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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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