Discrete Lyapunov inequality conditions for partial difference equations
type: Item
instance of: scholarly article
title: Discrete Lyapunov inequality conditions for partial difference equations
zbMATH Open document ID: 0709.39006
author: Sui Sun Cheng
author: Liang Yu Hsieh
author: Dale Z. T. Chao
published in: Hokkaido Mathematical Journal
publication date: 1990-01-01T00:00:00Z
review text: The authors obtain some necessary conditions for the existence of a nontrivial solution to some boundary value problems of second-order partial difference equations by means of a discrete maximum principle. The main result of the paper is embodied in the following Theorem 3.2. Given the boundary value problem \[ (1)\quad \Delta^ 2_ 1u(i-1,j)+\Delta^ 2_ 2u(i,j-1)+p(i,j)u(i,j)=0,\quad (i,j)\in S;\quad (2)\quad u(i,j)=0,\quad (i,j)\in \partial S, \] where S is a net, i.e., a connected and finite nonempty set of lattice points, \(\partial S\) is the exterior boundary of S, p(i,j)\(\geq 0\) for (i,j)\(\in S\), \(\Delta_ 1h(i,j)=h(i+1,j)-h(i,j)\), \(\Delta_ 2h(i,j)=h(i,j+1)-h(i,j)\). If (1)-(2) has a nontrivial solution then \[ \sum_{z\in S}p(z)\geq \mu (S)=\{\max_{z\in S}G(z| z)\}^{-1}, \] where G(z\(| w)\) with \(w=z\) being the Green function associated with (1)-(2), and the inequality is sharp. The above result is similar to a classical inequality proved by \textit{A. M. Lyapunov} [Problème général de a stabilité du mouvement (1947; M.R. 9, 34)] for boundary value problems of second-order ordinary differential equations. The maxima of the Green's functions are then calculated for nets of three different kinds. (Note that, Theorem 5.1 on page 235 should be Theorem 3.1, and in its statement the word ``circular'' should be replaced by ``straight''.) An open problem is suggested in the final section.
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 39A12
zbMATH DE Number: 4165553
zbMATH Keywords: discrete Lyapunov inequality
zbMATH Keywords: partial difference equations
zbMATH Keywords: lattice points
zbMATH Keywords: discrete maximum principle
zbMATH Keywords: Green's functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1381517357
OpenAlex ID: W1972069422
DOI: 10.14492/HOKMJ/1381517357
description: scientific article; zbMATH DE number 4165553