Fixed point theorems for compact multimaps on almost Γ-convex sets in generalized convex spaces

In this paper, we introduce the concept of the strict Kakutani multimap (KKM) property and investigate the fixed point problem for multimaps having this property on almost Γ-convex subsets of locally G-convex uniform spaces. Our new fixed point theorem generalizes the well-known Fan-Glicksberg fixed point theorem and partially extends the Himmelberg fixed point theorem. Its application to minimax theorem is also given.