KKM型定理及其在平衡问题中的应用
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摘要
在非线性分析领域中, KKM理论占据了重要的部分.在传统的KKM理论和应用中,凸性扮演了很重要的角色,但同时它也大大地限制了KKM理论的应用范围.如何削弱凸性,成为KKM理论研究的一个重要课题.基于KKM理论在平衡问题、对策论、数学规划、力学和物理、经济和金融、运筹与交通、优化和控制、变分不等式和相补问题等理论中有广阔的应用前景,近年来,有关KKM理论以及在各方面的应用还在进行广泛而深入的讨论.
本文主要分两部分引入和研究了新的KKM型定理和向量平衡问题.
第一部分在拓扑空间中引入了一类新的广义L-KKM映射:
第一章主要介绍了KKM理论的历史背景.
第二章介绍了一类新的广义L-KKM映射,同时在没有任何凸结构的拓扑空间中建立了一些新的广义L-KKM型定理.作为应用,我们在拓扑空间中建立了抽象广义向量平衡问题的平衡点定理.
第三章作为第二章的延伸,先建立了关于广义L-KKM映射新的非空交定理,随后证明了在拓扑空间中的集值不动点定理.作为应用,在拓扑空间中建立了上下界平衡问题和拟平衡问题的存在定理.
第二部分研究了FC-空间中KKM型定理和一些向量平衡问题:
第四章在非紧FC-空间中建立了新的重合点定理和一些关于较好容许集值映射的KKM型定理.作为应用,在非紧FC-空间中建立了几类广义向量平衡问题解的存在定理.
第五章在FC-空间中引入和研究了一类广义向量变分型不等式(GVVTIP),它包含了大多数向量平衡问题,向量变分不等式问题,广义向量平衡问题和广义向量变分不等式问题作为特殊情况.最后在非紧FC-空间中,建立了关于GVVTIP解的某些新的存在定理.
在第六章中,得到一个新的非空交定理.作为应用,我们在非紧的FC-空间中建立了上下界的平衡问题和拟平衡问题的存在定理.
第七章在乘积FC-空间中引入和研究一类新的广义混合向量拟平衡问题系统,利用极大元定理,得到了关于广义混合向量拟平衡问题系统解的一些存在性定理.
KKM theory is one of the important parts in nonlinear analysis. In most of knownKKM theorems and applications, the convexity assumptions play a crucial role whichstrictly restrict the applicable area of KKM principle. How to weaken convexity is theimportant subject of KKM principle. Since KKM theory has aroused continuous interestdue to its promising applications in many fields of applicable sciences, such as equi-librium problem, game theory, mechanics and physics, economics and finance, trans-portation and operations research, variational inequality and complementarity problem,optimization and control problem. For recent decades, the KKM theory and Its applica-tions have been extensively studied.
In this thesis, some new KKM type theorems and vector equilibrium problems areinvestigated. The thesis is composed of two parts.
Part I(2-3):KKM type theorems and Its applications in topological spacesChapter 1 mainly introduces the history of KKM theory.
Chapter 2 introduces a new concept of generalized L-KKM mapping and estab-lishes some new generalized L-KKM type theorems without any convexity structure intopological space. As application, an existence theorem of equilibrium point for abstractgeneralized vector equilibrium problem is proved in topological space.
Based on Chapter 2, Chapter 3 first establishes some new nonempty intersectiontheorems for generalized L-KKM mappings and proves some new fixed point theoremsfor set-valued mappings in topological spaces. As applications, an existence theorem foran equilibrium problem with lower and upper bounds and two existence theorems fora quasi-equilibrium problem with lower and upper bounds are obtained in topologicalspaces.
Part II(4-7): KKM type theorems and some vector equilibrium problems in FC-space.
A new coincidence theorem and some KKM type theorems for better admissibleset-valued mappings are established in Chapter 4. As applications, some new existencetheorems of solutions for several classes of generalized vector equilibrium problems areestablished in noncompact FC-space.
Chapter 5 studies a class of generalized vector variational-type inequality problems(in short, GVVTIP) in FC-space, which includes the most of vector equilibrium prob-lems, vector variational inequality problems, generalized vector equilibrium problemsand generalized vector variational inequality problem as special cases. As applications,some new existence results for GVVTIP are established in noncompact FC-space.
Chapter 6 obtains a new nonempty intersection theorem. As application, some newtheorems of equilibrium problems and quasi-equilibrium problems with lower and upperbounds are established.
