Regularity properties of set-valued solution mappings

Abstract

In the concept of a solution mapping for a problem dependent on parameters, whether formulated with equations or something broader like variational inequalities, we have always had to face the possibility that solutions might not exist, or might not be unique when they do exist. This goes all the way back to the setting of the classical implicit function theorem. In letting S(p) denote the set of all x satisfying f(p, x) = 0, where f is a given function from ${\mathbb{R}}^{d} \times {\mathbb{R}}^{n}$ to ${\mathbb{R}}^{m}$, we cannot expect to be defining a function S from ${\mathbb{R}}^{d}$ to ${\mathbb{R}}^{n}$, even when m = n. In general, we only get a set-valued mapping S. However, this mapping S could have a single-valued localization s with properties of continuity or differentiability. The study of such localizations, as “subfunctions” within a set-valued mapping, has been our focus so far, but now we open up to a wider view.