Global asymptotic stability for two recursive difference equations.
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 \(a\in [0,\infty )\).
scientific article; zbMATH DE number 2055869