Strong Convergence Theorems For Variational Inequalities Problem And Some Relative Researches

Because variational inequalities problem is closely related to the fixed-point problem, optimization problem, equilibrium problem and complementarity problem, and is widely used in the economics, finance and many other fields, it is increasingly attracting scholars'attention. In recent years, the method that we find the approximation solution of variational inequalities problem by iterative algorithms becomes an important tool to research and apply variational inequalities theories. In this paper, inspired and motivated by the previous scholars'work, we introduce several iterative algorithms for variational inequalities problem and equilibrium problem, and prove their corresponding convergence theorems.Our paper is divided into two chapters.The first chapter, we give three-step iterations for variational inequalities and nonexpansive mappings in a Hilbert space, and obtain strong convergence theorems under the condition that the intersection between the set of solutions of the variational inequality and the set of fixed points of a nonexpansive mapping is nonempty, which is discussed at the mappings ofα- inverse-strongly-monotone mapping, relaxed - (γ, r)- cocoercive mapping and so on.In chapter two, we give some new unified Halpern iterations for equilibrium problem in a Banach space, and obtain strong convergence theorems for equilibrium problem and quasi-φ-nonexpansive mapping.Our results are the generalizations and extensions of the corresponding results obtained in [7-10,19-22] and so on.