Estimates for the Norms of Solutions of Difference Systems with Several Delays

We derive explicit stability conditions for time-dependent difference equations with several delays in C n (the set of n complex vectors) and estimates for the size of the solutions. The growth rates obtained here are not necessarily decay rates. 1. Introduction. Stability of systems of difference equations with delays has been discussed by many authors, for example, see Gil' and Cheng [6], Zhang [11], Elaydi and Zhang [5], Pituk [10], Agarwal [1], and the references therein. In the stability literature, one can find two major trends; stability using the first approximation Lyapunov method and the direct Lyapunov functional method. For this latter trend, see Zhang and Chen [12], Crisci et al. [4], Lakshmikantham and Trigiante [7], and Agarwal and Wong [2]. By this method many very strong results are obtained. But finding Lyapunov functionals is usually a difficult task. In this note, we consider a class of perturbed difference equations with several delays and, by means of a Gronwall inequality and the recent estimates for the powers A k of a

In the stability literature, one can find two major trends; stability using the first approximation Lyapunov method and the direct Lyapunov functional method. For this latter trend, see Zhang and Chen [12], Crisci et al. [4], Lakshmikantham and Trigiante [7], and Agarwal and Wong [2]. By this method many very strong results are obtained. But finding Lyapunov functionals is usually a difficult task.
In this note, we consider a class of perturbed difference equations with several delays and, by means of a Gronwall inequality and the recent estimates for the powers A k of a constant matrix A established in Corduneanu [3], we derive explicit stability conditions. Further, we suppose that the unperturbed linear difference equations have a bounded growth. Actually, this work is an extension of Medina [8] to time-dependent difference equations with several delays.
In order to establish our main result, we will use the following discrete Gronwall-type inequality.
Theorem 2.1 [9]. Assume that provided that It is assumed that the unperturbed linear difference equation has a bounded growth, that is, there exist real constants γ ≥ 1 and α > 0 such that where Φ(k, l) = k−1 j=l A j , k > l, is the fundamental matrix solution of (2.9).
3. Main results. Now, we are in a position to establish our main results pertaining to the bounded growth and the zero convergence properties of the solutions of (2.1) subject to the conditions (2.2).
Case 1. If 0 < m ≤ 1 and C ≤ 1, then by Theorem 2.1(a) it follows that (3.9) and the proof of Case 1 is complete.
Remark 3.4. If A k ≡ A is a constant matrix, whose spectral radius is less than 1, then the zero solution of (2.9) is uniformly asymptotically stable. However, this result cannot be extended to nonautonomous equations (see [7,Theorem 4.4.1]).

Special cases.
If the system (2.9), has slowly varying coefficients, then the condition (2.10) concerning growth of the solutions can be avoided in the case On the other hand, Corduneanu [3] established that for any constant matrix A there exists a constant Γ ≥ 1, independent of the integers j = 0, 1, 2,... such that A j ≤ Γ ρ j (A), j = 0, 1,..., (4.3) where ρ(A) is the spectral radius of A.
In particular, if A = (a ij ) is a triangular constant matrix, then Γ = 1. Consider in C n the equation where A j (j = 0, 1,...) are n × n-complex matrices and g j , u j are vectors in C n .
Proof. Rewrite (4.4) as with a fixed integer l. The variation of parameters formula yields It follows from (4.2) and (4.3) that  (4.14) Now we are in a position to formulate the next result of this paper. Proof. It can be proved in a similar way to Theorem 4.1, so we will omit the proof.