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Vietnam Journal of Mathematics 37:4 (2009) 419-441

Applications of Fixed Point Theorems for Acyclic Maps -- A Review

Sehie Park

Abstract. We review applications of our fixed point theorems on compact compositions of acyclic maps. Our applications are mainly on acyclic polyhedra, locally convex topological vector spaces, admissible (in the sense of Klee) convex sets, and almost convex or Klee approximable sets in topological vector spaces. Those applications are concerned with general equilibrium problems like as (collective) fixed point theorems, the von Neumann type intersection theorems, the von Neumann type minimax theorems, the Nash type equilibrium theorems, cyclic coincidence theorems, best approximation theorems, (quasi-) variational inequalities, and the Gale-Nikaido-Debreu theorem. Finally, we briefly introduce some related results mainly appeared in other author's works.

2000 Mathematics Subject Classification: 47H10, 49J53, 54C60, 54H25, 55M20, 90A14, 91A13.

Keywords: Kakutani map, acyclic map, admissible set (in the sense of Klee), equilibrium problems, von Neumann minimax theorem, Nash equilibrium theorem.