积分凸性及其应用

该文在Banach空间中通过向量值函数的Bochner积分引进集合与泛函的积分凸性以及集合的积分端点等概念. 文章主要证明有限维凸集、开凸集和闭凸集均是积分凸集，下半连续凸泛函与开凸集上的上半连续凸泛函均是积分凸的, 非空紧集具有积分端点, 对紧凸集来说其积分端点集与端点集一致, 最后给出积分凸性在最优化理论方面的两个应用.     

Criteria for convexity in Banach spaces

In this paper two convexity criteria are proven. The first one characterizes compact convex sets in a locally convex space and extends a previous result by G. Aumann, while the second one characterizes closed bounded convex sets with the Radon-Nikodým property in a Banach space.

     

Banach spaces that admit support sets

It is shown that the existence of a closed convex set all of whose points are properly supported in a Banach space is equivalent to the existence of a certain type of uncountable ordered one-sided biorthogonal system. Under the continuum hypothesis, we deduce that this notion is weaker than the existence of an uncountable biorthogonal system.

     

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.     

On uniform convexity of Banach spaces

This paper gives some relations and properties of several kinds of generalized convexity in Banach spaces. As a result, it proves that every kind of uniform convexity implies the Banach-Sakes property, and several notions of uniform convexity in literature are actually equivalent.     

A note on integral-convexity in Banach spaces

In the present paper we focus on a generalization of the notion of integral convexity. This concept, introduced in [J.Y. Wang, Y.M. Ma, The integral convexity of sets and functionals in Banach spaces, J. Math. Anal. Appl. 295 (2004) 211-224] by replacing, in the definition of classical notion of convexity, the sum by the integral, has interesting applications in optimal control problems. By using, instead of Bochner integral, a more general vector integral, that of Pettis, we obtain some results on integral-extreme points of subsets of a Banach space stronger than those given in [J.Y. Wang, Y.M. Ma, The integral convexity of sets and functionals in Banach spaces, J. Math. Anal. Appl. 295 (2004) 211-224]. Finally, a natural example coming from measure theory is included, in order to reflect the relationships between different kinds of integral convexity.     

Envelopes of open sets and extending holomorphic functions on dual Banach spaces

We investigate certain envelopes of open sets in dual Banach spaces which are related to extending holomorphic functions. We give a variety of examples of absolutely convex sets showing that the extension is in many cases not possible. We also establish connections to the study of iterated weak^∗ sequential closures of convex sets in the dual of separable spaces.     

Near convexity,near smoothness and approximative compactness of half spaces in Banach spaces

The authors discuss the dual relation of nearly very convexity and property WS. By two kinds of near convexities and two kinds of near smoothness, the authors prove a series of characterizations such that every half-space in Banach space X and every weak* half-space in the dual space X* are approximatively weakly compact and approximatively compact. They show a sufficient condition such that a Banach space X is a Asplund space. Using upper semi-continuity of duality mapping, the authors also give two characterizations of property WS and property S.     

Equicompact sets of operators defined on Banach spaces

Let and be Banach spaces. We say that a set denotes the space of all compact operators from into ) is equicompact if there exists a null sequence in such that for all and all . It is easy to show that collectively compactness and equicompactness are dual concepts in the following sense: is equicompact iff is collectively compact. We study some properties of equicompact sets and, among other results, we prove: 1) a set is equicompact iff each bounded sequence in has a subsequence such that is a converging sequence uniformly for ; 2) if does not have finite cotype and is a maximal equicompact set, then, given and a finite set in , there is an operator such that for and all .

     

Convex integral functionals of regular processes

This article gives dual representations for convex integral functionals on the linear space of regular processes. This space turns out to be a Banach space containing many more familiar classes of stochastic processes and its dual can be identified with the space of optional random measures with essentially bounded variation. Combined with classical Banach space techniques, our results allow for a systematic treatment of stochastic optimization problems over BV processes and, in particular, yields a maximum principle for a general class of singular stochastic control problems.     

The level sets of the resolvent norm and convexity properties of Banach spaces

We construct a bounded linear operator on a separable, reflexive and strictly convex Banach space with the resolvent norm that is constant in a neighbourhood of zero.      

向量值函数的Mcshane积分

本文讨论了向量值函数的Ｍｃｓｈａｎｅ积分的性质．例如收敛定理和原函数的性质等．     

On a proximal point method for convex optimization in banach spaces

We analyze the behavior of a parallel proximal point method for solving convex optimization problems in reflexive Banach spaces. Similar algorithms were known to converge under the implicit assumption that the norm of the space is Hilbertian. We extend the area of applicability of the proximal point method to solving convex optimization problems in Banach spaces on which totally convex functions can be found. This includes the class of all smooth uniformly convex Banach spaces. Also, our convergence results leave more flexibility for the choice of the penalty function involved in the algorithm and, in this way, allow simplification of the computational procedure.     

Integral representation of holomorphic functions on Banach spaces

In this paper we discuss the problem of integral representation of analytic functions over a complex Banach space E. We obtain, for a wide class of functions, integral representations of the form

     

Test closures of sets in Banach spaces generated by linear functionals

Some criteria are proved for an element of a Banach space to belong to the test or Korovkin closure of a given set. The operations of closure in question are determined by linear functionals. The cases of arbitrary linear functionals and of positive functionals are considered. Some sufficient conditions are given for the test closure to coincide with the Korovkin closure. Bibliography: 33 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 114–145. Translated by S. Yu. Pilyugin.     

Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces

In this paper, first a new fixed point theorem is established, and then, by the use of it, the existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces are investigated. The results obtained in this paper generalize and improve the results corresponding to those obtained by others.     

Iterative equations in Banach spaces

Let X be a Banach space, let

     

Nonconvex variational problem with recursive integral functionals in Sobolev spaces: Existence and representation

The purpose of this paper is twofold. First, we present the existence theorem of an optimal trajectory in a nonconvex variational problem with recursive integral functionals by employing the norm-topology of a weighted Sobolev space. We show the continuity of the integral functional and the compactness of the set of admissible trajectories. Second, we show that a recursive integrand is represented by a normal integrand under the conditions guaranteeing the existence of optimal trajectories. We also demonstrate that if the recursive integrand satisfies the convexity conditions, then the normal integrand is a convex function. These results are achieved by the application of the representation theorem in Lp-spaces.