  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.
