Authors: Stević, Stevo 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: On the recursive sequence xn = 1 + ∑i=1 k αixn-pi/∑j=1 m βjxn-qj
Journal: Discrete Dynamics in Nature and Society
Volume: 2007
Issue: 1
Issue Date: 3-Apr-2007
Rank: M22
ISSN: 1026-0226
DOI: 10.1155/2007/39404
Abstract: 
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=1k αi = ∑j=1m βj = 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.
Publisher: Hindawi

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