Dynamics of a Higher-Order Nonlinear Difference Equation

Guo-Mei Tang,Xiu-Mei Jia,Lin-Xia Hu

doi:10.1155/2010/534947

Abstract

We consider the higher-order nonlinear difference equation , , where parameters are positive real numbers and initial conditions are nonnegative real numbers, . We investigate the periodic character, the invariant intervals, and the global asymptotic stability of all positive solutions of the abovementioned equation. We show that the unique equilibrium of the equation is globally asymptotically stable under certain conditions.

Highlights

Introduction and PreliminariesIn this paper, we will investigate the global behavior of solutions of the following nonlinear difference equation: xn 1 A α xn Bxn xn−k, n1.1 where parameters are positive real numbers and initial conditions x−k, . . . , x0 are nonnegative real numbers, k ≥ 2.In 2003, the authors in 1 considered the difference equation xn 1 α Bxn βxn Cxn−1Discrete Dynamics in Nature and Society with nonnegative parameters α, β, A, B, C and nonnegative initial conditions x−1, x0
We will investigate the global behavior of solutions of the following nonlinear difference equation: xn 1
For the sake of convenience, we recall some definitions and theorems which will be useful in the sequel

Summary

Introduction and Preliminaries

We will investigate the global behavior of solutions of the following nonlinear difference equation: xn 1. Discrete Dynamics in Nature and Society with nonnegative parameters α, β, A, B, C and nonnegative initial conditions x−1, x0 They obtained some global asymptotic stability results for the solutions of 1.2. For the sake of convenience, we recall some definitions and theorems which will be useful in the sequel. I The equilibrium y is called locally stable or stable if, for every ε > 0, there exists δ > 0 such that, for all y−k, . Iii The equilibrium y of 1.4 is called a global attractor if, for every y−k, . Vi The equilibrium y of 1.4 is called a source, or a repeller, if there exists r > 0 such that, for all y−k, . 1.15 has a unique equilibrium y ∈ a, b and every solution of 1.15 converges to y

Local Stability and Period-Two Solutions

Boundedness and Invariant Interval

Semicycles Analysis

Global Stability Proof