A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let α > 0 and let A be an α-inverse-strongly monotone mapping of C into H and let B be a maximal monotone operator on H. Let F be a maximal monotone operator on H such that the domain of F is included in C. Let 0 < k < 1 and let g be a k-contraction of H into itself. Let V be a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\gamma}}$$\end{document}-strongly monotone and L-Lipschitzian continuous operator with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\gamma} >0 }$$\end{document} and L > 0. Take \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu, \gamma \in \mathbb R}$$\end{document} as follows:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${0 < \mu < \frac{2\overline{\gamma}}{L^2}, \quad 0 < \gamma < \frac{\overline{\gamma}-\frac{L^2 \mu}{2}}{k}.}$$\end{document}In this paper, under the assumption \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(A+B)^{-1}0 \cap F^{-1}0 \neq \emptyset}$$\end{document}, we prove a strong convergence theorem for finding a point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${z_0\in (A+B)^{-1}0\cap F^{-1}0}$$\end{document} which is a unique solution of the hierarchical variational inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle (V-\gamma g)z_0, q-z_0 \rangle \geq 0, \quad \forall q\in (A+B)^{-1}0 \cap F^{-1}0.}$$\end{document}Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space which are useful in nonlinear analysis and optimization.