Error bounds for lower semicontinuous inequality systems

Let X be a Banach space and f:X →(-∞, ∞) be a proper and lower semicontinuous function, denoted as S={x ∈ X:f(x)≤0}, ds(x) :=inf{||x-s||:s ∈ S}. We say that the system f(x) ≤ 0 (or s) has a local (global) error bound if S is nonempty and, for some 0 < μ and ε ∈(0,+∞) (ε=+∞), ds(x) ≤ μf(x)+∀x ∈ X with f₊(x)<ε, where f(x)₊ := max{f(x),0}. We recall the concept of an abstract subdifferential ∂_ω subdifferential defined by Wu et al. (2001).