Generalized KKM theorems and common fixed point theorems

In this paper, we first prove a generalized KKM theorem, and then use this generalized KKM theorem to establish the generalized equi-KKM theorem, common fixed point theorems for a family of multivalued maps, and the Kakutani–Fan–Glicksberg fixed point theorem. We also show that an existence theorem of the common fixed point theorem is equivalent to the Kakutani–Fan–Glicksberg fixed point theorem.