SQUARE-ROOT METRIC REGULARITY AND RELATED STABILITY THEOREMS FOR SMOOTH MAPPINGS

Metric regularity and a stability theorem with a square-root distance estimate are investigated for smooth mappings which act in Hilbert spaces. The main result concerns a sufficient condition for this type of metric regularity and stability, which is formulated without a priori normality assumptions. As an application of the obtained conditions, Lyusternik's tangent cone theorem and the inverse function theorem in a neighborhood of abnormal point are derived. Examples are constructed which demonstrate the essence of the proposed assumptions.

Automated summary

This study investigates metric regularity and a stability theorem based on a square-root distance estimate for smooth mappings in Hilbert spaces. The main result provides a sufficient condition for this type of metric regularity and stability without the need for a priori normality assumptions. Applications include deriving Lyusternik's tangent cone theorem and the inverse function theorem in a neighborhood of abnormal points, with examples demonstrating the essence of the proposed assumptions.