  Our aim in the current article is to extend the developments in Kruger, Ngai
& Th\'era, SIAM J. Optim. 20(6), 3280-3296 (2010) and, more precisely, to
characterize, in the Banach space setting, the stability of the local and
global error bound property of inequalities determined by proper lower
semicontinuous under data perturbations. We propose new concepts of (arbitrary,
convex and linear) perturbations of the given function defining the system
under consideration, which turn out to be a useful tool in our analysis. The
characterizations of error bounds for families of perturbations can be
interpreted as estimates of the `radius of error bounds'. The definitions and
characterizations are illustrated by examples.
