Strongly Generated Banach Spaces and Measures of Noncompactness

To generalize the Hausdorff measure of noncompactness to other classes of bounded sets (like e. g. conditionally weakly compact or Asplund sets), we introduce Grothendieck classes. We deduce integral inequalities for quantities (called Grothendieck measures) related to these classes. As a by-product, we can answer a question concerning the measure of noncompactness for linear T : X → Y introduced in [14], and generalize a theorem about weak solutions of differential equations in Banach spaces.     

A Note on Ideal Spaces of Banach Algebras

In a previous paper, the second author introduced a compacttopology _r on the space of closed ideals of a unital Banachalgebra A. If A is separable, then _r is either metrizable orelse neither Hausdorff nor first countable. Here it is shownthat _r is Hausdorff if A is C¹[0, 1], but that if A is a uniformalgebra, then _r is Hausdorff if and only if A has spectral synthesis.An example is given of a strongly regular, uniform algebra forwhich every maximal ideal has a bounded approximate identity,but which does not have spectral synthesis. 1991 MathematicsSubject Classification 46H10.     

A Note on the Mann Iteration for k-Strict Pseudocontractions in Banach Spaces

We show that a variant of a previously defined function can be used to characterize 2-uniform smoothness. We then obtain greatly simplified proofs of two convergence theorems in the literature using a generalization of a lemma of and the aforementioned characterization. We also more easily obtain rates of asymptotic regularity corresponding to the studied iterations. Finally, we derive a way to relate two constants which are characteristic to 2-uniformly smooth spaces.     

Banach空间中余弦算子函数生成元的有界性

A是Banach空间X中余弦算子函数C（t),t∈R，和正弦算子函数S（t),t∈R，的生成元。本证明了，对每个f∈C（[0,T];X)，连续函数u,u(t)=∫-tS(t-s)f(s)ds,f∈[0,T]是二阶非齐次0初值问题，u″=Au f的强解的充要条件是：A是空间X中的有界算子。     

A Note on the Global Solutions of the Cauchy Problem in Banach Spaces

The aim of this paper is to describe the topological structure of global solutions set of the initial value problem in Banach spaces. More precisely, we prove that under suitable conditions this set is homeomorphic to the intersection of a decreasing sequence of compact absolute retracts (shortly: it is an R ₅ set). The main condition in our result is formulated in terms of the Kuratowski measure of noncompactness.     

A Note on Solvability of the Nonlinear Abstract Viscoelastic Problem in Banach Spaces

In this work we are concerned with the existence of integral solution for a nonlinear abstract viscoelastic problem in Banach space, where the operator is accretive dependent on time.     

Banach空间的弱*序列紧性

本文研究了Ｂａｎａｃｈ空间的弱序列紧性．Ｂａｎａｃｈ空间Ｘ称为有（ｗ）性质，如果Ｘ（Ｘ的共轭空间）的每个有界序列有弱收敛子列．我们证明了，如果Ｂａｎａｃｈ空间Ｘ有（ｗ）性质，那么ｌｐ（Ｘ）（１≤ｐ＜＋∞）与ｃ０（Ｘ）也有（ｗ）性质．     

Convergence of Sequential Parafirmly Nonexpansive Mappings in Reflexive Banach Spaces

In this paper, we study parafirmly nonexpansive mappings with Bregman distance and weak convergence of underrelaxed sequential parafirmly nonexpansive mappings.     

关于Banach空间中度量广义逆扰动定理的注记

Banach空间中线性算子的度量广义逆扰动定理具有重要应用,引入关注.在Acta Math Sinica English Series,2014(7)中对于从Banach空间X到Banach空间Y的有界线性算子T,在T的值域R(T)为切比雪夫子空间,T的零空间N(T)为切比雪夫子空间,且度量投影π_(N(T))为线性的条件下,得到二个有关非线性的广义逆扰动定理.本文证得:上述扰动定理结论的条件无需假定π_(N(T))的线性,只需假定N(T),R(T)分别为X,Y中的切比雪夫子空间即可.     

