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Global asymptotic stability of a nonlinear recursive sequence. (English) Zbl 1068.39014

Using a “semicycle analysis method” developed by the authors, they prove that the positive equilibrium of the nonlinear difference equation \[ x_{n+1}=\frac{x_nx_{n-1}^b+x_{n-2}^b+a}{x_{n-1}^b+x_nx_{n-2}^b+a}, \quad n\geq 0 \] is globally asymptotically stable for parameters \(a\geq 0\), \(b>0\) and initial values \(x_{-2},x_{-1},x_0>0\).

MSC:

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