Error bounds and gap functions for various variational type problems

In this work we study various gap functions for the generalized multivalued mixed variational-hemivariational inequality problems by using the (τM,σM)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\tau _{\mathscr {M}},\sigma _{\mathscr {M}})$$\end{document}-relaxed cocoercive mapping and Hausdorff Lipschitz continuity. Moreover, we establish global error bounds for such inequalities using the characteristic of the Clarke generalized gradient method. As application, we present a stationary nonsmooth semipermeability problem.