Iterative Algorithm For Solving Some Variational Inequalities And System Of Variational Inequalities

The main purpose of this paper is to study the iterative algorithm for solving some variational inequalities and system of variational inequalities. In the second chapter, we consider the proximal projected-like method for solving generalized variational inequalities in Rn spaces and Hilbert spaces. This method includes proximal method and projected-like method. At first, we obtain temporary iteration points through proximal method, and then by using the projected-like method, we project the temporary point onto the feasible set of generalized variational inequalities to get the next iterative point. Under the assumptions that the set-valued mapping is pseudo-monotone and upper semi-continuity in Rn, we prove that every accumulation point of the sequence is a solution of variational inequalities, and under the assumptions that the set-valued mapping is maximal monotone in Hilbert, we prove that every weal accumulation point of the sequence is a solution of variational inequalities.The main purpose of the third chapter is relax the r-strongly monotone assumption of T in Verma[20] to T being (η, r)-cocoercive and reduce the scope of the parameters. Finally, we prove that the sequence converges to the solution of a system of nonlinear variational inequalities.