一种求三维弹性力学基本解析解的特征方程解法

提出了一种简单的推导各向同性材料,三维弹性力学问题基本解析解的特征方程解法．应用三维问题控制微分方程的算子矩阵,通过计算其行列式可得到问题特征通解所需满足的特征方程．将满足各种不同简化特征方程的特征通解,代入到微分方程算子矩阵所对应的不同的缩减伴随矩阵,可推导得出相应的三维弹性力学问题的基本解析解,包括B-G解、修正的P-N（P-N-W）解和类胡海昌解．进一步对各类多项式形式的基本解析解的独立性进行了讨论．这些工作为构造数值方法中所需的完备独立的解析试函数奠定了基础．     

一类非线性变型方程广义解的存在唯一性

本文运用单调算子理论及紧性原理，证明了一类非线性变型方程初边值问题广义解的存在唯一性，推广了文［１］中的结论．     

带有有限个转移条件的高阶微分算子特征函数系的完备性北大核心CSCD

研究一类正则的具有混合边界条件并带有有限个转移条件的高阶不连续微分算子特征值问题以及特征函数系的完备性问题.通过结合转移条件定义的新的内积,把问题转换成一个新的Hilbert空间上的对称微分算子的特征值问题.使用分段定义的微分方程的基本解,给出了满足特征方程的特征值是一个整函数的零点,证明了问题的特征值至多可数,得到特征值的充要条件.在此基础上,结合紧算子的谱理论以及逆算子的相关性质,得到了Green函数,证明了特征函数系是完备的.     

带有有限个转移条件的高阶微分算子特征函数系的完备性

研究一类正则的具有混合边界条件并带有有限个转移条件的高阶不连续微分算子特征值问题以及特征函数系的完备性问题.通过结合转移条件定义的新的内积,把问题转换成一个新的Hilbert空间上的对称微分算子的特征值问题.使用分段定义的微分方程的基本解,给出了满足特征方程的特征值是一个整函数的零点,证明了问题的特征值至多可数,得到特征值的充要条件.在此基础上,结合紧算子的谱理论以及逆算子的相关性质,得到了Green函数,证明了特征函数系是完备的.     

正则预解算子族的乘积扰动

设K∈C（R＋）和B是一个有界线性算子．作者证明如果犃生成一个指数有界的A正则预解算子族，那么BA，AB或A（I＋B），（I＋B）A也生成一个指数有界的k-正则预解算子族．此外，作者也给出了k正则预解算子族的加法扰动的相应结果．     

非线性Lipschitz算子的Lipschitz对偶算子及其应用

在文山中我们对非线性Ｌｉｐｓｃｈｉｔｚ算子定义了其Ｌｉｐｓｃｈｉｔｚ对偶算子，并证明了任意非线性Ｌｉｐｓｃｈｉｔｚ算子的Ｌｉｐｓｃｈｉｔｚ对偶算子是一个定义在Ｌｉｐｓｃｈｉｔｚ对偶空间上的有界线性算子．本文还进一步证明：设Ｃ为 Ｂａｎａｃｈ空间 Ｘ的闭子集，Ｃ＊Ｌ为Ｃ的 Ｌｉｐｓｃｈｉｔｚ对偶空间，Ｕ为 Ｃ＊Ｌ上的有界线性算子，则当且仅当 Ｕ为 ｗ＊－ｗ＊连续的同态变换时，存在Ｌｉｐｓｃｈｉｔｚ连续算子Ｔ，使Ｕ为Ｔ的Ｌｉｐｓｃｈｉｔｚ对偶算子．这一结论的理论意义在于：它表明一个非线性Ｌｉｐｓｃｈｉｔｚ算子的可逆性问题可转化为有界线性算子的可逆性问题．作为应用，通过引入一个新概念──ＰＸ－对偶算子，在一般框架下给出了非线性算子半群的生成定理．     

微极各向同性弹性板中的Rayleigh-Lamb波

研究了无应力作用条件下，均匀、各向同性、圆柱形微极结构弹性板中波的传播．导出了对称和斜对称模式下波传播的特征方程．对短波这一极端情况，无应力圆板中对称和斜对称模态波的特征方程退化为Pmyle曲表面波频率方程．并得到薄板的计算结果．给出了位移和微转动分量，并绘制了相应图形．给出了若干特殊情况的研究结果及对称和斜对称模态特征方程的图示．     

算子权移位的紧性,正权和Cαβ分类

给出复可分Hilbert空间上任意重的算子权移位是紧算子的充要条件,重新证明了每个算子权移位酉等价于一个正算子权移位并讨论了算子权移位S～{Wk}与T～{|Wk|}的关系,给出了压缩的任意重算子权移位的Cαβ分类的充要条件.     

关于回复弱混合算子

本文研究了回复弱混合算子的一些性质．利用这些性质，得到了两个算子的积仍然是弱混合算子的一个充要条件．还获得了Banach空间上的算子与它在Banach空间的超空间上所诱导的算子之间回复弱混合性的关系．     

一类EPD－方程的不完全惠更斯现象和离散现象

在本文中，通过引入含两个参数的Ｒｉｅｓｚ分布，我们给出了ＥＰＤ一方程。Ｕｔｔ＋α／ｔｕ＿ｔ－ｕ＿（ｘｘ）＋β／ｔｕ＿ｘ＝０的一类Ｃａｕｃｈｙ问题的解．当α，β取某些离散值时，解具有不完全惠更斯现象．对这些离散值，波具有单向传播的性质．利用此现象，可解释一些具有重特征方程的定解问题的离散现象．     

