Asymptotic behavior of a sequence defined by iteration with applications

Stevo Stević

Colloquium Mathematicae, Tome 91 (2002), p. 267-276 / Harvested from The Polish Digital Mathematics Library

Résumé

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) $f(x,y) = px + (1-p)y - ∑_{s=m}^{∞}_{s}(x,y)$ uniformly in a neighborhood of the origin, where m > 1, $_{s}(x,y) = ∑_{i=0}^{s} a_{i,s}x^{s-i}y^{i}$; (c) $ₘ(1,1) = ∑_{i=0}^{m} a_{i,m} > 0$. 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: $xₙ ∼ ((2-p)/((m-1)∑_{i=0}^{m} a_{i,m}))^{1/(m-1)} 1/{\@root m-1 \of {n}}$.

Détail

Publié le : 2002-01-01

Zbl 1029.39006

EUDML-ID : urn:eudml:doc:286685