Boundary value problems for discrete equations
type: Item
instance of: scholarly article
title: Boundary value problems for discrete equations
zbMATH Open document ID: 0890.39001
DOI: 10.1016/S0893-9659(97)00064-5
published in: Applied Mathematics Letters
publication date: 1998-06-28T00:00:00Z
review text: The paper presents some general properties of solutions of a nonlinear discrete boundary value problem \[ \Delta^2 y(i-1)+\mu f(i,y(i))=0,\quad i\in N,\quad y(0)=0,\quad y(T+1)=0,\tag{1} \] where \(\mu\geq 0\), \(T\in\{1,2,\ldots,T\}\), \(N^+=\{0,1,\ldots,T+1\}\), \(y:N^+\to\mathbb{R}\), and \(f:N\times\mathbb{R}\to\mathbb{R}\) is continuous. The main result is a discrete version of the generalized Gelfand problem. The authors' aim here is to establish the existence of nonnegative solutions. The notion of upper and lower solutions of the boundary problem (1) is introduced. An easy argument (based on the Schauder fixed point theorem) guarantees that the problem has a solution which lies between the lower and the upper solution. The obtained results are the consequences of a new existence principle established in the first part of the paper.
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 1085391
zbMATH Keywords: nonlinear difference equations
zbMATH Keywords: boundary value problems
zbMATH Keywords: upper and lower solutions
zbMATH Keywords: nonnegative solutions
author: Ravi P. Agarwal
author: Donal O'Regan
reviewed by: Alexander Mednykh
MaRDI profile type: Publication
cites work: Q4003305
cites work: On boundary value problems for second order discrete systems
cites work: Q4421936
cites work: A fixed-point approach for nonlinear discrete boundary value problems
cites work: Q3293867
cites work: Q4015775
cites work: Theory of difference equations: Numerical methods and applications
cites work: A discrete boundary value problem
full work available at URL: s0893-9659(97)00064-5
OpenAlex ID: W1985066099
description: scientific article; zbMATH DE number 1085391