A variant of the Krein-Rutman theorem for Poincaré difference equations

Rafael Obaya, Mihály Pituk

Research output: Contribution to journalArticle

5 Citations (Scopus)

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 languageEnglish
Pages (from-to)1751-1762
Number of pages12
JournalJournal of Difference Equations and Applications
Volume18
Issue number10
DOIs
Publication statusPublished - 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

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