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

The global attractivity of the rational difference equation $y_n=A+\left (\frac {y_{n-k}}{y_{n-m}}\right )^p$
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by Kenneth S. Berenhaut, John D. Foley and Stevo Stević PDF

Proc. Amer. Math. Soc. 136 (2008), 103-110 Request permission

Abstract:

This paper studies the behavior of positive solutions of the recursive equation \begin{eqnarray} y_n=A+\left (\frac {y_{n-k}}{y_{n-m}}\right )^p,\quad n=0,1,2,\ldots , \nonumber \end{eqnarray} 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 $y_n$ tends to $A+1$. This complements several results in the recent literature, including the main result in K. S. Berenhaut, J. D. Foley and S. Stević, 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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