The Generalization Of Vector Variational Inequalities, Open Mapping Theorem And Conjugate Formulae

Vector variational inequalities,open mapping theorem and conjugate formulae play important roles in analysis.In this paper,we study these problems.This paper is divided into three chapters.It is organized as follows:In chapter 1,we consider existence of solutions to generalized strong vector variational-like inequalities for set-valued mappings in normed spaces.Firstly, we extend that the solvability for generalized strong vector variational-like inequalities with pseudomonotonicity assumption.Secondly,the solvability results for generalized strong vector variational-like inequalities without monotonicity assumption are also presented.In chapter 2,we generalize the open mapping theorem concerned with convex maps.Firstly,we prove that the open mapping theorem to situations where spaces are normed and the graphs of convex maps are complete.Secondly,in metrizable topological vector spaces,the results for linear maps with complete graphs is an open mapping are presented.In chapter 3,we study some generalizations concerned with the conjugate formulae for the composition of a proper convex lower semicontinuous function with a linear mapping in normed spaces.As applications,the closedness criterions for linear image of a closed convex set are discussed.