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The global attractivity of the rational difference equation $$y_n = \frac{y_{n-k}+y_{n-m}}{1+y_{n-k}y_{n-m}}$$. (English) Zbl 1131.39006
For the equation in the title with $$0< k< m$$ it is shown that all positive solutions converge to the equilibrium 1. Two conjectures concerning the asymptotic stability for positive solutions of more general difference equations are given.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations
##### Keywords:
asymptotic stability; symmetry; positive solutions
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##### References:
  Grove, E.A.; Ladas, G., Periodicities in nonlinear difference equations, (2004), Chapman and Hall/CRC Press Boca Raton, FL · Zbl 1068.39007  Kocić, V.; Ladas, G., Global behavior of nonlinear difference equations of higher order with applications, () · Zbl 0787.39001  Li, X.; Zhu, D., Two rational recursive sequences, Comput. math. appl., 47, 1487-1494, (2004) · Zbl 1072.39008  Li, X., Qualitative properties for a fourth-order rational difference equation, J. math. anal. appl., 311, 103-111, (2005) · Zbl 1082.39004  Li, X., Global behaviour for a fourth-order rational difference equation, J. math. anal. appl., 312, 555-563, (2005) · Zbl 1083.39007  K.S. Berenhaut, J.D. Foley, S. Stević, The global attractivity of the rational difference equation $$y_n = 1 + \frac{y_{n - k}}{y_{n - m}}$$, Proc. Amer. Math. Soc. (2006) (in press) · Zbl 1109.39004  Li, X.; Zhu, D., Global asymptotic stability in a rational equation, J. difference equ. appl., 9, 833-839, (2003) · Zbl 1055.39014  Camouzis, E.; DeVault, R.; Papaschinopoulos, G., On the recursive sequence, Adv. difference equ., 2005, 1, 31-40, (2005) · Zbl 1083.39005  Kalabušić, S.; Kulenović, M.R.S., Rate of convergence of solutions of rational difference equation of second order, Adv. difference equ., 2004, 2, 121-139, (2004) · Zbl 1079.39007  Camouzis, E.; Ladas, G.; Quinn, E.P., On third-order rational difference equations. VI, J. difference equ. appl., 11, 8, 759-777, (2005) · Zbl 1071.39502
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