The global attractivity of the rational difference equation 𝑦_{𝑛}=𝐴+(\𝑓𝑟𝑎𝑐{𝑦_{𝑛-𝑘}}𝑦_{𝑛-𝑚})^{𝑝}

Berenhaut, Kenneth; Foley, John; Stević, Stevo

doi:10.1090/S0002-9939-07-08860-0

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ISSN 1088-6826(online) ISSN 0002-9939(print)

The global attractivity of the rational difference equation $y_n=A+\left(\frac{y_{n-k}}{y_{n-m}}\right)^p$

Authors: Kenneth S. Berenhaut, John D. Foley and Stevo Stevic
Journal: Proc. Amer. Math. Soc. 136 (2008), 103-110
MSC (2000): Primary 39A10, 39A11
Posted: September 24, 2007
MathSciNet review: 2350394
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Abstract: This paper studies the behavior of positive solutions of the recursive equation

$\displaystyle y_n=A+\left(\frac{y_{n-k}}{y_{n-m}}\right)^p,\quad n=0,1,2,\ldots,$

with $y_{-s},y_{-s+1}, \ldots, y_{-1} \in (0, \infty)$ and $k,m \in \{1,2,3,4,\ldots\}$ , where $s=\max\{k,m\}$ . We prove that if $\mathrm{gcd}(k,m) = 1$ , and $p\leq \min\{1,(A+1)/2\}$ , then

tends to

. This complements several results in the recent literature, including the main result in K. S. Berenhaut, J. D. Foley and S. Stevic, The global attractivity of the rational difference equation $y_{n}=1+\frac{y_{n-k}}{y_{n-m}}$ , Proc. Amer. Math. Soc., 135 (2007) 1133-1140.

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