On a higher-order system of difference equations

Abstract

Here we study the following system of difference equations \begin{align} x_n&=f^{-1}\bigg(\frac{c_nf(x_{n-2k})}{a_n+b_n\prod_{i=1}^kg(y_{n-(2i-1)})f(x_{n-2i})}\bigg),\nonumber\\ y_n&=g^{-1}\bigg(\frac{\gamma_n g(y_{n-2k})}{\alpha_n+\beta_n \prod_{i=1}^kf(x_{n-(2i-1)})g(y_{n-2i})}\bigg),\nonumber \end{align} $n\in\mathbb{N}_0,$ where $f$ and $g$ are increasing real functions such that $f(0)=g(0)=0$, and coefficients $a_n,$ $b_n$, $c_n$, $\alpha_n$, $\beta_n$, $\gamma_n$, $n\in\mathbb{N}_0$, and initial values $x_{-i}$, $y_{-i}$, $i\in\{1,2,\ldots,2k\}$ are real numbers. We show that the system is solvable in closed form, and study asymptotic behavior of its solutions.

Keywords

• solvable system
• system of difference equations
• asymptotic behavior