On some discrete boundary value problem with parameters
type: Item
instance of: scholarly article
title: On some discrete boundary value problem with parameters
zbMATH Open document ID: 1264.39004
author: Marek Galewski
author: Joanna Smejda
published in: Applied Mathematics and Computation
publication date: 2012-09-13T00:00:00Z
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 6081866
zbMATH Keywords: fourth-order discrete boundary value problem
zbMATH Keywords: Green's function
zbMATH Keywords: sublinear problem
zbMATH Keywords: superlinear problem
zbMATH Keywords: existence of solution
zbMATH Keywords: multiplicity of solution
zbMATH Keywords: critical point theory
reviewed by: Petr Zemánek
MaRDI profile type: Publication
full work available at URL: j.amc.2012.04.040
OpenAlex ID: W2064067846
cites work: Q4033029
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cites work: On discrete fourth-order boundary value problems with three parameters
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cites work: Remarks on multiple nontrivial solutions for quasi-linear resonant problems
cites work: Q3787011
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cites work: On variational methods for periodic discrete problems
cites work: Existence results for discrete Sturm–Liouville problem via variational methods
cites work: Existence results for a nonlinear system with a parameter
cites work: Existence of solutions for some discrete boundary value problems with a parameter
cites work: Existence of solutions for a nonlinear system with a parameter
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DOI: 10.1016/J.AMC.2012.04.040
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review text: The authors study (with using the critical point theory) the existence and multiplicity of solutions of the discrete boundary value problem NEWLINE\[NEWLINE\begin{gathered}\Delta^4 u(k-2)+\eta \Delta^2 u(k-1)-\xi u(k)=\lambda f(k,u(k)),\quad k\in[a+1,b+1],\\ u(a)=\Delta^2 u(a-1)=0,\qquad u(b+2)=\Delta^2 u(b+1)=0,\end{gathered}NEWLINE\]NEWLINE where \(f:[a+1,b+1]\times\mathbb{R}\to\mathbb{R}\) is continuous and \(\eta,\xi,\lambda\) are real parameters such that NEWLINE\[NEWLINE\begin{gathered} \eta<8\sin^2 \frac{\pi}{2(b-a+2)}, \quad \eta^2+4\xi\geq0,\\ \xi+4\eta\sin^2\frac{\pi}{2(b-a+2)}<16\sin^4\frac{\pi}{2(b-a+2)},\quad \lambda>0.\end{gathered}NEWLINE\]NEWLINE The main results are obtained for the boundary value problem with superlinear (see Section 3) and sublinear (see Section 4) growth conditions imposed on the nonlinear terms. Finally, some improvements of results from \textit{Y. Yang} and \textit{J. Zhang} [Appl. Math. Comput. 211, No. 2, 293--302 (2009; Zbl 1169.39009)] are discussed in Section 5.
description: scientific article; zbMATH DE number 6081866