An iterative method for bilevel variational inequalities problem with g-fixed point constraints in Banach spaces

Abstract

This study investigates an inertial approach designed to address the bilevel variational inequality problem that incorporates a g-fixed point condition in real reflexive Banach spaces. The underlying mapping for the low-level variational inequality is continuous pseudomonotone, while the g-fixed point constraint is handled through a ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document}-generalized g-demimetric mapping. The requirement for the uniform continuity of the mapping is relaxed. We establish a strong convergence result under appropriate assumptions on control sequences. The applicability of our primary findings is demonstrated through problems in image processing and optimization. Additionally, a numerical example is included to illustrate the practical effectiveness of the proposed method. Our findings extend and generalize several existing results in the literature, transitioning from Hilbert spaces to the broader context of Banach spaces.