×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Grove, E.A.; Ladas, G., Periodicities in nonlinear difference equations, (2004), Chapman and Hall/CRC Press Boca Raton, FL · Zbl 1068.39007
[2] Kocić, V.; Ladas, G., Global behavior of nonlinear difference equations of higher order with applications, () · Zbl 0787.39001
[3] Li, X.; Zhu, D., Two rational recursive sequences, Comput. math. appl., 47, 1487-1494, (2004) · Zbl 1072.39008
[4] Li, X., Qualitative properties for a fourth-order rational difference equation, J. math. anal. appl., 311, 103-111, (2005) · Zbl 1082.39004
[5] Li, X., Global behaviour for a fourth-order rational difference equation, J. math. anal. appl., 312, 555-563, (2005) · Zbl 1083.39007
[6] 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
[7] Li, X.; Zhu, D., Global asymptotic stability in a rational equation, J. difference equ. appl., 9, 833-839, (2003) · Zbl 1055.39014
[8] Camouzis, E.; DeVault, R.; Papaschinopoulos, G., On the recursive sequence, Adv. difference equ., 2005, 1, 31-40, (2005) · Zbl 1083.39005
[9] 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
[10] 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
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.