Minimax theorems for set-valued maps without continuity assumptions

Abstract

We introduce several classes of set-valued maps with new generalized convexity properties. We also obtain minimax theorems for set-valued maps which satisfy these convexity assumptions and which are not continuous. Our method consists of the use of a fixed point theorem for weakly naturally quasiconcave set-valued maps, defined on a simplex in a topological vector space, or of a constant selection of quasiconvex set-valued maps.

Keywords:

• minimax theorems
• fixed point theorem
• weakly naturally quasiconcave set-valued map
• S-transfer -convex set-valued map
• transfer properly S-quasiconvex
• weakly z-convex set-valued map

AMS Subject Classifications:

• 49J35
• 90C47

Acknowledgements

The author would like to thank the referees for providing constructive comments and help in improving the contents of this paper.

Notes

No potential conflict of interest was reported by the author.