Banach空间范数的粗性与不可微性

本文讨论了粗范数的情形,所得结果表明:粗范数在概念上强于无处Fréchet可微而弱于一致不Fréchet可微,并给出若干范数粗性的不可微特征。     

Radon-Nikodym性质与范数的Gateaux可微性

本研究了Banach空间中集合的RNP与范数的Gateaux可微性之间的关系，给出了关于Banach空间中集合具RNP的一些充要条件．     

赋Orlicz范数的Orlicz空间中最佳逼近算子

在赋Ｏｒｌｉｃｚ范数的Ｏｒｌｉｃｚ空间中,给出最佳逼近算子单调性的一个充分条件和最佳逼近元存在定理．     

Banach空间上非Lipschitz映射的渐近行为

设Ｃ是具Ｆｒéｃｈｅｔ可微范数的一致凸Ｂａｎａｃｈ空间Ｅ的非空凸闭子集，Ｇ是定向集．｛Ｔ_ｔ：ｔ∈Ｇ｝是Ｃ上的渐近非扩张型映照．本文给出了｛Ｔ_ｔ：ｔ∈Ｇ｝的弱收敛定理．     

赋Orlicz范数的Orlicz序列空间的接近一致凸性

本文给出了赋Ｏｒｌｉｃｚ序列空间的Ｈ点的充分必要条件,进而得至了赋Ｏｒｌｉｃｚ范数的Ｏｒｌｉｃｚ序列空间具有（Ｈ）性质的判据．同时给出了赋Ｏｒｌｉｃｚ范数的Ｏｒｌｉｃｚ序列空间的一致ＵＫＫ和接近一致凸性的等价条件。     

赋Orlicz范数Musielak-Orlicz序列空间的一致Gateaux可微性

给出赋Orlicz范数的Musielak-Orlicz序列空间具有一致Gateaux可微性的充分必要条件.     

Musielak－Orlicz序列空间的一些凸性

本文给出了赋Ｏｒｌｉｃｚ范数的Ｍｕｓｉｅｌａｋ－Ｏｒｌｉｃｚ序列空间的局部一致凸性，（Ｈ）性质，（Ｓ）性质的判据．     

Musielak-Orlicz序列空间的一致Gateaux可微性

利用Musielak-Orlicz函数列的某些性质,给出了赋Luxemburg范数的Musielak-Orlicz序列空间是有一致Gateaux可微性的充要条件及赋Orlicz范数的Musielak-Orlicz序列空间是弱一致凸的判别准则.     

非线性波动方程解的Blow up

本文主要讨论高维空间非线性波动方程的Ｃａｕｃｈｙ问题整体解的非存在性，我们证明对Ｕｕ－△ｕ＝ｆ（ｕ），ｆ（ｕ）＝ｃ｜ｕ｜ｐ－１ｕ，当１＜ｐ≥时，若初始能量非正，则无论初值数据的ｃｋ－范数（连续空间范数）多么小，解按Ｃｋ－范数或按Ｈｓ（Ｒｎ）（Ｓ≥１）都在有限时间内Ｂｌｏｗｕｐ，并且有相同的生命跨度．     

一致凸Banach空间中渐近非扩张映象的几乎轨道的渐近行为

设Ｘ是有Ｆｒéｃｈｅｔ可微范数的一致凸Ｂａｎａｃｈ空间，Ｃ是Ｘ的有界闭凸子集，Ｔ：Ｃ→Ｃ是一个渐近非扩张映象．证明了，如果｛ｘ_ｎ：ｎ≥１｝是Ｔ的几乎轨道，则序列｛ｘ０｝弱几乎收敛到集合∩from∞to(ｎ＝１)ｃｏ｛ｘ_i：ｉ≥ｎ｝∩Ｆ（Ｔ）的唯一点，其中，Ｆ（Ｔ）是Ｔ的不动点集．     

The differentiability of real functions on normed linear space using generalized subgradients

The modification of the Clarke generalized subdifferential due to Michel and Penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space. The Gâteaux differentiability of any real function can be deduced from the Gâteaux differentiability of the norm if the function has a directional derivative which attains a constant related to its generalized directional derivative. For any distance function on a space with uniformly Gâteaux differentiable norm, the Clarke and Michel-Penot generalized subdifferentials at points off the set reduce to the same object and this generates a continuity characterization for Gâteaux differentiability. However, on a Banach space with rotund dual, the Fréchet differentiability of a distance function implies that it is a convex function. A mean value theorem for the modified generalized subdifferential has implications for Gâteaux differentiability.     

