The global attractivity of the rational difference equation 𝑦_{𝑛}=1+\frac{𝑦_{𝑛-𝑘}}𝑦_{𝑛-𝑚}

Berenhaut, Kenneth; Foley, John; Stević, Stevo

doi:10.1090/S0002-9939-06-08580-7

The global attractivity of the rational difference equation $y_{n}=1+\frac {y_{n-k}}{y_{n-m}}$
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by Kenneth S. Berenhaut, John D. Foley and Stevo Stević PDF

Proc. Amer. Math. Soc. 135 (2007), 1133-1140 Request permission

Abstract:

This paper studies the behavior of positive solutions of the recursive equation \begin{eqnarray} y_{n}=1+\frac {y_{n-k}}{y_{n-m}}, ~~ 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 $\mathrm {gcd}(k,m) = 1$, with $k$ odd, then $y_n$ tends to $2$, exponentially. When combined with a recent result of E. A. Grove and G. Ladas (Periodicities in Nonlinear Difference Equations, Chapman & Hall/CRC Press, Boca Raton (2004)), this answers the question when $y=2$ is a global attractor.

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