【作者】 韩敏；

【导师】 程曹宗；

【作者基本信息】 北京工业大学 ， 基础数学， 2003， 硕士

【摘要】 1928年Von Neumann证明了第一个极小极大定理，至今关于极小极大理论的研究已经取得了丰硕的成果。极小极大不等式是极小极大定理的另一种形式。在1929年波兰的三位数学家Knaster, Kuratowski和Mazurkiewcz提出并证明了一个关于单形的重要的定理，以后人们称之为KKM定理。Ky Fan于1961年将KKM定理推广为无穷维，称之为FKKM引理；Ky Fan并于1972年利用FKKM引理证明了第一个极小极大不等式。后来人们相继对Ky Fan极小极大不等式进行了很多推广，并将其应用于变分不等式、偏微分方程、不动点定理、位势论、截口问题、相补问题等诸多领域。和极小极大定理一样，极小极大不等式一般也是涉及三个假设条件：集合的空间结构，函数的连续性和函数的凹凸性。所不同的是极小极大不等式的条件要比极小极大定理的条件更弱。本文利用凸空间以及紧闭集的性质把Ky Fan极小极大不等式推广为拓扑向量空间和拓扑空间的乘积空间上两个函数的极小极大不等式，并由此得到一个拓扑向量空间和拓扑空间的乘积空间上一个函数的极小极大不等式。进一步得到一个新的截口定理，并证明了所得到的新的截口定理是我们得到的新的极小极大不等式的等价形式，是后者的几何形式。最后将得到的极小极大不等式应用于变分不等式之中，证明了两类变分不等式解的存在性。本文分为四章。第一章介绍极小极大理论的进展以及本文的背景；第二章给出并证明了一个定义在两个空间的乘积空间上的关于两个函数的极小极大不等式，并在此基础上得到一个定义在两个空间的乘积空间上的关于一个函数的极小极大不等式的推论；第三章给出了一个截口定理，并进一步阐述了此截口定理与第二章中的极小极大不等式的等价关系。第四章给出第二章中的极小极大不等式在变分不等式中的应用，证明了两种形式的变分不等式解的存在性更多还原

【Abstract】 Since Von Neumann proved the first minimax theorem in 1928, rich productions about minimax theory have been obtained. Minimax inequality is another form of minimax. In 1929, three mathematicians of Poland gave and proved an impotent theorem about simplex, which was called KKM theorem usually. Ky Fan generalized KKM theorem as infinite dimension in 1961, which was called FKKM lemma; Besides, Ky Fan proved the first minimax inequality by use of his own lemma in 1972. From then on, various generalizations of minimax inequalities have been obtained. Moreover, it has been applied to variational inequality, partial differential equation, fixed-point theorem, potential theory, section problem, complementarity problem, etc. Like minimax theorem, minimax inequality generally involves three assumptions: space structure, the continuity and concavity of functions. The differentia is that the assumptions of minimax inequality are weaker than that of minimax theorem.In this thesis, in the assist of convex space and compactable close set, we generalized Ky Fan minimax inequality to a minimax inequality about two functions in a cross space of a topological linear space and a topological space,and we also got a minimax inequality about one function with the same underline space as the inequality we have got previously. Ulteriorly, we got a new section theorem, and proved that the new section theorem we have got is the equivalent of the new minimax inequality we have got, what’s more the former is the geometrical form of the latter. At last, we put the minimax inequality we have got into the use of variational inequality, and proved the existence of keys of two kinds of variational inequalities. This thesis is composed of four chapters. In chapter one, we introduced the development of minimax theory and the background of this article. In chapter two, we got and proved a minimax inequality about two functions in two spaces. And on this basis we also got a deduction about one function in two spaces. In chapter three,<WP=5>we gave a section theorem, and also set forth the equivalent relation between it and the inequality in chapter two. In chapter four, we gave the application to variational inequality of the inequality in chapter two, and proved the existence of solutions about two forms of variational inequalities更多还原