| Variational inequality theory is proposed since the1960s. In recent years, variational inequality with applications in mathematical economics, mechanics, engineering, etc., have been further studied by many authors. The solvability of variational inequality problems is a basic problem, and has been studied by several classical methods:KKM-theorem, fixed-point theory and many other topological methods.Vector variational inequality was first introduced and studied by Giannessi in finite dimensional Euclidean spaces in1980. It is not the simple promotion of variational inequality. People often need to consider a number of evaluation indexes when making decisions in many practical situations. It is said that the evaluation index system is the vector form, and we need mathematical model in the vector form to evaluate. Vector variational inequality is proposed in such background. KKM-theorem is generally used to study the existence theorems of vector variational inequality.Coercivity conditions are generally used to ensure the existence of solution for vector variational inequality for unbounded set. In recent years, people use exceptional family of elements instead of coercivity conditions.Let X be reflexive Banach spaces, and K is a nonempty, closed and convex subset of X, let Y be a finite dimensional Euclidean space. Let C be a closed, convex and pointed cone in Y. F:K→2L(X,Y) is a set-valued mapping with nonempty compact convex values. In this dissertation, we focus on the weak set-valued vector variational inequality (short for ⅤⅥ(K,F)):Find x∈K, u∈F(x) such that {u,y-x)∈-intC,(?)y€K.In this paper, vector variational inequality is transformed into a class variational inequal-ity problem by constructing a new set-valued mapping, and the degree theory is employed to prove some existence theorems of solutions for set-valued vector variational inequality. At the same time, a new concept of exceptional family for vector variational inequality is established in reflexive Banach spaces to study the existence theorems of vector variational inequality. It is organized as follows:In Chapter1, we introduce the background of vector variational inequality, the develop-ment of degree theory and exceptional family, and list some basic conceptions and lemmas which are used is this dissertation.In Chapter2, vector variational inequality is transformed into a class of scalar varia-tional inequality problem. According to the conclusions of the degree theory for compact perturbation of monotonetype mappings studied by Hu and others, the degree theory for vector variational inequality is established in reflexive Banach spaces. And the degree theory is employed to prove some existence theorems of solutions for set-valued vector variational in-equality. Furthermore, some equivalent characterizations of nonemptiness and boundedness of the solution set for pseudomonotone vector variational inequality are also obtain.In Chapter3, a fixed-point theory of compact admissible mapping is established, and then an existence theorem for scalar variational inequality with weak to norm admissible mapping is proved on bounded sets. Basing on this existence theorem and combining the scalar technique, a new concept of exceptional family for vector variational inequality is established in reflexive Banach spaces. By using the method of exceptional family of ele-ments, the equivalence between the nonexistence of exceptional family of elements and the existence of solution for VVI(K, F) is established. In addition, some coercivity conditions for the solvability of VVI(K, F) is presented. And the solution set is proved to be nonempty and bounded if it is strictly feasible. |
