Asymptotic Analysis of Increasing Solutions of Cyclic Second-Order Difference Systems

Abstract

The existence and asymptotic behavior of positive increasing solutions of the cyclic second-order nonlinear difference system
\begin{equation*}\label{SEI}
\Delta(p_i(n)|\Delta x_i(n)|^{\alpha_i-1}\Delta x_i(n)) = q_i(n)|x_{i+1}(n+1)|^{\beta_i-1}x_{i+1}(n+1), \quad i = \overline{1,N},
\end{equation*}
are studied, where $x_{N+1} = x_1$, and the sequences $p_i = {p_i(n)}$ and $q_i = {q_i(n)}$ are positive for all $n\in\mathbb{N}$, while the constants $\alpha_i$ and $\beta_i$, $i = \overline{1,N}$, are positive and satisfy the sublinearity condition $\alpha_1 \alpha_2 \cdots \alpha_N > \beta_1 \beta_2 \cdots \beta_N$. We consider two types of positive increasing solutions: those converging to a positive constant and those diverging to infinity, whose associated quasi-differences tend to a positive constant. For both classes of solutions, necessary and sufficient conditions for existence are established using fixed point methods. In addition, under the assumption that the coefficient sequences are regularly varying, we investigate positive increasing solutions for which both the solution components and their quasi-differences tend to infinity. In this case, the corresponding existence conditions are also derived, along with precise asymptotic formulas, based on the theory of discrete regular variation.