The Connectedenss Of The Solution Sets For Vector Equilibrium Problems

Vector equilibrium problem is a kind of more extensive mathematical model,it contains the vector optimization problem,vector variational inequality problem and fixed point problem.It has wide applications in the fields of economy,finance,transportation,resource allocation and project management.In this paper,we study the connectedness of the strong efficient solution set to vector equilibrium problem and the weak connectedness of the weak efficient solution set to vector mixed variational inequality.The detailed arrangement of the dissertation is as follows:In chapter 1,we introduce the background and research status of vector equilibrium problems and the background and research status of vector variational inequalities.Moreover,we introduce the common symbols,basic concepts and lemmas used in this dissertation.In chapter 2,we investigate the connectedness of the strong efficient solution set for setvalued vector equilibrium problems in reflexive Banach Spaces.Firstly,we define a new nonlinear function,and use this nonlinear function to obtain the non-convex separation theorem,a neither convex nor closed set and a compact subset in a reflexive Banach Spaces can be strictly separated.Secondly,we prove that the strong efficient solution set for the set-valued strong vector equilibrium problem can be expressed as the union of the solution set of a series of nonlinear scalar problems.We present a sufficient condition for the connectedness of strong efficient solution set for vector equilibrium problems,and provide examples to show that the conditions of this dissertation are strictly weaker than those of the literature.In chapter 3,we investigate the weak connectedness of the weak efficient solution set for the vector mixed variational inequalities in reflexive Banach Spaces by using the connectedness of weakly*compact base of its dual cone and the Mosco-convergence convex function sequence.We obtain the necessary and sufficient condition for the nonemptiness and boundedness of the weak efficient solution set for the vector mixed variational inequalities in reflexive Banach Spaces by using the equivalent conditions for the nonemptiness and boundedness of the solution set for the scalar mixed variational inequalities.Ulilzing the weak compactness of the weak efficient solution set to the vector mixed variational inequalities,we prove the weak connectedness of the weak efficient solution set for the vector mixed variational inequalities,and present the weak path-connectedness of weak efficient solution set for the vector mixed variational inequalities when the mapping is strictly pseudomonotone.