## Global asymptotic stability for a nonlinear delay difference equation.(English)Zbl 1013.39003

The main result of this paper is that the positive equilibrium point of the nonlinear difference equation $x_{n+1}=\frac{x_n+x_{n-1}x_{n-2}+a}{x_nx_{n-1}+x_{n-2}+a},\quad n=0,1,\dots,$ is globally asymptotically stable provided $$a\geq 0$$ and the initial values $$x_{-2},x_{-1},x_0$$ are positive. Thereby the conjecture of G. Ladas [Open problems and conjectures. J. Differ. Equ. Appl. 4, No. 1, 95-97 (1998)] about global asymptotic stability of the positive equilibrium of the above equation with $$a=0$$ is confirmed.
Reviewer: Pavel Rehak (Brno)

### MSC:

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

### References:

 [1] Ladas, G., Open problems and conjectures, J. Differ. Equations Appl., 1998, 4(1):95–97. · Zbl 0920.92022 [2] Kocic, V.L., Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. · Zbl 0787.39001 [3] Li Xianyi, Tang Hengsheng, Liu Yachun, et al., A Conjecture by G. Ladas, Appl. Math. J. Chinese Univ. Ser B., 1998,13:39–44. · Zbl 0902.39003
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.