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Attractivity of the recursive sequence \(x_{n+1}=(\alpha-\beta x_n)F(x_{n-1},\dots,x_{n-k})\). (English) Zbl 1187.39023

Summary: We investigate the global attractivity of the recursive sequence \(x_{n+1}=(\alpha-\beta x_n)F(x_{n-1},\dots,x_{n-k})\), \(n=0,1\dots\), where \(\alpha ,\beta \geq 0\). We show that the unique positive equilibrium point of the equation is a global attractor with some basin. We apply this result to the rational recursive sequence \[ x_{n+1}=\frac{\alpha-\beta x_n}{\gamma +\sum_{i=1}^{k}a_{i}x_{n-i}+\sum_{i=1}^{k}b_{i}x_{n-i}^{2}}. \] where \(\alpha,\beta,a_i,b_i\geq 0\) and \(\gamma >0\), and prove that the positive equilibrium point of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients.

MSC:

39A30 Stability theory for difference equations
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