## Abstract

Sufficient conditions are provided for the boundedness of all positive solutions of the nonlinear difference equation x_{n+1} = x _{n}^{γ} f(x_{n-k}), where γ > 0 and f: [0,+∞) → [0,+∞) is a given function. In case k = 1 the classical Chebyshev polynomials of the second kind are used to obtain such sharp sufficient conditions. Also some convergence results are given. The results extend those given in [E. Camuzis, G. Ladas, I.W. Rodrigues and S. Northshield, The rational recursive sequence x_{n+1} = (βx_{n} ^{2})/(1 x+_{n-1}^{2}), Advances in difference equations, Comp. Math. Appl. 28 (1994), 37-43; E. Camuzis, E.A. Grove, G. Ladas and V.L. Kosić, Monotone unstable solutions of difference equations and conditions for boundedness, J. Differ. Equations Appl. 1 (1995), 17-44; George L. Karakostas, Asymptotic behavior of the solutions of the difference equation x_{n+1} = x_{n}^{2}f(x_{n-1}), J. Differ. Equations Appl. 9(6) (2001), 599-602; V.L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order and Applications, Kluwer Academic Publishers, Dordrecht, 1993; Wan-Tong Li, Hong-Rui Sun and Xing-Xue Yan, The asymptotic behavior of a higher order delay nonlinear difference equations, Indian J. Pure Appl. Math. 34(10) (2003), 1431-1441; D.C. Zhang, B. Shi and M.J. Gai, On the rational recursive sequence x^{n+1} = bx _{n}^{2}/(1 + x_{n-1}^{2}), Indian J. Pure Appl. Math. (2) 32(5) (2001), 657-663] concerning boundedness of the solutions.

Original language | English |
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Pages (from-to) | 941-948 |

Number of pages | 8 |

Journal | Journal of Difference Equations and Applications |

Volume | 10 |

Issue number | 11 |

DOIs | |

State | Published - Sep 2004 |

## Keywords

- Boundedness
- Chebyshev polynomials
- Difference equations
- Global attractivity
- Positive solution

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Applied Mathematics