Asymptotic Behavior of Equilibrium Point for a Class of Nonlinear Difference Equation

Chang-You Wang,Qi-Hong Shi,Fei Gong,Shu Wang,Lin-Rui Li

doi:10.1155/2009/214309

Abstract

We study the asymptotic behavior of the solutions for the following nonlinear difference equation where the initial conditions are arbitrary nonnegative real numbers, are nonnegative integers, , and are positive constants. Moreover, some numerical simulations to the equation are given to illustrate our results.

Highlights

Difference equations appear naturally as discrete analogues and in the numerical solutions of differential and delay differential equations having applications in biology, ecology, physics, and so forth 1
The study of nonlinear difference equations is of paramount importance in their own field but in understanding the behavior of their differential counterparts
There has been a lot of work concerning the globally asymptotic behavior of solutions of rational difference equations 2–6

Summary

Introduction

Difference equations appear naturally as discrete analogues and in the numerical solutions of differential and delay differential equations having applications in biology, ecology, physics, and so forth 1. There has been a lot of work concerning the globally asymptotic behavior of solutions of rational difference equations 2–6. Elabbasy et al 7 investigated the global stability and periodicity of the solution for the following recursive sequence: xn 1 axn. In 8 Elabbasy et al investigated the global stability, boundedness, and the periodicity of solutions of the difference equation: xn 1 αxn Axn βxn−1 Bxn−1 γ xn−2 Cxn−2. Yang et al 9 investigated the global attractivity of equilibrium points and the asymptotic behavior of the solutions of the recursive sequence: xn 1 axn−1 bxn−2 c dxn−1xn−2. The purpose of this paper is to investigate the global attractivity of the equilibrium point, and the asymptotic behavior of the solutions of the following difference equation xn 1. Some numerical simulations to the equation are given to illustrate our results. Some numerical simulations are given to illustrate our theoretical analysis

Some Preliminary Results

The Main Results and Their Proofs

Blj m0

Numerical Simulations