Openness Stability and Implicit Multifunction Theorems. Applications to Variational Systems

TL;DR

This paper establishes new general results on the stability of openness and the regularity of implicit multifunctions in variational systems, extending recent findings and providing deeper insights into their interrelations.

Contribution

It introduces two broad theorems on openness stability and metric regularity for set-valued maps and applies them to analyze variational systems, advancing current theoretical understanding.

Findings

Derived relations between regularity moduli of field maps and solutions.

Extended recent results on stability and regularity in variational analysis.

Provided a unified approach to analyze implicit multifunctions in variational systems.

Abstract

In this paper we aim to present two general results regarding, on one hand, the openness stability of set-valued maps and, on the other hand, the metric regularity behavior of the implicit multifunction related to a generalized variational system. Then, these results are applied in order to obtain, in a natural way, and in a widely studied case, several relations between the metric regularity moduli of the field maps defining the variational system and the solution map. Our approach allows us to complete and extend several very recent results in literature.