Global asymptotic stability for two recursive difference equations.(English)Zbl 1044.39006

The authors study the global asymptotic stability of the following two recursive difference equations $x_{n+1}=\frac{x_nx_{n-1}+x_{n-1}+a}{x_n+x_{n-1}x_{n-2}+a},\text{ } n=0,1,2,...,$ and $x_{n+1}=\frac{x_{n-1}+x_nx_{n-2}+a}{x_nx_{n-1}+x_{n-2}+a},\text{ } n=0,1,2,...,$ where $$a\in [0,\infty )$$ and the initial values $$x_{-2},x_{-1},x_0\in (0,\infty )$$. They prove that the positive equilibria of the two recursive difference equations are global asymptotically stable if $$a\in [0,\infty )$$.

MSC:

 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations
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References:

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