Algorithm Research On Splitting Feasibility Problems And Splitting Common Fixed Point Problems

In this paper,we study the split feasibility problem and the split common fixed point problem in Hilbert spaces and Banach spaces respectively,and establish an modified regularization method and a new viscosity iterative algorithm for asymptotically non-expansive mapping to approximate a solution of the split common fixed point problem.Under appropriate conditions,we prove the strong convergence of the two iterative algorithms,and the main results are applied to solve the variational inequality problems.The purpose of this article is to discuss the fixed point problems in the broader sense of metric spaces and to some extent,we also improve and popularize some results of other scholars,so the research of this article is profound.This article is mainly divided into four sections:In the first section,we mainly describe the research backgrounds and current status of domestic and foreign research on the split feasibility problem and the split common fixed point problem on the basis of fixed point theory,and introduce the main works and innovations of this article.In the second section,we use a modified regularization method to establish an iterative algorithm for asymptotic non-expansive mapping to solve the split feasibility problem under the framework of a real Hilbert space.Through the demonstration,we obtain the strong convergence of the algorithm sequence and apply the main results to solve the variational inequality problem in a Hilbert space.In the third section,we propose an new iterative algorithm to study the split common fixed point problem for asymptotically non-expansive mappings in the framework of two real Banach spaces,and obtain a new fixed point theorem with appropriate parameters and solve the variational inequality problem closely related to the optimization problem.In the fourth section,we mainly use the main results of Chapter 3 to solve the hierarchical variational inequality problem in Banach spaces,and give corresponding numerical examples to illustrate the scientificity and validity of the conclusions in this paper.