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Minimax theorems for scalar set-valued mappings with nonconvex domains and applications. (English) Zbl 1281.49023
Let \(X_{0}\) and \(Y_{0}\) be two nonempty compact subsets of the real Hausdorff topological vector spaces \(X\) and \(Y\), respectively, and \(F:X\times Y\rightarrow2^{\mathbb{R}}\) be a set-valued mapping. The main result of the paper is that, under suitable conditions, the following minimax equality holds: \[ \min {\bigcup \limits_{y\in Y_{0}}}\max F(X_{0},y)=\max {\bigcup \limits_{x\in X_{0}}}\min F(x,Y_{0}). \]
Using the above result and an argument involving the Ky Fan Lemma for KKM maps, the existence of a solution for a generalized vector valued equilibrium problem with set-valued maps is deduced. Other results obtained are a scalar and a vector version of a Ky Fan minimax inequality.

MSC:
49K35 Optimality conditions for minimax problems
49J53 Set-valued and variational analysis
90C47 Minimax problems in mathematical programming
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