Authors: Stević, Stevo 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Asymptotic periodicity of a higher-order difference equation
Journal: Discrete Dynamics in Nature and Society
Volume: 2007
Issue: 1
Issue Date: 1-Jan-2007
Rank: M22
ISSN: 1026-0226
DOI: 10.1155/2007/13737
We give a complete picture regarding the asymptotic periodicity of positive solutions of the following difference equation: xn = f(x n-p1,...,xn-pk,xn-q1,...,xn-qm), n ∈ ℕ0, where pi, i ∈ {1,...,k}, and qj, j ∈ {1,...,m}, are natural numbers such that p1 < p2 <... < pk, q1 < q2 <... < qm and gcd(p1,...,pk,q1,...,qm) = 1, the function f ∈ C[(0,∞)k+m, (α, ∞)], α > 0, is increasing in the first k arguments and decreasing in other m arguments, there is a decreasing function g ∈ C[(α, ∞),(α, ∞)] such that g(g(x)) = x, x ∈ (α,∞), (equation), x ∈ (α, ∞), limx→a+g (x) = +∞, and lim x→+∞g(x) = α. It is proved that if all p i, i ∈{1,...,k}, are even and all qj, j ∈{1,...,m} are odd, every positive solution of the equation converges to (not necessarily prime) a periodic solution of period two, otherwise, every positive solution of the equation converges to a unique positive equilibrium.
Publisher: Hindawi

Show full item record


checked on Jul 12, 2024

Page view(s)

checked on May 9, 2024

Google ScholarTM




Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.