Some nonempty intersection theorems in generalized interval spaces with applications (Q1916859)

A generalized interval space is defined and a parametric type of KKM theorems in generalized interval spaces are established. These results are applied to prove minimax theorems which contain results of Cheng and Lin, Lin and Quan, Brézis, Nirenberger and Stampacchia, Komornik, M. A. Geraghry and Lin, Stachó and Wu. They also include the famous von Neumann theorem. These results are also applied to the section problem and to variational inequality problems. They extend Ky Fan's famous theorem for implicit variational inequalities and Theorem 2 of Yen to the case of generalized interval spaces. Recently, \textit{B. Thompson} and \textit{X.-Z. Yuan} established topological intersection theorems for two set-valued mappings in [Numer. Funct. Anal. Optimization 17, 437-452 (1996; Zbl 0854.49009)]. But the results are quite different.