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**On the periodicity of a difference equation with maximum.**
*(English)*
Zbl 1149.39005

Summary: We investigate the periodic nature of solutions of the max difference equation

\[ x_{n+1}= \max\{x_n,A\}/(x_nx_{n-1}), \quad n=0,1,\dots, \] where \(A\) is a positive real parameter, and the initial conditions \(x_{-1}=A^{r_{-1}}\) and \(x_0=A^{r_0}\) such that \(r_{-1}\) and \(r_0\) are positive rational numbers. The results in this paper answer the open problem 6.2 posed by E. A. Grove and G. Ladas in their book “Periodicities in nonlinear difference equations”. Boca Raton (FL) (2005; Zbl 1078.39009).

\[ x_{n+1}= \max\{x_n,A\}/(x_nx_{n-1}), \quad n=0,1,\dots, \] where \(A\) is a positive real parameter, and the initial conditions \(x_{-1}=A^{r_{-1}}\) and \(x_0=A^{r_0}\) such that \(r_{-1}\) and \(r_0\) are positive rational numbers. The results in this paper answer the open problem 6.2 posed by E. A. Grove and G. Ladas in their book “Periodicities in nonlinear difference equations”. Boca Raton (FL) (2005; Zbl 1078.39009).

### MSC:

39A11 | Stability of difference equations (MSC2000) |

39A20 | Multiplicative and other generalized difference equations |

### Citations:

Zbl 1078.39009
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\textit{A. Gelisken} et al., Discrete Dyn. Nat. Soc. 2008, Article ID 820629, 11 p. (2008; Zbl 1149.39005)

### References:

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[20] | I. Yal\ccinkaya, C. Cinar, and A. Gelisken, “On the periodicity of a max-type difference equation,” to appear. |

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