Nonlinear resonance set for nonlinear matrix equations
type: Item
instance of: scholarly article
title: Nonlinear resonance set for nonlinear matrix equations
zbMATH Open document ID: 0944.15014
DOI: 10.1016/S0024-3795(99)00040-3
author: Caryl Margulies
author: William Margulies
published in: Linear Algebra and its Applications
publication date: 2000-03-19T00:00:00Z
review text: Given an \(n\times n\) real matrix \(A\), its Fučik spectrum \(A_{-1}\subset\mathbb{R}^2\) is the set of all \([a,b]^T \in\mathbb{R}^2\) such that the (nonlinear) equation \(Ax=ax^+-bx^-\) has a nontrivial solution. Here \(x^\pm\) has the elements \(x_i^\pm= \max\{\pm x_i,0\}\), where \(x_i\) are the elements of \(x\). The Fučik set is a closed subset of \(\mathbb{R}^2\) (in the Zariski topology). It is relevant in solving nonlinear equations of the form \(Ax=ax^+-bx^-+g(x)\), where \(g(x)=o(\|x\|)\) for \(\|x\|\to\infty\). The authors prove several results describing \(A_{-1}\). The case \(n=2\) is considered in detail.
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 1398469
zbMATH Keywords: nonlinear resonance
zbMATH Keywords: nonlinear matrix equations
zbMATH Keywords: Fučik spectrum
zbMATH Keywords: Fučik set
zbMATH Keywords: Zariski topology
reviewed by: Mihail M. Konstantinov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1398469