On a fourth order rational difference equation

Abstract

In this paper, we determine and study the behavior of all admissible solutions of the difference equation $$x_{n+1}=\frac{x_{n}x_{n-2}}{ax_{n-2}+ bx_{n-3}},\quad n=0,1,\ldots,$$ where $a,b$ are positive real numbers and the initial conditions $ x_{-3},x_{-2},x_{-1},x_0$ are real numbers. We show when $a=b=1$ that, every admissible solution converges to $0$.