tailieunhanh - RATE OF CONVERGENCE OF SOLUTIONS OF RATIONAL DIFFERENCE EQUATION OF SECOND ORDER ˇ ´ ´ S.

RATE OF CONVERGENCE OF SOLUTIONS OF RATIONAL DIFFERENCE EQUATION OF SECOND ORDER ˇ ´ ´ 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 + γxn−1 )/(A + Bxn + Cxn−1 ), 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. | 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 a fxn yxn-1 A Bxn Cxn-1 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 Poincare s 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 second-order rational difference equation a fxn yxn-1 A Bxn Cxn-1 n 0 1 . where the parameters a f y 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 . The study of these equations is quite challenging and is in rapid development. In this paper we will demonstrate the use of Poincare s theorem and an improvement of Perron s theorem to determine the precise asymptotics of solutions that converge to the equilibrium. We will concentrate on three special cases of namely for n 0 1 . xn 1 B xn xn-1 pxn xn-1 xn 1 _ _ qxn xn-1 pxn xn-1 xn 1 q xn-1 Copyright 2004 Hindawi Publishing Corporation AdvancesinDifferenceEquations2004 2 2004 121-139 2000 Mathematics Subject Classification 39A10 39A11 URL http S168718390430806X 122 Rate of convergence of rational difference equation where all the parameters are assumed to be positive and the initial conditions X-1 X0 are arbitrary positive real numbers. In 7 the second author and Ladas obtained both local and global stability results for and and found the region in the space of parameters where the equilibrium solution is globally asymptotically stable. In this paper