Approximation Spaces in the Numerical Analysis of Operator Equations

We show that generalized approximation spaces can be used to prove stability and convergence of projection methods for certain types of operator equations in which unbounded operators occur. Besides the convergence, we also get orders of convergence by this approach, even in case of non-uniformly bounded projections. We give an example in which weighted uniform convergence of the collocation method for an easy Cauchy singular integral equation is studied.     

新的齐型BMO, Lipschitz空间上的奇异积分算子

Let(X, d, μ) be a space of homogeneous type, BMO_A(X) and Lip_A(β,X) be the space of BMO type,lipschitz type associated with an approximation to the identity {A_t}_t0 and introduced by Duong,Yan and Tang, respectively. Assuming that T is a bounded linear operator on L~2(X), we find the sufficient condition on the kernel of T so that T is bounded from BMO(X) to BMO_A(X) and from Lip(β, X) to Lip_A(β, X). As an application, the boundedness of Calderón-Zygmund operators with nonsmooth kernels on BMO(R~n) and Lip(β, R~n) are also obtained.     

Weighted Estimates for the Maximal Commutator of Singular Integral Operator on Spaces of Homogeneous Type

Weighted estimates with general weights are established for the maximal operator associated with the commutator generated by singular integral operator and BMO function on spaces of homogeneous type, where the associated kernel satisfies the HSlder condition on the first variable and some condition which is fairly weaker than the Holder condition on the second variable.     

Banach空间中积分表示的一个反例

本文证明了存在一具有解析ＲＮＰ的Ｂａｎａｉｃｈ空间Ｘ，存在Ｘ上的Ｊｅｎｓｅｎ凸集Ｃ以及ｘ∈Ｃ，使得Ｘ上支撑在Ｃ的Ｊｅｎｓｅｎ边界点上的Ｊｅｎｓｅｎ测度均不以ｘ为重心．     

局部凸空间中一类非线性Volterra型积分方程解的存在性

首先利用局部凸空间非紧性测度得到一个新的不动点定理;并运用此定理研究了局部凸空间非线性V o lterra型积分方程解的存在性,然后应用到弱拓扑结构下对非线性V o lterra型积分方程解的存在性的讨论.推广了原有文献的结果.     

Toeplitz Type Operator Associated to Singular Integral Operator with Variable Kernel on Weighted Morrey Spaces

Suppose Tk,1 and Tk,2 are singular integrals with variable kernels and mixed homogeneity or ± I(the identity operator). Denote the Toeplitz type operator by Tb=∑Qk=1Tk,1MbTk,2,where Mbf=bf. In this paper, the boundedness of Tbon weighted Morrey space are obtained when b belongs to the weighted Lipschitz function space and weighted BMO function space, respectively.     

Janowski星形和凸函数族相关的非线性积分算子性质

In this paper, we consider a general nonlinear integral operator H_(αi),β_i(f_1,..., f_n;g_1,..., g_n)(z). Some results including coefficient problems, univalency condition and radius of convexity for this integral operator are given. Furthermore, we discuss the mapping properties between H_(αi),β_i(f_1,..., f_n; g_1,..., g_n)(z) and subclasses of analytic functions with bounded boundary rotation. The same subjects for some corresponding classes are shown upon specializing the parameters in our main results.     

序Banach空间中一类算子方程的可解性

利用改进的Mnch紧性条件,研究了一类算子方程Lx=Nx的解的存在性．本文结果并没有要求算子L和N的连续性．作为应用,讨论了右端项不连续的隐式椭圆偏微分方程的边值问题的解的存在性．     

非线性随机积分和微分方程组的解*

在本文中我们首先对具有随机定义域的连续随机算子组证明了Darbao型不动点定理。应用此定理我们给出了非线性随机Volterra积分方程组和非线性随机微分方程组的Cauchy问题解的存在性准则。这些随机方程组的极值随机解的存在性和随机比较结果也被获得。我们的定理改进和推广Tyaughn,Lakshmikantham,Lakshmikantham-Leela,DeBlast-Myjak和第一作者的相应结果。     

二维空间中具有积分型非线性Schrodinger方程组的初值问题

具有积分型非线性ｓｃｈｒｏｄｉｎｇｅｒ方程是在研究非线性Ｌａｎｇｍｕｉｒ波时考虑到离子惯性作用而导出的．本文讨论了二维空间中具有积分型非线性ｓｃｈｒｏｄｉｎｇｅｒ方程组的初值问题，用积分估计方法证明了整体解的存在唯一性．     

An Operatorial approach to singular integral equations of a modified type

In this article,the authors discuss a kind of modified singular integral equations on a disjoint union of closed contours or a disjoint union of open arcs.The authors introduce some singular integral operators associated with this kind of singular integral equations,and obtain some useful properties for them.An operatorial approach is also given together with some illustrated examples.     

