Analysis of the Convergence and Periodicity of a Rational Difference Equation

Abstract

The exact solutions of most difference equations cannot be obtained sometimes. This can be attributed to the fact that there is no a specific approach from which one can find the exact solution. Therefore, many researchers tend to study the qualitative behaviours of these equations. In this paper, we will investigate some qualitative properties such as local stability, global stability, periodicity and solutions of the following eighth order recursive equation \begin{eqnarray*} x_{n+1}=c_{1}x_{n-3}-\frac{c_{2}x_{n-3}}{c_{3} x_{n-3}- c_{4} x_{n-7}},\;\;\;n=0,1,..., \end{eqnarray*} {\Large \noindent }where the coefficients $c_{i},\ \textit{for all} \ i=1,...,4,$ are assumed to be positive real numbers and the initial conditions $x_{i} \ \textit{ for all} \ i=-7,-6,...,0, $ are arbitrary non-zero real numbers.

Keywords

• difference equation
• equilibria
• global attractivity
• local stability
• periodicity