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

Author(s): Kenneth S. Berenhaut; John D. Foley; Stevo Stevic
Journal: Proc. Amer. Math. Soc. 136 (2008), 103-110.
MSC (2000): Primary 39A10, 39A11
Posted: September 24, 2007
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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 $ 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. 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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Additional Information:

Kenneth S. Berenhaut
Affiliation: Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina 27109
Email: berenhks@wfu.edu

John D. Foley
Affiliation: Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina 27109
Email: folejd4@wfu.edu

Stevo Stevic
Affiliation: Mathematical Institute of The Serbian Academy of Science, Knez Mihailova 35/I 11000 Beograd, Serbia
Email: sstevic@ptt.yu, sstevo@matf.bg.ac.yu

DOI: 10.1090/S0002-9939-07-08860-0
PII: S 0002-9939(07)08860-0
Keywords: Rational difference equation, stability.
Received by editor(s): April 18, 2006 and in revised form, July 31, 2006
Posted: September 24, 2007
Additional Notes: The first author acknowledges financial support from a Sterge Faculty Fellowship.
Communicated by: Carmen C. Chicone
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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