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On the periodic character of some difference equations. (English) Zbl 1032.39004

The positive solutions \(\{ x_n\}_{n=-d}^\infty \) of the difference equation \[ x_n=\max \left\{ \frac{A}{x_{n-k}},\frac{B}{x_{n-m}}\right\} \qquad n=0,1,\dots \] where \(k,m\) are positive integers, \(d=\max\{ k,m\}\) and \(A,B\in (0,\infty)\), are investigated. It is shown that there exists a positive integer \(T\) such that every positive solution \(\{ x_n\}_{n=-d}^\infty \) is eventually periodic with period \(T\), that is \[ x_{n+T}=x_n \qquad\text{for }n\geq n_0 \] where \(n_0\) may depend on the initial conditions \(x_{-1},x_{-2},\dots ,x_{-d}\). In addition, the period \(T\) is determined in terms of the parameters \(A\), \(B\), \(k\), and \(m\).

MSC:

39A11 Stability of difference equations (MSC2000)
39B05 General theory of functional equations and inequalities
39A10 Additive difference equations
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