The Research Into General Nonconvey Variational Inequalities And Its Algorithm

Variational inequalities theory as a important branch of mathematical and engineering science, has a wide range of applications in industry, finance, economics, ecology, social and so on. But, it is worth mentioning that almost all the results regarding the existence and iterative schemes for variational inequalities are based on convex sets. This is because the projection operator over convex sets has some good properties, which may not hold in general, when the sets are nonconvex. Consequently, the research into variational inequalities which is based on nonconvex sets has a very important significance for theoretical and practical application.In this paper, we introduce and analyze a new class of variational inequalities, which is called the general nonconvex variational inequalities. We proof that the general nonconvex variational inequalities is equivalent to the fixed-point problems by using the projection technique. This equivalent formulation is used to discuss the existence of a solution of the general nonconvex variational inequalities. We also use this equivalent alternative formulation to establish a new iterative method for solving the general nonconvex variational inequalities. Finally, we also discuss the convergence of the iterative method under suitable conditions.We also establish the equivalence between the general nonconvex inequalities and Wiener-Hopf equation, which is based on the fixed-point problems. We use this equivalent alternative formulation to establish a new iterative method for solving the general nonconvex variational inequalities. We also discuss the convergence of the iterative method.