# Radius Theorems for Monotone Mappings

## Abstract

For a Hilbert space *X* and a mapping \(F: X\rightrightarrows X\) (potentially set-valued) that is maximal monotone locally around a pair \((\bar {x},\bar {y})\) 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 \((\bar {x},\bar {y}+B\bar {x})\) 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 \(\bar {x}\) and ∥*B*∥ is replaced by the Lipschitz modulus of *f* at \(\bar {x}\). 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.

## Keywords

Monotone mappings Maximal monotone Locally monotone Radius theorem Optimization problem Second-order sufficient optimality condition Newton method## Mathematics Subject Classification (2010)

47H05 49J53 90C31## Preview

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