Nonexistence and existence results for a 2\textit{n}th-order discrete Dirichlet boundary value problem
type: Item
instance of: scholarly article
title: Nonexistence and existence results for a 2\textit{n}th-order discrete Dirichlet boundary value problem
zbMATH Open document ID: 1303.39005
author: Xia Liu
author: Yuanbiao Zhang
author: Haiping Shi
published in: Kodai Mathematical Journal
publication date: 2014-09-09T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.kmj/1404393901
review text: The authors consider a second-order nonlinear difference equation of the form \[ \Delta^n( \gamma_{i-n+1}\Delta^n u_{i-n} ) = (-1)^n f(i,u_{i+1},u_i,u_{i-1}), \quad n\in \mathbb{Z}(1),\, i\in \mathbb{Z}(1,k), \] with boundary conditions \[ u_{1-n}=u_{2-n}= \dots= u_0 =0, \quad u_{k+1}=u_{k+2}=\dots=u_{k+n}=0, \] where \( \Delta \) is the forward difference operator, \( \gamma_i \) is nonzero and real-valued for each \( i\in \mathbb{Z}(2-n,k+1)\), \(f\in C(\mathbb{R}^4, \mathbb{R} ) \). Using the critical point theory they give some sufficient conditions for nonexistence and existence of the solutions of the boundary value problem (BVP), by using the transfer of existence of the solutions of the BVP to the existence of the critical points of some functional. The proof of the results is based on the mountain pass lemma in combination with a variational technique. The authors provide a large number of references related with the subject and interesting numerical examples which are illustrating the support of the introduced conditions.
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 6341713
zbMATH Keywords: nonexistence
zbMATH Keywords: existence
zbMATH Keywords: Dirichlet boundary value problem
zbMATH Keywords: mountain pass lemma
zbMATH Keywords: discrete vibrational theory
zbMATH Keywords: second-order nonlinear difference equation
zbMATH Keywords: critical point theory
reviewed by: Haydar Akca
MaRDI profile type: Publication
OpenAlex ID: W2088176503
DOI: 10.2996/KMJ/1404393901
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description: scientific article; zbMATH DE number 6341713