On a strong covering property of multivalued mappings

In this paper, a strong variant for multivalued mappings of the well-known property of openness at a linear rate is studied. Among other examples, a simply characterized class of closed convex processes between Banach spaces, which satisfies such a covering behaviour, is singled out. Equivalent reformulations of this property and its stability under Lipschitz perturbations are investigated in a metric space setting. Applications to the solvability of set-valued inclusions and to the exact penalization of optimization problems with set-inclusion constraints are discussed.