Minimax Inequalities And Section Theorems With Applications To Variational Inequalities

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,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...