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 mu such that the distance function d(S) associated with the solution set S of f(x) less than or equal to 0 satisfies d(S)(x) less than or equal to mu max{f(x),0} for each point x in a neighborhood of some point x(0) in X with f(x) < epsilon for some 0 < epsilon less than or equal to + infinity. 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.