##
**On the difference equation \(x_{n+1} = x_nx_{n-k}/(x_{n-k+1}(a + bx_nx_{n-k}))\).**
*(English)*
Zbl 1246.39011

Summary: We show that the difference equation \(x_{n+1} = x_nx_{n-k}/x_{n-k+1}(a + bx_nx_{n-k}), n \in \mathbb N_0\), where \(k \in \mathbb N\), the parameters \(a, b\) and initial values \(x_{-i}, i = \overline{0, k}\) are real numbers, can be solved in closed form considerably extending the results in the literature. By using obtained formulae, we investigate asymptotic behavior of well-defined solutions of the equation.

### MSC:

39A20 | Multiplicative and other generalized difference equations |

PDF
BibTeX
XML
Cite

\textit{S. Stević} et al., Abstr. Appl. Anal. 2012, Article ID 108047, 9 p. (2012; Zbl 1246.39011)

Full Text:
DOI

### References:

[1] | A. Andruch-Sobiło and M. Migda, “Further properties of the rational recursive sequence xn+1=axn-1/b+cxnxn-1,” Opuscula Mathematica, vol. 26, no. 3, pp. 387-394, 2006. · Zbl 1131.39003 |

[2] | A. Andruch-Sobiło and M. Migda, “On the rational recursive sequence xn+1=axn-1/b+cxnxn-1,” Tatra Mountains Mathematical Publications, vol. 43, pp. 1-9, 2009. · Zbl 1212.39008 |

[3] | I. Bajo and E. Liz, “Global behaviour of a second-order nonlinear difference equation,” Journal of Difference Equations and Applications, vol. 17, no. 10, pp. 1471-1486, 2011. · Zbl 1232.39014 |

[4] | L. Berg and S. Stević, “On difference equations with powers as solutions and their connection with invariant curves,” Applied Mathematics and Computation, vol. 217, no. 17, pp. 7191-7196, 2011. · Zbl 1260.39002 |

[5] | L. Berg and S. Stević, “On some systems of difference equations,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1713-1718, 2011. · Zbl 1233.39001 |

[6] | B. Iri\vcanin and S. Stević, “On some rational difference equations,” Ars Combinatoria, vol. 92, pp. 67-72, 2009. · Zbl 1224.39014 |

[7] | S. Stević, “More on a rational recurrence relation,” Applied Mathematics E-Notes, vol. 4, pp. 80-85, 2004. · Zbl 1069.39024 |

[8] | S. Stević, “A short proof of the Cushing-Henson conjecture,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 37264, 5 pages, 2006. · Zbl 1149.39300 |

[9] | S. Stević, “Existence of nontrivial solutions of a rational difference equation,” Applied Mathematics Letters, vol. 20, no. 1, pp. 28-31, 2007. · Zbl 1131.39009 |

[10] | S. Stević, “Global stability of a max-type difference equation,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 354-356, 2010. · Zbl 1193.39009 |

[11] | S. Stević, “On a system of difference equations,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3372-3378, 2011. · Zbl 1242.39017 |

[12] | S. Stević, “On the difference equation xn=xn-2/bn+cnxn-1xn-2,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4507-4513, 2011. · Zbl 1220.39011 |

[13] | S. Stević, “On a third-order system of difference equations,” Applied Mathematics and Computation, vol. 218, pp. 7649-7654, 2012. · Zbl 1243.39011 |

[14] | S. Stević, “On some solvable systems of difference equations,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5010-5018, 2012. · Zbl 1253.39011 |

[15] | S. Stević, “On the difference equation xn=xn-k/b+cxn-1 ... xn-k,” Applied Mathematics and Computation, vol. 218, no. 11, pp. 6291-6296, 2012. · Zbl 1246.39010 |

[16] | S. Stević, J. Diblík, B. Iri\vcanin, and Z. \vSmarda, “On a third-order system of difference equations with variable coefficients,” Abstract and Applied Analysis, vol. 2012, Article ID 508523, 22 pages, 2012. · Zbl 1242.39011 |

[17] | H. Levy and F. Lessman, Finite Difference Equations, The Macmillan Company, New York, NY, USA, 1961. · Zbl 0092.07702 |

[18] | H. El-Metwally and E. M. Elsayed, “Qualitative study of solutions of some difference equations,” Abstract and Applied Analysis, vol. 2012, Article ID 248291, 16 pages, 2012. · Zbl 1246.39007 |

[19] | E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, vol. 4, 2005. · Zbl 1078.39009 |

[20] | B. D. Iri\vcanin and S. Stević, “Some systems of nonlinear difference equations of higher order with periodic solutions,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 13, no. 3-4, pp. 499-507, 2006. · Zbl 1098.39003 |

[21] | R. P. Kurshan and B. Gopinath, “Recursively generated periodic sequences,” Canadian Journal of Mathematics. Journal Canadien de Mathématiques, vol. 26, pp. 1356-1371, 1974. · Zbl 0313.26019 |

[22] | G. Papaschinopoulos and C. J. Schinas, “On the behavior of the solutions of a system of two nonlinear difference equations,” Communications on Applied Nonlinear Analysis, vol. 5, no. 2, pp. 47-59, 1998. · Zbl 1110.39301 |

[23] | G. Papaschinopoulos and C. J. Schinas, “Invariants for systems of two nonlinear difference equations,” Differential Equations and Dynamical Systems, vol. 7, no. 2, pp. 181-196, 1999. · Zbl 0978.39014 |

[24] | G. Papaschinopoulos and C. J. Schinas, “Invariants and oscillation for systems of two nonlinear difference equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 46, no. 7, pp. 967-978, 2001. · Zbl 1003.39007 |

[25] | S. Stević, “Periodicity of max difference equations,” Utilitas Mathematica, vol. 83, pp. 69-71, 2010. · Zbl 1236.39018 |

[26] | S. Stević, “Periodicity of a class of nonautonomous max-type difference equations,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9562-9566, 2011. · Zbl 1225.39018 |

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.