Some Fixed Point Theorems And Vectorial Ekeland's Variational Principle On Ordered Metric Spaces

The class of ordered metric spaces is a new and widely applied space frame,which is more abstract and extensive than the kind of metric spaces.However,it is very rare of researches on fixed point theory and variational problems in ordered metric spaces.In this paper,we investigate fixed point problems of single valued and multivalued mappings,and provide a vectorial Ekeland's variational principle in ordered metric spaces.This paper is organized as follows:In chapter one,we introduce the background and some recent work of ordered metric spaces,fixed point theory and Ekeland's variational principle.Then we recall some concepts and properties in ordered vector spaces and ordered metric spaces.Also,we give the concept of order closed subset in ordered metric spaces.In chapter two,by changing the conditions,we extend some fixed point theorems of single valued mappings on metric spaces to the setting of ordered metric spaces from different angles.In chapter three,we introduce the concept of H-ordered metric in ordered metric spaces.By applying H-ordered metric,several fixed point and common fixed point theorems of multivalued mappings are provided.Meanwhile,we use the fixed point theorems to prove the existence of an equilibrium solution.In chapter four,by using some properties of Riesz spaces,and defining the concept of lower semicontinuous in ordered metric spaces,we prove a vectorial Ekeland's variational principle valued in Riesz spaces.