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.     

Growth of meromorphic solutions of linear difference equations

In this paper, the authors continue to study the growth of meromorphic solutions of homogeneous or non-homogeneous linear difference equations with entire coefficients, and obtain some results which are improvement and extension of previous results in Chiang and Feng (2008) [7] and Laine and Yang (2007) [19]. Examples are also given to illustrate the sharpness of our results.     

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.     

非齐次线性差分方程的Cm解[英文]

设c0,c1,…,cn均为实的常数，F（x)是个从R到R的C^m映射。本文讨论了非齐次线性差分方程∑i=1ncif(x i)=F(x)的C^m(m≥0）的存在性和唯一性。     

Growth and zeros of meromorphic solution of some linear difference equations

In this paper, we study growth and zeros of linear difference equations

Pn(z)f(z+n)+?+P₁(z)f(z+1)+P₀(z)f(z)=F(z)     

The existence of continuous invariants for linear difference equations

We study real continuous invariants for systems of linear difference equations. We shall prove a conjecture by Ladas about the existence of such invariants. In fact, necessary and sufficient conditions on existence of such invariants will be established. The invariants will be constructed when they exist.     

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.     

Remarks on oscillation and nonoscillation for second-order linear difference equations

In this remark, we shall show the main results of the earlier work [W.T. Li, S.S. Cheng, Remarks on two recent oscillation theorems for second-order linear difference equations, Appl. Math. Lett. 16 (2003) 161–163] are incorrect.     

Stability for non-hyperbolic fixed points of scalar difference equations

We give some new criteria to determine the stability of a non-hyperbolic fixed point of the scalar difference equation

where and f is a sufficient smooth function. Our results are based on higher order derivative at a fixed point of .     

Asyptotics for linear difference equations I:basic theory

We present here a unified treatment of asymptotic theory of linear difference equations. This is based on anadapted theory of discrete dichotomy. The obtained results narrow the gap between Poincarés Theorem and (the discrete analoue of) Levinson's Theorem.     

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.     

Exponential dichotomy and dichotomy radius for difference equations

The aim of this paper is to provide a new approach concerning the characterization of exponential dichotomy of difference equations by means of admissible pair of sequence spaces. We classify the classes of input and output spaces, respectively, and deduce necessary and sufficient conditions for exponential dichotomy applicable for a large variety of systems. By an example we show that the obtained results are the most general in this topic. As an application we deduce a general lower bound for the dichotomy radius of difference equations in terms of input-output operators acting on sequence spaces which are invariant under translations.     

Eigenvalue comparisons for boundary value problems for second order difference equations

We consider the eigenvalue problems for boundary value problems of second order difference equations

(1)

and

(2)

Comparison results for the eigenvalues of the problem (1) and the problem (2) are established.     

On the solutions of a class of difference equations

In this note we investigate the solutions of a class of difference equations and prove that Conjectures 4.8.2, 4.8.3, 5.4.6 and 6.10.3 proposed by M. Kulenovic and G. Ladas in [M. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures, Chapman & Hall/CRC Press, 2002] are true.     

On a class of third-order nonlinear difference equations

This paper studies the boundedness character of the positive solutions of the difference equation

     

Average implicit linear difference scheme for generalized Rosenau-Burgers equation

In this paper, a linear three-level average implicit finite difference scheme for the numerical solution of the initial-boundary value problem of Generalized Rosenau-Burgers equation is presented. Existence and uniqueness of numerical solutions are discussed. It is proved that the finite difference scheme is convergent in the order of O(τ² + h²) and stable. Numerical simulations show that the method is efficient.     

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.