Abstract
The rule of trajectory structure for fourth-order nonlinear difference equation , where and the initial values , is described clearly out in this paper. Mainly, the lengths of positive and negative semicycles of its nontrivial solutions are found to occur periodically with prime period 15. The rule is in a period. By utilizing this rule its positive equilibrium point is verified to be globally asymptotically stable.
Highlights
In this paper we consider the following fourth-order nonlinear difference equation: xn 1 xna−2 xn−3 xna−2xn−3 1, n1.1 where a ∈ 0, 1 and the initial values x−3, x−2, x−1, x0 ∈ 0, ∞
When a 0, 1.1 becomes the trivial case xn 1 1, n 0, 1, . . . . we will assume in the sequel that 0 < a < 1
First we analyze the structure of the semi-cycles of nontrivial solutions of 1.1
Summary
In this paper we consider the following fourth-order nonlinear difference equation: xn 1 xna−2 xn−3 xna−2xn−3 1 , n1.1 where a ∈ 0, 1 and the initial values x−3, x−2, x−1, x0 ∈ 0, ∞. In this paper we consider the following fourth-order nonlinear difference equation: xn 1 xna−2 xn−3 xna−2xn−3 1 There do not exist eventually positive non-oscillatory solutions of 1.1 .
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