THREE KINDS OF DENTABILITIES IN BANACH SPACES AND THEIR APPLICATIONS

In this paper,we study some dentabilities in Banach spaces which are closely related to the famous Radon-Nikodym property.We introduce the concepts of the weak^(*)-weak denting point and the weak^(*)-weak^(*)denting point of a set.These are the generalizations of the weak^(*)denting point of a set in a dual Banach space.By use of the weak^(*)-weak denting point,we characterize the very smooth space,the point of weak^(*)-weak continuity,and the extreme point of a unit ball in a dual Banach space.Meanwhile,we also characterize an approximatively weak compact Chebyshev set in dual Banach spaces.Moreover,we define the nearly weak dentability in Banach spaces,which is a generalization of near dentability.We prove the necessary and sufficient conditions of the reflexivity by nearly weak dentability.We also obtain that nearly weak dentability is equivalent to both the approximatively weak compactness of Banach spaces and the w-strong proximinality of every closed convex subset of Banach spaces.

Space Topologies and Their Dual Space Topologies for Conventional Functional Space Topologies

In this paper, we have studied the topology of some classical functional spaces. Among these spaces, there are standard spaces, spaces that can be metrizable and others that cannot be metrizable. But they are all topological vector spaces and it is in this context that we have chosen to present this work. We are interested in the topology of its spaces and in the topologies of their dual spaces. The first part, we presented the fundamental topological properties of topological vector spaces. The second part, we studied Frechet spaces and particularly the space S(Rn) of functions of class C∞ on Rn which are as well as all their rapidly decreasing partial derivatives. We have also studied its dual S'(Rn) the space of tempered distributions. The last part aims to define a topological structure on an increasing union of Frechet spaces called inductive limit of Frechet spaces. We study in particular the space D(Ω) of functions of class C∞ with compact supports on Ω as well as its dual D' (Ω) the space distributions over the open set Ω.

ATOMIC DECOATOMIC DECOMPOSITIONS AND DUALS OF WEAK HARDY SPACES OF B-VALUED MARTINGALES

In this article, several weak Hardy spaces of Banach-space-valued martingales are introduced, some atomic decomposition theorems for them are established and their duals are investigated. The results closely depend on the geometrical properties of the Banach space in which the martingales take values.

Atomic Decompositions of Weak Hardy Spaces of B-Valued Martingales

The atomic decompositions of weak Hardy spaces of Banach-space-valued martingales are given. With the help of the atomic decompositions, some inequalities for B-valued martingales are established in the case 0〈r≤1. Here the results are connected closely with the p-uniform smoothness and q-uniform convexity of Banach spaces which the martingales take values in.

WEAK ORLICZ SPACE AND ITS CONVERGENCE THEOREMS

In this article, a class of weak Orlicz function spaces is defined and their basic properties are discused. In particular, for the sequences in weak Orlicz space, we establish several basic convergence theorems including bounded convergence theorem, control convergence theorem and Vitali-type convergence theorem and so on. Moreover, the conditional compactness of its subsets is also discussed.

WEAK ATOMIC DECOMPOSITIONS OF MARTINGALE HARDY-LORENTZ SPACES AND APPLICATIONS

In this article, we establish some atomic decomposition theorems for martin- gale Hardy-Lorentz spaces. As applications, with the help of weak atomic decompositions, some interpolation theorems for martingale Hardy-Lorentz spaces are proved.

BOUNDEDNESS OF COMMUTATORS ON HOMOGENEOUS MORREY-HERZ SPACES OVER LOCALLY COMPACT VILENKIN GROUPS

This paper studies some boundedness results of commutators on a class of new spaces MKp,q^αλ （G） named as homogenous Morrey-Herz spaces over locally compact Vilenkin groups

CONTINUITY OF MARCINKIEWICZ INTEGRALS ON HOMOGENEOUS MORREY-HERZ SPACES

Let μΩ,b^m be the higher order commutator generated by Marcinkiewicz integral μΩ and a BMO（R^n） function b（x）. In this paper, we will study the continuity of μΩ and μΩ,b^m on homogeneous Morrey-Herz spaces.

CBMO ESTIMATES FOR MARCINKIEWICZ INTEGRALS ON HOMOGENEOUS MORREY-HERZ SPACES

The boundedness of the commutator μΩ,b generalized by Marcinkiewicz integral μΩ and a function b（x） ∈ CBMOq （Rn） on homogeneous Morrey-Herz spaces is established.

