Existence Of Equilibrium Pairs For Two Classes Of Games

In this article,using the theorem of hybrid best proximity points for a class of proximally mixed monotone set-valued maps,we mainly present the existences of equilibrium pairs for free n-person games in partially ordered Hausdorff topological vector spaces and for constrained generalized games in the partial linear metrics space.For this purpose,we first define a proximally ordered contraction,instead of the proximal distance contraction,for set-valued maps in posets to get rid of the fetter of continuity.Moreover,we introduce the OM complete concept.Then we prove the existence of hybrid best proximity points for proximally mixed monotone set-valued maps and of the best proximity point in OM complete partial order linear metric spaces.Our results do not require the continuity of the operator which is a necessary condition in many existing related literature.Finally,using the obtained results,we provide the existence theorems of equilibrium pairs for free n-person games and generalized constraint games in topological vector space.The full text is divided into five chapters:The first chapter is an introduction,in which we summarize the current research situation of the best proximally theory and the equilibrium pairs of games.At the same time,we summarize the main research issues of this paper.The second chapter is preparatory knowledge.In this chapter,some basic concepts are reviewed,such as the order theory in the metric space and the definition of the equilibrium pairs of the games.These will play a key role in this sequel.The third chapter is divided into three sections.In the first section,the existence theorem of the hybrid best proximally point in the partial metric space is given,which generalizes and improves some relevant conclusions.The second section the special case of the above theorem is given,ie,the existence theorem of the best proximally point in partially ordered metric spaces;the third section some examples are given to support our results.The fourth chapter is divided into two sections.Using the main results obtained in the previous chapter,in the first section,the existence of equilibrium pairs for the generalized constraint games is proved.In the second section,the partial order Hausdorff topological vector space is used as the platform.The sufficient conditions for the existence of equilibrium pairs for the free n-person games are given.The main tools are the results obtained in the previous chapter and the maximal element principle.In the fifth chapter,the full text is summarized,and the future work and the direction of efforts are analyzed.