On asymptotic equivalence of perturbed linear systems of differential and difference equations

Standard results on asymptotic integration of systems of linear differential equations give sufficient conditions which imply that a system is strongly asymptotically equivalent to its principal diagonal part. These involve certain dichotomy conditions on the diagonal part as well as growth conditions on the off-diagonal perturbation terms. Here, we study perturbations with a triangularly-induced structure and see that growth conditions can be substantially weakened. In addition, we give results for not necessarily triangular perturbations which in some sense “interpolate” between the classical theorems of Levinson and Hartman-Wintner. Some analogous results for systems of linear difference equations are also given.     

Asymptotic solutions and error estimates for linear systems of difference and differential equations

Classical results concerning the asymptotic behavior solutions of systems of linear differential or difference equations lead to formulas containing factors that are asymptotically constant, i.e., k+o(1) as t tends to infinity. Here we are interested in more precise information about the o(1) terms, specifically how they depend precisely on certain perturbation terms in the equation. Results along these lines were given by Gel'fond and Kubenskaya for scalar difference equations and we will both extend and generalize one of them as well as provide some corresponding results for differential equations.     

On the behavior of the solutions for linear autonomous neutral delay difference equations

A class of linear autonomous neutral delay difference equations is considered, and some new results on the asymptotic behavior and the stability are given, via a positive root of the correspondng characteristic equation.     

Transforms from differential equations to difference equations and vice-versa applied to computer control systems

This letter derives the transform relationship between differential equations to difference equations and vice-versa, applied to computer control systems. The key is to obtain the rational fraction transfer function model of a time-invariant linear differential equation system, using the Laplace transform, and to obtain the impulse transfer function model of a time-invariant linear difference equation, using the shift operator. Finally, we find the discrete-time models of the first-order, second-order and third-order systems from their continuous-time models and vice-versa and find the mapping relationship between the coefficients of discrete-time models and the continuous-time models using the bilinear transform. An example is provided to demonstrate the proposed model transform methods.     

On a nonautonomous difference equation with bounded coefficient

In this paper we investigate the boundedness, the persistence and the attractivity of the positive solutions of the nonautonomous difference equation

     

Oscillations of the system of linear nonautonomous difference equations

We consider the linear nonautonomous system of difference equations x_n+1−x_n+P(n)x_n−k=0, n=0,1,2,… , where k∈Z, P(n)∈R^rxr. We obtain sufficient conditions for the system to be oscillatory. The conditions based on the eigenvalues of the matrix coefficients of the system.     

Global behavior of a three-dimensional linear fractional system of difference equations

We investigate the global asymptotic behavior of solutions of the system of difference equations

where the parameters a, b, c, d, e, and f are in (0,∞) and the initial conditions x₀, y₀, and z₀ are arbitrary non-negative numbers. We obtain some global attractivity results for the positive equilibrium of this system for different values of the parameters.     

Asymptotic properties of solutions to linear non-autonomous neutral differential equations

In this article we study asymptotic properties of solutions to first order linear neutral differential equations with variable coefficients and constant delays. Results are stated in terms of the solution to a characteristic equation. By doing this, we extend some of the results obtained for delay equations in [J.G. Dix, Ch.G. Philos, I.K. Purnaras, An asymptotic property of solutions to linear non-autonomous delay differential equations, Electron. J. Differential Equations 2005 (2005) 1-9] to neutral equations.     

Asymptotic behaviour of the solutions of a nonautonomous linear delay difference system

It is proved under appropriate assumptions that the solutions of a linear system of nonautonomous delay difference equations have finite limit at infinity. The results are based on a transformation of the delay difference system into a first-step recursion, where the companion matrices are well treatable from our point of view. Our theory is illustrated by examples, including a class of linear delay difference equations with unbounded coefficients.     

On asymptotic equivalence of linear and quasilinear difference equations

The asymptotic equivalence of systems of difference equations of linear and quasilinear type is investigated. The first result on the asymptotic equivalence of linear systems is a discrete analog of an improved version of the Levinson's well-known theorem on asymptotic equivalence of linear differential equations, while the second one providing conditions for asymptotic equivalence of linear and quasilinear systems is related to that of Yakubovich in differential equations case.     

Asymptotic behavior of linear fractional stochastic differential equations with time-varying delays

In this paper we give a sufficient condition for the exponential asymptotic behavior of solutions of a general class of linear fractional stochastic differential equations with time-varying delays. Our obtained results allow us to employ the theories developed for the deterministic systems and to illustrate this, some examples are provided.     

Asymptotic behavior of solutions of difference equations in Banach spaces

In this paper we consider the first order difference equation

and give necessary and sufficient conditions so that there exist solutions which are asymptotically constant. These results generalize those given earlier by Popenda and Schmeidel. As an application we give necessary and sufficient conditions for the second order difference equation

to have asymptotically constant solutions.

     

Asymptotic behaviour and stability to linear non-autonomous neutral delay difference equations

This paper is concerned with the asymptotic behaviour and the stability of a class of linear neutral delay difference equations with variable coefficients and constant delays. Via an appropriate solution of the so-called generalized characteristic equation, an asymptotic criterion and a stability result are established.     

Trichotomy of a system of two difference equations

In this paper we study the boundedness and the asymptotic behavior of the positive solutions of the system of difference equations