Fixed point theorems and vector valued minimax theorems
The objective of this paper is the study of the following minimax relation: \[ \sup_{x\in X}\inf_{y\in Y}f(x,y)=\inf_{y\in Y}\sup_{x\in X}f(x,y) \] in a general setting, namely without convexity and compactness assumptions. Here, f: \(X\times Y\to E\) is a function with values in a partially ordered vector space and X,Y are topological spaces with a structure of H-space (a merely topological generalization of convexity). In order to show these minimax theorems, some H-space versions of fixed point theorems are proved which extend some previous results by \textit{K. Fan} [Inequalities III, Proc. 3rd Symp., Los Angeles 1969, 103-113 (1972; Zbl 0302.49019)] and \textit{M. Lassonde} [J. Math. Anal. 97,151-201 (1983; Zbl 0527.47037)].
scientific article; zbMATH DE number 4144814
scientific article