The Iterative Algorithm Of Split Variational Inclusion Problem And Monotone Variational Inclusion Problem

The problem of variational inequality is the hot research topic has caused widely concerned. As its important branches, split variational inclusion and monotone variational inclusion are favored naturally by scholars both domestic and abroad and lots of research results have been made. In this paper we do some discussion for split variational inclusion and monotone variational inclusion problems in Hilbert spaces,which base on the methods of viscosity iteration method, steepest descent method, and shrink algorithm. The main contents are as follows:Firstly, on the basis of the new iterative algorithm of Tian,We present a iterative algorithm to search for the common solution of the fixed point problem of the split variational inclusion problem and a single nonexpansive mapping, and prove the strong convergence of the algorithm.Secondly, under the enlightenment by the Yamada of the steepest descent algorithm and Zhou of the proposed algorithm, we present a new method with strong convergence for solving the problems of the split variational inclusion problem and finite nonexpansive mappings with common fixed point set of common solutions.Thirdly, based on mixed viscosity iteration algorithm, we improv the algorithm,which Kazmi and Rizvi have proposed. And we present new iterative algorithm for solving the split variational inclusion problem and public solutions of of the fixed point set of infinite nonexpansive mappings, and what most important is that if operator sequences are not satisfied the conditions of AKTT, we also can prove its strong strong convergence of the algorithm, and a numerical example is given to demonstrate the superiority of the proposed algorithm.In the end, we study the related problem of monotone variational inclusion by the shrink algorithm. We know many results of monotone variational inclusion problems are based on single valued mapping is strongly monotone or inverse stronglymonotone, but we weaken the operator f, if the operator f satisfies the conditions of monotone, we can get the result of weak convergence, and finally we prove the result.