The continuity of inf sup, with applications
Let X and Y be nonempty sets and a,b: \(X\times Y\to {\mathbb{R}}\). We give conditions under which the map on \({\mathbb{R}}^ 2_ +\) defined by \((\lambda,\mu)\to \inf _{\chi}\sup _{\gamma}(\lambda a+\mu b)\) is continuous. We deduce an infinite dimensional generalization of a result of Fan on the equilibrium value of a system of convex and concave functions. We also deduce an infinite dimensional generalization of a result of Aubin on eigenvalues of a multifunction. Our results have applications to matrix theory (the von Neumann-Kemeney theorem and the Perron-Frobenius theorem). We use the Hahn-Banach theorem and do not use any fixed-point related concepts.
scientific article; zbMATH DE number 3983974