Dynamics of the System of Difference Equations $x_{n+1} = A + \frac{y_{n-k}}{y_{n}}, \,\,\, y_{n+1} = B + \frac{x_{n-k}}{x_{n}}$

Abstract

In this paper, we solve open problem 1 in Gumuş (Differ Equ Appl 24(6):976–991 2018). We investigate the oscillatory behavior, boundedness, persistence of positive solutions and the global asymptotic stability of the unique positive equilibrium point of the system of two rational difference equations:

$$\begin{aligned} x_{n+1} = A + \frac{y_{n-k}}{y_{n}}, \qquad y_{n+1} = B + \frac{x_{n-k}}{x_{n}},\quad n = 0, 1, \ldots \end{aligned}$$

where the parameters $A> 0$ and $B> 0$, the initial conditions $x_{i}, y_{i}$ are arbitrary positive numbers for $i = -k,-k + 1, \ldots , 0$ and $k \in \mathbb {Z}^{+}$. Moreover, we give some numerical examples that support our analytical results.