© Hindawi Publishing Corp. ACCURATE SOLUTION ESTIMATES FOR VECTOR DIFFERENCE EQUATIONS

Accurate estimates for the norms of the solutions of a vector difference equation are derived. They give us stability conditions and bounds for the region of attraction of the stationary solution. Our approach is based on estimates for the powers of a constant matrix. We also discuss applications of our main results to partial reaction-diffusion difference equations and to a Volterra difference equation.

1. Introduction.Let C n be the set of n-complex vectors endowed with the Euclidean norm • .Consider the equation where A is an n × n complex matrix and f k : C n → C n are given functions.
A well-known result of Perron which dates back to 1929 (see [8], [12, page 270], and [10,Theorem 9.14]) states that (1.1) is asymptotically stable provided that A is stable (i.e., the spectral radius ρ(A) of A is less than 1) and f k (x) = f (x) = o( x ).Clearly, this kind of results for the perturbed equation (1.1) is purely local.It gives no information about the size of the region of asymptotic stability and the norms of solutions.
In this paper, we derive accurate estimates for the norms of solutions.They give us stability conditions for (1.1) and bounds for the region of attraction of the stationary solution.Our approach is based on recent estimates for the powers of a constant matrix, namely, Corduneanu [3] established that for any constant matrix A, there exists a constant N ≥ 1, independent of the integers k = 0, 1, 2,..., such that In particular, if A = (a ij ) is a triangular constant matrix, then N = 1.On the other hand, Gil' [5, Theorem 1.2.1] established a very sharp estimate for the powers A k of a constant matrix A. In order to establish Theorems 2.1 and 2.3, we use a method introduced by Gil' and Cheng [7] which deals with perturbed linear discrete dynamical systems.
We also discuss the applications of our main results to a partial reactiondiffusion difference equation and to the discrete analogous of an integrodifferential equation, respectively.

Main results.
For a positive number r ≤ ∞, denote the ball B r = {x ∈ C n : x ≤ r } and assume that there are constants q, µ ≥ 0 such that and put θ = N/(1 − ρ(A)).Now, we are in a position to formulate the main result of this paper.
Proof.By inductive arguments, it is easy to check that the unique solution {x k } ∞ k=0 of (1.1) under the initial condition x 0 is given by There are two cases to consider: r = ∞ and r < ∞.
Assuming first that r = ∞, hence Furthermore, since 0 < ρ(A) < 1, we obtain In view of (2.7) and the condition qθ < 1, it follows that Hence, it follows that (2.12) Next, we will consider the case r < ∞.Define the function satisfies the inequality From this, we infer that x k = x k for k = 0, 1, 2,..., and therefore (2.3) is satisfied, concluding the proof.
(b) If q = 0, then, by (2.4), every solution of (1.1) with the initial vector x 0 satisfying N x 0 +µθ ≤ r is bounded. (2.17) By using arguments similar to those in Theorem 2.1, the result follows, thus we will omit the proof.

Applications.
In this section, we will illustrate our main results by considering a partial difference equation and the discrete analogous of an integrodifferential equation, respectively.
Assume that the side conditions u (j) The existence and uniqueness of solutions of that problem is obvious provided the functions f j are well defined.
(3.5) Thus, we can write the considered problem as (1.1) with f j (x) = F j (x) + G j .Assume that there are nonnegative constants q 1 and µ 1 such that In addition, assume that Hence, condition (2.1) holds with µ = µ 1 + µ 2 .
We want to point out that if ac > 0, then the spectral radius ρ(A) is equal to |b|+2 √ ac cos(π /(n + 1)) and when ac < 0, then Then, as a direct consequence of Theorem 2.1, we get the following theorem.
We now consider the analogous of the integrodifferential equation Consider the discrete equation where A is an n × n complex matrix and f j : C n → C n (j = 0, 1, 2,...) are given functions, satisfying We are now in a position to establish the next theorem.
Proof.The variation of parameters formula yields
Assuming first that (3.15) is valid for r = ∞, hence Thus, following the lines of the proof of Theorem 2.3, the result follows.

Call for Papers
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This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.