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When does local asymptotic stability imply global attractivity in rational equations? (English) Zbl 1105.39001
The authors aim at drawing the attention of researchers to an important question that needs further investigating, namely
Given the difference equation \[ x_{n+1}=f(x_n,\dots,x_{n-k}),\quad n=0,1,\dots \] with an equilibrium solution \(\bar{x}\), how nice should \(f\) be so that, for \(\bar{x}\), local asymptotic stability implies global attractivity?
They also formulate conjectures and pose open problems in this regard in the context of rational difference equations of the form \[ x_{n+1}=\frac{\alpha+\sum_{i=0}^k \beta_i~ x_{n-i}}{A+\sum_{i=0}^k B_i ~x_{n-i}} \] with nonnegative parameters and with nonnegative initial conditions such that the denominator is always positive.
Interested readers are referred to the paper by U. Krause [A local-global stability principle for discrete systems and difference equations, Proceedings of the Sixth International Conference on Difference Equations, 167–180 (2004; Zbl 1065.39015)].

MSC:
39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
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References:
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