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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)
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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
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