## On the eventual periodicity of $$x_{n+1} = \max \{\frac{1}{x_n},\frac{A_n}{x_{n-1}}\}$$ with a period-four parameter.(English)Zbl 1095.39016

Summary: We study the max difference equation in the title when it is a constant sequence or period-four sequence of real numbers greater than one. In the former case, we provide an alternative proof to that already given by A. M. Amleh, J. Hoag, and G. Ladas [Comput. Math. Appl. 36, 401–404 (1998; Zbl 0933.39030)], and E. A. Grove and G. Ladas [Periodicities in nonlinear difference equations (2005; Zbl 1078.39009)], where we show that every solution is eventually periodic with period 4. We prove that every solution is eventually periodic with period 8 in the latter case.

### MSC:

 39A12 Discrete version of topics in analysis 39A20 Multiplicative and other generalized difference equations

### Citations:

Zbl 0933.39030; Zbl 1078.39009
Full Text:

### References:

 [1] DOI: 10.1016/S0898-1221(98)80040-0 · Zbl 0933.39030 [2] Briden W.J., Common Applications and Nonlinear Analysis 6 pp 31– (1999) [3] Briden W.J., Proceedings of the Third International Conference on Difference Equations and Applications pp 40– (1999) [4] Grove E.A., Fields Institute Communications 29 (2001) [5] Grove E.A., Periodicities in Nonlinear Difference Equations (2005) · Zbl 1078.39009
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.