A Kind Of Order Relation And Its Applications In The Fuzzy Quasi-metric Spaces

Fuzzy quasi metric is a kind of generalized fuzzy metric which does not need to satisfy symmetry.Due to the promotion of applications,the research of fuzzy quasi metric has been paid more and more attention by scholars at home and abroad in recent years,especially the GV fuzzy quasi metric space in the sense of Gregori and Romaguera(Fuzzy quasi metric spaces,applied general topology,2004,5(1): 129-136).In this paper,we study the optimization problem in GV fuzzy quasi metric space by using an order relation derived from GV fuzzy quasi metric.In the first part,by using the asymmetry of fuzzy quasi metric,a partial order relation is introduced in GV fuzzy quasi metric space.On this basis,some relations between the ordered structure and the topological structure in GV fuzzy quasi metric space are studied,such as the relations between open set,closed set and upper closed set,lower closed set,and the order property of convergence limit.In the second part,using the introduced partial order,the maximum and minimum points of the subset of GV fuzzy quasi metric spaces are defined.On this basis,a new concept of supremum and infimum of space subset is given,and it is proved that if there is a supremum and infimum for the upper(lower)semicontinuous mapping defined on compact subset and valued in GV fuzzy quasi metric space,then the supremum and infimum can be achieved.In the third part,we give a generalized partial order relation in GV fuzzy quasi metric spaces,and discuss some of its basic properties.On this basis,Caristi's fixed point theorem,Ekeland's variational principle and Takahashi's minimax principle are extended to GV fuzzy quasi metric space by using Brézis and Browder's principles on ordered sets,and their equivalence is proved.Fuzzy quasi metric spaces have important research value.The main purpose of this paper is to reveal its spatial characteristics,improve its theoretical system,and establish its application basis.The research results obtained in this paper are some useful explorations for this purpose.