ON THE RECURSIVE SEQUENCE E. CAMOUZIS, R. DEVAULT, AND G. PAPASCHINOPOULOS Received 12 January 2004 and in revised form 29 May 2004 Our aim in this paper is to investigate the boundedness, global asymptotic stability, and periodic character of solutions of the difference equation xn+1 = (γxn−1 + δxn−2 )/(xn + xn−2 ), n = 0,1,., where the parameters γ and δ and the initial conditions are positive real numbers. 1. Introduction Using an appropriate change of variables, we have that the recursive sequence xn+1 = (γxn−1 + δxn−2 )/(xn + xn−2 ) is equivalent to the difference equation xn+1 = γxn−1. | ON THE RECURSIVE SEQUENCE E. CAMOUZIS R. DEVAULT AND G. PAPASCHINOPOULOS Received 12 January 2004 and in revised form 29 May 2004 Our aim in this paper is to investigate the boundedness global asymptotic stability and periodic character of solutions of the difference equation xn 1 yxn-1 8xn-2 xn xn-2 n 0 1 . where the parameters y and 8 and the initial conditions are positive real numbers. 1. Introduction Using an appropriate change of variables we have that the recursive sequence xn 1 yxn-1 8xn-2 xn xn-2 is equivalent to the difference equation yxn- 1 xn-2 _ . xn 1 . 1 . 2 n 0 1 . xn xn-2 For all values of the parameter y has a unique positive equilibrium x y 1 2. When 0 y 1 the positive equilibrium x is locally asymptotically stable. In the case where y 1 the characteristic equation of the linearized equation about the positive equilibrium x 1 has three eigenvalues one of which is -1 and the other two are 0 and 1 2. In addition when y 1 possesses infinitely many period-two solutions of the form a a 2a - 1 a a 2a - 1 . for all a 1 2. When y 1 the equilibrium x is hyperbolic. The investigation of has been posed as an open problem in 1 2 . In this paper we will show that when 0 y 1 the interval y 1 is an invariant interval for and that every solution of falling into this interval converges to the positive equilibrium x. Furthermore we will show that when y 1 every positive solution xn f -2 of which is eventually bounded from below by 1 2 converges to a not necessarily prime period-two solution. Finally when y 1 we will prove that possesses unbounded solutions. We also pose some open questions for . We say that a solution xn f -k of a difference equation is bounded and persists if there exist positive constants P and Q such that P xn Q for n -k -k 1 . Copyright 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005 1 2005 31-40 DOI 32 On the recursive sequence 2. The case Y 1 In this .
