引理

It is organized as follows:In Chapter 1, we introduce proximal point algorithms" framework and current research situation, optimization problems and variational inequalities" background and current research situation, and some basic concepts and lemmas, which are used in this dissertation.

本文具体安排如下：第一章简要介绍了邻近点算法的基本框架和研究现状，最优化问题和变分不等式问题的背景和研究现状，以及本文需要用到的一些基本概念和引理。

In chapter one, we review the historical development of these branches. Some basic concepts and lemmas of set-valued mappings are briefly introduced, the main work of this thesis is outlined.

第一章综合论述这三个分支的历史发展状况，简单介绍集值映射的基本概念和本文所必需一些引理，以及本文所做的一些主要工作。

This paper has four parts. Induction gives the origin of problem and some definitions and some lemmas and the conditions for the later theorem.

全文共分四部分，引言介绍了问题的提出、一些定义、引理、以及后面定理所需要的条件。

Equidistance point and difference theory in theory of function approximation are studied. Meanwhile, the relation among difference, difference quotient and derivate is revealed. By drawing Lagrange's and Cauchy's theorem of mean on difference and Taylor's formula into difference function, four theorems, such as Lagrange's theorem of mean on difference, are concluded in simple way. On the basis of these conclusions, the asymptotic property of middle point is studied, a series of new conclusions are drawn and the discussions on the asymptotic property of middle point in differential mid-value are summarized.

对函数逼近论中等距节点和差分理论进行了研究，揭示了差分、差商与导数之间的联系；将Lagrange中值定理、Cauchy中值定理、Taylor公式引入到差分函数中，简明地推导出Lagrange差分中值定理等4个定理，并在此基础上对&中间点&的渐近性进行了研究，得出了一系列&中间点&的渐近性的结果，概括了有关文献对微分中值公式的&中间点&的渐近性的讨论；给出的引理改进了函数逼近论的证明方法，精简了函数逼近论中的一些内容。

In the second place, we introduce some relativeconceptions of topos and weak topos, including category, topos, middleobject, weak topos, power object and pullback lemma.

第二部分是预备知识，首先介绍了区间数的概念及其运算，在此基础上引入了区间值模糊集的概念、运算以及它与直觉模糊集之间的关系；然后介绍了topos和弱topos的相关概念，包括范畴、topos、中间元、弱topos、幂对象和拉回引理。

Sperner's lemma concerns the decomposition of a simplex into smaller simplices.

斯波纳引理涉及到把一个单纯形分解为较小的单纯形的问题。

On the base of the concept, we give the Schanual Lemma about weak injective modules and discuss the properties of the homomorphisms which can be extended in weak injective modules.

Kerβ的β∈Hom_R，都存在γ∈Hom_R使得β=γα成立，则称T为N-弱内射R-模，且称α为T与M的相伴同态。在此定义的基础上我们给出了弱内射模的Schanuel引理，讨论了弱内射模中能进行开拓的同态的性质。

This thesis is devoted to the study of 〓-type inequalities for martingales,such as the maximal inequalities,the inequalities for square functions,the inequali-ties for martingale transforms,the inequalities for singular integrals.

第三章证明了关于Hardy平均算子〓的双〓不等式、双〓凸引理，并通过考虑鞅的重排不等式，建立了鞅的双〓均方函数不等式、条件均方函数不等式，给出了这类不等式成立的一个充分必要条件。

In chapter two, we study the oscillations of a kind of secon...

第三章讨论了脉冲中立型方程的振动性，通过建立一些引理，得到了方程振动的一些充分条件。

In this paper, a concentration-compactness lemma for the problem of quasilinear elliptic equations is given, and the existence of nontrivial solutions is discussed by use of this lemma.

给出了相应的拟线性方程的定解问题的集中列紧引理，利用这一结果得到了方程在无界区域中非平凡解的存在性。

For a big chunk of credit-card losses; the number of filings (and thus charge-off rates) would be rising again, whether

年美国个人破产法的一个改动使得破产登记急速下降，而后引起了信用卡大规模的亏损。

Eph. 4:23 And that you be renewed in the spirit of your mind

弗四23 而在你们心思的灵里得以更新