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 d _S associated with the solution set S of f(x) ≤ 0 satisfies d _s(x) ≤ μ max{f(x), 0} for each point x in a neighborhood of some point X₀ in X with f(x) < ε for some 0 < ε ≤ + ∞. 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.