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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 $$x_{n+1} = \frac{px_{n-s} + x_{n-t}}{qx_{n-s} + x_{n-t}}, \ n = 0,1,\dots \tag1$$ 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 $p > q \ge 1,$ or $1 \ge p > q,$ or $(1 + 3q) / (1 - q) \ge p > 1 > q.$ (H2) Either $q > p \ge 1,$ or $1 \ge q > p,$ or $(1 + 3p) / (1 - p) \ge q > 1 > p.$

MSC:
39A11Stability of difference equations (MSC2000)
39A20Generalized difference equations
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References:
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