### Abstract

Consider the scalar kth order linear difference equation x(n + k) + p_{1}(n)x(n + k - 1) + ⋯ + p_{k}(n)x(n) = 0, where the limits q_{i} = lim_{n→∞} p_{i}(n) (i = 1,...,k) are finite. In this paper, we confirm the conjecture formulated recently by Elaydi. Namely, every nonzero solution x of (*) satisfies the same asymptotic relation as the fundamental solutions described earlier by Perron, i.e., ρ = lim sup_{n→∞} n√|x(n)| is equal to the modulus of one of the roots of the characteristic equation λ^{k} + q_{1}λ^{k-1} + ... + q_{k} = 0. This result is a consequence of a more general theorem concerning the Poincaré difference system x(n + 1) = [A + B(n)]x(n), where A and B(n) (n = 0, 1,....) are square matrices such that ∥B(n)∥ → 0 as n → ∞. As another corollary, we obtain a new limit relation for the solutions of (*).

Original language | English |
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Pages (from-to) | 201-216 |

Number of pages | 16 |

Journal | Journal of Difference Equations and Applications |

Volume | 8 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 1 2002 |

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### Keywords

- Asymptotic behavior
- Linear difference equations
- Perron's theorem
- Poincaré's theorem

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Applied Mathematics