The Refinement And Application Of The Generalized Variational Inequality Solution

In this thesis, we mainly research the refinements of stationary points to generalized variational inequalities. Firstly, we use the way of refining variational inequalities' stationary points to generalized variational inequalities, define stable stationary points and prove their existence. Secondly, we specialize the domain of generalized variational inequalities, define perfect stationary points of generalized variational inequalities and prove their existence results. Finally, we define robust stationary points of generalized variational inequalities and prove their existence.We arrange the skeleton as follows:Chapter 1, the development and research status of variational inequalities are introduced, specially the history and development of generalized variational inequalities. And we sketch out the main contents and results.Chapter 2, we introduce some basic concepts and results relatively, mainly containing the concepts of set-valued mappings and their continuity, the existence theorems of stationary points of variational inequalities and generalized variational inequalities,Ky-Fan's coincidence theorem and so on?Chapter 3, we introduce a refinement of stationary points to variational inequalities.we show the form of stationary points when the domain is specialized, then we introduce the perfect stationary points and robust stationary points.Chapter4, We generalize the refinement of variational inequalities' solutions to generalized variational inequalities' solutions, defining the concept of stable stationary points and proving their existence.Chapter5, when the domain is specialized, we present the form of the stationary points. then we introduce the perfect stationary points and robust stationary points and prove their existence. Moreover, we give an example to show the results of the refinements.Chapter6, We use refinement of generalized variational inequalities' solutions in vector set-valued variational inequalities, and we use it to research the refinement of noncooperative games' Nash equilibrium.Chapter7, we summarize content of this paper and expect our work in the future.