On positive solutions of a reciprocal difference equation with minimum (Q1767389)

The authors consider the behavior of positive solutions of the equation \[ x_{n+1}=\min\left\{\frac{A}{x_{n}x_{n-1}\cdots x_{n-k}}, \frac{B}{x_{n-(k+2)}\cdots x_{n-(2k+2)}}\right\}, \;\;n\geq0, \] where \(A\), \(B\) and the initial values \(x_{-(2k+2)},\ldots,x_0\) are positive real numbers. It is shown that if \(0<A\leq B\) then all positive solutions are eventually periodic with period \(k+2\). A detailed study of the case \(k=0\) is presented.