Uniform Regularity of Set-Valued Mappings and Stability of Implicit Multifunctions

We propose a unifying general (i.e. not assuming the mapping to have any particular structure) view on the theory of regularity and clarify the relationships between the existing primal and dual quantitative sufficient and necessary conditions including their hierarchy. We expose the typical sequence of regularity assertions, often hidden in the proofs, and the roles of the assumptions involved in the assertions, in particular, on the underlying space: general metric, normed, Banach or Asplund. As a consequence, we formulate primal and dual conditions for the stability properties of solution mappings to inclusions