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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
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[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
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