Estimates for the Norms of Solutions of Delay Difference Systems

We derive explicit stability conditions for delay difference equations in C n (the set of n complex vectors) and estimates for the size of the solutions are derived. Applications to partial difference equations, which model diffusion and reaction processes, are given. 1. Introduction. Stability of systems of difference equations with delays has been discussed by many authors, for example, see GiL' and Cheng [7], Zhang [11], Elaydi and Zhang [5], Pituk [10], Agarwal [1], and the references therein. In the stability literature, we 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 [8], and Agarwal and Wong [2]. By this method many very strong results are obtained. But finding Lyapunov's functionals is usually difficult. In this paper, 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 [6, Theorem 1.2.1] we derive explicit stability

In the stability literature, we 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 [8], and Agarwal and Wong [2].By this method many very strong results are obtained.But finding Lyapunov's functionals is usually difficult.
In this paper, 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 [6,Theorem 1.2.1] we derive explicit stability conditions.Further, we apply our main result to an abstract partial difference equation which models reaction and diffusion processes.

Preliminary facts.
Let C n be the set of n complex vectors endowed with a norm • .Let A be an n × n-complex matrix.Consider in C n the equation where p ≥ 1, and σ 1 ,σ 2 ,...,σ p are nonnegative integers such that 0 , and Z + is the set of nonnegative integers, f j maps C np into C n , for j = 0, 1, 2,... .We consider (2.1) subject to the initial conditions It is assumed that there are nonnegative sequences q l (l = 1, 2, 3,...,p) such that and m is a fixed positive real number.
Unlike differential equations, discrete equations with the given initial conditions always have a solution.
In order to establish our main result, we use the following discrete Gronwall type inequality.
Theorem 2.1 (see [9]).Assume that ) provided that Let λ 1 (A),...,λ n (A) be the eigenvalues of A, including their multiplicities.We will make use of the following quantity hereafter (see [6, Chapter 1]): where There are a number of properties of g(A) which are useful (see [6]).Here, we note (2.10) To facilitate the description of our main result, we adopt the convention that 0! = 1, 0 0 = 1, and that empty sums are zero.Further, the binomial coefficient C i j is given by As normal, but we also adopt the convention that C i j = 0 when j < 0 or j > i.We define (2.12) Note that , where ρ(A) is the spectral radius of A.
Case 1.If 0 < m ≤ 1 and L ≤ 1, then by (2.4) and Theorem 2.1(a) we have Thus, establishing that the solution u j is bounded for j = −σ p , −σ p + 1,...,0, and lim j→∞ u j = 0 whenever u 0 < δ, for δ > 0 small enough.Indeed, for all such a τ 0 , we have Consequently, for all τ such that τ 0 ≤ R, we have Therefore, we have the boundedness of the solution u j of (2.1), subject to the initial conditions (2.2), concluding the proof.

Application.
In this section, we illustrate our main result by considering an abstract partial difference equation, which models reaction and diffusion processes (see Cheng and Medina [3]).

Consider a simple three-level discrete reaction-diffusion equation of the form
defined on the set where q l (l = 1, 2,...,p) are nonnegative real sequences; u i,j are complex sequences, p ≥ 1; a, b, c are real numbers; and 0 For the sake of simplicity, Dirichlet boundary conditions of the form will be imposed.
Proof.The proof is a direct consequence of Theorem 3.1.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
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.
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