光滑Banach空间上扩张值函数的Frechet次微分

首先证明了Frechet光滑Banach空间上齐次函数的次微分的一个有用定理,然后利用下半连续函数和的次微分规则把Clarke-Ledyaev多方向中值不等式推广到多个函数的情形.     

Convex Optimization on Banach Spaces

Greedy algorithms which use only function evaluations are applied to convex optimization in a general Banach space \(X\). Along with algorithms that use exact evaluations, algorithms with approximate evaluations are treated. A priori upper bounds for the convergence rate of the proposed algorithms are given. These bounds depend on the smoothness of the objective function and the sparsity or compressibility (with respect to a given dictionary) of a point in \(X\) where the minimum is attained.     

Very Convex Banach Spaces

ＶｅｒｙＣｏｎｖｅｘＢａｎａｃｈＳｐａｃｅｓＴｅｇｕｓｉ（特古斯）Ｓｕｙａｌａｔｕ（苏雅拉图）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＩｎｎｅｒＭｏｎｇｏｌｉａＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｈｕｈｈｏｔ，０１００２２）ＬｉＹｏｎｇｊｉｎ...     

一致Banach空间中非扩张映象的弱收敛定理

设犈是一致凸Ｂａｎａｃｈ空间，满足Ｏｐｉａｌ条件或具有Ｆｒｅｃｈｅｔ可微范数，犆是犈的非空闭凸子集，且犜：犆→犆是非扩张映象．又设对任何初始数据狓１ ∈犆，序列｛狓狀｝由下列修改了的Ｉｓｈｉｋａｗａ迭代程序生成：狓狀＋１ ＝狋狀犜狀（狊狀犜狀狓狀＋ （１－狊狀）狓狀）＋ （１－狋狀）狓狀，　狀≥１， （Ｉ）其中，数列｛狋狀｝与｛狊狀｝满足下列条件（ｉ）和（ｉｉ）之一：（ｉ）狋狀∈ ［犪，犫］且狊狀∈ ［０，犫］；（ｉｉ）狋狀∈ ［犪，１］且狊狀∈ ［犪，犫］，这里，常数犪，犫满足０＜犪≤犫＜１．作者证明了，犜有不动点的充要条件是，｛狓狀｝ 弱收敛且｛‖狓狀－犜狓狀‖｝收敛到０．而且，由此即知，若犜有不动点，则｛狓狀｝弱收敛到犜的一个不动点．     

Mazur Intersection Properties and Differentiability of Convex Functions in Banach Spaces

It is proved that the dual of a Banach space with the Mazurintersection property is almost weak* Asplund. Analogously,the predual of a dual space with the weak* Mazur intersectionproperty is almost Asplund. Through the use of these arguments,it is found that, in particular, almost all (in the Baire sense)equivalent norms on l₁() and l() are Fréchet differentiableon a dense G subset. Necessary conditions for Mazur intersectionproperties in terms of convex sets satisfying a Krein–Milmantype condition are also discussed. It is also shown that, ifa Banach space has the Mazur intersection property, then everysubspace of countable codimension can be equivalently renormedto satisfy this property.     

Measure of Non-Radon–Nikodym Property and Differentiability of Convex Functions on Banach Spaces

We introduce and discuss measure of non-Radon–Nikodym property to investigate differentiability of continuous convex functions on Banach spaces. Using this new concept, we establish some results about Asplund spaces and generic Fréchet differentiability of continuous convex functions on non-Asplund spaces.Mathematics Subject Classifications (2000) 46B22, 49J50, 49J53.     

Weak and Strong Convergence of an Implicit Iteration Process in Uniformly Convex Banach Spaces

The purpose of this article is to introduce an implicit iteration process for approximating common fixed points of three finite families nonexpansive mappings and to prove weak and strong convergence theorems in uniformly convex Banach spaces.     

On the Weak Semi-continuity of Vector Functions and Minimum Problems

Lower semi-continuity from above or upper semi-continuity from below has been used by many authors in recent papers. In this paper, we first study the weak semi-continuity for vector functions having particular form as that of Browder in ordered normed vector spaces; we obtain several new results on the lower semi-continuity from above or upper semi-continuity from below for these vector functions. Our results generalize some well-known results of Browder in scalar case. Secondly, we study the minimum or maximum problems for vector functions satisfying lower semi-continuous from above or upper semi-continuous from below conditions; several new results on the existence of minimal points or maximal points are obtained. We also use these results to study vector equilibrium problems and von Neumann’s minimax principle in ordered normed vector spaces.     

