Equilibrium problems in generalized convex spaces (Q2706557)

In this interesting expository paper, the author shows how theorems on equilibrium problems in convex space and C-space are extended to generalized convex spaces. He points out that most of important theorems on KKM (Knaster-Kuratowski-Mazurkiewicz) theory still hold true without assuming the linearity in a topological vector space. As an examples of such theorems, the author introduces KKM principle, von Neumann type minimax theorems, Nash equilibrium theorem, Fan-Browder type fixed point theorems, maximal element theorem, Ky Fan type minimax inequalities, variational inequalities, best approximation theorems, fixed point theorems in a locally G-convex space, and existence theorems of solutions for generalized quasi-equilibrium problem. Since KKM theory was named by the author in 1991, hundreds of equivalent propositions were introduced and developed by many mathematicians. In particular, von Neumann, Nash, Fan, Browder, Brouwer, Schauder, Tychonoff, Hukuhara, Kakutani, Glicksberg, and Himmelberg as well as the author have contributed to develop the fixed point theorem.