Existence of a solution of discrete Emden-Fowler equation caused by continuous equation
type: Item
instance of: scholarly article
title: Existence of a solution of discrete Emden-Fowler equation caused by continuous equation
zbMATH Open document ID: 1486.39003
author: Josef Diblík
author: Evgeniya Korobko
author: I. V. Astashova
published in: Discrete and Continuous Dynamical Systems. Series S
publication date: 2022-01-03T00:00:00Z
review text: The authors consider a second-order nonlinear discrete equation of Emden-Fowler type: \[ \Delta^2 u(k)\pm k^{\alpha} u^m(k)=0, \tag{1}\] where \(u: \mathbb{N}(k_0)\longrightarrow \mathbb{R}\) is an unknown solution, \(k_0\) is a fixed integer, \(\Delta u(k)= u(k+1)-u(k)\) is the first-order forward difference, \(\Delta^2 u(k)= \Delta(\Delta u(k))\) is the second-order forward difference, and \(\alpha\), \(m\) are real numbers with \(m\neq 0,1\). Firstly, the authors construct an approximate solution of Equation (1) in power form, that is, an approximate function \(u_{app}: \mathbb{N}(k_0)\longrightarrow \mathbb{R}\), such that \[ u_{app}(k) \,\varpropto \frac{a_{\pm}}{k^s}+\frac{b_{\pm}}{k^{s+1}}, \] of order \(g(k)= k ^{s+3}\), where \[s=\frac{\alpha +2}{m-1}, \qquad a_{\pm}=[\mp s(s+1)]^{\frac{1}{m-1}}, \qquad b_{\pm}=\frac{a_{\pm}s(s+2)}{s+2-ms}. \] Secondly, they construct an equivalent system of difference equations, and then use them to prove their main result, which give conditions under which Equation (1) admits solutions whose asymptotic behavior satisfies conditions (36)--(38) in the paper. Some examples illustrate the obtained results. At the end of the paper, a comparison with the results obtained in [\textit{L. Erbe} et al., Ann. Mat. Pura Appl. (4) 191, No. 2, 205--217 (2012; Zbl 1259.34093)] is presented.
reviewed by: Nouressadat Touafek
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 39A22
Mathematics Subject Classification ID: 39A12
zbMATH DE Number: 7451493
zbMATH Keywords: discrete Emden-Fowler equation
zbMATH Keywords: exact solution
zbMATH Keywords: formal solution
zbMATH Keywords: asymptotic behavior
MaRDI profile type: Publication
full work available at URL: dcdss.2021133
OpenAlex ID: W3208114333
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DOI: 10.3934/DCDSS.2021133
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description: scientific article; zbMATH DE number 7451493