We consider the family of nonlinear difference equations: x n + 1 = ( ∑ i = 1 3 f i ( x n , … , x n − k ) + f 4 ( x n , … , x n − k ) f 5 ( x n , … , x n − k ) ) / ( f 1 ( x n , … , x n − k ) f 2 ( x n , … , x n − k ) + ∑ i = 3 5 f i ( x n , … , x n − k ) ) , n = 0 , 1 , … , where f i ∈ C ( ( 0 , + ∞ ) k + 1 , ( 0 , + ∞ ) ), for i ∈ { 1 , 2 , 4 , 5 } , f 3 ∈ C ( [ 0 , + ∞ ) k + 1 , ( 0 , + ∞ ) ) , k ∈ { 1 , 2 , … } and the initial values x − k , x − k + 1 , … , x 0 ∈ ( 0 , + ∞ ) . We give sufficient conditions under which the unique equilibrium x ¯ = 1 of these equations is globally asymptotically stable, which extends and includes corresponding results obtained in the cited references.