nonlinear Analysis
publication date: 2001-10-30T00:00:00Z
review text: The following Cauchy problem for singularly perturbed systems of differential inclusions in infinite-dimensional space is considered \[ x'(t) \in Ax(t)+ f_1\bigl(t,x(t), y(t)\bigr),\;\varepsilon y'(t)\in By(t)+f_2 \bigl(t, x(t),y(t)\bigr), \tag{1} \] \[ x(0)=x_0,\;y(0)=y_0,\tag{2} \] where \(A\) and \(B\) are infinitesimal generators of \(C^0\)-semigroups of linear operators \(\exp(At)\) and \(\exp(Bt)\). Moreover, \(x\in E_1\), \(y\in E_2\), where \(E_1,E_2\) are infinite-dimensional Banach spaces. The nonlinear operators \(f_1,f_2\) are multivalued with nonempty, convex and compact values. Conditions ensuring that the solution set to (1)--(2) is upper semicontinuous with respect to the parameter \(\varepsilon\geq 0\) are investigated. The main tool used in the proof is a suitable measure of noncompactness being a combination of the Hausdorff measure (strong) noncompactness and the De Blasi measure of weak noncompactness. Apart from that the topological degree theory is also applied.
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 34E15
zbMATH DE Number: 1545383
zbMATH Keywords: strong noncompactness
zbMATH Keywords: singularly perturbed systems of differential inclusions
zbMATH Keywords: infinitesimal generators
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: Blasi measure
zbMATH Keywords: weak noncompactness
reviewed by: Józef Banaś
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1545383