Infinite-dimensional topological degree for systems of nonconvex inclusions
type: Item
instance of: scholarly article
title: Infinite-dimensional topological degree for systems of nonconvex inclusions
zbMATH Open document ID: 0956.47028
DOI: 10.1007/BF01759366
author: Eugene Stepanov
author: Pietro Zecca
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2000-08-16T00:00:00Z
review text: The authors consider systems of functional inclusions of the type \[ u\in F(u,\lambda),\qquad 0\in D(u,\lambda), \] where \(u\in E\), \(\lambda\in \mathbb{R}^n\), \(F:E\times \mathbb{R}^n\to E\) and \(D:E\times \mathbb{R}^n\to \mathbb{R}^n\) are nonlinear set-valued maps, \(E\) is some (in general, infinite-dimensional) Banach space. In the case when the maps \(F\), \(D\) satisfy minimum smoothness requirements (i.e. upper-semicontinuous), one of the most powerful approaches to study solvability of such systems is the use of some extension of topological degree theory for set-valued maps. However, it can only be applied under some extra assumptions on the structure of images of the named multivalued operators. Such assumptions usually involve some geometrical (e.g. convexity) or, more generally, topological (contractibility, acyclicity) properties. In an order to get rid of the mentioned disadvantage of the reduction method, the authors develop a degree theoretic approach involving the study of the whole systems. As this necessarily requires to work in infinite-dimensional spaces, they would need in addition to the mentioned assumptions also some conditions on compactness of the respetive multivalued operators.
reviewed by: Ulrich Kosel
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 1463338
zbMATH Keywords: nonconvex inclusion
zbMATH Keywords: contractibility
zbMATH Keywords: systems of functional inclusions
zbMATH Keywords: nonlinear set-valued maps
zbMATH Keywords: topological degree theory
zbMATH Keywords: convexity
zbMATH Keywords: acyclicity
zbMATH Keywords: reduction method
zbMATH Keywords: compactness
MaRDI profile type: Publication
cites work: Q4087979
cites work: Q4321323
cites work: Q4774620
cites work: Q5690035
cites work: Turning points and eigenvalue problems
cites work: On solvability of some boundary value problems for differential equations with ``maxima''
cites work: On the solvability of systems of non-convex inclusions in Banach spaces
cites work: On the Solvability of Systems of Inclusions Involving Noncompact Operators
cites work: On the Leray-Schauder alternative
cites work: Multivalued analysis and operator inclusions
cites work: On the homotopy method in the fixed point index theory of multi-valued mappings of compact absolute neighborhood retracts
cites work: Q3996831
description: scientific article; zbMATH DE number 1463338