Ekeland Variational Principle In (1,q₂)-quasimetric Spaces And Applications

Ekeland variational principle plays a very important role in nonlinear functional analysis and provides a powerful tool for solving problems such as optimization,optimal control theory,nonlinear equations and so on.In this dissertation,some conditions of the existing Ekeland variational principle are generalized to study the Ekeland variational principle in complete weakly symmetric(l,q2)-quasimetric spaces,and some related applications are given.On the one hand,the conditions are generalized from the aspects of space and triangle inequality.On the other hand,the(generalized)Caristi-like conditions proposed by Arutyunov et al.are generalized from functions to bifunctions.In chapter 1,we introduce the research background and significance of(1,q2)-quasimetric spaces and Ekeland variational principle.Meanwhile,the current research status of Ekeland variational principle is also introduced.In chapter 2,some symbols,definitions and properties related to this dissertation are given.In particular,the generalized Caristi-like conditions related to bifunctions are given.In chapter 3,we generalize the classical triangle inequality and put forward(1,q2)f-triangle inequality.The main result in this chapter is proved in the complete weakly symmetric(1,q2)-quasimetric spaces,which is the Ekeland variational principle.Then it is proved that Ekeland variational principle is equivalent to Oettli-Thera theorem,Caristi-Kirk fixed point theorem(abbreviated as‘C-K Theorem')and Takahashi's non-convex minimization principle.In addition,the Ekeland variational principle of the system of equilibrium problems(not necessarily countable)are obtained in complete weakly symmetric(1,q2)-quasimetric spaces.In chapter 4,as an application,the Ekeland variational principle obtained in Chapter 3 is used to study the existence of solutions to the equilibrium problems.Firstly,under compactness and other appropriate conditions,the solutions of the(system of)quasi-equilibrium problems are obtained.Then,assuming that the bifunction satisfies the generalized Carist-like condition(1)or the generalized Carist-like condition(?),under certain conditions,a solution to the equilibrium problems is obtained.