A GENERALIZED SEMICONJUGACY IN DIFFERENCE EQUATIONS＊

Difference equations arise in many fields.This article is concerned to generalization of semiconjugacy in difference equations.In fact,H.Sedaghat in [7] investigated the semiconjugacy in difference equ...     

一类复差分方程组的亚纯解

许多作者研究了复差分方程解的存在性及增长性问题,得到了较多理想的结果.本文利用亚纯函数Nevanlinna值分布理论,研究了一类复高阶非线性差分方程解的表达式问题,将复差分方程的一结果推广至复差分方程组中.     

Difference Equations and Lettenmeyer's Theorem

Many classical results for ordinary differential equations have counterparts in the theory of difference equations, although, in general, the technical details for the difference versions are more involved than the corresponding ones for differential equations. This note surveys material related to a difference analogue of Lettenmeyer's theorem. The projection method of Harris et al. , developed to treat certain questions in the analytic theory of ordinary differential equations is used to obtain counterparts for linear difference equations as well as extensions to certain nonlinear differential and difference equations.     

几类非线性差分方程的对称和精确解

本文将微分方程的Lie变换群方法推广到差分方程,给出了三类非线性差分方程的不变变换,利用这种变换由差分方程的平凡解得到非平凡的单参数解族。     

Recursion operators,conservation laws,and integrability conditions for difference equations

We attempt to propose an algebraic approach to the theory of integrable difference equations. We define the concept of a recursion operator for difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. As in the case of partial differential equations, these canonical densities can serve as integrability conditions for difference equations. We obtain the recursion operators for the Viallet equation and all the Adler-Bobenko-Suris equations.     

On existence of solutions of differential‐difference equations

This paper applies Nevanlinna theory of value distribution to discuss existence of solutions of certain types of non‐linear differential‐difference equations such as (5) and (8) given in the succeeding paragraphs. Existence of solutions of differential equations and difference equations can be said to have been well studied, that of differential‐difference equations, on the other hand, have been paid little attention. Such mixed type equations have great significance in applications. This paper, in particular, generalizes the Rellich–Wittich‐type theorem and Malmquist‐type theorem about differential equations to the case of differential‐difference equations. Copyright © 2015 John Wiley & Sons, Ltd.     

On the difference approximations of overdetermined hyperbolic equations of classical mathematical physics

Issues concerning difference approximations of overdetermined systems of hyperbolic equations are examined. The formulations of extended overdetermined systems are given for hydrodynamics equations, magnetohydrodynamics equations, Maxwell equations, and elasticity equations. Some approaches to the construction of difference schemes are discussed for these systems.     

Perturbation of Volterra Difference Equations

A fixed point theorem is used to investigate nonlinear Volterra difference equations that are perturbed versions of linear equations. Sufficient conditions are established to ensure that the stability properties of linear Volterra difference equations are preserved under perturbation. The existence of asymptotically periodic solutions of perturbed Volterra difference equations is also proved.     

The Theory of Linear G-Difference Equations

We introduce the notion of difference equations defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants of the action of the symmetry group. Linear equations are modules over the skew group algebra, solutions are morphisms relating a given equation to other equations, symmetries of an equation are module endomorphisms, and conserved structures are invariants in the tensor algebra of the given equation.We show that the equations and their solutions can be described through representations of the isotropy group of the symmetry group of the underlying set. We relate our notion of difference equation and solutions to systems of classical difference equations and their solutions and show that out notions include these as a special case.     

一类复差分方程组的亚纯解(英文)

本文研究了一类复差分方程组的增长性的问题.利用亚纯函数的Nevanlinna值分布理论,得到了有关复差分方程组的亚纯解的一个重要结果,将复差分方程的某些结果推广到复差分方程组中.     

Asymptotic behavior of solutions of functional difference equations

For linear functional difference equations, we obtain some results on the asymptotic behavior of solutions, which correspond to a Perron-type theorem for linear ordinary difference equations. We also apply our results to Volterra difference equations with infinite delay.     

高阶非线性差分方程的导数收敛及其应用

首先研究高阶线性差分方程的整体收敛性，并证明了高阶非线性差分方程各阶导数的整体收敛；进而得到了关于高阶非线性差分方程整体收敛的一个定理，最后利用这个定理部分解决了Ladas提出的一个猜测。     

Estimates of N-function and m-function of meromorphic solutions of systems of complex difference equations

We apply Nevanlinna theory of the value distribution of meromorphic functions to study the properties of Nevanlinna counting function and proximity function of meromorphic solutions of a type of systems of complex difference equations. Our results can give estimates on the proximity function and the counting function of solutions of systems of difference equations. This implies that solutions have a relatively large number of poles. It extend some result concerning difference equations to the systems of difference equations.     

Oscillation of certain partial difference equations

In this paper we consider a class of nonlinear delay partial difference equations and a class of linear delay partial difference equations with variable coefficients, which may change sign. We obtain oscillation criteria for these equations. There are no results for the oscillation of these equations up to now.     

On difference Riccati equations and second order linear difference equations

In this paper, we treat difference Riccati equations and second order linear difference equations in the complex plane. We give surveys of basic properties of these equations which are analogues in the differential case. We are concerned with the growth and value distributions of transcendental meromorphic solutions of these equations. Some examples are given.     

Oscillation of solutions of second-order linear differential equations and corresponding difference equations

In this paper, we present conditions ensuring that solutions of linear second-order differential equations oscillate, provided solutions of corresponding difference equations oscillate. We also establish the converse result, namely, when oscillation of solutions of difference equations implies oscillation of solutions of corresponding differential equations.     

Modified classification of non-self-adjoint second-order difference equations

This paper is concerned with the classification of non-self-adjoint second-order difference equations. The relationship between the number of summable solutions of non-self-adjoint difference equations and that of the discrete linear Hamiltonian system is discussed. A classification for non-self-adjoint second-order difference equations is established, which is independent of the choice of the rotated half plane and the fixed point.     

Stability in Probability of Nonlinear Stochastic Volterra Difference Equations with Continuous Variable

Abstract

The general method of Lyapunov functionals construction, that was proposed by Kolmanovskii and Shaikhet and successfully used already for functional-differential equations, difference equations with discrete time, difference equations with continuous time, and is used here to investigate the stability in probability of nonlinear stochastic Volterra difference equations with continuous time. It is shown that the investigation of the stability in probability of nonlinear stochastic difference equation with order of nonlinearity more than one can be reduced to investigation of the asymptotic mean square stability of the linear part of this equation.     

Boundary value problems for certain classes of loaded differential equations and solving them by finite difference methods

Boundary value problems for loaded ordinary and partial differential equations are considered. A priori bounds are obtained for solutions to differential and difference equations. These bounds imply the stability and convergence of difference schemes for the equations under consideration.     

Stability in the first approximation of some volterra difference equations

volterra difference equations arise in the modeling of some real phenomena. In this paper stability problem of some volterra difference equations is investigated.Stability conditions are formulated in terms of the characteristics of equations.