Asymptotic behavior of a sequence defined by iteration with applications

• Autorzy: Stevo Stević
• Języki publikacji: EN

Abstrakty

EN

We consider the asymptotic behavior of some classes of sequences defined by a recurrent formula. The main result is the following: Let f: (0,∞)² → (0,∞) be a continuous function such that (a) 0 < f(x,y) < px + (1-p)y for some p ∈ (0,1) and for all x,y ∈ (0,α), where α > 0; (b) $f(x,y) = px + (1-p)y - ∑_{s=m}^{∞}𝒦_{s}(x,y)$ uniformly in a neighborhood of the origin, where m > 1, $𝒦_{s}(x,y) = ∑_{i=0}^{s} a_{i,s}x^{s-i}y^{i}$; (c) $𝒦ₘ(1,1) = ∑_{i=0}^{m} a_{i,m} > 0$. Let x₀,x₁ ∈ (0,α) and $x_{n+1} = f(xₙ,x_{n-1})$, n ∈ ℕ. Then the sequence (xₙ) satisfies the following asymptotic formula:
$xₙ ∼ ((2-p)/((m-1)∑_{i=0}^{m} a_{i,m}))^{1/(m-1)} 1/{\root{m-1}\of{n}}$.

Kategorie tematyczne

• 39A10: Difference equations, additive

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2002
• Tom: 93
• Numer: 2
• Strony: 267-276

Daty

wydano

2002

Twórcy

autor

Stevo Stević

• Matematički Fakultet, Studentski Trg 16, 11000 Beograd, Serbia, Yugoslavia

Identyfikatory

DOI

10.4064/cm93-2-6