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

Author(s): Kenneth S. Berenhaut; John D. Foley; Stevo Stevic
Journal: Proc. Amer. Math. Soc. 135 (2007), 1133-1140.
MSC (2000): Primary 39A10, 39A11
Posted: October 4, 2006
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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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A. M. AMLEH, E. A. GROVE, G. LADAS AND D. A. GEORGIOU, On the recursive sequence $ x\sb {n+1}=\alpha+x\sb {n-1}/x\sb n$. J. Math. Anal. Appl. 233 (1999), no. 2, 790-798. MR 1689579 (2000f:39002)

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E. CAMOUZIS, G. LADAS, AND H. D. VOULOV, On the dynamics of $ x\sb {n+1}={\alpha+\gamma x\sb {n-1}+\delta x\sb {n-2}\over A+x\sb {n-2}}$. Special Session of the American Mathematical Society Meeting, Part II (San Diego, CA, 2002). J. Differ. Equations Appl. 9 (2003), no. 8, 731-738. MR 1992906 (2004e:39005)

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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: 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
Posted: October 4, 2006
Additional Notes: The first author acknowledges financial support from a Sterge Faculty Fellowship.
Communicated by: Carmen C. Chicone
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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