Hilbert's projective metric and nonlinear elliptic problems
type: Item
instance of: scholarly article
title: Hilbert's projective metric and nonlinear elliptic problems
zbMATH Open document ID: 1215.47053
author: Min-Jei Huang
author: Chao-Ya Huang
author: Tzong-Mo Tsai
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-07-24T00:00:00Z
review text: The paper deals with eigenvalue problems for a class of positive nonlinear operators. Precisely, let \(K\) be a closed (not necessarily solid) cone in a real Banach space \(X,\) and let \(d\) be the Hilbert projective metric in \(K^+=K\setminus\{0\}\). For given \(f\in K^+\) and \(r>0,\) let \(K_f=\{x\in K^+:\;d(x,f)<\infty\}\) and \(E_{f,r}=\{x\in K_f:\;\|x\|=r\}\). The authors consider the eigenvalue problem \[ Tx=\lambda x,\quad x\in E_{f,r},\;\lambda>0, \] and the associated fixed point equation \(Tx=x\) for compact cone mappings \(T:\;K\to K\) which are of the form \(T=T_1+T_p,\) where \(T_j\) is increasing \(j\)-concave for \(j=1,p\) and \(0<p<1.\) Assuming \(T_1f\in K_f\) and \(T_pf\in K_f\) for some \(f\in K^+,\) and \(T(K^+)\subseteq K_f,\) it is proved that for each \(r>0\) there exists a unique pair \((x_r,\lambda_r)\in E_{f,r}\times(0,\infty)\) such that \(Tx_r=\lambda_rx_r.\) Moreover, the authors prove that \(T\) has a unique fixed point in \(K^+\) under the additional assumption that \(tf-T_1f\in K\) for some \(t<1.\) The results are applied to the study of nonlinear elliptic problems of the type \[ \begin{cases} \Delta u+a(|x|)\varphi(u)=0\quad \text{in}\;\Omega,\\ u=0\quad \text{on}\;\partial\Omega. \end{cases} \]
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 47J10
Mathematics Subject Classification ID: 47H07
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 5586792
zbMATH Keywords: projective metric
zbMATH Keywords: eigenvalue problem
zbMATH Keywords: fixed point
zbMATH Keywords: cone
zbMATH Keywords: concave operator
zbMATH Keywords: elliptic problem
zbMATH Keywords: positive solution
MaRDI profile type: Publication
full work available at URL: j.na.2009.01.093
OpenAlex ID: W1978293379
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DOI: 10.1016/J.NA.2009.01.093
description: scientific article; zbMATH DE number 5586792