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

The aim of this note is to show that the following difference equation:xn+1=α+xn−k∑i=0k−1cixn−i,n=0,1,…, where k∈Nk∈N, ci⩾0ci⩾0, i=0,…,k−1i=0,…,k−1, ∑i=0k−1ci=1, and α<−1α<−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 xn+1=α+xn−1xn, J. Math. Anal. Appl., in press.