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**Global behavior of the max-type difference equation \(x_{n+1}=\max\{1/x_{n},A_{n}/x_{n - 1}\}\).**
*(English)*
Zbl 1167.39303

Summary: We study global behavior of the following max-type difference equation \(x_{n+1}=\max\{1/x_{n},A_{n}/x_{n - 1}\}, n=0,1,\dots,\) where \(\{A_n\}_{n=0}^{\infty}\) is a sequence of positive real numbers with \(0\leq \text{inf} A_{n}\leq \text{sup} A_{n}<1\). The special case when \(\{A_n\}_{n=0}^{\infty}\) is a periodic sequence with period \(k\) and \(A_{n}\in (0,1)\) for every \(n\geq 0\) has been completely investigated by Y. Chen [J. Difference Equ. Appl. 11, No. 15, 1289–1294 (2005; Zbl 1086.39003)]. Here, we extend his results to the general case.

### MSC:

39A11 | Stability of difference equations (MSC2000) |

### Citations:

Zbl 1086.39003
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\textit{T. Sun} et al., Abstr. Appl. Anal. 2009, Article ID 152964, 10 p. (2009; Zbl 1167.39303)

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### References:

[1] | A. M. Amleh, J. Hoag, and G. Ladas, “A difference equation with eventually periodic solutions,” Computers & Mathematics with Applications, vol. 36, no. 10-12, pp. 401-404, 1998. · Zbl 0933.39030 |

[2] | K. S. Berenhaut, J. D. Foley, and S. Stević, “Boundedness character of positive solutions of a max difference equation,” Journal of Difference Equations and Applications, vol. 12, no. 12, pp. 1193-1199, 2006. · Zbl 1116.39001 |

[3] | W. J. Briden, E. A. Grove, G. Ladas, and L. C. McGrath, “On the nonautonomous equation xn+1=max {An/xn,Bn/xn - 1},” in Proceedings of the 3rd International Conference on Difference Equations, pp. 49-73, Gordon and Breach, Taipei, Taiwan, September 1997. · Zbl 0938.39012 |

[4] | W. J. Briden, E. A. Grove, G. Ladas, and C. M. Kent, “Eventually periodic solutions of xn+1=max {1/xn,An/xn - 1},” Communications on Applied Nonlinear Analysis, vol. 6, no. 4, pp. 31-43, 1999. · Zbl 1108.39300 |

[5] | Y. Chen, “Eventual periodicity of xn+1=max {1/xn,An/xn - 1} with periodic coefficients,” Journal of Difference Equations and Applications, vol. 11, no. 15, pp. 1289-1294, 2005. · Zbl 1086.39003 |

[6] | C. \cCinar, S. Stević, and I. Yal\ccinkaya, “On positive solutions of a reciprocal difference equation with minimum,” Journal of Applied Mathematics & Computing, vol. 17, no. 1-2, pp. 307-314, 2005. · Zbl 1074.39002 |

[7] | J. Feuer, “On the eventual periodicity of xn+1=max {1/xn,An/xn - 1} with a period-four parameter,” Journal of Difference Equations and Applications, vol. 12, no. 5, pp. 467-486, 2006. · Zbl 1095.39016 |

[8] | E. A. Grove, C. Kent, G. Ladas, and M. A. Radin, “On xn+1=max {1/xn,An/xn - 1} with a period 3 parameter,” Fields Institute Communication, vol. 29, pp. 161-180, 2001. · Zbl 0980.39012 |

[9] | E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, vol. 4 of Advances in Discrete Mathematics and Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2005. · Zbl 1078.39009 |

[10] | C. M. Kent and M. A. Radin, “On the boundedness nature of positive solutions of the difference equation xn+1=max {An/xn,Bn/xn - 1} with periodic parameters,” Dynamics of Continuous, Discrete & Impulsive Systems. Series B, supplement, pp. 11-15, 2003. |

[11] | W. T. Patula and H. D. Voulov, “On a max type recurrence relation with periodic coefficients,” Journal of Difference Equations and Applications, vol. 10, no. 3, pp. 329-338, 2004. · Zbl 1050.39017 |

[12] | S. Stević, “On the recursive sequence xn+1=A+(xnp/xn - 1r),” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 40963, 9 pages, 2007. · Zbl 1151.39011 |

[13] | S. Stević, “On the recursive sequence xn+1=max {c,xnp/xn - 1p},” Applied Mathematics Letters, vol. 21, no. 8, pp. 791-796, 2008. · Zbl 1152.39012 |

[14] | F. Sun, “On the asymptotic behavior of a difference equation with maximum,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 243291, 6 pages, 2008. · Zbl 1155.39008 |

[15] | I. Szalkai, “On the periodicity of the sequence xn+1=max {A0/xn, ... ,Ak/xn - k},” Journal of Difference Equations and Applications, vol. 5, no. 1, pp. 25-29, 1999. · Zbl 0930.39011 |

[16] | H. D. Voulov, “Periodic solutions to a difference equation with maximum,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2155-2160, 2003. · Zbl 1019.39005 |

[17] | H. D. Voulov, “On the periodic nature of the solutions of the reciprocal difference equation with maximum,” Journal of Mathematical Analysis and Applications, vol. 296, no. 1, pp. 32-43, 2004. · Zbl 1053.39023 |

[18] | J. Bibby, “Axiomatisations of the average and a further generalisation of monotonic sequences,” Glasgow Mathematical Journal, vol. 15, pp. 63-65, 1974. · Zbl 0291.40003 |

[19] | E. T. Copson, “On a generalisation of monotonic sequences,” Proceedings of the Edinburgh Mathematical Society. Series II, vol. 17, no. 2, pp. 159-164, 1971. · Zbl 0223.40001 |

[20] | S. Stević, “A note on bounded sequences satisfying linear inequalities,” Indian Journal of Mathematics, vol. 43, no. 2, pp. 223-230, 2001. · Zbl 1035.40002 |

[21] | S. Stević, “A generalization of the Copson’s theorem concerning sequences which satisfy a linear inequality,” Indian Journal of Mathematics, vol. 43, no. 3, pp. 277-282, 2001. · Zbl 1034.40002 |

[22] | S. Stević, “A global convergence result,” Indian Journal of Mathematics, vol. 44, no. 3, pp. 361-368, 2002. · Zbl 1034.39002 |

[23] | S. Stević, “Asymptotic behavior of a sequence defined by iteration with applications,” Colloquium Mathematicum, vol. 93, no. 2, pp. 267-276, 2002. · Zbl 1029.39006 |

[24] | S. Stević, “Asymptotic behaviour of a nonlinear difference equation,” Indian Journal of Pure and Applied Mathematics, vol. 34, no. 12, pp. 1681-1687, 2003. · Zbl 1049.39012 |

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