Bilevel minimax theorems for non-continuous set-valued mappings

We study new types for minimax theorems with a couple of set-valued mappings, and we propose several versions for minimax theorems in topological vector spaces setting. These problems arise naturally from some minimax theorems in the vector settings. Both the types of scalar minimax theorems and the set minimax theorems are discussed. Furthermore, we propose three versions of minimax theorems for the last type. Some examples are also proposed to illustrate our theorems. MSC: 49J35, 58C06.