Asymptotic estimates and exponential stability for higher-order monotone difference equations

Abstract

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.

Representation of solutions of linear discrete systems with constant coefficients and pure delay

“…Moreover results obtained can be useful in investigation of such asymptotic problems as describing the asymptotic behavior of solutions and the investigation concerning boundedness, convergence, or stability of solutions. With the aid of different methods (Liapunov type technique and retract principle), some of these problems have been investigated, for example, in the recent papers [2][3][4][5][6][7][8][9][11][12][13].…”

New stability conditions for linear difference equations using Bohl–Perron type theorems

“…where g l (n) ≤ n. Denote by Y (n, k) the fundamental function of equation (18). Corollary 3 Suppose (13) and at least one of the following conditions hold:…”

Stability of a delay difference system

“…A lot of explicit stability conditions for scalar equations (see, e.g., [2,11,4,12]) can be translated to the language of linear systems (1.1). However, we will indicate a new phenomenon.…”