Qualitative study of a higher order rational difference equation

Abstract

In this paper we study the behavior of the difference equation

                     $x_{n+1}$ = $\dfrac{\alpha x_nx_{n-l}}{\beta x_{n-m}+\gamma x_{n-l}}$,$\quad n=0,1,$ $\cdots$

where the initial conditions $x_{-r}$, $x_{-r+1}$, $\cdots$ ,$x_0$ are arbitrary non zero real numbers where $r=\max\{l,m\}$ is a non-negative integer and  $\alpha$, $\beta$ and $\gamma$ are constants: Also, we obtain the solutions of some special cases of this equation. At the end we present some numerical examples to support our theoretical discussion.

Keywords

• difference equation,stability,boundedness,global attractivity

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