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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Kenneth S. Berenhaut, John D. Foley and Stevo Stevic
Journal: Proc. Amer. Math. Soc. 135 (2007), 1133-1140
MSC (2000): Primary 39A10, 39A11
Published electronically: October 4, 2006
MathSciNet review: 2262916
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Abstract: This paper studies the behavior of positive solutions of the recursive equation

$\displaystyle y_{n}=1+\frac{y_{n-k}}{y_{n-m}}, ~~ n=0,1,2,\ldots,$      

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$, 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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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 Serbian Academy of Science, Knez Mihailova 35/I 11000 Beograd, Serbia
Email: sstevic@ptt.yu, sstevo@matf.bg.ac.yu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08580-7
PII: S 0002-9939(06)08580-7
Keywords: Difference equation, stability, exponential convergence, periodic solution.
Received by editor(s): September 7, 2005
Received by editor(s) in revised form: November 11, 2005
Published electronically: October 4, 2006
Additional Notes: The first author acknowledges financial support from a Sterge Faculty Fellowship.
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.