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

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.

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.