Generalized System for Relaxed Cocoercive Variational Inequalities and Projection Methods

Let K be a nonempty closed convex subset of a real Hilbert space H. The approximate solvability of a system of nonlinear variational inequality problems, based on the convergence of projection methods, is discussed as follows: find an element (x*, y*)∈K×K such that\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{gathered} \left\langle {\rho {\rm T}(y^* ,x^* ) + x^* - y^* ,x - x^* } \right\rangle \geqslant 0,{\text{ }}\forall x \in K{\text{ and }}\rho > 0, \hfill \\ \left\langle {\eta {\rm T}(x^* ,y^* ) + y^* - x^* ,x - x^* } \right\rangle \geqslant 0,{\text{ }}\forall x \in K{\text{ and }}\eta > 0, \hfill \\ \end{gathered}$$ \end{document}where T: K×K→H is a nonlinear mapping on K×K.