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Zbl 1220.39011
Berg, Lothar; Stević, Stevo
On the asymptotics of the difference equation $y_{n}(1 + y_{n - 1} \cdots y_{n - k + 1}) = y_{n - k}$.
(English)
[J] J. Difference Equ. Appl. 17, No. 4, 577-586 (2011). ISSN 1023-6198

The existence of positive solutions of the difference equation $$y_{n}=\frac{y_{n-k}}{1+y_{n-1}\dots y_{n-k+1}}~,\quad n\in\mathbb N_{0},\tag*$$ where $k\in\mathbb N\backslash \{1\}$, converging to zero is studied. The main result of this article is the following Theorem: For every $k\in\mathbb N\backslash \{1\}$, equation ($*$) has a positive solution with the following asymptotics $$y_{n}=\left( \frac{k}{(k-1)n}\right) ^{1/(k-1)}\left( 1+a\frac{\ln n}{n}+b\frac{\ln ^{2}n}{n^{2}}+o\left(\frac{\ln ^{2}n}{n^{2}}\right)\right),$$ where for $k=2p+1$, $$a=\frac{2p+1}{8p^{2}}\text{ and }b=\frac{(2p+1)^{3}}{128p^{4}},$$ while for $k=2p+2$, $$a=\frac{p+1}{(2p+1)^{2}}~\text{ and }b=\frac{(p+1)^{3}}{(2p+1)^{4}}.$$
[Fozi Dannan (Damascus)]
MSC 2000:
*39A20 Generalized difference equations
39A22

Keywords: rational difference equation; positive solution; convergence to zero; asymptotic behaviour

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