数学专业硕士论文—《论有关KKM定理与应用

发布时间:2019-09-26 00:54

数学专业硕士论文 —《论有关KKM定理与应用》
  
  摘要 5-6
  ABSTRACT 6
  目录 7-8
  第一章引言 8-16
  第二章预备知识 16-22
  2.1记号与用法 16
  2.2定义 16-22
  第三章H-度量空间KKM定理及其应用 22-32
  3.1KKM定理 22-23
  3.2KKM定理的应用 23-32
  3.2.1不动点定理 23
  3.2.2极大极小不等式 23-26
  3.2.3截口定理 26-27
  3.2.4平衡问题解的存在性定理 27-32
  第四章G-凸度量空间KKM定理及其应用 32-42
  4.1KKM定理 32-33
  4.2KKM定理的应用 33-42
  第五章结论 42-44
  参考文献 44-48
  致谢 48


【摘要】 非线性分析主要研究非线性问题,存在性的讨论是非线性分析的一个重要方面.KKM定理及由此产生的KKM技巧,在许多存在性问题的讨论中起着重要作用.本文主要对非线性分析中的几个热点问题进行分析和研究,对已有的结果进行推广.本文主要有两个部分:1.在H-度量空间中通过引入两个集值映像条件给出一类KKM定理(非空交定理),并在此基础上得到KKM定理的一些应用:H-度量空间中的不动点定理、极大极小不等式、极大元存在定理、截口定理、抽象广义矢量平衡问题平衡点的存在性定理及有上下界的平衡问题解的存在性定理.已有的成果都是在一个集值映像条件下给出,这里突破这一点,通过引入两个集值映像,并将条件由紧闭值到有限度量紧闭值,得到进一步的结果.2.将上述结论推广到G-凸度量空间,我们给出G-凸度量空间中的一类KKM定理(非空交定理),并在此基础上可以得到KKM定理的一些应用:G-凸度量空间中不动点定理、极大极小不等式、极大元存在定理、截口定理、抽象广义矢量平衡问题平衡点的存在性定理及有上下界的平衡问题解的存在性定理.条件由有限度量紧闭值到关于Γ有限度量紧闭值,得到进一步的结果.


【Abstract】 Nonlinear analysis mainly studies on the nonlinear questions in some fields. The discussion of existence is an important aspect of nonlinear analysis. The theorem of KKM and its skill has an important role in the discussion of the existence. This paper mainly focuses on hot questions in nonlinear functional analysis, and generalizes some results in previous studies. This paper has two main parts:1. Under a set-valued mapping condition, the previous KKM theorem is given in H-metric space. A general KKM theorem (Non-empty intersection theorem), in this paper, are given in H-metric space with two extreme conditions. Some applications are obtained on the basis of KKM theorem. Fixed point theorems in H-metric space, minimax inequalities, existence theorems of maximal element, section theorems, existence theorems for abstract generalized vector balance equilibrium point, and existence theorems of upper and lower bounds of the balance. The former results were given under one extreme conditions. The paper obtains the further results by changing one extreme condition into two extreme conditions.2.The above-mentioned theory can be extended to the G-convex metric space by giving the generalized KKM theorem in G-convex metric space. Some applications in G-convex metric space are gotten based on KKM theorem. New fixed point theorems in the G-convex metric space, minimax inequalities, new maximal element existence theorems, new section theorems which are mapping in H-metric space can be obtained under some conditions. The advanced results are obtained by changing conditions in H-metric space.

硕士论文-【关键词】 H-度量空间; G-凸度量空间; 极大极小不等式; 截口定理; 平衡问题;

【Key words】 H-metric space; G-convex metric space; Minimax Inequality; Section theorem; Balance;//

 

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