The difference equation xn + 1 = α + frac(xn - k, ∑i = 0k - 1 ci xn - i) has solutions converging to zero

Abstract

The aim of this note is to show that the following difference equation:xn + 1 = α + frac(xn - k, ∑i = 0k - 1 ci xn - i), n = 0, 1, ..., where k ∈ N, ci ≥ 0, i = 0, ..., k - 1, ∑i = 0k - 1 ci = 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 xn + 1 = α + frac(xn - 1, xn), J. Math. Anal. Appl., in press.