Double solutions of boundary value problems for 2\(m^{\text{th}}\)-order differential equations and difference equations
type: Item
instance of: scholarly article
title: Double solutions of boundary value problems for 2\(m^{\text{th}}\)-order differential equations and difference equations
zbMATH Open document ID: 1070.34036
DOI: 10.1016/S0898-1221(03)00063-4
author: John R. Graef
author: Johnny Henderson
publication date: 2004-08-06T00:00:00Z
review text: The authors prove the existence of two positive solutions for the two-point boundary value problem \[ (-1)^m\,y^{(2m)}=f(y),\qquad 0\leq t\leq 1,\quad y^{(2i)}(0)=y^{(2i+1)},\qquad i=0,1,\dots,m-1, \] where \(f:\mathbb{R}\to [0,\,\infty)\) is continuous. The arguments are based on a double fixed-point theorem due to \textit{R. I. Avery} and \textit{J. Henderson} in [Commun. Appl. Nonlinear Anal. 8, 27--36 (2001; Zbl 1014.47025)]. They also apply the double fixed-point theorem to the boundary value problem for the finite difference equation \[ \begin{aligned}&(-1)^m\,\Delta^{2m}u(k)=g(u(k)),\qquad k=0,1,\dots,N,\\ &\Delta^{2i}u(0)=\Delta^{2i+1}u(N+1),\qquad i=0,1,\dots,m-1, \end{aligned} \] where \(g:\mathbb{R}\to [0,\,\infty)\) is continuous, and obtain an analogous existence result of two positive solutions.
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B27
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 39A12
zbMATH DE Number: 2082201
zbMATH Keywords: boundary value problem
zbMATH Keywords: Green function
zbMATH Keywords: multiple solutions
MaRDI profile type: Publication
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full work available at URL: s0898-1221(03)00063-4
OpenAlex ID: W1966959738
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 2082201