Asymptotic behavior of a sequence defined by iteration with applications

Stevo Stević. "Asymptotic behavior of a sequence defined by iteration with applications." Colloquium Mathematicae 93.2 (2002): 267-276. <http://eudml.org/doc/286685>.

@article{StevoStević2002,
abstract = {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: $xₙ ∼ ((2-p)/((m-1)∑_\{i=0\}^\{m\} a_\{i,m\}))^\{1/(m-1)\} 1/\{\@root m-1 \of \{n\}\}$.},
author = {Stevo Stević},
journal = {Colloquium Mathematicae},
keywords = {second order nonlinear difference equation; asymptotic behavior; homogeneous polynomial; ecology},
language = {eng},
number = {2},
pages = {267-276},
title = {Asymptotic behavior of a sequence defined by iteration with applications},
url = {http://eudml.org/doc/286685},
volume = {93},
year = {2002},
}

TY - JOUR
AU - Stevo Stević
TI - Asymptotic behavior of a sequence defined by iteration with applications
JO - Colloquium Mathematicae
PY - 2002
VL - 93
IS - 2
SP - 267
EP - 276
AB - 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: $xₙ ∼ ((2-p)/((m-1)∑_{i=0}^{m} a_{i,m}))^{1/(m-1)} 1/{\@root m-1 \of {n}}$.
LA - eng
KW - second order nonlinear difference equation; asymptotic behavior; homogeneous polynomial; ecology
UR - http://eudml.org/doc/286685
ER -