Báo Cáo Báo cáo hóa học RATE OF CONVERGENCE OF SOLUTIONS OF RATIONAL DIFFERENCE EQUATION OF SECOND ORDER

RATE OF CONVERGENCE OF SOLUTIONS OF RATIONAL DIFFERENCE EQUATION OF SECOND ORDER S. KALABUSIC AND M. R. S. KULENOVIC Received 13 August 2003 and in revised form 7 October 2003

We investigate the rate of convergence of solutions of some special cases of the equation xn+1 = (α + βxn + γxnư1 )/(A + Bxn + Cxnư1 ), n = 0,1, ., 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 Poincar´ ’s e theorem and an improvement of Perron’s theorem.

1. Introduction and preliminaries We investigate the rate of convergence of solutions of some special types of the secondorder rational difference equation xn+1 = α + βxn + γxnư1 , A + Bxn + Cxnư1 n = 0,1, .,

(1.1) where the parameters α, β, γ, A, B, and C are positive real numbers and the initial conditions xư1 , x0 are arbitrary nonnegative real numbers. Related nonlinear