Existence of a solution of discrete Emden-Fowler equation caused by continuous equation
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.
scientific article; zbMATH DE number 7451493
scientific article