Existence of positive solutions for a semipositone boundary value problem on the half-line
type: Item
instance of: scholarly article
title: Existence of positive solutions for a semipositone boundary value problem on the half-line
zbMATH Open document ID: 1200.34024
author: Bao Zhu
author: Ruyun Ma
publication date: 2010-06-28T00:00:00Z
review text: The paper discusses the existence of single and twin positive solutions to the following boundary value problem \[ (P)\quad\begin{cases} x''(t)-k^2(t)x(t)+\lambda m(t)f(t,x(t))=0,\qquad t\in(0,+\infty),\\ x(0)=0,\qquad x(+\infty)=0, \end{cases} \] where \(k: [0,\infty)\longrightarrow(0,\infty)\) is continuous, bounded, \(f: [0,\infty)\longrightarrow\mathbb{R}\) is continuous and semi-positone, \(m\) is continuous and positive, and \(\lambda\) is a positive parameter. After defining a suitable Green's function, the authors formulate Problem (P) as a fixed point problem for a compact nonlinear mapping. Under some growth assumptions and some restrictions on \(\lambda\), the authors then prove the existence of a single or twin solutions in a positive cone of a Banach space via the classical Krasnosel'skii's fixed point theorem of cone compression and expansion. Existence theorems extend some known results for the constant case \(k>0\) (see e.g. [\textit{S. Djebali, K. Mebarki}, Comput. Math. Appl. 55, No.~12, 2940--2952 (2008; Zbl 1142.34316)] and the references therein).
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B40
Mathematics Subject Classification ID: 34B27
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 5727369
zbMATH Keywords: positive solutions
zbMATH Keywords: boundary value problem on the half-line
zbMATH Keywords: fixed point
zbMATH Keywords: cone
reviewed by: Smail Djebali
MaRDI profile type: Publication
full work available at URL: j.camwa.2009.07.005
OpenAlex ID: W2095034759
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DOI: 10.1016/J.CAMWA.2009.07.005
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published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 5727369