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On a max type recurrence relation with periodic coefficients. (English) Zbl 1050.39017

The authors consider the difference equation \[ x_{n+1}=\max\{A_{n}/x_{n},B_{n}/x_{n-2}\} \] with \(A_{n}\geq 0, B_{n}\geq 0\) and having period 3; positive initial conditions \(x_{-2},x_{-1},x_{0}\) are considered hence the solutions are positive. It is shown, using the properties of the solutions of the auxiliary equation \[ z_{n+1}=\max\{1,E_{n}(z_{n}/z_{n-1})\} \] that every positive solution of the basic equation is eventually periodic provided there exists \(m\in\{0,1,2\}\) such that \[ A_{m+1}=A_{m+2}=\max\{A_m,B_m\}\geq \max\{B_{m+1},B_{m+2}\} \]

MSC:

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