Set-valued mappings specified by regularization of the Schrödinger equation with degeneration

The Cauchy problem for the Schrödinger equation with an operator degenerating on a half-line and a family of regularized Cauchy problems with uniformly elliptic operators, whose solutions approximate the solution to the degenerate problem, are considered. A set-valued mapping is investigated that takes a bounded operator to a set of partial limits of values of its quadratic form on solutions of the regularized problems when the regularization parameter tends to zero. The dynamics of quantum states are determined by applying an averaging procedure to the set-valued mapping.     

Non-compact generalized variational inequalities for quasi-monotone and hemi-continuous operators with applications

Some results are obtained for non-compact cases in topological vector spaces for the existence problem of solutions for some set-valued variational inequalities with quasi-monotone and lower hemi-continuous operators, and with quasi-semi-monotone and upper hemi-continuous operators. Some applications are given in non-reflexive Banach spaces for these existence problems of solutions and for perturbation problems for these set-valued variational inequalities with quasi-monotone and quasi-semi-monotone operators. This revised version was published online in June 2006 with corrections to the Cover Date.     

Minimum selections and fixed points of set-valued operators in Banach spaces with some uniform convexity

A new continuity theorem of minimum selection is presented for a continuous set-valued operator from a topological space into a Banach space with some uniform convexity. As applications, some problems concerning minimum right inverses for linear operators and minimum fixed points for condensing set-valued nonlinear operators are discussed. Also, the existence of minimum solutions for an integral inclusion is proved.     

Approximations of Set-Valued Functions by Metric Linear Operators

In this paper we introduce new approximation operators for univariate set-valued functions with general compact images in Rⁿ. We adapt linear approximation methods for real-valued functions by replacing linear combinations of numbers with new metric linear combinations of finite sequences of compact sets, thus obtaining "metric analogues" of these operators for set-valued functions. The new metric linear combination extends the binary metric average of Artstein to several sets and admits any real coefficients. Approximation estimates for the metric analogue operators are derived. As examples we study metric Bernstein operators, metric Schoenberg operators, and metric polynomial interpolants.     

Regularization of quasi-variational inequalities

An ill-posed quasi-variational inequality with contaminated data can be stabilized by employing the elliptic regularization. Under suitable conditions, a sequence of bounded regularized solutions converges strongly to a solution of the original quasi-variational inequality. Moreover, the conditions that ensure the boundedness of regularized solutions, become sufficient solvability conditions. It turns out that the regularization theory is quite strong for quasi-variational inequalities with set-valued monotone maps but restrictive for generalized pseudo-monotone maps. The results are quite general and are applicable to ill-posed variational inequalities, hemi-variational inequalities, inverse problems, and split feasibility problem, among others.     

向量集值映射的共轭对偶

借助抽象算子将共轭映射的概念的到抽象空间,引入了集值映射的共轭映射和次梯度,据此讨论了集值映射共轭对偶的全局稳定性。     

Steiner选择和算子的集值扩张(Ⅰ)——理论结果

提出了一种从连续函数空间到连续集值函数空间中扩张算子的方法.这种扩张是通过Steiner集值函数选择法得到的.此外,还研究了其性质及该类算子序列的收敛特性.在文章的第(Ⅱ)部分,还将给出在逼近理论中的一些应用.     

Approximate computation of eigenvalues and eigenfunctions of Sturm-Liouville differential operators by applying the theory of regularized traces

This paper overviews studies dealing with the approximate computation of eigenvalues and eigenfunctions of Sturm-Liouville differential operators by applying methods of the theory of regularized traces. Dorodnicyn’s method and its development in the form of the theory of regularized traces of differential operators are described.     

Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators

We propose a primal-dual splitting algorithm for solving monotone inclusions involving a mixture of sums, linear compositions, and parallel sums of set-valued and Lipschitzian operators. An important feature of the algorithm is that the Lipschitzian operators present in the formulation can be processed individually via explicit steps, while the set-valued operators are processed individually via their resolvents. In addition, the algorithm is highly parallel in that most of its steps can be executed simultaneously. This work brings together and notably extends various types of structured monotone inclusion problems and their solution methods. The application to convex minimization problems is given special attention.     

Steiner选择和算子的集值扩张(Ⅱ)——对逼近理论的应用

在(Ⅰ)的基础上,得出对集值函数逼近理论的某些应用:Korovkin型定理,一种将经典逼近算子扩张到集值族的方法,以及Jackson估算.     

Representations of regularized determinants of exponentials of differential operators by functional integrals

Representations of regularized determinants of elements of one-parameter operator semigroups whose generators are second-order elliptic differential operators by Lagrangian functional integrals are obtained. Such semigroups describe solutions of inverse Kolmogorov equations for diffusion processes. For self-adjoint elliptic operators, these semigroups are often called Schrödinger semigroups, because they are obtained by means of analytic continuation from Schrödinger groups. It is also shown that the regularized determinant of the exponential of the generator (this exponential is an element of a one-parameter semigroup) coincides with the exponential of the regularized trace of the generator.     

Existence Theorems for Set-valued Operators in Banach Spaces

In this paper, we study the problem of existence of zero points for set-valued operators in Banach spaces. We present a simple condition which can be used to verify that under some conditions, set-valued operators have zero points. Furthermore, we derive the surjectivity result and the Kakutani fixed point theorem by using this condition.      

An extension of Tarski's fixed point theorem and its application to isotone complementarity problems

Tarski's fixed point theorem is extended to the case of set-valued mappings, and is applied to a class of complementarity problems defined by isotone set-valued operators in a complete vector lattice.     

Ideals of compact operators with the Orlicz norms

The ideal of compact operators in a Hilbert space with the Orlicz type norm is investigated. Upper and lower bounds for the regularized determinants of operators from that ideal are derived.     

Set-valued variational inclusions with T-accretive operators in Banach spaces

In this work, a new class of set-valued variational inclusions involving $T$-accretive operators in Banach spaces is introduced and studied. And a new iterative algorithm for this class of set-valued variational inclusions and its convergence result are established.     

集值测度的Radon-Nikodym定理及集值算子的Pettis-Aumann积分表示

本文建立了 Ｂａｎａｃｈ空间集值测度的 Ｒａｄｏｎ－Ｎｉｋｏｄｙｍ定理,并给出了两类集值算子的Ｐｅｔｔｉｓ－Ａｕｍａｎｎ积分表示．     

Generalized Monotone Operators on Dense Sets

In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalized convexity of a real valued function, which is obtained out of its local counterpart on some dense sets.     

集值测度的Radon-Nikodym定理及集值算子的Pettis-Aumann积分表示

本文建立了 Ｂａｎａｃｈ空间集值测度的 Ｒａｄｏｎ－Ｎｉｋｏｄｙｍ定理,并给出了两类集值算子的Ｐｅｔｔｉｓ－Ａｕｍａｎｎ积分表示．     

集值测度的RadOn-Nikodym定理及集值算子的Pettis-Aumanri积分表示

本文建立了Banach空间集值测度的Radon-Nikodym定理,并给出了两类集值算子的Pettis-Aumann积分表示.