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

Abstract

Abstract The article investigates a second-order nonlinear difference equation of Emden-Fowler type Δ 2 u ( k ) ± k α u m ( k ) = 0 , {\Delta }^{2}u\left(k)\pm {k}^{\alpha }{u}^{m}\left(k)=0, where k k is the independent variable with values k = k 0 , k 0 + 1 , … k={k}_{0},{k}_{0}+1,\ldots \hspace{0.33em} , u : { k 0 , k 0 + 1 , … } → R u:\left\{{k}_{0},{k}_{0}+1,\ldots \hspace{0.33em}\right\}\to {\mathbb{R}} is the dependent variable, k 0 {k}_{0} is a fixed integer, and Δ 2 u ( 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 ∈ R m\in {\mathbb{R}} , m ≠ 1 m\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 ) ± x α y m ( 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.