A New Class Of Ekeland’s Variational Principle

Ekeland’s variational principle has been widely used in the filed of nonlinear analysis and so on. In this paper, we first deal with bifunctions defined on complete metric spaces and with valued in locally convex spaces ordered by closed convex cone. This aim is to provide a version of Ekeland’s variational principle for the system of vector equilibrium problems in the complete metric spaces. To provide this-principle, a weak notion of continuity of vector valued function is considered, and some of its properties are present. Via the vector Ekeland’s variational principle, we also get an equivalent result, and existence results for the system of vector equilibrium problems are proved in both compact and noncompact sets. It is turn out that functions satisfies vectorial of Ekeland variational principle and they are upper semicontinuous, then the set of solution of the system of vector equilibrium problems is nonempty. Because the system of vector equilibrium problems and multi-objective game, we also get some conclusions.In the third chapter, based on equivalent formulations types of Ekeland variational principle, we introduce the notion of (e, K)-lower semicontinuity. By using the nonlinear scalarilization method, vector optimization problems are transformed into scalar optimiza-tion problems, we get Ekeland variational principle in the setting of complete quasi-metric spaces with a W-distance.