Convergence Analysis of a Power Penalty Approach for a Class of Nonlocal Double Phase Complementarity Systems

In the present paper, we consider a nonlinear complementarity problem (NCP, for short) with a nonlinear and nonhomogeneous partial differential operator (called double phase differential operator), a convection term (i.e., a reaction depending on the gradient), a generalized multivalued boundary condition, and two nonlocal terms which appear in the domain and on the boundary. We employ a power penalty method to NCP for introducing an approximating problem associated with NCP which is a nonlinear and nonlocal elliptic equation with mixed boundary value conditions. Denoting by S∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}_\infty $$\end{document} the solution set of NCP and by Sρ(ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}_\rho (\varepsilon )$$\end{document} the solution set of the approximating problem corresponding to penalty parameter ρ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho >0$$\end{document} and regularized parameter ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document}, we establish a critical convergence result in which S∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}_\infty $$\end{document} can be approached by the solution sets Sρ(ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}_\rho (\varepsilon )$$\end{document} of approximating problems in the sense of Kuratowski when ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} goes to zero, i.e., the following convergence relation holds ∅≠w-lim supρ→0Sρ(ε)=s-lim supρ→0Sρ(ε)⊂S∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\emptyset \ne w\text {-}\limsup \nolimits _{\rho \rightarrow 0}{\mathcal S}_\rho (\varepsilon )=s\text {-}\limsup \nolimits _{\rho \rightarrow 0}{\mathcal S}_\rho (\varepsilon )\subset {\mathcal {S}}_\infty ,$$\end{document} where w-lim supρ→0Sρ(ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\limsup _{\rho \rightarrow 0}{\mathcal {S}}_\rho (\varepsilon )$$\end{document} and s-lim supρ→0Sρ(ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\limsup _{\rho \rightarrow 0}{\mathcal {S}}_\rho (\varepsilon )$$\end{document} are the weak and the strong Kuratowski upper limits of Sρ(ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal S_\rho (\varepsilon )$$\end{document}, respectively.