Abstract
This paper studies the boundedness, global asymptotic stability, and periodicity of positive solutions of the equationxn=f(xn−2)/g(xn−1),n∈ℕ0, wheref,g∈C[(0,∞),(0,∞)]. It is shown that iffandgare nondecreasing, then for every solution of the equation the subsequences{x2n}and{x2n−1}are eventually monotone. For the case whenf(x)=α+βxandgsatisfies the conditionsg(0)=1,gis nondecreasing, andx/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, thenf(x)=c1/xandg(x)=c2x, for some positivec1andc2.
Highlights
There has been great interest in the study of nonlinear and rational difference equations cf. 1–35 and the references therein .In this paper, we study the boundedness, global asymptotic stability, and periodicity for positive solutions of the equation xn f g xn−2 xn−1, n ∈ N0, 1.1 where f, g ∈ C 0, ∞, 0, ∞ .Abstract and Applied Analysis2
We present the proof of the theorem for the case of 2.3, for the benefit of the reader, since the proof is instructive
If every solution of 3.2 is periodic with period p 2, it must hold that xhyfx, x, y ∈ 0, ∞, 3.3 that is x/f x h y, which implies that f x cx and h y c for some positive constant c
Summary
Otherwise, {ri}i≥0 itself is a constant sequence and the result again follows. Note that Theorem 2.1 guarantees only the eventually monotonicity of the sequences {x2n} and {x2n 1}. For example, 27, Theorem 1 , where it was shown that for the case f x g x xp, p ≥ 1, 1.1 has unbounded solutions.
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