A Generalized Variational Principle And Its Application

Ekeland's variational principle proves the existence of approximate minimum points and the construction of approximate minimum points for a very wide range of functional classes,which has a profound impact on the research and development of optimization problems and approximate solution theory.Through the continuous exploration of many scholars And research,the theorem is widely used in many different fields,such as nonlinear analysis,convex analysis,game theory,optimal control theory,critical point theory,vector optimization problems,and dynamic systems.Based on the existing conclusions,this paper develops and generalizes the Ekeland variational principle,discusses the generalization of the Zhong-type Ekeland variational principle in the metric space,and proves the Ekeland variational principle in the partial metric space.The existence of the system equilibrium solution and the fixed points of the mapping.The first chapter mainly introduces the research background and significance of Ekeland's variational principle,the research status at home and abroad,and the arrangement of the content of this article.The second chapter reviews some basic concepts and properties in the metric space,gives the Ekeland variational principle popularized by the predecessors,and proves our main conclusion in this chapter-the Ekeland variational principle of equilibrium form,and using this The conclusion proves the existence of equilibrium solution and Caristi type fixed point theorem.The third chapter introduces the concept of partial metric space and the definition and properties of the space.By proving the Cantor intersection theorem in partial metric space,the Ekeland variational principle in this space is derived,and Caristi type The moving point theorem,and the number and construction of fixed points in this space,as well as the fixed points of compression maps are discussed.