Asymptotic representation of intermediate solutions to a cyclic systems of second-order difference equations with regularly varying coefficients - Aleksandra Kapesic (Electronic Journal of Qualitative Theory of Differential Equations, 2018)
The cyclic system of second-order difference equations \begin{equation*} Δ(p_i(n)|Δ x_i(n)|^{α_i-1}Δ x_i(n)) = q_i(n)|x_{i+1}(n+1)|^{β_i-1}x_{i+1}(n+1), \end{equation*} for $i=\overline{1,N}$ where $x_{N+1}=x_1,$ is analysed in the framework of discrete regular variation. Under the assumption that $α_i$ and $β_i$ are positive constants such that $α_1α_2…α_N>β_1β_2…β_N$ and $p_i$ and $q_i$ are regularly varying sequences it is shown that the situation in which this system possesses regularly varying intermediate solutions can be completely characterized. Besides, precise information can be acquired about the asymptotic behavior at infinity of these solutions.