Geometric conditions of differentiability for a regular locally Lipschitz function

The differentiability of a norm of a Banach space may be characterized by its unit sphere. This paper generalizes these geometric conditions of norm's differentiability to the case of a regular locally Lipschitz function. Supported by the National Natural Science Foundation of China     

Uniform Gateaux differentiability and weak uniform rotundity of Musielak–Orlicz function spaces

Criteria for uniform Gateaux differentiability and weak uniform rotundity of Musielak–Orlicz function spaces equipped with the Orlicz norm as well as the Luxemburg norm are presented. Interpretations of the criteria in the case of Orlicz spaces are given. It is worth noticing that criteria for the Luxemburg norm are different than for the Orlicz norm.     

On Differentiability of Metric Projections onto Moving Convex Sets

We consider properties of the metric projections onto moving convex sets in normed linear spaces. Under certain conditions about the norm, directional differentiability of first and higher order of the metric projections at boundary points is characterized. The conditions are formulated in terms of differentiability of multifunctions and properties of the set-derivatives are shown.     

Banach空间范数的k-点态粗性和k-粗性

对Banach空间范数引入了k-点态粗和k-粗的概念,利用Banach空间理论的方法,给出了x∈S(X)为范数的k-粗糙点和X的范数是k-粗的一些充分必要条件,证明了(k+1)-粗糙点是k-粗糙点以及k-粗糙点与Fréchet可微性的一些结果.特别地,在k=1的情形下蕴含了关于范数的粗糙点、点态粗范数和粗范数的相应结果.     

A new proof of Fréchet differentiability of Lipschitz functions

We give a relatively simple (self-contained) proof that every real-valued Lipschitz function on ℓ₂ (or more generally on an Asplund space) has points of Fréchet differentiability. Somewhat more generally, we show that a real-valued Lipschitz function on a separable Banach space has points of Fréchet differentiability provided that the w ^* closure of the set of its points of Gateaux differentiability is norm separable. Received May 31, 1999 / final version received February 16, 2000?Published online April 19, 2000     

Uniform Gateaux differentiability and weak uniform rotundity in Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm

Criteria for weak uniform rotundity and reflexivity of Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm are given. Although, criteria for uniform Gateaux differentiability and weak uniform rotundity of Musielak-Orlicz function spaces of real functions equipped with the Luxemburg norm were known, they can be also easily deduced from our main results.     

On norm continuity,differentiability and compactness of perturbed semigroups

Semigroup Forum - The main purpose of this paper is to treat semigroup properties like norm continuity, compactness and differentiability for perturbed semigroups in Banach spaces. In particular,...     

Supporting Functions and The Differentiabilities of the Norms on Banach Spaces

ＳｕｐｐｏｒｔｉｎｇＦｕｎｃｔｉｏｎｓａｎｄＴｈｅＤｉｆｆｅｒｅｎｔｉａｂｉｌｉｔｉｅｓｏｆｔｈｅＮｏｒｍｓｏｎＢａｎａｃｈＳｐａｃｅｓ￥ＷａｎｇＪｉａｎｇｅｎ（王建根）（ＬｕｏｙａｎｇＴｅａｃｈｅｒ＇ｓＣｏｌｌｅｇｅ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓ...     

Strong bounds on perturbations

This paper provides strong bounds on perturbations over a collection of independent random variables, where ‘strong’ has to be understood as uniform w.r.t. some functional norm. Our analysis is based on studying the concept of weak differentiability. By applying a fundamental result from the theory of Banach spaces, we show that weak differentiability implies norm Lipschitz continuity. This result leads to bounds on the sensitivity of finite products of probability measures, in norm sense. We apply our results to derive bounds on perturbations for the transient waiting times in a G/G/1 queue. This research is supported by the Technology Foundation STW, applied science division of NWO and the technology programme of the Ministry of Economic Affairs.