Sufficient conditions for error bounds

For a lower semicontinuous (l.s.c.) inequality system on a Banach space, it is shown that error bounds hold, provided every element in an abstract subdifferential of the constraint function at each point outside the solution set is norm bounded away from zero. A sufficient condition for a global error bound to exist is also given for an l.s.c. inequality system on a real normed linear space. It turns out that a global error bound closely relates to metric regularity, which is useful for presenting sufficient conditions for an l.s.c. system to be regular at sets. Under the generalized Slater condition, a continuous convex system on Rⁿ is proved to be metrically regular at bounded sets.