# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Amleh A., Journal of Difference Equations and Applications (2006) [2] DOI: 10.1016/j.jmaa.2005.05.008 · Zbl 1090.39004 · doi:10.1016/j.jmaa.2005.05.008 [3] DOI: 10.1080/10236190500272897 · Zbl 1090.39005 · doi:10.1080/10236190500272897 [4] DOI: 10.1080/10236190410001726430 · Zbl 1055.39500 · doi:10.1080/10236190410001726430 [5] Camouzis, E. and Ladas, G., Dynamics of Third Order Rational Difference Equations; With Open Problems and Conjectures, Chapman and Hall/CRC Press (to appear in 2007). · Zbl 1129.39002 [6] DOI: 10.1080/10236190500035328 · Zbl 1228.39002 · doi:10.1080/10236190500035328 [7] DOI: 10.1080/10236190410001726449 · doi:10.1080/10236190410001726449 [8] DOI: 10.1080/10236190410001726430 · Zbl 1055.39500 · doi:10.1080/10236190410001726430 [9] Camouzis E., Journal of Difference Equations and Applications (2006) · Zbl 1099.39003 [10] DOI: 10.1080/10236190500044197 · Zbl 1071.39502 · doi:10.1080/10236190500044197 [11] Grove E.A., Periodicities in Nonlinear Difference Equations (2005) · Zbl 1078.39009 [12] Kocic V.L., Global Asymptotic Behavior of Nonlinear Difference Equations of Higher Order with Applications (1993) · Zbl 0787.39001 [13] DOI: 10.1201/9781420035384 · doi:10.1201/9781420035384 [14] Kulenović, M.R.S. and Merino, O., Convergence to a period-two solution of a class of second order rational difference equations (to appear) · Zbl 1128.39006 [15] DOI: 10.1080/10236190500410109 · Zbl 1099.39007 · doi:10.1080/10236190500410109 [16] DOI: 10.1080/10236190410001726458 · Zbl 1057.39505 · doi:10.1080/10236190410001726458 [17] DOI: 10.1080/10236190512331319352 · Zbl 1071.39017 · doi:10.1080/10236190512331319352 [18] Wang Q., Journal of Difference Equations and Applications (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.