具连续变量和最大值项的二阶差分方程的振动性

研究了一类具有最大值项和连续变量的非线性二阶中立型时滞差分方程的振动性,利用Banach空间的不动点原理和一些不等式技巧,得到了这类方程存在最终正解的充分条件,并得到了该方程振动的一些判别准则.     

小参数在高阶导数项的椭圆型方程第一边值问题的一致收敛差分格式

本文讨论了曲边区域上小参数ε在高阶导数项的椭圆型方程第一边值问题,从一致收敛的必要条件出发构造了特殊的差分格式,证明了差分方程问题解的一致收敛性,估计了收敛的阶数,并讨论了差分方程解的渐近性态.     

Optimum Runge-Kutta-Fehlberg Methods for Second-order Differential Equations

^* Presently at Deparment of Mathematics, Indian Institute of Technology, Madras, India. The optimum Runge-Kutta method of a particular order is theone whose truncation error is minimum. In this paper, we havederived optimum Runge-Kutta mehtods of 0(h^m+4), 0(h^m+5) and0(h^m+6) for m = 0(1)8, which can be directly used for solvingthe second order differential equation yⁿ = f(x, y, y'). Thesemethods are based on a transformation similar to that of Fehlbergand require two, three and four evaluations of f(x, y, y') respectively,for each step. The numercial solutions of one example obtainedwith these methods are given. It has been assumed that f(x,y, y')is sufficiently differentiable in the entire region ofintegration.     

一类二阶泛函微分方程解的渐近性

对各类二阶微分方程解的性质,自1971年Hammett以来已有许多讨论,如[1]—[10]本文讨论二阶时滞泛函微分方程 (r(t)x′(t))′+sum from i=0 to n (P_i(t)g_i′(x(t-τ_i(t))))+sum from i=0 to n (q_i(t)g_i(x(t-τ_i(t))))=f(t) (1)的解的渐近性质,其中;r(t)、q_i(t)、g_i(x)、τ_i(t)、f(t)连续;p_i(t)连续可微;当p_i(t)不恒为0时,g_i(x)连续可微;当x≠0,xg_i(x)>0;g_i(x)关于x单调不减;F(u)=integral from n=to to u (|f(s)|ds)<∞;g_0(x)=x,τ_0(t)=0。     

Asymptotic Expansion of Solutions of Parabolic Equations with a Small Parameter

The heat equation with a small parameter, $\left( {1 + \varepsilon ^{ - m} \chi \left( {\frac{x}{\varepsilon }} \right)} \right)ut = u_{xx} $ , is considered, where ε ∈ (0, 1), m < 1 and χ is a finite function. A complete asymptotic expansion of the solution in powers ε is constructed.     

Projection-Iterative Methods for a Class of Difference Equations

We develop a theoretical framework for projection-iterative methods to solve operator equations of the form Au + Bu = f, where A is a Toeplitz operator in a Banach space , B is considered as a perturbation (of general form) of A, and f is a given element in this space. The methods are adopted for application to general situations, in particular, to the equations in which A need not be a Fredholm operator. The idea to involve iteration procedures and the technique which we apply allow to obtain conditions on perturbations for convergence and effective error estimates in terms of some weighted spaces (without any restrictions on the norms for perturbations). Based on established evaluations we derive further information about decaying properties of the solutions. The obtained results are illustrated by considering concrete classes of equations as, for instance, equations corresponding to Jacobi type operators.      

一类滞后差分方程解的渐近性

考虑时滞差分方程xn－xn-1＝F（－f（xn）＋g（xn－k）），这里k是正整数，F，f，g是R→R的连续函数，F和f在R上单调增加，且对所有的u≠0，uF（u）＞0.我们证明了如果对所有的y∈R，有f（y）≥g（y）（f（y）≤g（y）），则方程的每个解趋于一个常数或－∞（∞）．进一步，如果对所有的y∈R，有f（y）≡g（y）；则方程的每个解当n→∞时趋于常数．     

Local Asymptotic Normality for Linear Homogeneous Difference Equations with Non-Gaussian Noise

This paper considers the problem of estimation of drift parameter for linear homogeneous stochastic difference equations. The Local Asymptotic Normality (LAN) for the problem is proved. LAN implies the Hajek–Le Cam minimax lower bound. In particular, it is shown that the Fisher's information matrix for the problem can be expressed in terms of the stationary distribution of an auxiliary Markov chain on the projective space P(^d).     

