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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 N. Kruse and T. Nesemann [J. Math. Anal. Appl. 235, 151–158 (1999; Zbl 0933.37016)]. In addition, using an inclusion theorem due to 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:
 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations
Citations:
Zbl 0933.37016; Zbl 1056.39003
Full Text:
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), Dt. Verlag Wiss. Berlin · 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
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