Abstract and Applied Analysis
Volume 2008 (2008), Article ID 653243, 8 pages
doi:10.1155/2008/653243

The behavior of positive solutions of a nonlinear second-order difference equation

Stevo Stević1 and Kenneth S. Berenhaut2

1Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 35/I, Beograd 11000, Serbia
 2Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109, USA

Abstract

This paper studies the boundedness, global asymptotic stability, and periodicity of positive solutions of the equation xn=f(xn−2)/g(xn−1), n∈ℕ0, 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.