Chapter 7 introduces and studies a new system of generalized mixed vector quasi-equilibrium problems, which includes as special cases the system of generalized vectorvariational-like inequality problems, the system of generalized quasi-equilibrium prob-lems, and the system of generalized equilibrium problems. By using the maximal ele-ment theorem, some existence results of solutions for the system of generalized mixedvector quasi-equilibrium problems are given.
本文主要分两部分引入和研究了新的KKM型定理和向量平衡问题.
第一部分在拓扑空间中引入了一类新的广义L-KKM映射:
第一章主要介绍了KKM理论的历史背景.
第二章介绍了一类新的广义L-KKM映射,同时在没有任何凸结构的拓扑空间中建立了一些新的广义L-KKM型定理.作为应用,我们在拓扑空间中建立了抽象广义向量平衡问题的平衡点定理.
第三章作为第二章的延伸,先建立了关于广义L-KKM映射新的非空交定理,随后证明了在拓扑空间中的集值不动点定理.作为应用,在拓扑空间中建立了上下界平衡问题和拟平衡问题的存在定理.
第二部分研究了FC-空间中KKM型定理和一些向量平衡问题:
第四章在非紧FC-空间中建立了新的重合点定理和一些关于较好容许集值映射的KKM型定理.作为应用,在非紧FC-空间中建立了几类广义向量平衡问题解的存在定理.
第五章在FC-空间中引入和研究了一类广义向量变分型不等式(GVVTIP),它包含了大多数向量平衡问题,向量变分不等式问题,广义向量平衡问题和广义向量变分不等式问题作为特殊情况.最后在非紧FC-空间中,建立了关于GVVTIP解的某些新的存在定理.
在第六章中,得到一个新的非空交定理.作为应用,我们在非紧的FC-空间中建立了上下界的平衡问题和拟平衡问题的存在定理.
第七章在乘积FC-空间中引入和研究一类新的广义混合向量拟平衡问题系统,利用极大元定理,得到了关于广义混合向量拟平衡问题系统解的一些存在性定理.
KKM theory is one of the important parts in nonlinear analysis. In most of knownKKM theorems and applications, the convexity assumptions play a crucial role whichstrictly restrict the applicable area of KKM principle. How to weaken convexity is theimportant subject of KKM principle. Since KKM theory has aroused continuous interestdue to its promising applications in many fields of applicable sciences, such as equi-librium problem, game theory, mechanics and physics, economics and finance, trans-portation and operations research, variational inequality and complementarity problem,optimization and control problem. For recent decades, the KKM theory and Its applica-tions have been extensively studied.
In this thesis, some new KKM type theorems and vector equilibrium problems areinvestigated. The thesis is composed of two parts.
Part I(2-3):KKM type theorems and Its applications in topological spacesChapter 1 mainly introduces the history of KKM theory.
Chapter 2 introduces a new concept of generalized L-KKM mapping and estab-lishes some new generalized L-KKM type theorems without any convexity structure intopological space. As application, an existence theorem of equilibrium point for abstractgeneralized vector equilibrium problem is proved in topological space.
Based on Chapter 2, Chapter 3 first establishes some new nonempty intersectiontheorems for generalized L-KKM mappings and proves some new fixed point theoremsfor set-valued mappings in topological spaces. As applications, an existence theorem foran equilibrium problem with lower and upper bounds and two existence theorems fora quasi-equilibrium problem with lower and upper bounds are obtained in topologicalspaces.
Part II(4-7): KKM type theorems and some vector equilibrium problems in FC-space.
A new coincidence theorem and some KKM type theorems for better admissibleset-valued mappings are established in Chapter 4. As applications, some new existencetheorems of solutions for several classes of generalized vector equilibrium problems areestablished in noncompact FC-space.
Chapter 5 studies a class of generalized vector variational-type inequality problems(in short, GVVTIP) in FC-space, which includes the most of vector equilibrium prob-lems, vector variational inequality problems, generalized vector equilibrium problemsand generalized vector variational inequality problem as special cases. As applications,some new existence results for GVVTIP are established in noncompact FC-space.
Chapter 6 obtains a new nonempty intersection theorem. As application, some newtheorems of equilibrium problems and quasi-equilibrium problems with lower and upperbounds are established.
Chapter 7 introduces and studies a new system of generalized mixed vector quasi-equilibrium problems, which includes as special cases the system of generalized vectorvariational-like inequality problems, the system of generalized quasi-equilibrium prob-lems, and the system of generalized equilibrium problems. By using the maximal ele-ment theorem, some existence results of solutions for the system of generalized mixedvector quasi-equilibrium problems are given.
引文
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