An induction theorem and nonlinear regularity models

TL;DR

This paper develops a nonlinear regularity model for set-valued mappings in metric spaces, revising an induction theorem and applying it to derive criteria for regularity and openness properties, offering alternatives to classical variational principles.

Contribution

It introduces a new nonlinear regularity model using an induction theorem, providing novel criteria for regularity and openness in set-valued mappings, and offers a substitution for the Ekeland variational principle.

Findings

Revised induction theorem for regularity analysis.

Derived basic estimates for regularity and openness.

Established criteria for local and global regularity properties.

Abstract

A general nonlinear regularity model for a set-valued mapping $F:X\times R_+\rightrightarrows Y$, where $X$ and $Y$ are metric spaces, is considered using special iteration procedures, going back to Banach, Schauder, Lusternik and Graves. Namely, we revise the induction theorem from Khanh, J. Math. Anal. Appl., 118 (1986) and employ it to obtain basic estimates for studying regularity/openness properties. We also show that it can serve as a substitution of the Ekeland variational principle when establishing other regularity criteria. Then, we apply the induction theorem and the mentioned estimates to establish criteria for both global and local versions of regularity/openness properties for our model and demonstrate how the definitions and criteria translate into the conventional setting of a set-valued mapping $F:X\rightrightarrows Y$.