On the difference equation ${ x_{n+1}=\frac{ax_{n}^{2}+bx_{n-1}x_{n-k}}{cx_{n}^{2}+dx_{n-1}x_{n-k}}}$
DOI:
https://doi.org/10.5644/SJM.04.2.09Keywords:
Stability, periodic solutions, difference equationsAbstract
In this paper we investigate the global convergence result, boundedness, and periodicity of solutions of the recursive sequence
\begin{equation*}
x_{n+1}=\frac{ax_{n}^{2}+bx_{n-1}x_{n-k}}{cx_{n}^{2}+dx_{n-1}x_{n-k}},\;\;\;n=0,1,\dots
\end{equation*}
where the parameters $a,b,c$ and $d$ are positive real numbers and the initial conditions $ x_{-k},x_{-k+1},\dots,x_{-1}$ and $x_{0}$ are arbitrary positive numbers.
2000 Mathematics Subject Classification. 39A10
Downloads
References
E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the periodic nature of some max-type difference equation, Int. J. Math. Sci., 14 (2005), 2227–2239.
E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, On the difference equation $x_{n+1}=frac{alpha
x_{n-k}}{beta +gamma prod_{i=0}^{k}x_{n-i}},$, J. Conc. Appl. Math., 5 (2) (2007), 101–113.
H. El-Metwally, E. A. Grove, G. Ladas, and H. D.Voulov, On the global attractivity and the periodic character of some difference equations, J. Difference Equ. Appl., 7 (2001), 837–850.
H. El-Metwally, E. A. Grove, G. Ladas, and L. C. McGrath, On the Difference Equation $y_{n+1}=frac{y_{n-(2k+1)}+p}{
y_{n-(2k+1)}+qy_{n-2l}}$, Proceedings of the 6th ICDE, Taylor and Francis, London, 2004.
V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.
M. R. S. Kulenovic, G. Ladas and N. R. Prokup, On a rational difference equation, Comput. Math. Appl., 41 (2001), 671–678.
M. R. S. Kulenovic, G. Ladas and N. R. Prokup, On the recursive sequence $;x_{n+1}=frac{ax_{n}+bx_{n-1}}{1+x_{n}},$, J. Difference Equ. Appl., 6 (5) (2001), 563–576.
M. R. S. Kulenovic, G. Ladas and W. S. Sizer, On the recursive sequence $;x_{n+1}=frac{ax_{n}+bx_{n-1}}{gamma x_{n}+delta x_{n-1}},$, Math. Sci. Res. Hot-Line 2 (5) (1998), 1–16.
