Gurarii的凸性模与正规结构

本文研究了Gurarii的凸性模与正规结构的联系.利用关于该模的不等式得出了如果存在ε,1≤ε≤2,使得β(ε)＞ε-1.则空间X具有一致正规结构.     

Normal structure and some geometric parameters related to the modulus of U-convexity in banach spaces

We present a short and simple proof of the recent result of Yang and Wang [12]. Stimulated by their idea, two geometric parameters U~ax(ε) and βx (ε), both related to Gao’s modulus of U-convexity of a Banach space X, are introduced. Their properties and the relationships with normal structure are studied. Some existing results involving normal structure and fixed points for non-expansive mappings in Banach spaces are improved.     

On a class of Köthe sequence spaces with normal structure

In this note, we investigate the generalized modulus of convexity δ ( λ ) and the generalized modulus smoothness ρ ( λ ) . We obtain some estimates of these moduli for X = lp . We obtain inequalities between WCS coefficient of a K¨othe sequence space X and δ ( λ ) X . We show that, for a wide class of K¨othe sequence spaces X, if for some ε∈ (0, 9 10 ] holds δ X (ε) > 1 3 1 √ 3 2 ε, then X has normal structure.     

The characteristic of convexity of a Banach space and normal structure

We present some sufficient conditions for normal structure of Banach spaces and their dual spaces in terms of the characteristic of convexity, the James constant, and the coefficient of weak orthogonality. Many known results are improved and strengthened. We also show that some of our results are sharp.     

On generalized convexity and duality with a square root term

We extend the duality theorems for a class of nondifferentiable problems with Mond-Weir type duals.     

Generalized monotonicity and generalized convexity

Generalized monotonocity of bifunctions or multifunctions is a rather new concept in optimization and nonsmooth analysis. It is shown in the present paper how quasiconvexity, pseudoconvexity, and strict pseudoconvexity of lower semicontinuous functions can be characterized via the quasimonotonicity, pseudomonotonicity, and strict pseudomonotonicity of different types of generalized derivatives, including the Dini, Dini-Hadamard, Clarke, and Rockafellar derivatives as well.This research was supported by the National Science Foundation of Hungary, Grant No. OTKA 1313/1991.     

On the weak uniform convexity of

We will discuss the geometry of the unit sphere in the Banach space of integrable holomorphic quadratic differentials on a Riemann surface and answer some questions posed by L.R. Goldberg (Proc. Amer. Math. Soc. 118 (1993), 1179--1185).

     

Banach空间的正规结构与对径点

假设S（X）是Banach空间X的单位球面,作引进了四个新的几何参数：Jε（X）=sup{βε（x）,x∈S（X）},jε（X）=inf{βε（x）,x∈S（X）},Gε（X）=sup{αε（x）,x∈S（X）},gε（X）=inf{αε（x）,x∈S（S）},其中≤ε≤1,βε（x）=sup{min{‖x εy‖,‖x-εy‖,y∈S（X）}},αε（x）=inf{max{‖x εy‖,‖x-εy‖,y∈S（X）}},讨论了这些参数的性质,本主要结果是：如果主要结果是：如果有一个ε,0≤ε≤1,使得Jε（X）＜1 ε/2或gε（X）＞1 ε/3,那末X有一至正规结构。     

Normal structure and Pythagorean approach in Banach spaces

Summary Let <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"13"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>X$ be a real Banach space and $S(X) = \{x \in X: \|x\| = 1\}$ be the unit sphere of $X$. The parameters $E_{\epsilon}(X)=\sup\{\alpha_{\epsilon}(x): x \in S(X)\}$, $e_{\epsilon}(X)=\inf\{\alpha_{\epsilon}(x): x \in S(X)\}$, $F_{\epsilon}(X)=\sup\{\beta_{\epsilon}(x): x \in S(X)\}$, and $f_{\epsilon}(X)=\inf\{\beta_{\epsilon}(x): x \in S(X)\}$, where $\alpha_{\epsilon}(x) = \sup\{\| x + \epsilon y \|^{2}+ \| x - \epsilon y \|^{2}: y \in S(X)\}$ and $\beta_{\epsilon}(x) = \inf\{\| x + \epsilon y \|^{2}+ \| x - \epsilon y \|^{2}: y \in S(X)\}$, are defined and studied. The main result is that a Banach space $X$ with $E_{\epsilon}(X) < 2 + 2\epsilon +\frac{1}{2}\epsilon^{2}$ for some $0\leq \epsilon \leq 1$ has uniform normal structure.     

