The Existence Of Solutions And Iterative Algorithm For Two Kinds Of Variational Inequality Systems

Variational inequality theory is a vital research area in applied mathematics. This theory has a wide range of applications in many academic area,such as nonlinear optimization theory, game theory, control theory, di?erential equation and so on. As a significant extension of variational inequalities, variational inequality system has attracted a lot of attention in recent years. In this paper, two special variational inequality systems in Hilibert space are studied. One of them is a new class of generalized nonlinear variational inequality system, and the other is a general nonconvex variational inequality system which is defined on uniformly prox-regular sets in Hilbert space with four nonlinear operators. The existence of solutions and iterative algorithm for these two types of systems are considered. We now proceed the contends of this paper as follows:In Chapter 1, the background and developments of variational inequality theory are briefly introduced, and the outline of the research work in this thesis is given.In Chapter 2, some basic theories of variational inequalities are given, including some common variational inequality problem, the existence and uniqueness of solution for variational problem, the algorithms for solving variational inequalities and so on.In Chapter 3, a kind of general nonlinear variational inequality systems in Hilbert spaces, are considered, which are denoted by SGNLVIP. The equivalence between the SGNLVIP and the fixed point problem is established. Moreover, the existence of solutions for SGNLVIP is proven and some new explicit parallel iterative algorithms are proposed by using the resolvent operator technique. The convergence analysis of these algorithms is also illuminated with suitable assumptions.In Chapter 4, we consider a general nonconvex variational inequality system which is defined on uniformly prox-regular sets in Hilbert space with four nonlinear operators.This is denoted by SGNCVIP.The equivalence between the SGNCVIP and the fixed point problems is established and a parallel projection iterative algorithm for solving SGNCVIP is proposed based on the equivalence. What’s more, the existence and approximation of solutions of SGNCVIP are proven as well by using this algorithm with some suitable conditions. Finally, We suggest another new parallel projection algorithm for solving SGNCVIP. It converges to the intersection between the solution set of SGNCVIP and two Lipschitzian mappings. The convergence of this algorithm is also given with suitable conditions.In Chapter 5, we briefly summarize the conclusion of the research in this article and make a prospect for further studying.