一类具有转移条件高阶微分算子的J-自伴性

本文研究了一类具有转移条件的高阶复系数微分算子的J-自伴性,利用J-对称微分算式的拉格朗日双线性型、J-自伴算子的定义及矩阵表示的方法,证明了这类微分算子是J-自伴的,且对应于不同特征值的特征向量和特征子空间都是C-正交的.     

具有对数函数系数的J-自伴微分算子谱是离散的充分条件

运用算子直和分解、Lidskii定理和二次型比较法,研究了一类具有对数函数系数的J-自伴微分算子谱的离散性,得到了这类J-自伴微分算子谱离散的若干充分条件.     

一类系数中含有指数函数和幂函数的J-自伴微分算子谱的离散性

运用算子直和分解法和二次型比较法研究了由2n阶复系数中含有指数函数和幂函数的微分算式所生成的J-自伴微分算子谱的离散性,得到了一类系数中含有指数函数和幂函数的J-自伴微分算子谱是离散的若干充分条件.     

关于强J-空间与JHB-空间

引入了两个很有联系的空间类JHB-空间与强J HB-空间，分别推广了J-空间与强J-空间.讨论了J-空间、强J-空间、J HB-空间及强JHB-空间类间的相包含关系及此四空间类逆包含的条件，还得到了JHB-空间的内部刻画，并证明了若对每个α∈S，Xα.都是非紧的连通空间，则积空间∏α∈S Xα是强J-空间。     

Π_1空间上J-正常算子的J-酉等价

对于Π_1空间上J-正常算子的J-酉等价问题进行讨论.针对不同情况,给出了Π_1空间上两个J-正常算子J-酉等价的充要条件.这将有助于研究Π_1空间上交换J-von Neumann代数之间的J-酉等价.     

2n阶J-自伴向量微分算子的本质谱

利用算子直和分解的方法研究了2n阶J-自伴向量微分算子的本质谱,得到了这类算子的本质谱的分布范围.     

｜AX—B｜min的J-中心对称解

研究方程｜AX-B｜min的J-中心对称矩阵解,给出通解的一般形式．讨论了最小J-中心对称解及其扰动界,给出J-中心对称解的逼近程度等相关问题的一些结论．     

偶阶非对称微分算子离散谱准则

本文研究了由2n阶复系数J-对称微分算式生成的J-自伴微分算子谱的离散性,分别得到了一类J-自伴微分算子谱离散的充分条件与必要条件,为判断一类微分算子谱的离散性提供了若干准则．     

Krein空间上J-正常算子的可定化性

对于Krein空间上J-正常算子 的各种可定化性进行了研究. 利用可定化J-正常算子的谱函数, 给出了临界线的概念, 得到了可定化的J-正常算子成为强可定化算子和一致可定化算子的充要条件.     

J-中心对称矩阵方程的解及其最佳逼近

利用J-中心对称矩阵的结构和约化性质,本文研究了J-中心对称矩阵方程的通解、最小二乘解,然后考虑了方程解集合中对给定矩阵的最佳逼近问题,并给出了唯一最佳逼近解的表达式。     

关于J－对称微分算子的J－自伴扩张的若干注记

本文给出了一条解析描述Ｊ－对称微分算子Ｊ－自伴扩张的新途径．我们借助方程Ｔ（ｙ）＝λｏｙ的解，而不是如文［３］利用方程＋（ｙ）＇＝－ｙ的解来描述Ｊ－对称微分算式的所有Ｊ－自伴域在奇异端点的边条件，不过我们假设生成的最小算子具非空正则域．我们对主要定理给出了若干有趣的注，得到了二阶极限圆情形的有趣结果．     

复系数2n阶微分算子的谱

本文研究了复系数２ｎ阶微分算式（２．１）生成的Ｊ－自伴微分算子谱,对两类微分算子的本质谱,离散谱作了定性研究,得到了所生成微分算子本质谱的存在范围,以及所生成微分算子的谱是离散的充分条件．     

J—自共轭微分算子谱的定性分析

本文对J－自共轭微分算子谱理论研究情况做一些概要性的介绍,第一部分简要回顾了J－自共轭微分算子理论研究的发展过程,第二,三部分介绍了J－自共轭微分算子的本质谱和离散谱定性分析的主要方法和结论;第四部分扼要叙述J－自共轭微分算子其它方面的一些工作,以及J－自共轭微分算子谱理论研究中尚待解决的问题。     

