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On the difference equation \(x_{n+1}= \frac{px_{n-s} + x_{n-t}}{qx_{n-s} + x_{n-t}}\). (English) Zbl 1162.39015

The paper deals with the difference equation
\[ x_{n+1} = \frac{px_{n-s} + x_{n-t}}{qx_{n-s} + x_{n-t}}, \;n = 0,1,\dots \tag{1} \]
with positive initial conditions where \(s, t\) are distinct nonnegative integers, \(p > 0, q >0, p \not= q.\) The authors prove that the positive equilibrium of eq. (1) is globally asymptotically stable if one of the following two conditions is satisfied:
(H1) Either \(p > q \geq 1,\) or \(1 \geq p > q,\) or \((1 + 3q) / (1 - q) \geq p > 1 > q.\)
(H2) Either \(q > p \geq 1,\) or \(1 \geq q > p,\) or \((1 + 3p) / (1 - p) \geq q > 1 > p.\)

MSC:

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

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