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Berenhaut, Kenneth S.; Stević, Stevo

The behaviour of the positive solutions of the difference equation \(x_n = A + (\frac{x_{n-2}}{x_{n-1}})^p\). (English) Zbl 1111.39003

J. Difference Equ. Appl. 12, No. 9, 909-918 (2006).
For the difference equation in the title with positive parameters and \(p\neq 1\) it is shown that there exist unbounded solutions if \(p>1\), that all positive solutions converge to a 2-periodic solution it \((A+1)/2<p<1\), and that all solutions converge to \(A+1\) in the other cases of \(p\). The case \(p=1\) was already treated by A. M. Amleh, E. A. Grove, G. Ladas and D. A. Georgiou [J. Math. Anal. Appl. 233, No. 2, 790–798 (1999; Zbl 0962.39004)].
Reviewer: Lothar Berg (Rostock)

Cited in 44 Documents

MSC:

39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations

Keywords:

rational difference equation; stability; boundedness; period two solution; unbounded solutions; positive solutions

Citations:

Zbl 0962.39004
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\textit{K. S. Berenhaut} and \textit{S. Stević}, J. Difference Equ. Appl. 12, No. 9, 909--918 (2006; Zbl 1111.39003)
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References:

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