First-Order and Second-Order Conditions for Error Bounds

For a lower semicontinuous function f on a Banach space X, we study the existence of a positive scalar μ such that the distance function dS associated with the solution set S of f(x) ≤ 0 satisfies dS(x) ≤ μmax{f(x), 0} for each point x in a neighborhood of some point x0 in X with f(x) < for some 0 < ≤ +∞. We give several sufficient conditions for this in terms of an abstract subdifferential and the Dini derivatives of f . In a Hilbert space we further present some second-order conditions. We also establish the corresponding results for a system of inequalities, equalities, and an abstract constraint set.