More on Poincaré's and Perron's theorems for difference equations

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Abstract

Consider the scalar kth order linear difference equation x(n + k) + p1(n)x(n + k - 1) + ⋯ + pk(n)x(n) = 0, where the limits qi = limn→∞ pi(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 supn→∞ n√|x(n)| is equal to the modulus of one of the roots of the characteristic equation λk + q1λk-1 + ... + qk = 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 languageEnglish
Pages (from-to)201-216
Number of pages16
JournalJournal of Difference Equations and Applications
Volume8
Issue number3
DOIs
Publication statusPublished - 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

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