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ASYMPTOTIC ESTIMATES AND EXPONENTIAL STABILITY FOR HIGHER-ORDER MONOTONE DIFFERENCE EQUATIONS ´

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 ,xn−1. | 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 Xn-1 . Xn-k n e 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 Tn 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 e Rk 1. 1.2 As usual Tn denotes the nth iterate of T for n 1 and T0 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 Tn 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 C x X0 X1 . Xk e Rk 1 I Xk 0 Xi ỊLXi 1 i 0 1 . k - 1 1.4 Copyright 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005 1 2005 41-55 DOI 10.1155 ADE.2005.41 42 Monotone difference equations where p 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 .