On the dynamics of a rational difference equation x n +1

Abstract

In this paper, we will investigate the dynamics of a nonlinear rational difference equation of a higher order. Our concentration is on invariant intervals, periodic character, the character of semi-cycles and global asymptotic stability of all positive solutions of xn+1 = α + βxn + γ xn−k Bxn + Cxn−k , n = 0, 1, . . . It is worth to mention that our results solve partially some of the open problems proposed by Kulenvic and Ladas in their monographs [17], [18].