Study On The Existence Of Solutions Of Mixed Vector Variational Inequalities

Mixed vector variational inequality is a kind of more extensive mathematical model,in-cluding variational inequality problem,optimization problem and vector variational inequality problem.It has a wide range of applications in mechanics,game theory,economics.In this paper,we study the existence of solutions of noncoercive mixed vector variational inequality and the stability of solution set of mixed vector variational inequality.The detailed arrangement of the dissertation is as followsIn chapter 1,we introduce the historical background and research status of vector varia-tional inequalities,the development of exceptional family of elements and the research status of the stability of solution sets of vector variational inequalities.Moreover,we introduce the common symbols,basic concepts and lemmas used in this dissertationIn chapter 2,we investigate the existence of weakly efficient solutions of noncoercive mixed vector variational inequalities in reflexive Banach Spaces by using the exceptional family of elements.Firstly,we show that the nonexistence of an exceptional family of elements is a nec-essary condition for the solvability of mixed vector variational inequality problems.By using the asymptotic mappings of vector-valued mappings,we present a sufficient condition for the nonexistence of an exceptional family of elements for mixed vector variational inequality prob-lems in reflexive Banach spaces,we obtain some existence results for weakly efficient solutions for mixed vector variational inequality problems.Secondly,when the operator is affine and copositive,we present a sufficient condition for the nonexistence of an exceptional family of el-ements for mixed affine vector variational inequality problems,establish some existence results for weakly efficient solutions of mixed affine vector variational inequality problems,and provide some sufficient conditions for solution sets of mixed affine vector variational inequality problems to be nonempty and compactIn chapter 3,we study the stability of solution set of mixed vector variational inequality with respect to constraint set perturbation and operator perturbation by using generalized.f-projection operator.Firstly,we study the convergence of the generalized f-projection operator when the constraint set is perturbed and the operator is perturbed,and obtain the upper semi-continuity of the solution set of the mixed variational inequality.We transform the mixed vector variational inequality problem into a fixed point problem,and construct a Mann type iterative algorithm by using the fixed point iterative algorithm,so that the sequence generated by the algorithm has subsequences convergent to the solution of the mixed vector variational inequality We obtain the upper semi-continuity of the solution set of the mixed vector variational inequality.