A more complete version of a minimax theorem

Abstract

In this paper, we present a more complete version of the minimax theorem established in [7]. As a consequence, we get, for instance, the following result: Let $X$ be a compact, not singleton subset of a normed space $(E,\Vert \cdot \Vert )$ and let $Y$ be a convex subset of $E$ such that $X\subseteq \overline{Y}$. Then, for every convex set $S\subseteq Y$ dense in $Y$, for every upper semicontinuous bounded function $\gamma :X\to \mathbf{R}$ and for every $\lambda >\frac{4{\sup }_X|\gamma |}{diam(X)}$, there exists $y^{\ast }\in S$ such that the function $x\to \gamma (x)+\lambda \Vert x-y^{\ast }\Vert$ has at least two global maxima in $X$.

arXiv admin note: text overlap with arXiv:1810.08957

References in corpus (3)

• On a minimax theorem: an improvement, a new proof and an overview of its applications
• An alternative theorem for gradient systems
• Another multiplicity result for the periodic solutions of certain systems