MATHEMATICA BOHEMICA, Vol. 135, No. 3, pp. 319-336, 2010

On the rational recursive sequence $ x_{n+1}=\dfrac{\alpha_0x_n+\alpha_1x_{n-l}+\alpha_2x_{n-k}} {\beta_0x_n+\beta_1x_{n-l}+\beta_2x_{n-k}} $

E. M. E. Zayed, M. A. El-Moneam

E. M. E. Zayed, Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt. Present address: Mathematics Department, Faculty of Science, Taif University, El-Taif, El-Hawiyah, P. O. Box 888, Kingdom of Saudi Arabia, e-mail: emezayed@hotmail.com; M. A. El-Moneam, Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt. Present address: Mathematics Department, Faculty of Science and Arts, Farasan, Jazan University, Jazan, Kingdom of Saudi Arabia, e-mail: mabdelmeneam2004@yahoo.com

Abstract: The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation
x_{n+1}=\frac{\alpha_0x_n+\alpha_1x_{n-l}+\alpha_2x_{n-k}} {\beta_0x_n+\beta_1x_{n-l}+\beta_2x_{n-k}}, \quad n=0,1,2,\dots
where the coefficients $\alpha_i,\beta_i\in(0,\infty)$ for $ i=0,1,2,$ and $l$, $k$ are positive integers. The initial conditions $ x_{-k}, \dots, x_{-l}, \dots, x_{-1}, x_0 $ are arbitrary positive real numbers such that $l<k$. Some numerical experiments are presented.

Keywords: difference equation, boundedness, period two solution, convergence, global stability

Classification (MSC 2010): 39A10, 39A99, 34C99


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