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:
xn+1 =
xnxn−1xn−3 + xn + xn−1 +xn−3 +a
xnxn−1 + xnxn−3 +xn−1xn−3 +1+ a
, n= 0, 1, 2, . . . ,
where a ∈ [0,∞) and the initial values x−3, x−2, x−1, x0 ∈ (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.
2005 Elsevier Inc. All rights reserved.
