Composite Viscosity Midpoint Iterative Algorithms For Nonlinear Mappings With Applications In Systems Of Variational Inequalitues

The theory of variational inequalities has played an important role in the study of pure and applied sciences,and it has been attracting the attention and effort of many scholars.More and more scholars have used various methods to conduct in-depth research on it under appropriate conditions.Variational inequalities are widely applied in differential equations,optimal control,optimization,mathematical programming,mechanics,the economics,etc.Therefore,how to solve variational inequalities has always been a hot topic in the research work.In this paper,we study the problem of finding a common element of the solution set of the variational inequality problem and the fixed point set of the asymptotically nonexpansive mapping in a real Hilbert space.Also,we consider the problem of finding a common solution of a general system of the variational inequalities and a common fixed point problem of an asymptotically nonexpansive mapping and countably many nonexpansive mappings.We design some iterative algorithms.Some strong convergence theorems are established under certain suitable assumptions imposed on the parameters.The outline of this paper is organized as follows:In Chapter 1,we introduce the research background of the variational inequality problem and fixed point problem and the development status at home and abroad.and we specify the main research work of this paper.In Chapter 2,the basic definitions,propositions and related lemmas are presented.In Chapter 3,we studied the common element of the solution set of the variational inequality problem and the fixed-point set of the asymptotically nonexpansive mapping,and established the implicit midpoint iteration method.Under suitable assumptions,it is proven that the sequence generated by the iterative algorithm strongly converges to the common element of the solution set of the variational inequality problem and the fixed point set.In Chapter 4,we study the problem of finding a common solution of a general system of the variational inequalities and a common fixed point problem of an asymptotically nonexpansive mapping and countably many nonexpansive mappings.With the help of the technique of the midpoint iteration method,some strong convergence theorems are established under certain suitable assumptions imposed on the parameters.In Chapter 5,we give the conclusion and outlook of this paper.