Locally uniform convexity in Musielak-Orlicz function spaces of Bochner type endowed with the Luxemburg norm

Criteria for locally uniform convexity of Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm are given. We also prove that, in Musielak-Orlicz function spaces generated by locally uniformly convex Banach space, locally uniform convexity and strict convexity are equivalent.     

On the moduli of convexity

It is known that, given a Banach space , the modulus of convexity associated to this space is a non-negative function, non-decreasing, bounded above by the modulus of convexity of any Hilbert space and satisfies the equation for every , where is a constant. We show that, given a function satisfying these properties then, there exists a Banach space in such a way its modulus of convexity is equivalent to , in Figiel's sense. Moreover this Banach space can be taken to be two-dimensional.

     

Optimality and duality with generalized convexity

Hanson and Mond have given sets of necessary and sufficient conditions for optimality and duality in constrained optimization by introducing classes of generalized convex functions, called type I and type II functions. Recently, Bector defined univex functions, a new class of functions that unifies several concepts of generalized convexity. In this paper, optimality and duality results for several mathematical programs are obtained combining the concepts of type I and univex functions. Examples of functions satisfying these conditions are given.     

On weak uniform normal structure in weighted Orlicz sequence spaces

It is proved that a weighted Orlicz sequence space ?M(w), equipped with Luxemburg or Amemiya norm has weak uniform normal structure iff ?M(w)≅hM(w) for wide class of weight sequences . An example is constructed, where M has not Δ₂-condition but by choosing a suitable weight sequence limn→∞wn=∞ we get that ?M(w) has weak uniform normal structure.     

On a new geometric constant related to the modulus of smoothness of a Banach space

We shall introduce a new geometric constant A(X) of a Banach space X,which is closely related to the modulus of smoothness ρX(τ),and investigate it in relation with the constant A2(X) by Baronti et al.,the von Neumann–Jordan constant CNJ(X) and the James constant J(X).A sequence of recent results on these constants as well as some other geometric constants will be strengthened and improved.     

Higher-order duality for nondifferentiable minimax programming problem with generalized convexity

In this paper, we are concerned with a class of nondifferentiable minimax programming problems and its two types of higher-order dual models. We establish weak, strong and strict converse duality theorems in the framework of generalized convexity in order to relate the optimal solutions of primal and dual problems. Our study improves and extends some of the known results in the literature.     

The uniform convexity of the Edmunds-Triebel logarithmic spaces

We show how uniform convexity can be preserved in the logarithmic spaces A_θ(logA)_b,p. Estimates are given for the moduli of convexity of A_θ(logA)_b,p in terms of the moduli of A₀ and A₁, when one or both of them are uniformly convex.     

The uniform modulus of continuity of iterated Brownian motion

LetX be a Brownian motion defined on the line (withX(0)=0) and letY be an independent Brownian motion defined on the nonnegative real numbers. For allt0, we define theiterated Brownian motion (IBM),Z, by setting . In this paper we determine the exact uniform modulus of continuity of the process Z.Research supported by NSF grant DMS-9122242.     

Functions related to convexity and smoothness of normed spaces

We introduce a few functions related to convexity and smoothness of normed spaces. Those functions turn out to be moduli of convexity or smoothness or play an intermediate role. We calculate the exact formulas for introduced functions in some classical Banach spaces. An application to geometry of normed spaces is also indicated.     

关于随机一致凸性

致力于随机一致凸性概念的进一步探讨．首先,通过一个特殊的层次剖分指出对任意的随机赋范模而言随机凸性模都有良好定义,从而改进了近期的文献中许多已知的结果．然后,提出并研究了一种与随机一致凸性密切相关的新性质,从一个新的角度阐述了随机一致凸性的复杂性．     

Second order duality for nondifferentiable minimax programming problems with generalized convexity

In this paper, we are concerned with a class of nondifferentiable minimax programming problem and its two types of second order dual models. Weak, strong and strict converse duality theorems from a view point of generalized convexity are established. Our study naturally unifies and extends some previously known results on minimax programming.