| Variational inequality is a very important field of research in mathematics, a lot of basic questions in mathematics can be attributed to a variational inequality problem. As we all know, set-valued variational inequality is an important branch in variational inequality. It is widely applied to mathematic, economic, mechanical and control theory and so on. Discussing the existence of solution is an basic problem in researching variational inequality and non-linear complementarity problem. The thesis mainly studies the existence of solution to set-valued variational inequality on the aspects of algorithm and theory.This article mainly includes four parts:In chapter1, we introduce the development of set-valued variational inequality and some necessary lemmas and theorems; In chapter2, we discuss the relationship between exceptional family and the existence of solution, in the paper, we introduce the development of exceptional family and give the relationship between exceptional family and the existence of solution in reflexive uniformly convex Banach space and reflexive Banach space, respectively; In chapter3, the paper mainly describe the relationship between the existence of zeros of mapping and the existence of solutions to set-valued variational inequality, in the paper, we prove that the equivalence between the existence of zeros of mapping and variational inequality has solution on closed convex subset, we also prove fixed-point theory in Hilbert space by the existence of proposed monotonous mapping of zero, and spread the equivalent relationship between the existence of solutions to set-valued variational inequality and fixed-point; In chapter4, we mainly use the auxiliary problem to study a kind of set-valued variational inequality.We study the iterative algorithm of problem and prove the convergence of the algorithm. The study of this problem is a promotion for those original using this iterative algorithm to prove the existence of solutions. |
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