Banach空间中全纯映射的若干性质

本文将全纯映射的若干性质推广到Ｂａｎａｃｈ空间，并应用这些性质研究有界域中的凸映射与星形映射．     

赋序列范数的矢值Banach序列空间ss(E)的凸性

本文对赋序列范数的矢值Banach序列空间ss(E)的一些凸性进行了讨论,得到的主要结果如下: 1.ss(E)是局部一致凸的当且仅当ss和E是局部一致凸的; 2.ss(E)是强凸的当且仅当ss和E是强凸的; 3.设ss和ss*具有AK性质,则ss(E)是非常凸的当且仅当ss和E是非常凸的.     

Banach空间上凸泛函的某些对偶性质

本语文对Ｂａｎａｃｈ空间上的凸泛函建立了若干对偶性质,通过对偶的手段,建立了非常简洁的凸泛函极小化序列弱收敛的特征定理。     

Banach空间中二阶脉冲积分方程周期边值问题的注记

通过建立Ｂａｎａｃｈ空间二阶非线性脉冲微分 积分方程周期边值问题新的比较定理，给出了其最大解和最小解的存在性．     

Summability Properties for Multiplication Operators on Banach Function Spaces

Consider a couple of Banach function spaces X and Y over the same measure space and the space X ^Y of multiplication operators from X into Y. In this paper we develop the setting for characterizing certain summability properties satisfied by the elements of X ^Y . At this end, using the “generalized Köthe duality” for Banach function spaces, we introduce a new class of norms for spaces consisting of infinite sums of products of the type xy with ${x \in X}Consider a couple of Banach function spaces X and Y over the same measure space and the space X ^Y of multiplication operators from X into Y. In this paper we develop the setting for characterizing certain summability properties satisfied by the elements of X ^Y . At this end, using the “generalized K?the duality” for Banach function spaces, we introduce a new class of norms for spaces consisting of infinite sums of products of the type xy with x ? X{x \in X} and y ? Y{y \in Y} .     

On Sequential Properties of Spaces of Measures

Mathematical Notes -     

关于sn度量空间的注记

The main purpose of this paper is to establish the mapping relation between metric spaces and sn-metric spaces.     

A Note on Sobolev Spaces

In the theory of averaging and other branches of mathematical physics, a crucial role is played by Sobolev spaces connected with a Borel measure defined in the Euclidean space ℝ^d. These spaces are defined as closures of the sets of smooth functions in an appropriate norm. In this paper, we propose another (dual) definition of Sobolev spaces and give an example of using this definition.__________Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 10, Suzdal Conference-4, 2003.     

关于Banach空间的无条件基

给出具有唯一无条件基的无穷维Banach空间,并给出其无条件基的若干性质.     

Banach 空间中von Neumann-Jordan 型常数的一些性质

该文主要给出von Neumann-Jordan型常数的一些性质. 首先得到利用von Neumann-Jordan型常数来刻划一致非 方的等价条件. 其次介绍了von Neumann-Jordan型常数与正规结构的关系. 最后介绍了C_1'(X)与其他空间常数之间的关系, 例如C_1'(X)与C₁(X)之间的关系等等.     

Extremal Properties of Contraction Semigroups on Hilbert and Banach Spaces

Let f be a unit vector and T = {T(t) = e^tA: t 0} be a (C₀)contraction semigroup generated by A on a complex Hilbert spaceX. If |T(t)f,f| 1 as t then f is an eigenvector of A correspondingto a purely imaginary eigenvalue. If one allows X to be a Banachspace, the same situation can be considered by replacing T(t)f,fby (T(t)f) where is a unit vector in X^* dual to f. If |(T(t)f)| 1, as t , is f an eigenvector of A? The answer is sometimesyes and sometimes no.