关于解析凸性的注记

本文研究了解析一致凸性Ｂａｎａｃｈ空间,证明了重赋范定理并且给出了具有解析一致凸性的一些具体空间。     

On some generalizations of denting points

In this paper a local version is given of a well-known theorem for uniform rotund renorming of superreflexive Banach spaces.     

Average locally uniform rotundity and a class of nonlinear maps

We consider some topological characterizations of dual Banach spaces that admit an equivalent dual average locally uniformly rotund norm and provide a criterion for such renorming which involves the class of σ-slicely continuous maps.     

Uncountable equilateral sets in Banach spaces of the form <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)

The paper is concerned with the problem whether a nonseparable Banach space must contain an uncountable set of vectors such that the distances between every two distinct vectors of the set are the same. Such sets are called equilateral. We show that Martin’s axiom and the negation of the continuum hypothesis imply that every nonseparable Banach space of the form C(K) has an uncountable equilateral set. We also show that one cannot obtain such a result without an additional set-theoretic assumption since we construct an example of nonseparable Banach space of the form C(K) which has no uncountable equilateral set (or equivalently no uncountable (1+ε)-separated set in the unit sphere for any ε > 0) making another consistent combinatorial assumption. The compact K is a version of the split interval obtained from a sequence of functions which behave in an anti-Ramsey manner. It remains open if there is an absolute example of a nonseparable Banach space of the form different than C(K) which has no uncountable equilateral set. It follows from the results of S. Mercourakis and G. Vassiliadis that our example has an equivalent renorming in which it has an uncountable equilateral set. It remains open if there are consistent examples of nonseparable Banach spaces which have no uncountable equilateral sets in any equivalent renorming but it follows from the results of S. Todorcevic that it is consistent that every nonseparable Banach space has an equivalent renorming in which it has an uncountable equilateral set.     

Every weakly compact set can be uniformly embedded into a reflexive Banach space

Based on an application of the Davis-Figiel-Johnson-Pelzyski procedure, this note shows that every weakly compact subset of a Banach space can be uniformly embedded into a reflexive Banach space. As its application, we present the recent renorming theorems for reflexive spaces of Odell- Schlumprecht and Hajek-Johanis can be extended and localized to weakly compact convex subsets of an arbitrary Banach space.     

Characterization of reflexivity by equivalent renorming

A new rotundity property of Day's norm on c₀(Γ) is introduced. This property provides in particular a renorming characterization of the class of all reflexive Banach spaces.     

Every real Banach space can be renormed to satisfy the denseness of numerical radius attaining operators

First we show that every real Banach space satisfying a certain property, calledβ (used by Lindenstrauss and Partington) verifies the denseness of the numerical radius attaining operators. Using this result and a renorming theorem by Partington we conclude that every Banach space is isomorphic to a new one satisfying the denseness of the numerical radius attaining operators.     

对带误差项的Banach空间中完全广义集值拟变分包含的摄动迭代算法

研究了一类新的实Banach空间中的广义集值拟变分包含:f∈N(x，y) M(z，v) W(g(u)-h(w)，u)，它包含了近几年许多作者所作的变分包含同题．在买Banach空间中，利用极大增生算子的性质，建立了Banach空间中的广义集值拟变分包含和不动点问题间的等价性．利用这种等价性，建立了一些摄动迭代算法，并证明了近似解序列强收敛于精确解．本文的算法和结果改进和一般化了最近许多文章中相应的算法和结果。     

Characterization of inner product spaces in V- and L-spaces

In [4], Freese and Murphy introduce a new class of spaces, the V-spaces, which include Banach spaces, hyperbolic spaces, and other metric spaces. In this class of spaces they investigate conditions which are equivalent to strict convexity in Banach spaces, and extend some of the Banach space results to this new class of spaces. It is natural to ask if known characterizations of real inner product spaces among Banach spaces can also be extended to this larger class of spaces. In the present paper it will be shown that a metrization of an angle bisector property used in [3] to characterize real inner product spaces among Banach spaces also characterizes real inner product spaces among V-spaces, and among another class of spaces, the L-spaces, which include hyperbolic spaces and strictly convex Banach spaces. In the process it is shown that in a complete, convex, externally convex metric space M, if the foot of a point on a metric line is unique, then M satisfies the monotone property, thus answering a question raised in [4].     

Banach空间中一类广义混合非线性隐拟变分包含解的三步迭代

本文引入并研究了实Banach空间中一类新的广义混合非线性隐拟变分包含 ,通过对实Banach空间中的m 增生映象运用Nadler定理和Michael选择定理 ,构建了这类新的广义变分包含解的三步迭代算法 ,并证明了其解的存在性和由迭代算法生成的迭代序列的收敛性     

Banach空间的上带松弛共强制的广义隐式变分包含组

本文在Banach空间上引入和研究了一类带松弛共强制的广义隐式变分包含组（SNSIVI）．使用M增值算子的预解算子技术，我们构造了一类新的迭代算法逼近这类隐式变分包含组，且在q-一致平滑Banach空间上证明了这类迭代算法的收敛性．我们的结果推广和改进了最近的相关工作．     

A renorming of some nonseparable Banach spaces with the Fixed Point Property

We prove that for every Banach space which can be embedded in c₀(Γ) (for instance, reflexive spaces or more generally spaces with M-basis) there exists an equivalent renorming which enjoys the (weak) Fixed Point Property for non-expansive mappings. As a consequence, we solve a longtime open question in Metric Fixed Point Theory: Every reflexive Banach can be renormed to satisfy the Fixed Point Property. Furthermore, this norm can be chosen arbitrarily closed to the original norm.     

