Ioffe-Type Criteria in Extended Quasi-Metric Spaces

We study various regularity properties, including subregularity and semiregularity, of set-valued mappings acting in extended quasi-metric spaces. It turns out that this abstract framework allows to unify criteria for the usual (sub/semi) regularity as well as their directional and Hölder counterparts. Ioffe-type critera are obtained by applying a general version of the Ekeland's variational principle. We provide a self-contained material gathering and extending the existing theory on the topic. We (briefly) illustrate the importance of these criteria for applications. For example, we consider local convergence of a directional version of a Newton-type method for solving a generalized equation which may be applied when the usual (non-directional) regularity does not hold.