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A concave-convex Ky Fan minimax inequality. (English) Zbl 1333.49013

Summary: In the form most frequently used in scalar generalizations of the celebrated Ky Fan minimax inequality, some topological hypotheses and one convexity condition are assumed. In this work, we introduce a Ky Fan minimax inequality where the convexity assumption is replaced by a concavity-convexity requirement: a slightly restrictive concavity property on one variable and a new convexity condition, necessary for the validity of such an inequality, in the another one. Our result is different from the aforementioned extensions of the Ky Fan minimax inequality and moreover avoids their vectorial setting.

MSC:

49J35 Existence of solutions for minimax problems
49K35 Optimality conditions for minimax problems
26B25 Convexity of real functions of several variables, generalizations
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