Global asymptotic stability for quadratic fractional difference equation

Consider the difference equation x n + 1 = α + ∑ i = 0 k a i x n − i + ∑ i = 0 k ∑ j = i k a i j x n − i x n − j β + ∑ i = 0 k b i x n − i + ∑ i = 0 k ∑ j = i k b i j x n − i x n − j , n = 0 , 1 , … , where all parameters α , β , a i , b i , a i j , b i j , i , j = 0 , 1 , … , k , and the initial conditions x i , i ∈ { − k , … , 0 } , are nonnegative. We investigate the asymptotic behavior of the solutions of the considered equation. We give simple explicit conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation.