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SOOCHOW JOURNAL OF MATHEMATICS

Volume 33, No. 4, pp. 861-873, October 2007




QUALITATIVE BEHAVIOR OF HIGHER ORDER

DIFFERENCE EQUATION


BY


E. M. ELABBASY, H. EL-METWALLY AND E. M. ELSAYED




Abstract. In this paper we investigate some qualitative behavior of the solutions of the recursive sequence

$\displaystyle x_{n+1}=\frac{dx_{n-l}x_{n-k}}{cx_{n-s}-b}+a,\ \ n=0, 1, \ldots
$

where the initial conditions $ x_{-r},x_{-r+1},x_{-r+2},\ldots$, $ x_{0}$ are arbitrary positive real numbers with $ x_{i}\not
=\frac{b}{c}$ for $ i=-r,-r+1,\ldots,0$, $ a>b/c$, $ r=\max\{l,k,s\}$ is nonnegative integer and $ a,b,c,d$ are positive constants. Also, we study two special cases of this equation.

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AMS Subject Classification. 39A10.

Key words. difference equations, stability, periodic solutions.