Asymptotic Periodicity of a Higher-Order Difference Equation
We give a complete picture regarding the asymptotic periodicity of positive solutions of the following difference equation: xn=f(xn−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,...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2007-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2007/13737 |
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| Summary: | We give a complete picture regarding the asymptotic periodicity of positive
solutions of the following difference equation:
xn=f(xn−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,∞),
x=f(x,…,x︸k,g(x),…,g(x)︸m), x∈(α,∞), limx→α+g(x)=+∞, and limx→+∞g(x)=α. It is proved that if all
pi, 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. |
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| ISSN: | 1026-0226 1607-887X |