Dynamics for Nonlinear Difference Equation xn+1=(αxn−k)/(β+γxn−lp)

Dongmei Chen,Xianyi Li,Yanqin Wang

doi:10.1155/2009/235691

Abstract

We mainly study the global behavior of the nonlinear difference equation in the title, that is, the global asymptotical stability of zero equilibrium, the existence of unbounded solutions, the existence of period two solutions, the existence of oscillatory solutions, the existence, and asymptotic behavior of non-oscillatory solutions of the equation. Our results extend and generalize the known ones.

Highlights

Consider the following higher order difference equation: xn 1 β αxn−k γ xnp−l, n1.1 where k, l, ∈ {0, 1, 2, . . .}, the parameters α, β, γ and p, are nonnegative real numbers and the initial conditions x− max{k, l}, . . . , x−1 and x0 are nonnegative real numbers such that β γ xnp−l > 0, ∀n ≥ 0.It is easy to see that if one of the parameters α, γ, p is zero, the equation is linear
In the sequel we always assume that the parameters α, β, γ, and p are positive real numbers
Our aim in this paper is to extend and generalize the work in 3

Summary

Introduction

0, 1, . . . , 1.1 where k, l, ∈ {0, 1, 2, . . .}, the parameters α, β, γ and p, are nonnegative real numbers and the initial conditions x− max{k, l}, . . . , x−1 and x0 are nonnegative real numbers such that β γ xnp−l > 0, ∀n ≥ 0. In the sequel we always assume that the parameters α, β, γ, and p are positive real numbers. The linearized equation of 1.3 about the equilibrium point y1 0 is zn 1 rzn−k, n 0, 1, . The linearized equation of 1.3 about the positive equilibrium point y2 has the form zn 1. We will investigate the global behavior of 1.1 , including the global asymptotical stability of zero equilibrium, the existence of unbounded solutions, the existence of period two solutions, the existence of oscillatory solutions, the existence and asymptotic behavior of nonoscillatory solutions of the equation. The linearized equation of 1.11 associated with the equilibrium point x is yn 1 k ∂F i 0 ∂ui x, . For the related investigations for nonlinear difference equations, see 7–11 and the references cited therein

Global Asymptotic Stability of Zero Equilibrium

Existence of Eventual Period Two Solution

Existence of Oscillatory Solution

Existence of Unbounded Solution

Existence and Asymptotic Behavior of Nonoscillatory Solution