Asymptotic properties of solutions to difference equations of Emden-Fowler type

We study the higher order difference equations of the following form \[ \Delta^m x_n=a_nf(x_{\sigma(n)})+b_n. \] We are interested in the asymptotic behavior of solutions $x$ of the above equation. Assuming $f$ is a power type function and $\Delta^m y_n=b_n$, we present sufficient conditions that guarantee the existence of a solution $x$ such that \[ x_n=y_n+\mathrm{o}(n^s), \] where $s\leq 0$ is fixed. We establish also conditions under which for a given solution $x$ there exists a sequence $y$ such that $\Delta^m y_n=b_n$ and $x$ has the above asymptotic behavior