×

zbMATH — the first resource for mathematics

On the oscillation and periodic character of a third order rational difference equation. (English) Zbl 1014.39010
The authors prove that every positive solution of the following difference equation \[ x_n=1+{x_{n-2}\over x_{n-3}},\;n=0,1,\dots, \] converges to a periodic solution. As to some related results, see El-Metwally, E. A. Grove, G. Ladas and H. D. Voulov [J. Difference Equ. Appl. 7, No. 6, 837-850 (2001; Zbl 0993.39008)].

MSC:
39A11 Stability of difference equations (MSC2000)
39B05 General theory of functional equations and inequalities
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] R. DeVault, G. Ladas, and S. W. Schultz, On the recursive sequence \?_\?+1=\?/\?_\?+1/\?_\?-2, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3257 – 3261. · Zbl 0904.39012
[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. · Zbl 0971.39004 · doi:10.1006/jmaa.2000.6747 · doi.org
[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. · Zbl 0993.39008
[4] George Karakostas, Asymptotic 2-periodic difference equations with diagonally self-invertible responses, J. Differ. Equations Appl. 6 (2000), no. 3, 329 – 335. · Zbl 0963.39020 · doi:10.1080/10236190008808232 · doi.org
[5] George Karakostas, Convergence of a difference equation via the full limiting sequences method, Differential Equations Dynam. Systems 1 (1993), no. 4, 289 – 294. · Zbl 0868.39002
[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. · Zbl 1057.39505
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.