Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type

Abstract

The article investigates a second-order nonlinear difference equation of Emden-Fowler type Δ2u(k)±kαum(k)=0,{\Delta }^{2}u\left(k)\pm {k}^{\alpha }{u}^{m}\left(k)=0, where kk is the independent variable with values k=k0,k0+1,…k={k}_{0},{k}_{0}+1,\ldots \hspace{0.33em}, u:{k0,k0+1,…}→Ru:\left\{{k}_{0},{k}_{0}+1,\ldots \hspace{0.33em}\right\}\to {\mathbb{R}} is the dependent variable, k0{k}_{0} is a fixed integer, and Δ2u(k){\Delta }^{2}u\left(k) is its second-order forward difference. New conditions with respect to parameters α∈R\alpha \in {\mathbb{R}} and m∈Rm\in {\mathbb{R}}, m≠1m\ne 1, are found such that the equation admits a solution asymptotically represented by a power function that is asymptotically equivalent to the exact solution of the nonlinear second-order differential Emden-Fowler equation y″(x)±xαym(x)=0.{y}^{^{\prime\prime} }\left(x)\pm {x}^{\alpha }{y}^{m}\left(x)=0. Two-term asymptotic representations are given not only for the solution itself but also for its first- and second-order forward differences as well. Previously known results are discussed, and illustrative examples are considered.

Keywords

• discrete equation
• emden-fowler equation
• equation of the second-order
• asymptotic behavior
• nonlinearity
• 39a22
• 39a12