Oscillatory and Asymptotic Behavior for Second Order Quasilinear Difference Equations

1 IntroductionConsider the second order quasilinear difference equationA(g(Ay.--l)) + f(n,y.) = 0, for n E N(no), (l'l)where A is defined by Ay. = Vn+1--yn, n E N(no) = {no, no + 1,'' }, nO E N = {l, 2,'. }.The following hold throughout the paPer:(H0) (i) g: R-R is a continuous increasing fUnction with propertiessgng(y) = sgny) g(R) = R;(il) f: N(no) x R--+ R is continuous as a function of y E R;(iii) yf(n,y) > 0 for n E N and y / 0.By a solution of the equation (1.l) we mean a non…     

二阶拟线性差分方程的振动性与渐近性

This paper is concerned with the oscillatory(and nonoscillatory)behavior of solutions of second oder quasilinear difference equations of the type Δ(g(Δyn-1)) f(n,yn)=0.Some necessary and sufficient conditions are given for the equation to admit oscillatory and nonocillatory solutions with special asymptotic properties.These results generalize and improve some konown results.     

扰动的中立型时滞差分方程的一致渐近稳定性

本文研究扰动的一维中立型时滞差分方程△(x(n)-cx(n-k))=f(n,xn)+g(n,xn),n∈Z+的稳定性问题.证明了在某些条件下,无扰动方程△(x(n)-cx(n-k))=f(n,xn),n∈Z+零解的一致渐近稳定性蕴涵着上述扰动方程零解的一致渐近稳定性.本文的结果推广并改进了已有的结果.     

Asymptotic Expansions for Second-Order Linear Difference Equations,II

Infinite asymptotic expansions are derived for the solutions to the second-order linear difference equation where p and q are integers, a(n) and b(n) have power series expansions of the form for large values of n, and a₀ ≠ 0, b₀ ≠ 0. Recurrence relations are also given for the coefficients in the asymptotic solutions. Our proof is based on the method of successive approximations. This paper is a continuation of an earlier one, in which only the special case p ≤ 0 and q = 0 is considered.     

Asymptotic Behavior of a Class of Nonlinear Delay Difference Equations

In this paper, we shall study the asymptotic behavior of solutions of difference equations of the form x n +1 = x n p f ( x n m k 1 , x n m k 2 ,…, x n m k r ), n =0,1,…, where p is a positive constant and k 1 ,…, k r are (fixed) nonnegative integers. In particular, permanence and global attractivity will be discussed.     

含最大值项二阶中立型差分方程的渐近性

考虑含最大值项二阶中立型差分方程其中{an},{pn}和{qn}为实数列,k和■为整数且k≥1,■≥0,我们研究了方程(*)非振动解的渐近性．通过例子说明了含最大值项的方程和相应的不含最大值项方程之间的区别．     

Parabolic Equations with Large Parameter. Inverse Problems

Mathematical Notes - For an abstract parabolic equation with initial condition and multidimensional parabolic initial-boundary value problem with absolute terms rapidly oscillating in time, inverse...     

Multitime Methods for Systems of Difference Equations

Systems of difference equations containing small parameters are studied by a constructive perturbation scheme analogous to the one developed by the authors for the study of differential equations. The method results in an averaging procedure for difference equations, and it is particularly well suited to certain highly oscillatory, nonlinear systems. The method is applied to problems from population genetics, pattern recognition, and the numerical analysis of stiff differential equations     

常差分方程奇异摄动问题的渐近方法

在本文中,我们讨论如下差分方程问题(P_ε):(L.y)_k≡εy(k+1)+a(k,ε)y(k)+b(k,ε)y(k-1)=f(k,ε)(1≤k≤N-1)B₁y≡-y(0)+c₁y(1)=a,B₂y≡-c₂y(N-1)+y(N)=β这里ε是一个小参数,c₁,c₂,a,β为常数,a(k,ε),b(k,ε),f(k,ε)(1≤k≤N)是k和ε的函数.首先,我们讨论了常系数的情形;接着引进伸长变换对变系数的情形进行了讨论,给出了解的一致渐近展开式;最后给出了一个数值例子.