Some properties of singular integral operators over an open curve

In this paper, the stability properties, the endpoint behavior and the invertible relations of Cauchy-type singular integral operators over an open curve are discussed. If the endpoints of the curve are not special, this type of operators are proved to be stable. At the endpoints, either the singularity or smoothness of the operators are exactly described. And the function sets or spaces on which the operators are invertible as well as the corresponding inverted operators are given. Meanwhile, some applications for the solution of Cauchy-type singular integral equations are illustrated. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10471048)     

Some properties of singular integral operators over an open curve Dedicated to the 85th birthday of Prof. Lu Jianke

In this paper, the stability properties, the endpoint behavior and the invertible relations of Cauchy-type singular integral operators over an open curve are discussed. If the endpoints of the curve are not special, this type of operators are proved to be stable. At the endpoints, either the singularity or smoothness of the operators are exactly described. And the function sets or spaces on which the operators are invertible as well as the corresponding inverted operators are given. Meanwhile, some applications for the solution of Cauchy-type singular integral equations are illustrated.     

Some properties of singular integral operators over an open curve Dedicated to the 85th birthday of Prof.Lu Jinake

In this paper, the stability properties, the endpoint behavior and the invertible relations of Cauchy-type singular integral operators over an open curve are discussed. If the endpoints of the curve are not special, this type of operators are proved to be stable. At the endpoints, either the singularity or smoothness of the operators are exactly described. And the function sets or spaces on which the operators are invertible as well as the corresponding inverted operators are given. Meanwhile, some applications for the solution of Cauchy-type singular integral equations are illustrated.     

On Fatou Type Convergence of Convolution Type Double Singular Integral Operators

In this paper, some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type(T_λf)(x, y) = ∫_a~b ∫_a~b f(t, s)K_λ(t-x,s-y)dsdt, x,y ∈(a,b), λ∈Λ  [0,∞),(0.1)are given. Here f belongs to the function space L_1( a,b ~2), where a,b is an arbitrary interval in R. In this paper three theorems are proved, one for existence of the operator(T_λf)(x, y) and the others for its Fatou-type pointwise convergence to f(x_0, y_0), as(x,y,λ) tends to(x_0, y_0, λ_0). In contrast to previous works, the kernel functions K_λ(u,v)don't have to be 2π-periodic, positive, even and radial. Our results improve and extend some of the previous results of [1, 6, 8, 10, 11, 13] in three dimensional frame and especially the very recent paper [15].     

含奇线(奇面)二阶线性偏微分方程的解在奇线(奇面)附近的性质

本文的第一部分研究了含奇线方程的解在奇线附近的性质;引进了“指数”的概念,从而给出了关于这类方程的“奇型郭西问题”的正确提法;并且通过一种特殊的积分-征分方程的研究,证明了这种“奇型郭西问题”的解的存在性,并且给出其近似解法;最后,就一般的情形,给出了方程一般解的表达式,从而说明了在β+β′<0时,郭西问题的多解性。本文的第二部分研究了空间含奇面方程(?)其中 A_σ是任一祇与变元σ=(σ_1…,σ_n)有关的算子,并且关于(15.5)的奇型郭西问题的解可以用关于方程(不合奇面)(?)(15.6)的郭西问题的解表示出来。同样的方法可用来解决空间却普里金方程(17.1)的郭西问题。     

一维$P$-Laplace二阶差分系统奇异边值问题正解的存在性

应用锥压缩锥拉伸不动点定理和Leray-Schauder 抉择定理研究了一类具有P-Laplace算子的奇异离散边值问题$$\left\{\begin{array}{l}\Delta[\phi (\Delta x(i-1))]+ q_{1}(i)f_{1}(i,x(i),y(i))=0, ~~~i\in \{1,2,...,T\}\\\Delta[\phi (\Delta y(i-1))]+ q_{2}(i)f_{2}(i,x(i),y(i))=0,\\x(0)=x(T+1)=y(0)=y(T+1)=0,\end{array}\right.$$的单一和多重正解的存在性,其中$\phi(s) = |s|^{p-2}s, ~p>1$,非线性项$f_{k}(i,x,y)(k=1,2)$在$(x,y)=(0,0)$具有奇性.     

On a class of singular boundary value problems with singular perturbation

In this paper we investigate a class of singular second order differential equations with singular perturbation subject to three-point boundary value conditions, whose solution exhibits a couple of boundary layers at two endpoints. We first establish a lower–upper solutions theorem by using the Schauder fixed point theorem. By the asymptotic expansions and the lower–upper solutions theorem we obtain the existence, asymptotic estimates and uniqueness for the proposed problem. Several examples are given for illustrating our results.