Applying twice a minimax theorem

Abstract

Here is one of the results obtained in this paper: Let $X,Y$ be two convex sets each in a real vector space, let $J:X\times Y\to \mathbf{R}$ be convex and without global minima in $X$ and concave in $Y$, and let $\Phi :X\to \mathbf{R}$ be strictly convex. Also, assume that, for some topology on $X$, $\Phi$ is lower semicontinuous and, for each $y\in Y$ and $\lambda >0$, $J(\cdot ,y)$ is lower semicontinuous and $J(\cdot ,y)+\lambda \Phi (\cdot )$ is inf-compact. Then, for each $r\in ]{\inf }_X\Phi ,{\sup }_X\Phi [$ and for each closed set $S\subseteq X$ satisfying $${\Phi }^{-1}(r)\subseteq S\subseteq {\Phi }^{-1}(]-\infty ,r]),$$ one has $$\underset{Y}{\sup }\underset{S}{\inf }J=\underset{S}{\inf }\underset{Y}{\sup }J.$$

References in corpus (1)

• On a minimax theorem: an improvement, a new proof and an overview of its applications

Cited by in corpus (2)

• An alternative theorem for gradient systems
• An improvement of a saddle point theorem and some of its applications