二元样条的积分表示及分层三角剖分下二次样条空间的维数

本文通过引入一个积分协调条件，首次给出了二元样条的一个积分表示．文中还定义了平面单连通多边形区域的所谓分层三角剖分，并确定了此剖分下二次样条空间的维数．     

广义Hammerstain型非线性奇异积分方程的可解性定理

本文所讨论的广义Hammerstain型非线性奇异积分方程:(*)是以流体弹性力学和断裂力学的边值问题为背景。我们分别对k>0,k=0,k<0这三种情况研究了方程(＊)的可解性,用Newton-Kantorovic方法和Banach不动点原理证明了方程(*)在不同条件下解的存在唯一性定理,从而推广了的工作。     

Banach空间中非线性脉冲Volterra型积分方程组的可解性

在较弱的条件下，利用锥理论和单调迭代方法首先建立了Banach空间中一类非线性算子方程组最小最大解的存在性定理；然后作为应用，利用一个新的比较结果，得到了Banach空间中非线性脉冲Volterra型积分方程组的整体解，改进了最近的许多结果．     

A Probabilistic Approach for Nonlinear Equations Involving the Fractional Laplacian and a Singular Operator

We consider a class of nonlinear integro-differential equations involving a fractional power of the Laplacian and a nonlocal quadratic nonlinearity represented by a singular integral operator. Initially, we introduce cut-off versions of this equation, replacing the singular operator by its Lipschitz continuous regularizations. In both cases we show the local existence and global uniqueness in L¹L^p. Then we associate with each regularized equation a stable-process-driven nonlinear diffusion; the law of this nonlinear diffusion has a density which is a global solution in L¹ of the cut-off equation. In the next step we remove the cut-off and show that the above densities converge in a certain space to a solution of the singular equation. In the general case, the result is local, but under a more stringent balance condition relating the dimension, the power of the fractional Laplacian and the degree of the singularity, it is global and gives global existence for the original singular equation. Finally, we associate with the singular equation a nonlinear singular diffusion and prove propagation of chaos to the law of this diffusion for the related cut-off interacting particle systems. Depending on the nature of the singularity in the drift term, we obtain either a strong pathwise result or a weak convergence result. Mathematics Subject Classifications (2000) 60K35, 35S10.     

弱拓扑下的非线性随机积分和微分方程组的解*

在本文中,我们首先对具有随机定义域的弱连续随机算子组证明了一个Darbo型随机不动点定理.利用这一定理,我们对Banach空间中关于弱拓扑的非线性随机Volterra积分方程组给出了随机解的存在性准则.作为应用,我们得到了非线性随机微分方程组的Canchy问题弱随机解的存在定理.也得到了这些随机方程组在Banach空间中关于弱拓扑的极值随机解的存在性和随机比较结果.我们的定理改进和推广了Szep,Mitchell－Smith,Cramer－Lakshmikantham,Lakshmikantham-Leela和丁的相应结果.     

On the Divergence of Collocation Solutions in Smooth Piecewise Polynomial Spaces for Volterra Integral Equations

In this paper we present a characterization of those smooth piecewise polynomial collocation spaces that lead to divergent collocation solutions for Volterra integral equations of the second kind. The key to these results is an equivalence result between such collocation solutions and collocation solutions in slightly smoother spaces for initial-value problems for ordinary differential equations. For the latter problems Mülthei (1979/1980) established a complete divergence (and convergence) theory. Our analysis can be extended to furnish divergence results for smooth collocation solutions to Volterra integral equations of the first kind. AMS subject classification (2000) 65R20, 65L20, 65L60.Received May 2004. Accepted September 2004. Communicated by Tom Lyche.Hermann Brunner: This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).     

Pseudodifferential Multi-Product Representation of the Solution Operator of a Parabolic Equation

By using a time slicing procedure, we represent the solution operator of a second-order parabolic pseudodifferential equation on ? ⁿ as an infinite product of zero-order pseudodifferential operators. A similar representation formula is proven for parabolic differential equations on a compact Riemannian manifold. Each operator in the multi-product is given by a simple explicit Ansatz. The proof is based on an effective use of the Weyl calculus and the Fefferman-Phong inequality.     

E2类二阶椭圆组一般形式的非线性边值问题

研究了E2类二阶椭圆型方程组相当广泛的一类非线性边值问题，通过引进一种代换把它化为一类非线性广义Riemann-Hilbert边值问题，再引进奇异积分算子，建立与该问题等价的非线性奇异积分方程。应用奇异积分算子性质和泛函分析与函数论方法，在一定的假设条件下，证得了该问题的可解性。