Chain Rules for Linear Openness in Metric Spaces. Applications to Parametric Variational Systems

TL;DR

This paper develops chain rules for linear openness of set-valued mappings in metric spaces, extending classical results like Lyusternik-Graves, and applies these to analyze the stability of solutions in parametric variational systems.

Contribution

It introduces a general theorem for chain rules in metric spaces and derives new and classical results, including sharp estimates for regularity moduli in variational systems.

Findings

Established a general chain rule theorem for linear openness.

Derived new results extending the Lyusternik-Graves Theorem.

Provided sharp estimates for regularity moduli in variational systems.

Abstract

In this work we present a general theorem concerning chain rules for linear openness of set-valued mappings acting between metric spaces. As particular cases, we obtain classical and also some new results in this field of research, including the celebrated Lyusternik-Graves Theorem. The applications deal with the study of the well-posedness of the solution mappings associated to parametric variational systems. Sharp estimates for the involved regularity moduli are given.