Homogeneous differential-operator equations in locally convex spaces

We describe a general method that allows us to find solutions to homogeneous differential-operator equations with variable coefficients by means of continuous vector-valued functions. The “homogeneity” is not interpreted as the triviality of the right-hand side of an equation. It is understood in the sense that the left-hand side of an equation is a homogeneous function with respect to operators appearing in that equation. Solutions are represented as functional vector-valued series which are uniformly convergent and generated by solutions to a kth order ordinary differential equation, by the roots of the characteristic polynomial and by elements of a locally convex space. We find sufficient conditions for the continuous dependence of the solution on a generating set. We also solve the Cauchy problem for the considered equations and specify conditions for the existence and the uniqueness of the solution. Moreover, under certain hypotheses we find the general solution to the considered equations. It is a function which yields any particular solution. The investigation is realized by means of characteristics of operators such as the order and the type of an operator, as well as operator characteristics of vectors, namely, the operator order and the operator type of a vector relative to an operator. We also use a convergence of operator series with respect to an equicontinuous bornology.     

Potential bounds in problems on quasilinear elliptic equations

We study quasilinear elliptic equations with strong nonlinear terms and systems of such equations. The methods developed by the authors in [1], [2] are used to prove the existence of solutions for boundary—value problems using some information on behavior of potential bounds for nonlinearities; the L–characteristics of elliptic operators and their fractional powers play an important role. New conditions are suggested for the existence of classical solutions of quasilinear second order elliptic equations.     

分析力学中的高阶Nielsen算子和高阶Euler算子

本文提出与研究完整约束系统和高阶非完整约束系统相关的高阶Nielsen算子和高阶Euler算子的定义,建立表示两类算子之间关系的若干定理,应用这些定理得到高阶约束系统的若干新型运动微分方程,最后举例说明新方程的应用。     

Constructive methods for certain Bergman operators

This paper is concerned with the class of linear partial differential equations of second order such that there exist Bergman operators with polynomial kernels (cf, [12]). In an earlier paper [ll] the authors have shown that these equations also admit differential operators as introduced by K. W. Bauer [I]. In the present paper, relations between different types of representations of solutions are investigated. These representations are of interest in developing a function theory of solutions; cf., for instance, K. W. Bauer [I] and S. Ruscheweyh [19]. They are also essential to global extensions of local results obtained by means of Bergman operators of the first kind. The inversion problem for those operators is solved, and it is shown that all solutions of equations of that class which are holomorphic in a domain of C² can be represented by operators with polynomial kernels. Furthermore, a construction principle for deriving the equations investigated by K. W. Bauer [2] is obtained; this yields corresponding representations of solutions by differential and integral operators in a systematic fashion     

Generalization of the Lagrange Method to the Case of Second-Order Linear Differential Equations with Constant Operator Coefficients in Locally Convex Spaces

The well-known Lagrange method for linear inhomogeneous differential equations is generalized to the case of second-order equations with constant operator coefficients in locally convex spaces. The solutions are expressed in terms of uniformly convergent functional vector-valued series generated by a pair of elements of a locally convex space. Sufficient conditions for the continuous dependence of solutions on the generating pair are obtained. The solution of the Cauchy problem for the equations under consideration is also obtained and conditions for its existence and uniqueness are given. In addition, under certain conditions, the so-called general solution of the equations (a function of most general form from which any particular solution can be derived) is obtained. The study is carried out using the characteristics (order and type) of an operator and of a sequence of operators. Also, the convergence of operator series with respect to equicontinuous bornology is used.     

Local solvability of semilinear partial differential equations

Summary It is proved an abstract theorem for local solvability of semilinear equations with complex coefficients. This result is illustrated by nonlinear perturbations of complex principal type and multiple complex characteristics operators.

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Riassunto Si dimostra un teorema astratto per la risolvibilità locale di equazioni semilineari di coefficienti complessi. Il risultato ottenuto viene illustrato mediante perturbationi nonlineari di operatori di tipo principale complesso e caratteristiche compless multiple.

     

The generalization of bivariate MKZ operators by multiple generating functions

In the present paper, we study approximation properties of multiple generating functions type bivariate Meyer-König and Zeller (MKZ) operators with the help of Volkov type theorem. We compute the order of convergence of these operators by means of modulus of continuity and the elements of modified Lipschitz class. Finally, we give application to partial differential equations.     

Multiple fixed points of a sum operator and applications

This paper considers existence of multiple positive fixed points for some nonlinear operators, a particular case of the operators is sum of an e-concave operator and an e-convex operator. Then we apply the results to nonlinear integral equations.     

Approximation of weighted local mean operators

Recently, weighted local mean operators are widely used in image processing, compressive sensing and other areas. A weighted local mean operator changes its characteristics depending on a function content within a local area in order to preserve the function features. The directional diffusion filter and Yaroslavsky neighbourhood filter (also called the sigma filter) are discrete versions of such operators. Although these operators are not convolution ones, due to their sparsity, the corresponding numerical algorithms have simple structure and fast performance. In this paper, we study the approximate properties of the weighted local mean operators, particularly focus on their asymptotic expansions, which are related to non-linear diffusion equations.     

Fourth Order Conservative Exponential Methods for Linear Evolution Equations

A few numerical methods for linear evolution equations are developed and analyzed in this paper. These fourth order exponential methods reproduce the exact solutions for equations with time-independent evolution operators. For highly oscillatory problems with evolution operators that vary slowly in time, these methods are often more efficient than the traditional methods, since large step sizes can be used. The methods developed in this paper are also conservative for equations such as the Schrödinger equation, where the evolution operator is skew-selfadjoint.