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。     

Very Convex Banach Spaces

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Integral equations in reflexive Banach spaces and weak topologies

The Schauder Tychonoff theorem in a locally convex topological space is used to establish existence results for Volterra-Hammerstein and Hammerstein integral equations in a reflexive Banach space.

     

On complemented subspaces of sums and products of Banach spaces

It is proved that there exist complemented subspaces of countable topological products (locally convex direct sums) of Banach spaces which cannot be represented as topological products (locally convex direct sums) of Banach spaces.

     

Locally Asplund Preduals of Spaces of Holomorphic Functions

Defant [5] introduced the local Radon–Nikodym property for duals of locally convex spaces. This is a generalization of Asplund spaces as defined in Banach space theory. In this paper we generalise Dunford"s Theorem [7] to Banach spaces with Schauder decompositions and apply this result to spaces of holomorphic functions on balanced domains in a Banach space. This revised version was published online in June 2006 with corrections to the Cover Date.     

σ半凸性与Ekeland变分原理

本文引进了σ半凸性,并给出了局部凸空间中一般的Ekeland变分原理.     

Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces

In this paper, we introduce the concept of τ-function which generalizes the concept of w-distance studied in the literature. We establish a generalized Ekeland's variational principle in the setting of lower semicontinuous from above and τ-functions. As applications of our Ekeland's variational principle, we derive generalized Caristi's (common) fixed point theorems, a generalized Takahashi's nonconvex minimization theorem, a nonconvex minimax theorem, a nonconvex equilibrium theorem and a generalized flower petal theorem for lower semicontinuous from above functions or lower semicontinuous functions in the complete metric spaces. We also prove that these theorems also imply our Ekeland's variational principle.     

Vectorial variational principle with variable set-valued perturbation

We give a general vectorial Ekeland's variational principle, where the objective function is defined on an F-type topological space and taking values in a pre-ordered real linear space. Being quite different from the previous versions of vectorial Ekeland's variational principle, the perturbation in our version is no longer only dependent on a fixed positive vector or a fixed family of positive vectors. It contains a family of set-valued functions taking values in the positive cone and a family of subadditive functions of topology generating quasi-metrics. Hence, the direction of the perturbation in the new version is a family of variable subsets which are dependent on the ob jective function values. The general version includes and improves a number of known versions of vectorial Ekeland's variational principle. From the general Ekeland's principle, we deduce the corresponding versions of Caristi–Kirk's fixed point theorem and Takahashi's nonconvex minimization theorem. Finally, we prove that all the three theorems are equivalent to each other.     

局部凸空间中的Drop定理,Phelps引理和Ekeland变分原理的推广

在局部凸空间框架下,我们利用Drop定理,Phelps引理和Ekeland变分原理的赋范线性空间的形式对其分别进行了推广.并且阐述了这些定理之间以及和它们赋范线性空间的形式之间是等价的.     

The strong Ekeland variational principle

In this paper, we consider the strong Ekeland variational principle due to Georgiev [P.G. Georgiev, The strong Ekeland variational principle, the strong drop theorem and applications, J. Math. Anal. Appl. 131 (1988) 1–21]. We discuss it for functions defined on Banach spaces and on compact metric spaces. We also prove the τ-distance version of it.     

Uniqueness of <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroups on a general locally convex vector space and an application

The main purpose is to generalize a theorem of Arendt about uniqueness of C ₀-semigroups from Banach space setting to general locally convex vector spaces. More precisely, we show that cores are the only domains of uniqueness for C ₀-semigroups on locally convex spaces. As an application, we find a necessary and sufficient condition for that the mass transport equation has one unique L ¹(ℝ ^d ,dx) weak solution.