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On the global attractivity and the periodic character of some difference equations. (English) Zbl 0993.39008
Asymptotic properties of solutions of the $$k$$-th order difference equation $x_{n+1}=\frac{A_0}{x_n}+\frac{A_1}{x_{n-1}}+\dots+ \frac{A_{k-1}}{x_{n-k+1}},\quad n\in \mathbb N=\{0,1,\dots\} \tag{*}$ are investigated. It is shown that under some restrictions on the numbers $$A_0,\dots,A_{k-1}$$ every positive solution of (*) converges to a $$p$$-periodic solution, where the period $$p$$ is determined in terms of the coefficients $$A_0,\dots,A_{k-1}$$. The main result of the paper reads as follows.
Theorem. Let $$A_0,\dots,A_{k-1}$$ be nonnegative real numbers and suppose that the set $$J=\{j\geq 1:\;A_{j-1}>0\}$$ is nonempty. Set $$L=\{i+j: i,j\in J\}$$, and let $$p=2(\langle L\rangle +1)-\langle L\rangle/\langle J\rangle$$, where $$\langle \cdot\rangle$$ denotes the greatest common divisor of the elements of the set indicated. Then every positive solution of (*) converges to a periodic solution of (*) with (not necessarily prime) period $$p$$. Moreover, there exist solutions of (*) which are periodic with prime period $$p$$.

##### MSC:
 39A12 Discrete version of topics in analysis 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 39A11 Stability of difference equations (MSC2000) 39A10 Additive difference equations
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