The difference equation xn+1=α+xn−k∑i=0k−1cixn−i has solutions converging to zero

Abstract The aim of this note is to show that the following difference equation: x n + 1 = α + x n − k ∑ i = 0 k − 1 c i x n − i , n = 0 , 1 , … , where k ∈ N , c i ⩾ 0 , i = 0 , … , k − 1 , ∑ i = 0 k − 1 c i = 1 , and α − 1 , has solutions which monotonically converge to zero. This result shows the existence of such solutions which was not shown in the recently accepted paper: A.E. Hamza, On the recursive sequence x n + 1 = α + x n − 1 x n , J. Math. Anal. Appl., in press.