Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition -- a non-monotone case
type: Item
instance of: scholarly article
title: Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition -- a non-monotone case
zbMATH Open document ID: 1269.34029
author: Zhiming Guo
author: Xingfu Zou
author: Zhichun Yang
published in: Communications on Pure and Applied Analysis
publication date: 2013-04-30T00:00:00Z
review text: The authors are concerned with the existence and uniqueness of positive solutions to the boundary value problem \[ -w''+k^2w=\int_0^\pi f_\alpha(x,y)b(w(y))\,dy,\quad 0<x<\pi, \] subject to the Dirichlet boundary conditions \(w(0)=w(\pi)=0\), where the nonlinearity \(b\) represents the birth function and satisfies some biological sign, monotonicity and boundedness assumptions; the kernel \(f_\alpha\) is the sum of a sine series. The problem corresponds to searching steady state solutions for a reaction-diffusion equation. The authors consider separately the monotone and the non-monotone cases; then using the super-sub solutions, they prove existence and uniqueness of positive solutions. Two examples of applications illustrate the obtained results.
reviewed by: Smail Djebali
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B10
zbMATH DE Number: 6159400
zbMATH Keywords: nonlocal second-order differential equations
zbMATH Keywords: Dirichlet boundary condition
zbMATH Keywords: positive solution
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
MaRDI profile type: Publication
full work available at URL: cpaa.2012.11.1825
OpenAlex ID: W2315469403
Wikidata QID: Q115219307
DOI: 10.3934/CPAA.2012.11.1825
description: scientific article; zbMATH DE number 6159400