Global asymptotic stability and oscillation of a family of difference equations. (English) Zbl 1055.39017

The authors are concerned with the oscillatory behavior of the positive solutions of the family of difference equations \[ x_{n+1}= \Biggl(\sum^k_{\substack{ i=0\\ i\neq j,j-1}} x_{n-i}+ x_{n-j+1} x_{n-j}+ 1\Biggr)\Biggl/\sum^k_{i=0} x_{n-j},\quad j= 1,\dots,k, \] where \(n\in \{0,1,\dots\}\), \(k\in \{1,2,\dots\}\) and the initial values \(x_{-k}, x_{-k+1},\dots, x_0\) are positive numbers. It is also proved that the unique equilibrium \(\overline x= 1\) is globally asymptotically stable.


39A11 Stability of difference equations (MSC2000)
Full Text: DOI


[1] Amleh, A. M.; Kruse, N.; Ladas, G., On a class of difference equations with strong negative feedback, J. Differ. Equations Appl., 5, 497-515 (1999) · Zbl 0951.39002
[2] Kocic, V. L.; Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (1993), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0787.39001
[3] Kruse, N.; Nesemann, T., Global asymptotic stability in some discrete dynamical systems, J. Math. Anal. Appl., 235, 151-158 (1999) · Zbl 0933.37016
[4] Xianyi, L.; Deming, Z., Global asymptotic stability in a rational equation, J. Differ. Equations Appl., 9, 833-839 (2003) · Zbl 1055.39014
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.