Dynamics of a higher order rational difference equation

Abstract

In this paper, we will investigate a nonlinear rational difference equation of higher order. Our concentration is on invariant intervals, periodic character, the character of semicycles and global asymptotic stability of all positive solutions of x n + 1 = β x n + γ x n - k Bx n + Cx n - k , n = 0 , 1 , … It is worth to mention that our results solve the open problem proposed by Kulenvic and Ladas in their monograph [Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002].