# Global behavior of the difference equation $x_{n+1}=\frac{ax_{n-3} }{b+ cx_{n-1}x_{n-3}}$

## Raafat Abo-Zeid

Address: Department of Basic Science, The Valley Higher Institute of Engineering & Technology, Cairo, Egypt

E-mail: abuzead73@yahoo.com

Abstract: In this paper, we introduce an explicit formula and discuss the global behavior of solutions of the difference equation
\[ x_{n+1}=\frac{ax_{n-3} }{b+ cx_{n-1}x_{n-3}}\,,\qquad n=0,1,\dots \]
where $a,b,c$ are positive real numbers and the initial conditions $x_{-3}$, $x_{-2}$, $x_{-1}$, $x_0$ are real numbers.

AMSclassification: primary 39A20; secondary 39A21, 39A23, 39A30.

Keywords: difference equation, periodic solution, convergence.

DOI: 10.5817/AM2015-2-77