Variational Inequalities, System Of Variational Inclusion And KKM Theorem

In this paper, we mainly study the solutions to variational inequalities and a system of variational inclusion and the iterative algorithms to compute the approximate solution. It is proved that the iterative algorithms converge strongly to their solutions. We also analyze the convergence of the iterative algorithms by use a computer. The R-KKM selections and the R-KKM mappings are also introduced.The first, an extra-gradient projection method for general variational inequalities on Rn is proposed, which has a better performance in computational experience than some known in the literature. Under some mild assumptions, global convergence is proved and convergence rate is analyzed.The second, by using a new class of monotone operator H-monotone operator and the resolvent operator associated with this H-monotone operator, the resolvent technique defined with H-monotone operator, this paper introduces a system of set-valued variational inclusions. An iterative algorithm to compute the approximate solution is provided. It is proved that the iterative algorithm converges strongly to the solution of the systems of generalized set-valued variational inclusions.