Authors: Berenhaut, Kenneth
Stević, Stevo 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: The behaviour of the positive solutions of the difference equation x n = A + (xn-2/xn-1)p
Journal: Journal of Difference Equations and Applications
Volume: 12
Issue: 9
First page: 909
Last page: 918
Issue Date: 1-Sep-2006
Rank: M21
ISSN: 1023-6198
DOI: 10.1080/10236190600836377
Abstract: 
This paper studies the boundedness, global asymptotic stability and periodicity for solutions of the equation xn = A + (x n-2/xn-1)p, n = 0, 1,..., with p, A ∈ (0, ∞), p ≠ 1 and x-2, x-1 ∈ (0, ∞). It is shown that: (a) all solutions converge to the unique equilibrium, x̄ = A + 1, whenever p ≤ min{1, (A + 1)/2}; (b) all solutions converge to period two solutions whenever (A + 1)/2 < p < 1; and (c) there exist unbounded solutions whenever p > 1. These results complement those for the case p = 1 in A.M. Amleh et al., On the recursive sequence yn+1 = α + (yn-1/yn, Journal of Mathematical Analysis and Applications 233 (1999), 790-798.
Keywords: Boundedness | Period two solution | Rational difference equation | Stability
Publisher: Taylor & Francis

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