On the difference equation Xn+1 = α+ Xpn/Xpn−1

Abstract

The dynamics of the difference equation Xn+1 = α+ Xpn/Xpn−1 , with p ∈ (0, 1) and α ∈ [0, ∞) was studied first. Our aim was to study the asymptotic behaviour of the rational difference equation Xn+1 = α+ Xp/pn Xn-k

We find conditions for the global asymptotic stability of the unique negative equilibrium y ¼ 1 þ A of the equation ynþ1 ¼ A þ yn ynk ; ð0:1Þ where yk,yk+1,...,y0 2 (0,1), A < 0 and k 2 {1, 2, 3, 4,...}.