Attractivity of the recursive sequence

In this paper we investigate the global attractivity of the recursive sequence x n + 1 = ( α − β x n ) F ( x n − 1 , … , x n − k ) , n = 0 , 1 , … where α , β ≥ 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 = α − β x n γ + ∑ i = 1 k a i x n − i + ∑ i = 1 k b i x n − i 2 , n = 0 , 1 , … where α , β , a i , b i ≥ 0 and γ > 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.