Generalized Vector Equilibrium Problems in Topological Ordered Spaces with Applications

The generalized vector equilibrium has become important in social economy system and engineering technology. Several problems in optimization, fixed point theory, game theory, complementarity, minimax inequalities, and variational inequalities can be written as special cases of the generalized vector equilibrium. In 1961, Fan generalized the classical KKM theorem from finite dimensional spaces to infinite dimensional spaces, and since then, this theorem has become a very versatile tool in nonlinear analysis. The main purpose of this paper is to use an O-KKM type theorem to obtain some existence theorems of solutions for abstract generalized vector equilibrium problems in the framework of topological ordered spaces. Furthermore, these existence theorems can be applied to derive some Nash equilibrium existence theorems for non-cooperative games in topological ordered spaces without linear structure.