Banach spaces of universal disposition

In this paper we present a method to obtain Banach spaces of universal and almost-universal disposition with respect to a given class M of normed spaces. The method produces, among others, the only separable Banach space of almost-universal disposition with respect to the class F of finite-dimensional spaces (Gurari? space G); or the only, under CH, Banach space with density character the continuum which is of universal disposition with respect to the class S of separable spaces (Kubis space K). We moreover show that K is isomorphic to an ultrapower of the Gurari? space and that it is not isomorphic to a complemented subspace of any C(K)-space. Other properties of spaces of universal disposition are also studied: separable injectivity, partially automorphic character and uniqueness.     

Sum theorems for monotone operators and convex functions

In this paper, we derive sufficient conditions for the sum of two or more maximal monotone operators on a reflexive Banach space to be maximal monotone, and we achieve this without any renorming theorems or fixed-point-related concepts. In the course of this, we will develop a generalization of the uniform boundedness theorem for (possibly nonreflexive) Banach spaces. We will apply this to obtain the Fenchel Duality Theorem for the sum of two or more proper, convex lower semicontinuous functions under the appropriate constraint qualifications, and also to obtain additional results on the relation between the effective domains of such functions and the domains of their subdifferentials. The other main tool that we use is a standard minimax theorem.

     

On uniformly convex functions

Non-convex functions that yet satisfy a condition of uniform convexity for non-close points can arise in discrete constructions. We prove that this sort of discrete uniform convexity is inherited by the convex envelope, which is the key to obtain other remarkable properties such as the coercivity. Our techniques allow to retrieve Enflo's uniformly convex renorming of super-reflexive Banach spaces as the regularization of a raw function built from trees. Among other applications, we provide a sharp estimation of the distance of a given function to the set of differences of Lipschitz convex functions. Finally, we prove the equivalence of several possible ways to quantify the super weakly noncompactness of a convex subset of a Banach space.     

Universal mappings for certain classes of operators and polynomials between Banach spaces

A well‐known result of J. Lindenstrauss and A. Pełczyński (1968) gives the existence of a universal non‐weakly compact operator between Banach spaces. We show the existence of universal non‐Rosenthal, non‐limited, and non‐Grothendieck operators. We also prove that there does not exist a universal non‐Dunford–Pettis operator, but there is a universal class of non‐Dunford–Pettis operators. Moreover, we show that, for several classes of polynomials between Banach spaces, including the non‐weakly compact polynomials, there does not exist a universal polynomial.     

Locally uniformly rotund renorming and fragmentability

In this paper we characterize those Banach spaces E that admita locally uniformly rotund renorming by means of a linear topologicalcondition that is a particular case of the notion, introducedby J. E. Jayne, I. Namioka and C. A. Rogers, called countablecover by sets of small local diameter. This allows us to developa Decomposition Method and a Transfer Technique to obtain locallyuniformly rotund renormings. 1991 Mathematics Subject Classification:46B20.     

Universal extrapolation spaces for \hbox {C}_{0}-semigroups

The classical theory of Sobolev towers allows for the construction of an infinite ascending chain of extrapolation spaces and an infinite descending chain of interpolation spaces associated with a given \(C_0\) -semigroup on a Banach space. In this note we first generalize the latter to the case of a strongly continuous and exponentially equicontinuous semigroup on a complete locally convex space. As a new concept—even for \(C_0\) -semigroups on Banach spaces—we then define a universal extrapolation space as the completion of the inductive limit of the ascending chain. Under mild assumptions we show that the semigroup extends to this space and that it is generated by an automorphism of the latter. Dually, we define a universal interpolation space as the projective limit of the descending chain. We show that the restriction of the initial semigroup to this space is again a semigroup and always has an automorphism as generator.     

A New Double-Projection Method for Solving Variational Inequalities in Banach Spaces

In this paper, we study the variational inequalities involving monotone and Lipschitz continuous mapping in Banach spaces. A new and simple iterative method, which combines Halpern’s technique and the subgradient extragradient idea, is given. Under mild and standard assumptions, we establish the strong convergence of our algorithm in a uniformly smooth and convex Banach spaces. We also present a modification of our method using a line-search approach, this enable to obtain strong convergence in real and reflexive Banach spaces, without the prior knowledge of the Lipschitz constant. Numerical experiments illustrate the performances of our new algorithm and provide a comparison with related algorithms. Our results generalize and extend some of the existing works in Hilbert spaces to Banach spaces as well as provide an extension from weak to strong convergence.     

On Banach Spaces Whose Unit Sphere Determines Polynomials

In this paper we study the problem of characterizing the real Banach spaces whose unit sphere determines polynomials, i.e., if two polynomials coincide in the unit sphere, is this sufficient to guarantee that they are identical？ We show that, in the frame of spaces with unconditional basis, non- reflexivity is a sufficient, although not necessary, condition for the above question to have an affirmative answer. We prove that the only lp^n spaces having this property are those with p irrational, while the only lp spaces which do not enjoy it are those with p an even integer. We also introduce a class of polynomial determining sets in any real Banach space.