Research Of Iterative Algorithms For Variational Inequality Problems And Its Related Problems

The variational inequality problem is a very important research field in mathematics.It is widely used in the fields of mechanics problem,differential equation,economic decision-making,cybernetics,signal processing and image restoration.Recently,the split feasibility problem has been extensively studied.It has been widely used in signal processing,image reconstruction and medically adjustable intensity radiation therapy.In this paper,we propose several iterative algorithms for variational inequalitie problems and related problems in Hilbert spaces,and obtain the convergence theorems and related corollaries.The specific research in this paper mainly includes the following contents:First,we study the variational inequality problems involved inverse strongly monotone in Hilbert spaces.Because the computation of metric projection on a general nonempty closed convex subset is not easy to achieve,we propose an iterative algorithm without projection operators combined with Mann iteration and hybrid steepest descent method,and give convergence theorem and proof.Secondly,we study the split variational inequality problems in Hilbert spaces.Using the extragradient method,the subgradient extragradient method,the CQ algorithm and the viscosity approximation method,we construct iterative methods for solving split variational inequality problems under weaker conditions.Finally,we study the bilevel variational inequality problems with split fixed point constraints.We propose several iterative algorithms,and give the convergence theorems and proofs under certain conditions.