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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| Format: | Article |
| Language: | English |
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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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| _version_ | 1832549996280414208 |
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| author | Stevo Stevic |
| author_facet | Stevo Stevic |
| author_sort | Stevo Stevic |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-4f774b484dd04d5297709df089fe79b7 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2007-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-4f774b484dd04d5297709df089fe79b72025-02-03T06:07:58ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2007-01-01200710.1155/2007/1373713737Asymptotic Periodicity of a Higher-Order Difference EquationStevo Stevic0Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 35/I, Beograd 11000 , SerbiaWe 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.http://dx.doi.org/10.1155/2007/13737 |
| spellingShingle | Stevo Stevic Asymptotic Periodicity of a Higher-Order Difference Equation Discrete Dynamics in Nature and Society |
| title | Asymptotic Periodicity of a Higher-Order Difference Equation |
| title_full | Asymptotic Periodicity of a Higher-Order Difference Equation |
| title_fullStr | Asymptotic Periodicity of a Higher-Order Difference Equation |
| title_full_unstemmed | Asymptotic Periodicity of a Higher-Order Difference Equation |
| title_short | Asymptotic Periodicity of a Higher-Order Difference Equation |
| title_sort | asymptotic periodicity of a higher order difference equation |
| url | http://dx.doi.org/10.1155/2007/13737 |
| work_keys_str_mv | AT stevostevic asymptoticperiodicityofahigherorderdifferenceequation |