  Although the property of strong metric subregularity of set-valued mappings
has been present in the literature under various names and with various
definitions for more than two decades, it has attracted much less attention
than its older "siblings", the metric regularity and the strong metric
regularity. The purpose of this paper is to show that the strong metric
subregularity shares the main features of these two most popular regularity
properties and is not less instrumental in applications. We show that the
strong metric subregularity of a mapping F acting between metric spaces is
stable under perturbations of the form f + F, where f is a function with a
small calmness constant. This result is parallel to the Lyusternik-Graves
theorem for metric regularity and to the Robinson theorem for strong
regularity, where the perturbations are represented by a function f with a
small Lipschitz constant. Then we study perturbation stability of the same kind
for mappings acting between Banach spaces, where f is not necessarily
differentiable but admits a set-valued derivative-like approximation. Strong
metric q-subregularity is also considered, where q is a positive real constant
appearing as exponent in the definition. Rockafellar's criterion for strong
metric subregularity involving injectivity of the graphical derivative is
extended to mappings acting in infinite-dimensional spaces. A sufficient
condition for strong metric subregularity is established in terms of
surjectivity of the Frechet coderivative. Various versions of Newton's method
for solving generalized equations are considered including inexact and
semismooth methods, for which superlinear convergence is shown under strong
metric subregularity.
