Research On Several Kinds Of Equilibrium Problems And Variational Inclusion Problems

Equilibrium problem was first proposed by Blum and Oettli in the 1990 s.Because it is closely related to fixed point problem,variational inequality problem and differential variational inequality problem,it has deep application in many fields such as mathematical economy,physics,cybernetics and so on.In recent decades,it has attracted extensive attention and research from many scholars at home and abroad.In the 1960 s,Browder,Ky Fan,Lions et al.put forward variational inequalities and variational inclusion problems.In the 1966 s,Hartman and Stampacchia bring up the famous H-S variational inequalities.Many studies have been carried out on variational inequalities and variational inclusions.It is widely used in economy,operational research,mechanics,differential equation and traffic network modeling.The above problems provide us with an effective nonlinear theoretical framework and become a powerful tool for solving many practical and applied problems.The main contents of this thesis are arranged as follows.In Chapter1,we introduce the preparatory knowledge,research background and present situation of equilibrium problems,variational inequalities and variational inclusion problems.In Chapter2,we study two kinds of set-valued vector equilibrium problems and Browder variational inclusion problems.Using Ky Fan Lemma,the existence results of solutions for this kind of set-valued vector equilibrium problems are obtained in the case of cones,and are applied to the study of Browder variational inclusions.Compared with the literature,our conclusions and research methods are new.In Chapter3,we mainly study two kinds of generalized set-valued vector equilibrium problems and mixed variational problems,and use Himmelberg fixed point theorem to study the existence of solutions.For generalized monotone mappings,by KKM techniques,the existence and behavior of solutions for mixed variational problems are studied.We change the compactness condition in this kind of research to paracompact,which improves and generalizes the results of recent relevant literature.In Chapter4,we first time introduce the differential mixed equilibrium problem.By using semigroup theory,Filippov implicit function theorem and related fixed point theorems,we prove some results on the existence of solutions for differential mixed equilibrium problems.