局部凸空间中的Drop定理和Phelps引理以及Ekeland变分原理

本文将Drop定理,Phelps引理,Ekeland变分原理和基于Isac的Pareto有效性定理推广到局部完备的局部凸Hausdorff空间,而且证明了它们彼此是等价的．     

Local completeness, drop theorem and Ekeland's variational principle

By using a very general drop theorem in locally convex spaces we obtain some extended versions of Ekeland's variational principle, which only need assume local completeness of some related sets and improve Hamel's recent results. From this, we derive some new versions of Caristi's fixed points theorems. In the framework of locally convex spaces, we prove that Daneš' drop theorem, Ekeland's variational principle, Caristi's fixed points theorem and Phelps lemma are equivalent to each other.     

The density of extremal points in Ekeland's variational principle

In this paper, we investigate the density of extremal points appeared in Ekeland's variational principle. By introducing radial intersections of sets, we give a very general result on the density of extremal points in the framework of locally convex spaces. This solves a problem proposed by G. Isac in 1997. From the general result we deduce several convenient criterions for judging the density of extremal points, which extend and improve a result of F. Cammaroto and A. Chinni. Using the equivalence between Ekeland's variational principle and Caristi's fixed point theorem, we obtain some density results on Caristi's fixed points.     

关于局部凸空间中向量Ekeland变分原理的等价性

基于各种Ekeland变分原理的等价形式, 主要研究局部凸空间中给定有界凸子集乘以距离函数为扰动的单调半连续映射的向量Ekeand变分原理的等价性问题. 首先利用局部凸空间中的向量Ekeland变分原理证明了向量Caristi-Kirk不动点定理,向量 Takahashi非凸极小化定理和向量Oettli-Th\'{e}ra定理. 进一步研究了向量Ekeland变分原理与向量Caristi-Kirk不动点定理,向量Takahashi非凸极小化定理和向量Oettli-Th\'{e}ra定理的等价性.     

Ekeland's variational principle in locally convex spaces and the density of extremal points

In this paper, we prove a general version of Ekeland's variational principle in locally convex spaces, where perturbations contain subadditive functions of topology generating seminorms and nonincreasing functions of the objective function. From this, we obtain a number of special versions of Ekeland's principle, which include all the known extensions of the principle in locally convex spaces. Moreover, we give a general criterion for judging the density of extremal points in the general Ekeland's principle, which extends and improves the related known results.     

A Frobenius Theorem for Locally Convex Global Analysis

 We prove a Frobenius theorem for Banach space complemented subbundles of the tangent bundle of a manifold modelled on locally convex spaces. The proof is based on an implicit function theorem for maps from locally convex spaces to Banach spaces proved in a recent paper of the author.     

A Frobenius Theorem for Locally Convex Global Analysis

 We prove a Frobenius theorem for Banach space complemented subbundles of the tangent bundle of a manifold modelled on locally convex spaces. The proof is based on an implicit function theorem for maps from locally convex spaces to Banach spaces proved in a recent paper of the author. (Received 15 March 1999; in revised form 2 June 1999)     

A drop theorem without vector topology

Daneš' drop theorem is extended to bornological vector spaces. An immediate application is to establish Ekeland-type variational principle and its equivalence, Caristi fixed point theorem, in bornological vector spaces. Meanwhile, since every locally convex space becomes a convex bornological vector space when equipped with the canonical von Neumann bornology, Qiu's generalization of Daneš' work to locally convex spaces is recovered.     

A property of Dunford-Pettis type in topological groups

The property of Dunford-Pettis for a locally convex space was introduced by Grothendieck in 1953. Since then it has been intensively studied, with especial emphasis in the framework of Banach space theory.

In this paper we define the Bohr sequential continuity property (BSCP) for a topological Abelian group. This notion could be the analogue to the Dunford-Pettis property in the context of groups. We have picked this name because the Bohr topology of the group and of the dual group plays an important role in the definition. We relate the BSCP with the Schur property, which also admits a natural formulation for Abelian topological groups, and we prove that they are equivalent within the class of separable metrizable locally quasi-convex groups.

For Banach spaces (or for metrizable locally convex spaces), considered in their additive structure, we show that the BSCP lies between the Schur and the Dunford-Pettis properties.

     

Diestel-Faires定理在局部凸空间中的推广

通过Banach 空间与局部凸空间的对比,将Banach 空间上的Diestel-Faires 定理在局部凸空间上进行推广。进一步给出了局部凸空间上的Orlicz-Pettis定理与推论。