When does local asymptotic stability imply global attractivity in rational equations? (English) Zbl 1105.39001

Camouzis, E.; Ladas, G.

J. Difference Equ. Appl. 12, No. 8, 863-885 (2006).

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)].

Reviewer: Raghib Abu-Saris (Sharjah)

MSC:

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

Keywords:

local asymptotic stability; global attractivity; rational difference equations

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