On the Recursive Sequence xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj

We give a complete picture regarding the behavior of positive solutions of the following important difference equation: xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj, n∈ℕ0, where αi, i∈{1,…,k}, and βj, j∈{1,…,m}, are positive numbers such that ∑i=1kαi=∑j=1mβj=1, and pi, i∈{1,…,k}, and qj, j∈{1,…,m}, are natural nu...

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Bibliographic Details
Main Author: Stevo Stevic
Format: Article
Language:English
Published: Wiley 2007-01-01
Series:Discrete Dynamics in Nature and Society
Online Access:http://dx.doi.org/10.1155/2007/39404
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Summary:We give a complete picture regarding the behavior of positive solutions of the following important difference equation: xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj, n∈ℕ0, where αi, i∈{1,…,k}, and βj, j∈{1,…,m}, are positive numbers such that ∑i=1kαi=∑j=1mβj=1, and pi, i∈{1,…,k}, and qj, j∈{1,…,m}, are natural numbers such that p1<p2<⋯<pk and q1<q2<⋯<qm. The case when gcd(p1,…,pk,q1,…,qm)=1 is the most important. For the case we prove that if all pi, i∈{1,…,k}, are even and all qj, j∈{1,…,m}, are odd, then every positive solution of this equation converges to a periodic solution of period two, otherwise, every positive solution of the equation converges to a unique positive equilibrium.
ISSN:1026-0226
1607-887X