The Iterative Algorithm Of Monotone Variational Inclusion Problem In Hilbert Space

In recent years,the variational inequality theory is an important part of the nonlinear theory by more and more scholars research and promotion,and has made a great progress.The monotone variational inclusion theory as its important branch,plays an important role in modern mathematics,and has been widely used in various fields,such as mechanics,control theory,mathematical economics,optimization theory,nonlinear programming.In this paper,the variational inequality problems and monotone variational inclusions are studied by the viscosity iterative algorithm,the proximal contraction algorithm and the inertial proximal contraction algorithm.Firstly,basing on the projection contraction algorithm of Dong,we give the proximal contraction algorithm to find the solution of the monotone variational inclusion problem.Under the appropriate conditions,the weak convergence of the algorithms are proved,and two numerical examples are used to verify the feasibility of our algorithm.Secondly,based on the proximal contraction algorithm,an algorithm of inertial proximal and multi-step inertial proximal algorithm are proposed for solving the monotone variational inclusion problem,and the weak convergence of the algorithms are proved.The superiority of the algorithms are verified by numerical examples.Finally,based on the viscosity iterative algorithm,we have improved the algorithm of Aoyama and Kohsaka.For quasi-nonexpansive mappings,we propose an iterative algorithm for solving variational inequality problems,obtain strong convergence theorem,and illustrate the feasibility of our algorithm with numerical examples.