On an Open Question Concerning Product-Type Difference Equations

In Yang et al. (Acta Math Univ Comenianae LXXX(1):63–70, 2011), Yang, Chen, and Shi examined the system of difference equations: $$\begin{aligned} x_n=\frac{a}{y_{n-p}},\qquad y_n=\frac{by_{n-p}}{x_{n-q}y_{n-q}},\qquad n=0,1,\ldots , \end{aligned}$$ where q is a positive integer with \(p < q\), \(p \not \mid q\), and \(p \ge 3\) is an odd number, both a and b are nonzero real constants, and the initial values \(x_{-q+1},x_{-q+2},\ldots ,\) \(x_0,y_{-q+1},y_{-q+2},\ldots ,y_0\) are nonzero real numbers. At the end of their note, they posted a question regarding the behavior of solutions of the given system when p is even. More precisely, they wondered what the solutions of the system may be if p is even. In this note we answer this question raised by the authors. Particularly, we show that the system may or may not admit a periodic solution depending on the coprimality of the parameters p and q and on the parity of the integer \(p/\gcd (p,q)\).