ON THE RECURSIVE SEQUENCE $x_{n+1}=x_{n-1}/g(x_n)$

Abstract

In [5] the following problem was posed. Is there a solution of the following difference equation $$ x_{n+1}=\displaystyle\frac{\beta x_{n-1}}{\beta+x_n},\quad x_{-1},x_0\gt 0,\ \beta \gt 0, \quad n=0,1,2,... $$ such that $x_n\to 0$ as $n\to\infty.$ We prove a result which, as a special case, solves the above problem.