The rule of semicycle and global asymptotic stability for a fourth-order rational difference equation

In this paper, the rule for the lengths of positive and negative semicycles of nontrivial solutions of the following fourth-order rational difference equation, where a, b ε [0, ∞) and the initial values x−3,x−2,x−1,x0 ε (0, ∞), to successively occur is found to be …, 4+, 3−, 1+, 2−, 2+, 1−, 1+, 1−, 4+, 3−, 1+, 2−, 2+, 1−, 1+, 1−, 4+, 3−, 1+, 2−, 2+, 1−, 1+, 1−, …, by which the positive equilibrium point of the equation is verified to be globally asymptotically stable.