A proof of the Dancer's conjecture on fixed point indices of mappings in cones.
type: Item
instance of: scholarly article
title: A proof of the Dancer's conjecture on fixed point indices of mappings in cones.
zbMATH Open document ID: 1059.47059
author: Dongsheng Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-03-14T00:00:00Z
review text: Let \(C\) be a total cone in a real Banach space \(E\), \(A:C\to C\) be a compact and \(C^1\)-operator, \(y\in C\), \(C_y=\{v\in E:y+tv\in C \text{ for some } t>0\}\) and \(S_y=\{v:\pm v\in \overline{C_y} \}\). Denote by \(\widetilde{S_y}\) the quotient space \(E/S_y\) and by \(\widetilde{A}\) a quotient mapping. Moreover, let \(r(\widetilde{A})\) denote the spectral radius of \(\widetilde{A}\). \textit{E. N. Dancer} [J. Reine Angew. Math. 371, 46--66 (1986; Zbl 0597.47034)] proposed the following conjecture: If \(y\) is an isolated fixed point of \(A\) in \(C\) and \(r(\widetilde{A})>1\), then \(i_C(A,y)=0\). The above conjecture is proved in section 4 of the present paper.
reviewed by: Mirosława Zima
Mathematics Subject Classification ID: 47H07
zbMATH DE Number: 2056387
zbMATH Keywords: cone
zbMATH Keywords: positive operator
zbMATH Keywords: fixed point index
Wikidata QID: Q122978255
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.10.012
OpenAlex ID: W2019613478
cites work: Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces
cites work: On the indices of fixed points of mappings in cones and applications
cites work: Multiple fixed points of positive mappings.
cites work: Positive solutions for a three-species competition system with diffusion—I. General existence results
cites work: Solution branches for mappings in cones, and applications
cites work: A priori estimates and existence of positive solutions of semilinear elliptic equations
cites work: Q4080169
cites work: Q3222981
cites work: On the Existence of Positive Solutions of Semilinear Elliptic Equations
cites work: Some global results for nonlinear eigenvalue problems
cites work: Q5613420
cites work: Q3706042
DOI: 10.1016/J.JMAA.2003.10.012
description: scientific article; zbMATH DE number 2056387