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Global stability and asymptotics of some classes of rational difference equations. (English) Zbl 1090.39009
The author proves that the equilibrium solution $\bar{x}=1$ is globally asymptotically stable for the difference equations $$x_{n+3}=\frac{x_{n+j}+x_{n+i}~x_{n+k}+a}{x_{n+i}+x_{n+j}~x_{n+k}+a},\quad n=0,1,2,\dots$$ where the initial values $x_{-2},x_{-1},x_0$ are positive, the parameter $a$ is nonnegative, $i,j\in\{0,1,2\}$ but different from each other, and $k=3-i-j$. In his proof he utilizes a global convergence result due to {\it N. Kruse and T. Nesemann} [J. Math. Anal. Appl. 235, 151--158 (1999; Zbl 0933.37016)]. In addition, using an inclusion theorem due to {\it L. Berg} [J. Difference Equ. Appl. 10, 399--408 (2004; Zbl 1056.39003)], he finds asymptotics of some solutions of the above difference equations.

MSC:
39A11Stability of difference equations (MSC2000)
39A20Generalized difference equations
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Full Text: DOI
References:
[1] Amleh, A. M.; Kruse, N.; Ladas, G.: On a class of difference equations with strong negative feedback. J. difference equ. Appl. 5, 497-515 (1999) · Zbl 0951.39002
[2] Berg, L.: Asymptotische darstellungen und entwicklungen. (1968) · Zbl 0165.36901
[3] Berg, L.: On the asymptotics of nonlinear difference equations. Z. anal. Anwendungen 21, 1061-1074 (2002) · Zbl 1030.39006
[4] Berg, L.: Inclusion theorems for non-linear difference equations with applications. J. difference equ. Appl. 10, 399-408 (2004) · Zbl 1056.39003
[5] Berg, L.: Corrections to ”inclusion theorems for non-linear difference equations with applications,” from [3]. J. difference equ. Appl. 11, 181-182 (2005) · Zbl 1080.39002
[6] Kruse, N.; Nesemann, T.: Global asymptotic stability in some discrete dynamical systems. J. math. Anal. appl. 235, 151-158 (1999) · Zbl 0933.37016
[7] Xianyi, L.; Deming, Z.: Global asymptotic stability for two recursive difference equations. Appl. math. Comput. 150, 481-492 (2004) · Zbl 1044.39006
[8] Xianyi, L.; Deming, Z.: Global asymptotic stability of a nonlinear recursive sequence. Comput. math. Appl. 17, 833-838 (2004) · Zbl 1068.39014