Asymptotic behaviour of nonlinear difference equations

In this paper, we investigate the growth/decay rate of solutions of a class of nonlinear Volterra difference equations. Our results can be applied for the case when the characteristic equation of an associated linear difference equation has complex dominant eigenvalue with higher than one multiplicity. Illustrative examples are given for describing the asymptotic behaviour of solutions in a class of linear difference equations and in several discrete nonlinear population models.     

On the asymptotic integration of systems of linear differential equations with oscillatory decreasing coefficients

A system of linear differential equations with oscillatory decreasing coefficients is considered. The coefficients have the formt ^-α a(t), α > 0 wherea(t) is a trigonometric polynomial with an arbitrary set of frequencies. The asymptotic behavior of the solutions of this system ast → ∞ is studied. We construct an invertible (for sufficiently larget) change of variables that takes the original system to a system not containing oscillatory coefficients in its principal part. The study of the asymptotic behavior of the solutions of the transformed system is a simpler problem. As an example, the following equation is considered:

Asymptotic behaviour of solutions of the difference Schrödinger equation

We use the method of averaging and the discrete analogue of Levinson's theorem to construct the asymptotics for solutions of the difference Schrödinger equation. Moreover, we present the general form for the averaging change of variable.     

The remainder term in asymptotic summation of linear difference systems

Quantitative estimates for the remainder terms in the asymptotic summation of linear difference systems are derived. For example, it is shown that any decay in excess of summability is passed on to the remainder.     

中立型抛物方程解的振动性与渐近性

本文研究了,具有分布滞量的非线性中立型抛物方程（E）解的振动性与渐近性,我们的结果推广了Bainov和Petrov[1]对方程的一些结果。     

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 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 behavior of solutions of linear homogeneous second-order difference equations

We analyze and study the asymptotic behavior (asn→∞) of the general solutionx _n of the equationx _n+2 =Ax _n +Bx _n+1 ,A≠0,n=0,1,2,..., for various possible values of coefficients and initial data. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 211–215, August, 99.     

Asymptotic behavior of a linear difference equation with continuous time

We consider a class of linear homogeneous difference equations with constant coefficients and commensurate delays. We prove an asymptotic formula for the solutions as t → ∞ under the assumption of the existence of a simple “dominant” real characteristic value. Research supported by FONDECYT Grant, Chile, No. 1.070.980.     

The generalized Cauchy problem for singularly perturbed impulsive systems in the critical case

We construct an asymptotic expansion for solutions to nonlinear singularly perturbed systems of impulsive differential equations. We successively determine all terms of the asymptotic expansion by means of pseudoinverse matrices and orthoprojections.     

Asymptotic behaviour of the energy for electromagnetic systems with memory

Some linear evolution problems arising in the theory of hereditary electromagnetism are considered here. Making use of suitable Liapunov functionals, existence of solutions as well as asymptotic behaviour, are determined for rigid conductors with electric memory. In particular, we show the polynomially decay of the solutions, when the memory kernel decays exponentially or polynomially. Copyright © 2004 John Wiley & Sons, Ltd.     

Asymptotic analysis of certain classes of singularly perturbed problems on the semiaxis

We propose a new method for asymptotic integration of certain classes of singularly perturbed Cauchy problems on the semiaxis for nonhomogeneous systems of linear ordinary differential equations; this method is a further development of the ideas of the regularization method. This method enables us to prove the existence of a unique bounded (as ε→+0) solution of such problems and leads to a simpler and more constructive algorithm for obtaining the asymptotic expansion of the solution and singling out all of its singularities in closed analytic form (including the critical case in which the spectral points of the limit operator may touch the imaginary axis). The proposed method supplements and sharpens earlier results. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 111–117, July, 1997. Translated by N. K. Kulman     

出现在化学反应器理论中的奇摄动边值问题的渐近解

研究了一类出现在化学反应器理论中的奇摄动边值问题.在适当的条件下,用合成展开法构造出该问题的形式近似式,并应用微分不等式理论证明了解的存在性及其渐近性质.