### Abstract

Let,x _{n}, n ∈ ℕ, be a non-vanishing solution of the Poincaré difference equationwhere A _{n}, n ∈ ℕ,are K × k real matrices such that the limit A=lim _{n rarr; ∞}A _{n} exists (entrywise). According to a Perron type theorem, the limit limit p=lim _{n rarr; ∞} ^{n}√{pipe}X _{n}{pipe} exists and is equal to the modulus of one of the eigenvalues of A. In this paper, we show that if the solution belongs to a given order cone K in ℝ ^{k}, then p is an eigenvalue of A with an eigenvector in K. In the case of constant coefficients, this result implies the finite-dimensional version of the Krein-Rutman theorem.

Original language | English |
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Pages (from-to) | 1751-1762 |

Number of pages | 12 |

Journal | Journal of Difference Equations and Applications |

Volume | 18 |

Issue number | 10 |

DOIs | |

Publication status | Published - Oct 1 2012 |

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### Keywords

- Krein-Rutman theorem
- Poincaré difference equation
- order cone
- quasilinear equation

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Applied Mathematics