On the dynamics of a rational difference equation xn+1=α+βxn+γxn−kBxn+Cxn−k

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 x n +1 = α+ βx n + γx n −k Bx n + Cx n −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].