On an Open Question Concerning Product-Type Difference Equations

Abstract

In [Acta Math. Univ. Comenianae Vol. LXXX, 1 (2011), pp. 63--70], Yang, Chen and Shi examined the system of difference equations \[ x_n=\frac{a}{y_{n-p}},\qquad y_n=\frac{by_{n-p}}{x_{n-q}y_{n-q}},\qquad n=0,1,\ldots, \] where $q$ is a positive integer with $p < q$, $p \nmid q$, $p \geq 3$ 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 behaviour of solutions of the given system when $p$ is even. More precisely, they asked what the solutions of the system may look like 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 $p/\gcd(p,q)$.