Authors: Stević, Stevo 
Berenhaut, Kenneth
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: The behavior of positive solutions of a nonlinear second-order difference equation
Journal: Abstract and Applied Analysis
Volume: 2008
Issue Date: 1-Apr-2008
Rank: M22
ISSN: 1085-3375
DOI: 10.1155/2008/653243
This paper studies the boundedness, global asymptotic stability, and periodicity of positive solutions of the equation xn =f( x n-2)/g( x n-1), n∈0, where f,g∈C[(0,∞),(0,∞)]. It is shown that if f and g are nondecreasing, then for every solution of the equation the subsequences {x2n} and {x2n-1} are eventually monotone. For the case when f(x)=α+βx and g satisfies the conditions g(0)=1, g is nondecreasing, and x/g(x) is increasing, we prove that every prime periodic solution of the equation has period equal to one or two. We also investigate the global periodicity of the equation, showing that if all solutions of the equation are periodic with period three, then f(x)=c1/x and g(x)=c2x, for some positive c1 and c2.
Publisher: Hindawi

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