On the recursive sequence \(x_{n+1} = \max \left\{ c, \frac {x_n^p}{x_{n-1}^p} \right\}\). (English) Zbl 1152.39012

Stević, Stevo

Appl. Math. Lett. 21, No. 8, 791-796 (2008).

This work studies the boundedness and global attractivity for the posivitive solutions of the difference equation \(x_{n+1}= \max\{c,{x^p_n\over x^p_{n-1}}\}\), \(n\in\mathbb{N}_0\) with \(p,c\in(0,+\infty)\).
It is shown that: (a) there exist unbounded solutions whenever \(p\geq 4\); (b) all positive solutions are bounded when \(p\in(0, 4)\); (c) every positive solution is eventually equal to 1 when \(p\in(0, 4)\) and \(c\geq 1\); (d) all positive solutions converge to 1 whenever \(p,c\in(0,1)\).

Reviewer: Stefan Balint (Timişoara)

MSC:

39A11

Stability of difference equations (MSC2000)

Keywords:

max type difference equations; boundedness; difference equation; global attractivity

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