Rate of convergence of solutions of rational difference equation of second order

S Kalabušić,Mrs Kulenović

doi:10.1155/s168718390430806x

Abstract

We investigate the rate of convergence of solutions of some special cases of the equation , with positive parameters and nonnegative initial conditions. We give precise results about the rate of convergence of the solutions that converge to the equilibrium or period-two solution by using Poincare's theorem and an improvement of Perron's theorem.

Highlights

Introduction and preliminariesWe investigate the rate of convergence of solutions of some special types of the secondorder rational difference equation xn+1 = α + βxn A + Bxn + γxn−1 + Cxn−1, n = 0, 1, . . ., (1.1)where the parameters α, β, γ, A, B, and C are positive real numbers and the initial conditions x−1, x0 are arbitrary nonnegative real numbers.Related nonlinear second-order rational difference equations were investigated in [2, 5, 6, 7, 8, 9, 10]
We investigate the rate of convergence of solutions of some special cases of the equation xn+1 = (α + βxn + γxn−1)/(A + Bxn + Cxn−1), n = 0, 1, . . . , with positive parameters and nonnegative initial conditions
Equation (1.3) was studied in detail in [7, 10], where we have found the region of parameters for which the equilibrium is globally asymptotically stable and the region where the equation has a unique period-two solution which is locally asymptotically stable

Summary

Introduction

We investigate the rate of convergence of solutions of some special cases of the equation xn+1 = (α + βxn + γxn−1)/(A + Bxn + Cxn−1), n = 0, 1, . This equation was considered in [7], where the method of full limiting sequences was used to prove that the equilibrium is globally asymptotically stable for all values of parameters B and C. This equation was considered in detail in [7, 10], where it was proved that the equilibrium is globally asymptotically stable for values of parameters p and q that satisfy p

Results

Conclusion