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Zbl 1090.39009
Stević, Stevo
Global stability and asymptotics of some classes of rational difference equations. (English)
[J] J. Math. Anal. Appl. 316, No. 1, 60-68 (2006). ISSN 0022-247X

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.
[Raghib Abu-Saris (Sharjah)]

MSC 2000:: *39A11 Stability of difference equations
39A20 Generalized difference equations

Keywords: Rational difference equation; Global asymptotic stability; Equilibrium solutions; positive solution

Citations: Zbl 0933.37016; Zbl 1056.39003

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