The global attractivity of the rational difference equation

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper studies the behavior of positive solutions of the recursive equation

with and , where . We prove that if , with odd, then tends to , 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 is a global attractor.

**1.**R. M. Abu-Saris and R. DeVault,*Global stability of 𝑦_{𝑛+1}=𝐴+\frac{𝑦_{𝑛}}𝑦_{𝑛-𝑘}*, Appl. Math. Lett.**16**(2003), no. 2, 173–178. MR**1962312**, 10.1016/S0893-9659(03)80028-9**2.**A. M. Amleh, E. A. Grove, G. Ladas, and D. A. Georgiou,*On the recursive sequence 𝑥_{𝑛+1}=𝛼+𝑥_{𝑛-1}/𝑥_{𝑛}*, J. Math. Anal. Appl.**233**(1999), no. 2, 790–798. MR**1689579**, 10.1006/jmaa.1999.6346**3.**K. S. BERENHAUT, J. D. FOLEY AND S. STEVIC, Quantitative bounds for the recursive sequence ,*Appl. Math. Lett.*, in press, (2005).**4.**E. Camouzis, G. Ladas, and H. D. Voulov,*On the dynamics of 𝑥_{𝑛+1}=𝛼+𝛾𝑥_{𝑛-1}+𝛿𝑥_{𝑛-2}\over𝐴+𝑥_{𝑛-2}*, J. Difference Equ. Appl.**9**(2003), no. 8, 731–738. Special Session of the American Mathematical Society Meeting, Part II (San Diego, CA, 2002). MR**1992906**, 10.1080/1023619021000042153**5.**H. El-Metwally, E. A. Grove, G. Ladas, and H. D. Voulov,*On the global attractivity and the periodic character of some difference equations*, J. Differ. Equations Appl.**7**(2001), no. 6, 837–850. On the occasion of the 60th birthday of Calvin Ahlbrandt. MR**1870725**, 10.1080/10236190108808306**6.**C. H. Gibbons, M. R. S. Kulenović, G. Ladas, and H. D. Voulov,*On the trichotomy character of 𝑥_{𝑛+1}=(𝛼+𝛽𝑥_{𝑛}+𝛾𝑥_{𝑛-1})/(𝐴+𝑥_{𝑛})*, J. Difference Equ. Appl.**8**(2002), no. 1, 75–92. MR**1884593**, 10.1080/10236190211940**7.**E. A. Grove and G. Ladas,*Periodicities in nonlinear difference equations*, Advances in Discrete Mathematics and Applications, vol. 4, Chapman & Hall/CRC, Boca Raton, FL, 2005. MR**2193366****8.**V. L. Kocić and G. Ladas,*Global behavior of nonlinear difference equations of higher order with applications*, Mathematics and its Applications, vol. 256, Kluwer Academic Publishers Group, Dordrecht, 1993. MR**1247956****9.**W. T. Patula and H. D. Voulov,*On the oscillation and periodic character of a third order rational difference equation*, Proc. Amer. Math. Soc.**131**(2003), no. 3, 905–909 (electronic). MR**1937429**, 10.1090/S0002-9939-02-06611-X**10.**Stevo Stević,*Behavior of the positive solutions of the generalized Beddington-Holt equation*, Panamer. Math. J.**10**(2000), no. 4, 77–85. MR**1801533****11.**Stevo Stević,*A note on periodic character of a difference equation*, J. Difference Equ. Appl.**10**(2004), no. 10, 929–932. MR**2079642**, 10.1080/10236190412331272616

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
39A10,
39A11

Retrieve articles in all journals with MSC (2000): 39A10, 39A11

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

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.