On the oscillation and periodic character of a third order rational difference equation

Authors:
W. T. Patula and H. D. Voulov

Journal:
Proc. Amer. Math. Soc. **131** (2003), 905-909

MSC (2000):
Primary 39A10

Published electronically:
July 17, 2002

MathSciNet review:
1937429

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that every positive solution of the following difference equation:

converges to a period two solution.

**1.**R. DeVault, G. Ladas, and S. W. Schultz,*On the recursive sequence 𝑥_{𝑛+1}=𝐴/𝑥_{𝑛}+1/𝑥_{𝑛-2}*, Proc. Amer. Math. Soc.**126**(1998), no. 11, 3257–3261. MR**1473661**, 10.1090/S0002-9939-98-04626-7**2.**H. El-Metwally, E. A. Grove, and G. Ladas,*A global convergence result with applications to periodic solutions*, J. Math. Anal. Appl.**245**(2000), no. 1, 161–170. MR**1756582**, 10.1006/jmaa.2000.6747**3.**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. Diff. Eqn. Appl.***7**(2001), 837-850.**4.**George Karakostas,*Asymptotic 2-periodic difference equations with diagonally self-invertible responses*, J. Differ. Equations Appl.**6**(2000), no. 3, 329–335. MR**1785059**, 10.1080/10236190008808232**5.**George Karakostas,*Convergence of a difference equation via the full limiting sequences method*, Differential Equations Dynam. Systems**1**(1993), no. 4, 289–294. MR**1259169****6.**M.R.S. Kulenovic and G. Ladas,*Dynamics of Second-Order Rational Difference Equations with Open Problems and Conjectures*, Chapman and Hall/CRC, 2002.**7.**G. Ladas, Open problems and conjectures, AMS Joint Math. Meetings, January 2001 (New Orleans), Program #364.

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Additional Information

**W. T. Patula**

Affiliation:
Department of Mathematics, Southern Illinois University Carbondale, Carbondale, Illinois 62901-4408

Email:
wpatula@math.siu.edu

**H. D. Voulov**

Affiliation:
Department of Mathematics, Southern Illinois University Carbondale, Carbondale, Illinois 62901-4408

Email:
voulovh@yahoo.com

DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06611-X

Keywords:
Periodic solution,
semicycles,
oscillation

Received by editor(s):
May 28, 2001

Received by editor(s) in revised form:
October 22, 2001

Published electronically:
July 17, 2002

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2002
American Mathematical Society