Research On Several Types Of Variational Inclusions

Variational inclusion problems,originating from variational inequalities,have great research significance and been widely used in differential equations,economic models,optimization theory and other fields.This paper mainly studies two types of nonlinear variational inclusions in infinite dimensional spaces,i.e.the set-valued equilibrium problem and the coupled systems of variational inclusions.The set-valued equilibrium problem is an equilibrium problem involving set-valued mapping.It is a special type of variational inclusions.Under certain appropriate assumptions,this paper proves the existence of solutions for two class of set-valued equilibrium problems via using the non-compactness measures.Our work improves some existing results in the literature.The coupled system of variational inclusions is a coupled problem consisting of two or more variational inclusions.In order to study its solvability,we first introduce a coupling optimal approximation problem in a real normed linear space.Then,the existence of solutions to this approximation problem is proved by KKM’s lemma.Finally,we establish an existence theorem of the coupled system of variational inclusions,and it is used to solve the problems of coupled coincidence points and coupled fixed points.In addition,a class of coupled system with convex subdifferential mappings is considered in this thesis.By introducing a variational inequality equivalent to the coupled system,and combining the Minty and KKM lemmas,we shows the existence of solutions to this variational inequality.Then,an existence theorem of solutions to the original coupled system is established.The work in this part is one of the latest results in the field of coupled systems of variational inclusions.