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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
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[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
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