Stability Results of Variational Systems Under Openness with Respect to Fixed Sets

In this paper we present the notions of (U; V )-openness and (U; V )- metric regularity for a set-valued map, proving their equivalence. By using different approaches we show the stability, with respect to the sum of maps, of the (U; V )-openness property, both in the setting of Banach spaces, and of metric spaces. Finally, we infer the regularity of the map solving a generalized parametric equation defined via a parametric map that is, in its turn, perturbed by the sum with another map.