Stability of Nonhyperbolic Equilibrium Solution of Second Order Nonlinear Rational Difference Equation

S Atawna,E S Ismail,R Abu-Saris,I Hashim

doi:10.1155/2015/486985

Abstract

This is a continuation part of our investigation in which the second order nonlinear rational difference equation xn+1=(α+βxn+γxn-1)/(A+Bxn+Cxn-1), n=0,1,2,…, where the parameters A≥0 and B, C, α, β, γ are positive real numbers and the initial conditions x-1, x0 are nonnegative real numbers such that A+Bx0+Cx-1>0, is considered. The first part handled the global asymptotic stability of the hyperbolic equilibrium solution of the equation. Our concentration in this part is on the global asymptotic stability of the nonhyperbolic equilibrium solution of the equation.

Highlights

In part 1 of this investigation [1], we have established the global stability of the hyperbolic equilibrium solution of the second order rational difference equation: xn+1
Where the parameters A ≥ 0 and B, C, α, β, γ are positive real numbers and the initial conditions x−1, x0 are nonnegative real numbers such that A + Bx0 + Cx−1 > 0. Our aim in this part is on the global attractivity of the nonhyperbolic equilibrium solution of (1)
Equation (4) has a unique positive equilibrium given by 1 + p − z + √(1 + p − z)2 + 4r (q + 1)

Summary

Introduction

In part 1 of this investigation [1], we have established the global stability of the hyperbolic equilibrium solution of the second order rational difference equation: xn+1. A systematic study of the second order rational difference equation of form (1), where the parameters α, β, γ, A, B, C and the initial conditions x−1, x0 are nonnegative real numbers, was considered in the monograph of Kulenovic and Ladas [4]. They presented the known results up to 2002. Kulenovic and Ladas [4] derived several ones on the boundedness, the global stability, and the periodicity of solutions of all rational difference equations of form (1) They posed several open problems and conjectures related to this equation and its functional generalization.

Preliminaries

Existence of Nonhyperbolic Equilibrium Solution

Global Stability Analysis

Numerical Examples

Findings

Conclusion