Radius Theorems for Monotone Mappings

For a Hilbert space X and a mapping F:X⇉X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F: X\rightrightarrows X$\end{document} (potentially set-valued) that is maximal monotone locally around a pair (x̄,ȳ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\bar {x},\bar {y})$\end{document} in its graph, we obtain a radius theorem of the following kind: the infimum of the norm of a linear and bounded single-valued mapping B such that F + B is not locally monotone around (x̄,ȳ+Bx̄)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\bar {x},\bar {y}+B\bar {x})$\end{document} equals the monotonicity modulus of F. Moreover, the infimum is not changed if taken with respect to B symmetric, negative semidefinite and of rank one, and also not changed if taken with respect to all functions f : X → X that are Lipschitz continuous around x̄\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\bar {x}$\end{document} and ∥B∥ is replaced by the Lipschitz modulus of f at x̄\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\bar {x}$\end{document}. As applications, a radius theorem is obtained for the strong second-order sufficient optimality condition of an optimization problem, which in turn yields a lower bound for the radius of quadratic convergence of the smooth and semismooth versions of the Newton method. Finally, a radius theorem is derived for mappings that are merely hypomonotone.