International Journal of Advances in Mathematics
International Journal of Advances in Mathematics
2456-6098
Volume 2018
Number 4
2018
July
02
On the higher order rational difference equation $x_{n+1}=\dfrac{%
\alpha +\beta x_{n-k}}{A+Bx_{n-k}}$}
67
79
EN
Abdul
Khaliq
Mathematics Department, Riphah Institute of Computing and \\
Applied Sciences, Riphah International University, Lahore, Pakistan.
khaliqsyed@gmail.com
Sk.
Sarif Hassan
Department of Mathematics, Pingla Thana Mahavidyalaya,\\
Paschim Medinipur, West Bengal, India.
khaliqsyed@gmail.com
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*}%
{\Large \noindent }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.
Global asymptotic stability, periodic solutions, equilibrium point, difference equations.
http://adv-math.com/higher-order-rational-difference-equation-x_n1dfrac-alpha-beta-x_n-kabx_n-k/
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