Minimax Principle Of Two Functions

Minimax principle deals with the relation between the minimax value and maximin value of a function. It originates from game theory. The first mathematical formulation was established by von Neumann in 1928. Since then, various forms of the results of this principle have been obtained. The study of the subject is very brisk and resultful. As time goes on, minimax principle not only is the fundamental principle in the theory of games but also becomes an important content of nonlinear analysis and an object of study in their own right. On the other side, along with the development of minimax theory, many application of this principle have been found. For instance, it can be applied in game theory, mathematical economics, optimization theory, variational inequalities, differential equations, fixed-point theory, potential theory, section problem, etc.. In fact, since the book Theory of games and economic behavior by von Neumann and O. Morgenestem was published, this mathematical method has become an important tool hi modem economic theory.hi this paper, we shall study minimax principle involving two functions which is a generalized form of one-function minimax principle. Since the method dealing with minimax principle of two functions is different from the method dealing with one-function minimax principle and minimax results of one function have not direct two functions extension, so far there are no many minimax results of two functions. In Section 3 of this paper, we shall answer some fundamental questions on two-function minimax theorems such as the ways of establishing two-function minimax theorems, the classification on two-function minimax theorems and the relation between two-function results and one-function results, hi Section 4-7, we shall obtain some two-function quantitative minimax theorems, two-function topological minimax theorems, two-function topological-quantitative minimax theorems and some other types of two-function minimax theorems. By these theorems, we extend some formerly important results and answer some open questions. In Section 8, we shall discuss two-function minimax inequalities and their applications to variational inequalities and to fixed-point theory.