Newton and Bouligand derivatives of the scalar play and stop operator

TL;DR

This paper establishes the existence and formulas of Newton and Bouligand derivatives for the scalar play and stop operators, including enhanced remainder estimates and a chain rule, ensuring measurability of the derivatives.

Contribution

It provides the first explicit formulas and enhanced remainder estimates for derivatives of the play and stop operators, with a focus on measurability and chain rule development.

Findings

Derivatives of play and stop operators are proven to exist.

Explicit formulas for Newton and Bouligand derivatives are derived.

Enhanced remainder estimates and a chain rule are established.

Abstract

We prove that the play and the stop operator possess Newton and Bouligand derivatives, and exhibit formulas for those derivatives. The remainder estimate is given in a strenghtened form, and a corresponding chain rule is developed. The construction of the Newton derivative ensures that the mappings involved are measurable.