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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| 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/10.1155/2007/39404 |
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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 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. |
| format | Article |
| id | doaj-art-ef072159589749338b8c89b8407c4bf3 |
| 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-ef072159589749338b8c89b8407c4bf32025-02-03T06:11:47ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2007-01-01200710.1155/2007/3940439404On the Recursive Sequence xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qjStevo Stevic0Mathematical Institute of the Serbian Academy of Sciences and Arts , Knez Mihailova 35/I, Belgrade 11001, SerbiaWe 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.http://dx.doi.org/10.1155/2007/39404 |
| spellingShingle | Stevo Stevic On the Recursive Sequence xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj Discrete Dynamics in Nature and Society |
| title | On the Recursive Sequence xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj |
| title_full | On the Recursive Sequence xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj |
| title_fullStr | On the Recursive Sequence xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj |
| title_full_unstemmed | On the Recursive Sequence xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj |
| title_short | On the Recursive Sequence xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj |
| title_sort | on the recursive sequence xn 1 ∑i 1kαixn pi ∑j 1mβjxn qj |
| url | http://dx.doi.org/10.1155/2007/39404 |
| work_keys_str_mv | AT stevostevic ontherecursivesequencexn1i1kaixnpij1mbjxnqj |