Báo Cáo Báo cáo hóa học ASYMPTOTIC ESTIMATES AND EXPONENTIAL STABILITY FOR HIGHER-ORDER MONOTONE DIFFERENCE

ASYMPTOTIC ESTIMATES AND EXPONENTIAL STABILITY FOR HIGHER-ORDER MONOTONE DIFFERENCE EQUATIONS ´ EDUARDO LIZ AND MIHALY PITUK Received 21 May 2004 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 ,xnư1 , .,xnưk , 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 , .,xkư1 , x = x0 ,x1 , .,xk ∈