Minimax theorems for set-valued maps without continuity assumptions

TL;DR

This paper establishes minimax theorems for set-valued maps with generalized convexity properties, extending results to non-continuous maps using fixed point theorems in topological vector spaces.

Contribution

It introduces new classes of set-valued maps with generalized convexity and proves minimax theorems without requiring continuity.

Findings

Minimax theorems for non-continuous set-valued maps established.

New classes of set-valued maps with generalized convexity introduced.

Fixed point theorem applied to prove main results.

Abstract

We introduce several classes of set-valued maps with generalized convexity. We obtain minimax theorems for set-valued maps which satisfy the introduced properties and are not continuous, by using a fixed point theorem for weakly naturally quasi-concave set-valued maps defined on a simplex in a topological vector space.