More on P-Stable Convex Sets in Banach Spaces

We study the asymptotic behavior and limit distributions for sums S _n =b_n ^-1 _i=1 ⁿ _i,where _i, i 1, are i.i.d. random convex compact (cc) sets in a given separable Banach space B and summation is defined in a sense of Minkowski. The following results are obtained: (i) Series (LePage type) and Poisson integral representations of random stable cc sets in B are established; (ii) The invariance principle for processes S _n(t) =b_n ^-1 _i=1 ^[nt] _i, t[0, 1], and the existence of p-stable cc Levy motion are proved; (iii) In the case, where _i are segments, the limit of S _n is proved to be countable zonotope. Furthermore, if B = R ^d , the singularity of distributions of two countable zonotopes Y_{p 1}, ₁,Y_{p 2}, ₂, corresponding to values of exponents p ₁, p ₂ and spectral measures ₁, ₂, is proved if either p ₁ p ₂ or _{1 2}; (iv) Some new simple estimates of parameters of stable laws in R ^d , based on these results are suggested.     

The Geometry of Convex Transitive Banach Spaces

Throughout this paper, X will denote a Banach space, S=S(X)and B=B(X) will be the unit sphere and the closed unit ballof X, respectively, and G=G(X) will stand for the group of allsurjective linear isometries on X. Unless explicitly statedotherwise, all Banach spaces will be assumed to be real. Nevertheless,by passing to real structures, the results remain true for complexspaces. 1991 Mathematics Subject Classification 46B04, 46B10,46B22.     

Banach空间上凸集的凸性

以Banach空间的一般凸集为研究对象,将Banach空间的凸性研究推广到了内部非空的凸集上．打破了从单位球出发研究Banach空间几何的具有局限性的研究方法,给出了严格凸集的若干特征刻画及性质,并得到了严格凸集和光滑集之间的对偶定理．     

Two Characterizations of Super-Reflexive Banach Spaces by the Behaviour of Differences of Convex Functions

A function defined on a Banach space X is called Δ-convex if it can be represented as a difference of two continuous convex functions. In this work we study the relationship between some geometrical properties of a Banach space X and the behaviour of the class of all Δ-convex functions defined on it. More precisely, we provide two new characterizations of super-reflexivity in terms Δ-convex functions.     

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。     

Coarse Embedding into Uniformly Convex Banach Spaces

In this paper, the author studies the coarse embedding into uniformly convex Banach spaces. The author proves that the property of coarse embedding into Banach spaces can be preserved under taking the union of the metric spaces under certain conditions. As an application, for a group G strongly relatively hyperbolic to a subgroup H,the author proves that B(n) = {g ∈ G | |g|S∪H≤ n} admits a coarse embedding into a uniformly convex Banach space if H does.     

Metric Entropy of Convex Hulls in Banach Spaces

The paper presents diverse methods for estimating the coveringnumber of a precompact subset of a Banach space when the entropyof the set of its extremal points is already known. In the caseof a Hilbert space, the Gelfand diameters of the subset arealso estimated.     

对偶空间上凸函数的逼近

设Banach空间E具有等价二次严格凸范数, f为其对偶空间E^*上的w^*下半连续Lipschitz凸函数, 该文证明了E^*上存在w^*下半连续且很光滑点集稠密(从而在稠子集上Gateaux可微)的Lipschitz 凸函数的单调序列{f_n}在有界集上一致逼近f.     

Convex Transitive Norms on Spaces of Continuous Functions

A norm on a Banach space X is called maximal when no equivalentnorm has a larger group of isometries. If, besides this, thereis no equivalent norm with the same isometries (apart from itsscalar multiples), the norm is said to be uniquely maximal,which is equivalent to the convex-transitivity of X: the convexhull of the orbits under the action of the isometry group onthe unit sphere is dense in the unit ball of X. The main resultof the paper is that the complex C₀() is convex-transitive inits natural supremum norm if is a connected manifold (withoutboundary). As a complement, it is shown that if is a connectedmanifold of dimension at least two, then the diameter norm isconvex transitive on the corresponding space of real functions.2000 Mathematics Subject Classification 46B15, 47B99     

B-空间上凸函数的若干指数性质

本文引入且用次梯度集该划Ｂａｎａｃｈ空间上凸函数的上、下指数,得到共轭凸函数上、下指数间的完整共轭关系、正齐次凸函数是Ｍｉｎｋｏｖｓｋｉ规函数的幂等结果,已有的一维相关结果是本文的特例。