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Minimax Theory and its Applications 02 (2017), No. 1, 099--152
Copyright Heldermann Verlag 2017

On a Minimax Theorem: an Improvement, a New Proof and an Overview of its Applications

Biagio Ricceri
Dip. di Matematica e Informatica, UniversitÓ di Catania, Viale A. Doria 6, 95125 Catania, Italy


Theorem 1 of the author in {\it ``Multiplicity of global minima for parametrized functions''} [Rend. Lincei Mat. Appl. 21 (2010) 47--57], a minimax result for functions $f:X\times Y\to {\bf R}$, where $Y$ is a real interval, was partially extended to the case where $Y$ is a convex set in a Hausdorff topological vector space as Theorem 3.2 in {\it A strict minimax inequality criterion and some of its consequences} [Positivity 16 (2012) 455--470]. As a key tool in the proof a partial extension of the same result for the case where $Y$ is a convex set in ${\bf R}^n$ of S. J. N. Mosconi [Theorem 4.2 in {\it ``A differential characterisation of the minimax inequality''}, J. Convex Analysis 19 (2012) 185--199] was used. In the present paper, we first obtain a full extension of the first result mentioned above by means of a new proof fully based on the use of the result itself via an inductive argument. Then, we present an overview of the various and numerous applications of these results.

Keywords: Minimax; quasi-concavity; inf-compactness; global minimum; multiplicity.

MSC: 49J27, 49J35, 49J45, 49K35, 90C47; 90C25, 46A55, 46B20, 46C05, 35J20

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