A coincidence theory involving Fredholm operators of nonnegative index
type: Item
instance of: scholarly article
title: A coincidence theory involving Fredholm operators of nonnegative index
zbMATH Open document ID: 0971.47046
DOI: 10.12775/TMNA.2000.004
author: Dorota Gabor
author: Wojciech Kryszewski
published in: Topological Methods in Nonlinear Analysis
publication date: 2001-11-04T00:00:00Z
review text: This article deals with the analysis of the problem \(F\circ p=q\), where \(\Gamma\) is a metric space, \(X\) and \(Y\) Banach spaces, \(p:\Gamma\to X\) is a Vietoris mapping (``a proper surjection with acyclic fibres''), \(F: X\to Y\), \(q:\Gamma\to Y\) a Fredholm operator of index \(i(F)\geq 0\), \(q: \Gamma\to Y\). The basic construction is connected with the passing from the equation \(F\circ p(\xi)= q(\xi)\) to the inclusion \(F(x)\in q(p^{-1}(x))\); then the generalized degree \(\deg((p,q), U,0)\) is defined as an element of \(\pi^n(S^m)\) in the finite-dimensional case and as an element of the corresponding stable homotopic group \(\Pi_k\) in the infinite-dimensional one; this degree has natural properties, in particular, the properties of homotopy invariance, additivity, and nontriviality; in general, the generalized degree coincides with the ``algebraic count'' of solutions to the inclusion \(F(x)\in q(p^{-1}(x))\). The considerations cover the set-valued setting yet.
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 55Q55
zbMATH DE Number: 1545386
zbMATH Keywords: algebraic count
zbMATH Keywords: Vietoris mapping
zbMATH Keywords: Fredholm operator
zbMATH Keywords: homotopy invariance
zbMATH Keywords: additivity
zbMATH Keywords: nontriviality
zbMATH Keywords: set-valued setting
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1545386