Existence And Algorithm Of Solutions For Variational Inequalities

Variational inequalities have been developed into an important branch of applicable mathematics with a wide range of applications in nonlinear optimization theory, di?erential equation, control problem, game theory and social economic equilibrium theory. One of the basic problem in variational inequalities is the existence of solutions problem and the research of the iterative algorithm for solutions.For these research problems,there are two types variational inequalities:(1)the mixed variational inequalities;it is a useful and important generalization of variational inequalities with containing a nonlinear term.(2)nonconvex variational inequalities;it’s a new class of variational inequality problem that is proposed by Noor et al.In this paper, on the basis of mixed variational inequalities and nonconvex variational inequalities, two new system of variational inequalities generalized mixed variational inequalities system problems and generalized regular nonconvex variational inequalities system problems are proposed. And we also consider the two variational inequalities system problems by using the resolvent operator technique and projection operator technique respectively.The rest of the paper is organized as follows:In Chapter 1, we introduce the historical background and developments of variational inequalities theory, and briefly describes some of the research work of this thesis.In Chapter 2, we introduces some relevant knowledge for this paper, including some definitions and conclusions.In Chapter 3, we study the following system of generalized mixed variational inequalities, i.e.,(SGMVIP), which involving six di?erent nonlinear operators:existence and approximation of its solution using the resolvent operator technique. We establish the equivalence between the new system of general mixed variational inequalities by using the resolvent operator technique. We prove the existence of(SGMVIP) and using this equivalence we also propose an parallel iterative algorithm and consider its convergence analysis. We also construct a relaxed iterative algorithm for solving the problem(SGMVIP) and study the convergence of the iterative sequence generated by the algorithm. We further propose an common element algorithm which converges to its solution and common fixed points of two Lipschitzian mappings. The results presented in this paper are more general, improve and extend the previously known results for the variational inequalities.In Chapter 4, we study the following system of generalized regularized non-convex variational inequalities, i.e.,(SGRNCVIP), which involving four di?erent nonlinear operators:existence and approximation of its solution using the project operator technique. We also establish the equivalence between the new system of generalized regularized non-convex variational inequalities by using the project operator technique. We prove the existence of solutions of(SGRNCVIP) under some suitable conditions. Furthermore, we propose a parallel iterative algorithm, a relaxed iterative algorithm for problem(SGRNCVIP) by this equivalence. Using this algorithm, we consider the approximation solvability of the problem(SGRNCVIP).In Chapter 5, we summarizes the research work of this article and made a prospect for the fourther research work.