Minimax Theorems And Its Proofs

Since Von Neumann proved the first minimax theorem in 1928, rich fruits have been obtained about research on minimax theory. Along with the development of minimax theory, it had been applied in game theory, mathematical economics, optimization theory, variational inequalities, fixed-point theory, potential theory, section problem,etc. A minimax theorem generally involves three assumption conditions: space structures on sets X and Y, the continuity of the functions and the concavity and convexity of functions. Thereinto the concavity and convexity of functions is important condition in minimax theory. Sometimes different combinations in the concavity and convexity of functions may construct a new theorem. According to the forms of minimax theorems, minimax theorems may be devided into minimax theorems for a function, minimax theorems for two or more functions, minimax theorems endowed with differential structure,etc.. Because inminimax theorems for two functions let g = f, we can receive the correspondingminimax theorems for a function, so, minimax theorems for two functions becomes the focal point studied at present. This thesis was organized according to the following way. Chapter one has introduced the background and classification of minimax theorems; Chapter two summarizes several proof method of minimax theorems, which are illustrated with examples; Chapter three has explained the development general situation of minimax theorems for a function and for two functions with chapter four respectively, and according to the classification of the theorem, has illustrated some important conclusionses in quantitative minimax theorems, topological minimax theorems and quantitative-topological minimax theorems separately. Chapter five has introduced some results of minimax inequalities briefly. Through reading a large amount of documents, the author surveys the formulation background and evolution of minimax theorems.