Structure of positive solution sets of sub-linear semi-positone operator equations
type: Item
instance of: scholarly article
title: Structure of positive solution sets of sub-linear semi-positone operator equations
zbMATH Open document ID: 1270.47049
author: Xian Xu
author: Donal O'Regan
published in: Positivity
publication date: 2013-01-02T00:00:00Z
review text: In this paper, the authors consider the nonlinear operator equation \(x=G(x,\lambda)\) in a real Banach space \(E\). Here, \(\lambda\in(0,+\infty)\) and \(x\in P\), where \(P\) is an order cone. Moreover, \(G(x,\lambda):=\lambda KFx+e_0\), where \(K : P\to P\) is a linear, completely continuous operator, \(F : P\to E\) is a bounded nonlinear continuous operator, and \(e_0\in(-P)\). Due to this latter fact (i.e., that \(e_0\in(-P)\)), it follows that the authors study the so-called semipositone problem. The authors employ classical degree theory as well as Krein-Rutman theory. After studying the aforementioned, more abstract problem, the authors provide some results for the boundary value problem \(\left(p(t)u'\right)'+\lambda f(t,u)+q(t)=0\) subject to the boundary conditions \(au(0)-bp(0)u'(0)=0\) and \(cu(1)+dp(1)u'(1) =0\). By imposing standard hypotheses on \(f\) (e.g., \(\frac{f(t,u)}{u}\to 0\), uniformly for \(t\in[0,1]\), as \(u\to+\infty\)), the authors deduce the existence of at least one positive solution to the above boundary value problem for certain values of the parameter \(\lambda\). The paper is well written and concludes with an example of a boundary value problem to which their theory is applied. Unfortunately, no additional examples are provided to explicate the more abstract theory.
reviewed by: Christopher Goodrich
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 34B09
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34B18
zbMATH DE Number: 6121311
zbMATH Keywords: structure of positive solutions set
zbMATH Keywords: sub-linear operator
zbMATH Keywords: global bifurcation theories
zbMATH Keywords: eigenvalue
zbMATH Keywords: boundary value problem
MaRDI profile type: Publication
full work available at URL: s11117-011-0131-y
OpenAlex ID: W1968008482
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DOI: 10.1007/S11117-011-0131-Y
description: scientific article; zbMATH DE number 6121311