# On the higher order rational difference equation $x_{n+1}=\frac{ \alpha +\beta x_{n-k}}{A+Bx_{n-k}}$ -by Abdul Khaliq and Sk.Sarif Hassan

**Abstract:** In this paper, we have investigated a nonlinear rational difference equation of higher order. Our concentration is on invariant intervals, periodic

character, the character of semicycles and global asymptotic stability of all positive solutions of

\begin{equation*}

x_{n+1}=\dfrac{\alpha +\beta x_{n-k}}{A+Bx_{n-k}},\;\;\;n=0,1,...,

\end{equation*}

where the parameters $\alpha $, $\beta $ and $A$, $B$ and the initial conditions $x_{k},\ ...,\ x_{1},\ x_{0}$ are positive real numbers $k=\{1,2,3,...\}$. It is shown that the equilibrium point is globally asymptotically stable under the condition \ $\beta \leq A$, and the unique positive solution is also globally asymptotically stable under the conditions $\beta $ $\leq $ $A$ $\leq $ $\beta $. It is shown that there does not exists any periodic solution for any positive parameters but it is shown computationally that there are periodic solutions with low as well as high periods.

**Key words:** Global asymptotic stability, periodic solutions, equilibrium point, difference equations.

**How to cite this article:**

Abdul Khaliq and Sk.Sarif Hassan, On the higher order rational difference equation $x_{n+1}=\frac{ \alpha +\beta x_{n-k}}{A+Bx_{n-k}}$, International Journal of Advances in Mathematics, Volume 2018, Number 4, Pages 67-79, 2018.