Asymptotic estimates and exponential stability for higher-order monotone difference equations
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Asymptotic estimates are established for higher-order scalar difference equations and inequalities the right-hand sides of which generate a monotone system with respect to the discrete exponential ordering. It is shown that in some cases the exponential estimates can be replaced with a more precise limit relation. As corollaries, a generalization of discrete Halanay-type inequalities and explicit sufficient conditions for the global exponential stability of the zero solution are given. 1. Introduction Consider the higher-order scalar difference equation
xn+1 = f xn ,xn−1 ,...,xn−k ,
n ∈ N = {0,1,2,...},
(1.1)
where k is a positive integer and f : Rk+1 → R. With (1.1), we can associate the discrete dynamical system (T n )n≥0 on Rk+1 , where T : Rk+1 → Rk+1 is defined by
T(x) = f (x),x0 ,x1 ,...,xk−1 ,
x = x0 ,x1 ,...,xk ∈ Rk+1 .
(1.2)
As usual, T n denotes the nth iterate of T for n ≥ 1 and T 0 = I, the identity on Rk+1 . It follows by easy induction on n that if (xn )n≥−k is a solution of (1.1), then
xn ,xn−1 ,...,xn−k = T n x0 ,x−1 ,...,x−k ,
n ≥ 0.
(1.3)
Therefore, the dynamical system (T n )n≥0 contains all information about the behavior of the solutions of (1.1). In a recent paper [7], motivated by earlier results for delay differential equations due to Smith and Thieme [13] (see also [12, Chapter 6]), Krause and the second author have introduced the discrete exponential ordering on Rk+1 , the partial ordering induced by the convex closed cone
Cµ = x = x0 ,x1 ,...,xk ∈ Rk+1 | xk ≥ 0, xi ≥ µxi+1 , i = 0,1,...,k − 1 , Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:1 (2005) 41–55 DOI: 10.1155/ADE.2005.41
(1.4)
42
Monotone difference equations
where µ ≥ 0 is a parameter. In [7], it has been shown that T is monotone (order preserving) under appropriate conditions on f . As a consequence of monotonicity, necessary and sufficient conditions have been given for the boundedness of all solutions and for the local and global stability of an equilibrium of (1.1) (see [7, Section 4]). In this paper, we give further consequences of the monotonicity of T for (1.1) and for the corresponding difference inequality
yn+1 ≤ f yn , yn−1 ,..., yn−k ,
n ≥ 0,
(1.5)
under the additional assumption that the nonlinearity f is positively homogeneous (of degree one) on the generating cone Cµ , that is, f (λx) = λ f (x) for λ ≥ 0, x ∈ Cµ .
(1.6)
An example of (1.1) with property (1.6) is the max type difference equation xn+1 =
k i=0
Ki xn−i + b max xn ,xn−1 ,...,xn−r ,
(1.7)
where k and r are positive integers and the coefficients Ki and b are constants. For other examples of higher-order difference equations with a positively homogeneous right-hand side, see, for example, [6]. Using the monotonicity of T and a simple comparison theorem, we give upper exponential estimates for the solutions of (1.5) in terms of the largest positive root of the characteristic equation
λk+1 = f λk ,λk−1 ,...,1 .
(1.8)
As a corollary for the difference
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