Rate of convergence of solutions of rational difference equation of second order
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e 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´e’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 secondorder rational difference equation xn+1 =
α + βxn + γxn−1 , A + Bxn + 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]. The study of these equations is quite challenging and is in rapid development. In this paper, we will demonstrate the use of Poincar´e’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 (1.1), namely, for n = 0,1,..., B C + , xn xn−1 pxn + xn−1 xn+1 = , qxn + xn−1 pxn + xn−1 xn+1 = , q + xn−1 xn+1 =
Copyright © 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:2 (2004) 121–139 2000 Mathematics Subject Classification: 39A10, 39A11 URL: http://dx.doi.org/10.1155/S168718390430806X
(1.2) (1.3) (1.4)
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 (1.2), (1.3), and (1.4) and found the region in the space of parameters where the equilibrium solution is globally asymptotically stable. In this paper, we will precisely determine the rate of convergence of all solutions in this region by using Poincar´e’s theorem and an improvement of Perron’s theorem. We will show that the asymptotics of solutions that converge to the equilibrium depends on the character of the roots of the characteristic equation of the linearized equation evaluated at the equilibrium. The results on asymptotics of (1.2), (1.3), and (1.4) will show all the complexity of the asymptotics of the general equation (1.1). Here we give some necessary definitions and results that we will use later. Let I be an interval of real numbers and let f ∈ C 1 [I × I,I]. Let x¯ ∈ I be an equilibrium point of the difference equation
xn+1 = f xn ,xn−1 ,
n = 0,1,...,
(1.5)
∂f ¯ (¯x, x) ∂v
(1.6)
¯ that is, x¯ = f (¯x, x). Let s=
∂f ¯ (¯x, x), ∂u
t=
denote the partial derivatives of f (u,v) evaluated at an equilibrium x¯ of (1.5). Then the equation yn+1 = syn + t yn−1 ,
n = 0,1,...,
(1.7)
is called the linearized equation associated with (1.5) about the equilibrium point x¯ . Theorem 1.1 (linearized stability). (a) If both roots of the qu
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