Existence Of Solutions For Vector Variational Inequalities And Exceptional Family Of Elements

fn this dissertation, we focus on the existence of solutions to the vector variation inequality, vector optimization problems and generalized vector variational inequality. The disserta-tion is trying to substitute exceptional family to study the existence of solution to vector variational inequality and optimization problems. Meanwhile, we are trying to apply the fixed point theorem to study the existence of solution to the generalized vector variational inequality problem. It is organized as follows:In Chapter1, we introduce the background of vector variational inequality and the vec-tor optimization problems, the development of exceptional family and some basic conceptions and lemmas which are used is this dissertation.In Chapter2, we study the following vector variational inequality VVI(K, T):find c0∈K such that.<T(x0), y-x0>(?)-intC(x0),(?)y∈K.The existence of solution for variational inequality by the exception family of ele-ments have obtained many good conclusion, according this idea, we study the existence of solution for vector variational inequality in real Banach space by the exception fam-ily of elements. Firstly, we introduce a conception of the exception family of elements for VVI(K, T) problem, and obtain VVI(K, T) problem equivalence VVI(K, Ts) problem without T being C_monotone, hemicontinuous on K. Secondly, according to concern be-tween VVI(K, Ts) problem with the exception family of elements, we study the existence of solution for VVI(K, Ts) problem if only if there not exist the exception family of elements, and show that some sufficient conditions for the solvability of VVI(K, T) problem. Finally, we show the theorem about the two equivalent problem, according to this theorem, we study the bounded solvability of VVI(K. T) problem.In Chapter3, we study the following vector variational inequality (VOP):find x0∈K such that <T(x), y-x>(?)-intC,(?)y∈K.Firstly, we define a conception of the exceptional family of elements for the vector optimiza-tion problems in the Banach space. Secondly, we show a sufficient condition for the existence of exceptional family of elements, we prove the existence of solution for (VOP) problem if only if there not exist the exception family of elements. Meanwhile, show that some sufficient conditions for the solvability of (VOP) problem. Finally, applying the notion of exceptional family and the lemma, we study the bounded solvability of (VOP).In chapter4, we discuss generalized vector variational inequality (GVVIP):find x0∈K such that (A(x0. x0),y-x0)+f(y)-f(x0)∈-intC,(?)y∈K. we obtain some existence theorems of the solution, applying the fixed point theorem, and generalize some conclusion.