Qualitative properties for a fourth-order rational difference equation ✩

Abstract In this paper, we use a method different from the known literature to investigate the qualitative properties of the following fourth-order rational difference equation: x n + 1 = x n x n − 1 x n − 3 + x n + x n − 1 + x n − 3 + a x n x n − 1 + x n x n − 3 + x n − 1 x n − 3 + 1 + a , n = 0 , 1 , 2 , … , where a ∈ [ 0 , ∞ ) and the initial values x −3 , x −2 , x −1 , x 0 ∈ ( 0 , ∞ ) . The successive lengths of positive and negative semicycles of nontrivial solutions of the above equation is found to periodically occur, that is, … , 3 + , 2 − , 1 + , 1 − , 3 + , 2 − , 1 + , 1 − , 3 + , 2 − , 1 + , 1 − , 3 + , 2 − , 1 + , 1 − , … , or, … , 2 + , 1 − , 1 + , 3 − , 2 + , 1 − , 1 + , 3 − , 2 + , 1 − , 1 + , 3 − , 2 + , 1 − , 1 + , 3 − , 2 + , 1 − , 1 + , 3 − , … . By using the rule, the positive equilibrium point of the equation is verified to be globally asymptotically stable.