COMMON FIXED POINT OF GENERALIZED WEAKLY CONTRACTIVE MAPS IN PARTIAL METRIC SPACES

In this paper, using the context of complete partial metric spaces, some common fixed point results of maps that satisfy the generalized （ψ, Ф）-weak contractive conditions are obtained. Our results generalize, extend, unify, enrich and complement many existing results in the literature. Example are given showing the validaty of our results.

ODEs IN BANACH SPACES AND FIXED POINTS OF WEAKLY INWARD MAPS

This paper is an attempt to investigate systematically fixed points of weakly inward maps by using some basic results from differential equations in Banach spaces. By investigating the Poincare operators for such differential equations, we establish a fixed point index theory for two classes of weakly inward maps.

Unique Common Fixed Points for Two Weakly C~*-contractive Mappings on Partially Ordered 2-metric Spaces

In this paper, we give existence theorems of common fixed points for two mappings with a weakly C*-contractive condition on partially ordered 2-metric spaces and give a sufficient condition under which there exists a unique common fixed point.

Weakly δ-continuous Correspondences on Super-space

In the paper,we define weakly δ-continuous correspondences on super-space,On the basis of δ-open(closed) sets,θ-open(closed) sets and regular open(closed) sets in topological space,some equivalent conditions of this kind of correspondences are obtained,and some applications of subset nets and convergence nets are given.

Common Fixed Point Theorems for Compatible and Weakly Compatible Maps in G-Metric Spaces

In this paper first we prove common fixed point theorems for compatible and weakly compatible maps. Secondly, we prove common fixed point theorems for weakly compatible maps along with property (E.A.) and (CLRg) property respectively.

Common Fixed Point Theorems for Weakly Compatible Mappings in Fuzzy Metric Spaces Using (JCLR) Property

In this paper, we prove a common fixed point theorem for a pair of weakly compatible mappings in fuzzy metric space using the joint common limit in the range property of mappings called (JCLR) property. An example is also furnished which demonstrates the validity of main result. We also extend our main result to two finite families of self mappings. Our results improve and generalize results of Cho et al. [Y. J. Cho, S. Sedghi and N. Shobe, “Generalized fixed point theorems for compatible mappings with some types in fuzzy metric spaces,” Chaos, Solitons & Fractals, Vol. 39, No. 5, 2009, pp. 2233-2244.] and several known results existing in the literature.

Weak KKM theorems in generalized finitely continuous space with applications

Some new weak Knaster-Kuratouski-Mazurkiewicz （KKM） theorems are proved under the noncompact situation in the generalized finitely continuous space （GFC-space） without any convexity. As applications, the minimax inequalities of the Ky Fan type are also given under some suitable conditions. The results unify and generalize some known results in recent literatures.

Boundedness of Multilinear Oscillatory Singular Integral on Weighted Weak Hardy Spaces

In this paper, by using the atomic decomposition of the weighted weak Hardy space WH;（R;）, the authors discuss a class of multilinear oscillatory singular integrals and obtain their boundedness from the weighted weak Hardy space WH;（R;） to the weighted weak Lebesgue space WL;（R;） for ω∈A;（R;）.

Boundedness of Multilinear Commutators with Rough Kernels on Morrey-Herz Spaces

This paper is concerning the commutators generated by the multilinear singular integral with rough kernels and BMO functions. The boundedness of the multilinear commutators Tb→ （f→） is established on the Morrey-Herz space by using the John-Nirenberg inequality.

Boundedness of Some Singular Integral Operators on Morrey-Herz Spaces over Locally Compact Vilenkin Groups

In this paper the boundedness results for some fractional and singular integral operators on the homogeneous Morrey-Herz spaces over locally compact Vilenkin groups are shown.

Characterization of Type p Banach Spaces by the Weak Law of Large Numbers

For weighted sums of the form ?j = 1kn anj Xnj\sum {_{j = 1}^{k_n } } a_{nj} X_{nj} where {a nj , 1 ?j?k n ↑∞,n?1} is a real constant array and {X aj , 1≤j≤k n, n≥1} is a rowwise independent, zero mean, random element array in a real separable Banach space of typep, we establishL r convergence theorem and a general weak law of large numbers respectively, conversely, we characterize Banach spaces of typep in terms of convergence inr-th mean and probability for such weighted sums.