MATHEMATICA BOHEMICA, Vol. 133, No. 2, pp. 133-147 (2008)

# On the difference equation $x_{n+1}=\dfrac {a_{0}x_{n}+a_{1}x_{n-1}+\dots +a_{k}x_{n-k}}{b_{0}x_{n}+b_{1}x_{n-1}+\dots +b_{k}x_{n-k}}$

## E. M. Elabbasy, H. El-Metwally, E. M. Elsayed

E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt, e-mail: emelabbasy@mans.edu.eg, helmetwally@mans.edu.eg, emelsayed@mans.edu.eg

Abstract: In this paper we investigate the global convergence result, boundedness and periodicity of solutions of the recursive sequence \begin {equation*} x_{n+1}=\frac {a_{0}x_{n}+a_{1}x_{n-1}+\dots +a_{k}x_{n-k}}{b_{0}x_{n}+b_{1}x_{n-1}+\dots +b_{k}x_{n-k}},   n=0,1,\dots  \^^M\end {equation*} where the parameters $a_{i}$ and $b_{i}$ for $i=0,1,\dots ,k$ are positive real numbers and the initial conditions $x_{-k},x_{-k+1},\dots ,x_{0}$ are arbitrary positive numbers.

Keywords: stability, periodic solution, difference equation

Classification (MSC2000): 39A10

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