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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
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##### 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
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